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 Articles in journal or book chapters
 2022
1. M.A. Al-Radhawi and E.D. Sontag. Analysis of a reduced model of epithelial-mesenchymal fate determination in cancer metastasis as a singularly-perturbed monotone system. In C.A. Beattie, P. Benner, M. Embree, S. Gugercin, and S. Lefteriu, editors, Realization and model reduction of dynamical systems. Springer-Verlag, 2022. Note: To appear (see preprint in arXiv:1910.11311). [PDF] Keyword(s): epithelial-mesenchymal transition, miRNA, singular perturbations, monotone systems, oncology, cancer, metastasis, chemical reaction networks, systems biology.
Abstract:
 Metastasis can occur after malignant cells transition from the epithelial phenotype to the mesenchymal phenotype. This transformation allows cells to migrate via the circulatory system and subsequently settle in distant organs after undergoing the reverse transition. The core gene regulatory network controlling these transitions consists of a system made up of coupled SNAIL/miRNA-34 and ZEB1/miRNA-200 subsystems. In this work, we formulate a mathematical model and analyze its long-term behavior. We start by developing a detailed reaction network with 24 state variables. Assuming fast promoter and mRNA kinetics, we then show how to reduce our model to a monotone four-dimensional system. For the reduced system, monotone dynamical systems theory can be used to prove generic convergence to the set of equilibria for all bounded trajectories. The theory does not apply to the full model, which is not monotone, but we briefly discuss results for singularly-perturbed monotone systems that provide a tool to extend convergence results from reduced to full systems, under appropriate time separation assumptions.

2. M.A. Al-Radhawi, M. Sadeghi, and E.D. Sontag. Long-term regulation of prolonged epidemic outbreaks in large populations via adaptive control: a singular perturbation approach. IEEE Control Systems Letters, 6:578-583, 2022. [PDF] Keyword(s): epidemiology, COVID-19, COVID, systems biology.
Abstract:
 In order to control highly-contagious and prolonged outbreaks, public health authorities intervene to institute social distancing, lock-down policies, and other Non-Pharmaceutical Interventions (NPIs). Given the high social, educational, psychological, and economic costs of NPIs, authorities tune them, alternatively tightening up or relaxing rules, with the result that, in effect, a relatively flat infection rate results. For example, during the summer of 2020 in parts of the United States, daily COVID-19 infection numbers dropped to a plateau. This paper approaches NPI tuning as a control-theoretic problem, starting from a simple dynamic model for social distancing based on the classical SIR epidemics model. Using a singular-perturbation approach, the plateau becomes a Quasi-Steady-State (QSS) of a reduced two-dimensional SIR model regulated by adaptive dynamic feedback. It is shown that the QSS can be assigned and it is globally asymptotically stable. Interestingly, the dynamic model for social distancing can be interpreted as a nonlinear integral controller. Problems of data fitting and parameter identifiability are also studied for this model. This letter also discusses how this simple model allows for a meaningful study of the effect of population size, vaccinations, and the emergence of second waves.

3. M. Bin, J. Huang, A. Isidori, L. Marconi, M. Mischiati, and E. D. Sontag. Internal models in control, bioengineering, and neuroscience. Annual Review of Control, Robotics, and Autonomous Systems, 5:20.1â€“20.25, 2022. [PDF] Keyword(s): feeedback, internal model principle, nonlinear systems.
Abstract:
 Internal models are nowadays customarily used in different domains of science and engineering to describe how living organisms or artificial computational units embed their acquired knowledge about recurring events taking place in the surrounding environment. This article reviews the internal model principle in control theory, bioengineering, and neuroscience, illustrating the fundamental concepts and theoretical developments of the few last decades of research.

4. Eduardo D. Sontag. Remarks on input to state stability of perturbed gradient flows, motivated by model-free feedback control learning. Systems and Control Letters, 161:105138, 2022. [PDF] Keyword(s): iss, input to state stability, data-driven control, gradient systems, steepest descent, model-free control.
Abstract:
 Recent work on data-driven control and reinforcement learning has renewed interest in a relatively old field in control theory: model-free optimal control approaches which work directly with a cost function and do not rely upon perfect knowledge of a system model. Instead, an "oracle" returns an estimate of the cost associated to, for example, a proposed linear feedback law to solve a linear-quadratic regulator problem. This estimate, and an estimate of the gradient of the cost, might be obtained by performing experiments on the physical system being controlled. This motivates in turn the analysis of steepest descent algorithms and their associated gradient differential equations. This note studies the effect of errors in the estimation of the gradient, framed in the language of input to state stability, where the input represents a perturbation from the true gradient. Since one needs to study systems evolving on proper open subsets of Euclidean space, a self-contained review of input to state stability definitions and theorems for systems that evolve on such sets is included. The results are then applied to the study of noisy gradient systems, as well as the associated steepest descent algorithms.

 2021
1. D. Angeli, M.A. Al-Radhawi, and E.D. Sontag. A robust Lyapunov criterion for non-oscillatory behaviors in biological interaction networks. IEEE Transactions on Automatic Control, 2021. Note: In press. Preprint in arXiv.2009.10702, 2020. [PDF] Keyword(s): oscillations, dynamical systems, enzymatic cycles, systems biology.
Abstract:
 We introduce the notion of non-oscillation, propose a constructive method for its robust verification, and study its application to biological interaction networks (also known as, chemical reaction networks). We begin by revisiting Muldowney's result on non-existence of periodic solutions based on the study of the variational system of the second additive compound of the Jacobian of a nonlinear system. We show that exponential stability of the latter rules out limit cycles, quasi-periodic solutions, and broad classes of oscillatory behavior. We focus then on nonlinear equations arising in biological interaction networks with general kinetics, and we show that the dynamics of the aforementioned variational system can be embedded in a linear differential inclusion. We then propose algorithms for constructing piecewise linear Lyapunov functions to certify global robust non-oscillatory behavior. Finally, we apply our techniques to study several regulated enzymatic cycles where available methods are not able to provide any information about their qualitative global behavior.

2. T. Chen, M. Ali Al-Radhawi, C.A. Voigt, and E.D. Sontag. A synthetic distributed genetic multi-bit counter. iScience, 24:103526, 2021. [PDF] Keyword(s): counters, synthetic biology, transcriptional networks, gene networks, boolean circuits, boolean gates, systems biology.
Abstract:
 A design for genetically-encoded counters is proposed via repressor-based circuits. An N-bit counter reads sequences of input pulses and displays the total number of pulses, modulo $2^N$. The design is based on distributed computation, with specialized cell types allocated to specific tasks. This allows scalability and bypasses constraints on the maximal number of circuit genes per cell due to toxicity or failures due to resource limitations. The design starts with a single-bit counter. The N-bit counter is then obtained by interconnecting (using diffusible chemicals) a set of N single-bit counters and connector modules. An optimization framework is used to determine appropriate gate parameters and to compute bounds on admissible pulse widths and relaxation (inter-pulse) times, as well as to guide the construction of novel gates. This work can be viewed as a step toward obtaining circuits that are capable of finite-automaton computation, in analogy to digital central processing units.

3. J Greene and E D Sontag. Minimizing the infected peak utilizing a single lockdown: a technical result regarding equal peaks. medRxiv, 2021. [WWW] [doi:10.1101/2021.06.26.21259589]
Abstract:
 Due to the usage of social distancing as a means to control the spread of the novel coronavirus disease COVID-19, there has been a large amount of research into the dynamics of epidemiological models with time-varying transmission rates. Such studies attempt to capture population responses to differing levels of social distancing, and are used for designing policies which both inhibit disease spread but also allow for limited economic activity. One common criterion utilized for the recent pandemic is the peak of the infected population, a measure of the strain placed upon the health care system; protocols which reduce this peak are commonly said to { extquoteleft}flatten the curve." In this work, we consider a very specialized distancing mandate, which consists of one period of fixed length of distancing, and addresses the question of optimal initiation time. We prove rigorously that this time is characterized by an equal peaks phenomenon: the optimal protocol will experience a rebound in the infected peak after distancing is relaxed, which is equal in size to the peak when distancing is commenced. In the case of a non-perfect lockdown (i.e. disease transmission is not completely suppressed), explicit formulas for the initiation time cannot be computed, but implicit relations are provided which can be pre-computed given the current state of the epidemic. Expected extensions to more general distancing policies are also hypothesized, which suggest designs for the optimal timing of non-overlapping lockdowns.

4. J. Hanson, M. Raginsky, and E.D. Sontag. Learning recurrent neural net models of nonlinear systems. Proc. of Machine Learning Research, 144:1-11, 2021. [PDF] Keyword(s): machine learning, empirical risk minimization, recurrent neural networks, dynamical systems, continuous time, system identification, statistical learning theory, generalization bounds.
Abstract:
 This paper considers the following learning problem: given sample pairs of input and output signals generated by an unknown nonlinear system (which is not assumed to be causal or time-invariant), one wishes to find a continuous-time recurrent neural net, with activation function tanh, that approximately reproduces the underlying i/o behavior with high confidence. Leveraging earlier work concerned with matching derivatives up to a finite order of the input and output signals the problem is reformulated in familiar system-theoretic language and quantitative guarantees on the sup-norm risk of the learned model are derived, in terms of the number of neurons, the sample size, the number of derivatives being matched, and the regularity properties of the inputs, the outputs, and the unknown i/o map.

5. H. Hong, J. Kim, M.A. Al-Radhawi, E.D. Sontag, and J. K. Kim. Derivation of stationary distributions of biochemical reaction networks via structure transformation. Communications Biology, 4:620-, 2021. [PDF] Keyword(s): stationary distribution, chemical reaction networks, network translation, biochemical reaction networks, chemical master equation, stochastic, probabilistic, systems biology.
Abstract:
 Long-term behaviors of biochemical reaction networks (BRNs) are described by steady states in deterministic models and stationary distributions in stochastic models. Unlike deterministic steady states, stationary distributions capturing inherent fluctuations of reactions are extremely difficult to derive analytically due to the curse of dimensionality. Here, we develop a method to derive analytic stationary distributions from deterministic steady states by transforming BRNs to have a special dynamic property, called complex balancing. Specifically, we merge nodes and edges of BRNs to match in- and out-flows of each node. This allows us to derive the stationary distributions of a large class of BRNs, including autophosphorylation networks of EGFR, PAK1, and Aurora B kinase and a genetic toggle switch. This reveals the unique properties of their stochastic dynamics such as robustness, sensitivity, and multimodality. Importantly, we provide a user-friendly computational package, CASTANET, that automatically derives symbolic expressions of the stationary distributions of BRNs to understand their long-term stochasticity.

6. K. Johnson, G. Howard, D. Morgan, E. Brenner, A. Gardner, R. Durrett, W. Mo, A. Al'Khafaji, E.D. Sontag, A. Jarrett, T. Yankeelov, and A. Brock. Integrating transcriptomics and bulk time course data into a mathematical framework to describe and predict therapeutic resistance in cancer. Physical Biology, 18:016001, 2021. [PDF] Keyword(s): oncology, cancer, chemoresistance, resistance, intratumor heterogeneity, population dynamics, DNA barcoding, evolution, systems biology.
Abstract:
 The development of resistance to chemotherapy is a major cause of treatment failure in cancer. Intratumoral heterogeneity and phenotypic plasticity play a significant role in therapeutic resistance. Individual cell measurements such as flow and mass cytometry and single cell RNA sequencing (scRNA-seq) have been used to capture and analyze this cell variability. In parallel, longitudinal treatment-response data is routinely employed in order to calibrate mechanistic mathematical models of heterogeneous subpopulations of cancer cells viewed as compartments with differential growth rates and drug sensitivities. This work combines both approaches: single cell clonally-resolved transcriptome datasets (scRNA-seq, tagging individual cells with unique barcodes that are integrated into the genome and expressed as sgRNA's) and longitudinal treatment response data, to fit a mechanistic mathematical model of drug resistance dynamics for a MDA-MB-231 breast cancer cell line. The explicit inclusion of the transcriptomic information in the parameter estimation is critical for identification of the model parameters and enables accurate prediction of new treatment regimens.

7. M. Sadeghi, J.M. Greene, and E.D. Sontag. Universal features of epidemic models under social distancing guidelines. Annual Reviews in Control, 51:426-440, 2021. Note: Also in bioRxiv, 2020, https://www.biorxiv.org/content/10.1101/2020.06.21.163931v2.[WWW] [PDF] [doi:https://doi.org/10.1016/j.arcontrol.2021.04.004] Keyword(s): epidemiology, COVID-19, COVID, systems biology.
Abstract:
 Different epidemiological models, from the classical SIR system to more sophisticated ones involving population compartments for socially distanced, quarantined, infection aware, asymptomatic infected, and other individuals, share some remarkable dynamic characteristics when contact rates are subject to periodic or one-shot changes. In simple pulsed isolation policies, a linear relationship is found among optimal start time and duration for reduction of the infected peak. If a single interval social distancing starts too early or too late it will be ineffective with respect to decreasing the peak of infection. On the other hand, the nonlinearity of epidemic models leads to non-monotone behavior of the peak of infected population under periodic relaxation policies. This observation led us to hypothesize that an additional single interval social distancing at a proper time can significantly decrease the infected peak of periodic policies, and we verified this improvement.

8. E.D. Sontag. An explicit formula for minimizing the infected peak in an SIR epidemic model when using a fixed number of complete lockdowns. International Journal of Robust and Nonlinear Control, Special Issue on Control-Theoretic Approaches for Systems in the Life Sciences, pp 1-24, 2021. [PDF] Keyword(s): epidemiology, COVID-19, COVID, systems biology.
Abstract:
 Careful timing of NPIs (non-pharmaceutical interventions) such as social distancing may avoid high second waves'' of infections of COVID-19. This paper asks what should be the timing of a set of K complete-lockdowns of prespecified lengths (such as two weeks) so as to minimize the peak of the infective compartment. Perhaps surprisingly, it is possible to give an explicit and easily computable rule for when each lockdown should commence. Simulations are used to show that the rule remains fairly accurate even if lockdowns are not perfect.

9. A.P. Tran, M. Ali Al-Radhawi, E. Ernst, and E.D. Sontag. Optimization of heuristic logic synthesis by iteratively reducing circuit substructures using a database of optimal implementations. 2021. Note: Submitted. Keyword(s): Heuristic logic minimizer, Boolean circuit reduction, optimal synthesis, logic optimization, synthetic biology.
Abstract:
 Minimal synthesis of Boolean functions is an NP-hard problem, and heuristic approaches typically give suboptimal circuits. However, in the emergent field of synthetic biology, genetic logic designs that use even a single additional Boolean gate can render a circuit unimplementable in a cell. This has led to a renewed interest in the field of optimal multilevel Boolean synthesis. For small numbers (1-4) of inputs, an exhaustive search is possible, but this is impractical for large circuits. In this work, we demonstrate that even though it is challenging to build a database of optimal implementations for anything larger than 4-input Boolean functions, a database of 4-input optimal implementations can be used to greatly reduce the number of logical gates required in larger heuristic logic synthesis implementations. The proposed algorithm combines the heuristic results with an optimal implementation database and yields average improvements of 5.16% for 5-input circuits and 4.54% for 6-input circuits on outputs provided by the logic synthesis tool extit{ABC}. In addition to the gains in the efficiency of the implemented circuits, this work also attests to the importance and practicality of the field of optimal synthesis, even if it cannot directly provide results for larger circuits. The focus of this work is on circuits made exclusively of 2-input NOR gates but the presented results are readily applicable to 2-input NAND circuits as well as (2-input) AND/NOT circuits. In addition, the framework proposed here is likely to be adaptable to other types of circuits. An implementation of the described algorithm, HLM (Hybrid Logic Minimizer), is available at https://github.com/sontaglab/HLM/.

10. A.P. Tran, J.H. Meldon, and E.D. Sontag. Transient diffusion into a bi-layer membrane with mass transfer resistance: Exact solution and time lag analysis. Frontiers in Chemical Engineering, 2:25, 2021. [PDF] Keyword(s): Bi-layer membrane, transient diffusion, heat conduction, mass transfer resistance.
Abstract:
 Exact analytical and closed-form solutions to a problem involving transient diffusion in a bi-layer membrane with external transfer resistance are presented. In addition to the solutions of the transient response, the lead and lag times that are often of importance in the characterization of membranes and arise from the analysis of the asymptotic behavior of the mass permeated through the membrane are also provided. The solutions presented here are also compared to previously derived limiting cases of the diffusion in a bi-layer with an impermeable wall and constant concentrations at the upstream and downstream boundaries. Analysis of the time lag shows that this membrane property is independent of the direction of flow. Finally, an outline is provided of how these solutions, which characterize the response to a step function increase in concentration, can be also used to derive more complex input conditions. Adequately handling boundary layer effects has a wide array of potential applications such as the study of bi-layer undergoing phenomena of heat convection, gas film resistance, and absorption/desorption.

11. N. Trendel, P. Kruger, S. Gaglione, J. Nguyen, J. Pettmann, E.D. Sontag, and O. Dushek. Perfect adaptation of CD8+ T cell responses to constant antigen input over a wide range of affinity is overcome by costimulation. Science Signaling, 14:eaay9363, 2021. [PDF] Keyword(s): immunology, cell signaling, T cells, systems biology.
Abstract:
 Maintaining and limiting T cell responses to constant antigen stimulation is critical to control pathogens and maintain self-tolerance, respectively. Antigen recognition by T cell receptors (TCRs) induces signalling that activates T cells to produce cytokines and also leads to the downregulation of surface TCRs. In other systems, receptor downregulation can induce perfect adaptation to constant stimulation by a mechanism known as state-dependent inactivation that requires complete downregulation of the receptor or the ligand. However, this is not the case for the TCR, and therefore, precisely how TCR downregulation maintains or limits T cell responses is controversial. Here, we observed that in vitro expanded primary human T cells exhibit perfect adaptation in cytokine production to constant antigen stimulation across a 100,000-fold variation in affinity with partial TCR downregulation. By directly fitting a mechanistic model to the data, we show that TCR downregulation produces imperfect adaptation, but when coupled to a switch produces perfect adaptation in cytokine production. A pre diction of the model is that pMHC-induced TCR signalling continues after adaptation and this is confirmed by showing that, while costimulation cannot prevent adaptation, CD28 and 4-1BB signalling reactivated adapted T cells to produce cytokines in a pMHC-dependent manner. We show that adaptation also applied to 1st generation chimeric antigen receptor (CAR)-T cells but is partially avoided in 2nd generation CARs. These findings highlight that even partial TCR downregulation can limit T cell responses by producing perfect adaptation rendering T cells dependent on costimulation for sustained responses.

12. A.L. Williams, J.E. Fitzgerald, F. Ivich, E.D. Sontag, and M. Niedre. Comment on In vivo flow cytometry reveals a circadian rhythm of circulating tumor cells. npg Light: Science & Applications, 10:188, 2021. [PDF] Keyword(s): circulating tumor cells, liquid biopsy, cancer, oncology, multiple myeloma, systems biology.
Abstract:
 Correspondence regarding circulating tumor cell detection

 2020
1. E.D. Sontag. Input-to-State Stability. In J. Baillieul and T. Samad, editors, Encyclopedia of Systems and Control, pages 1-9. Springer-Verlag, 2020. [PDF] Keyword(s): input to state stability, integral input to state stability, iISS, ISS, input to output stability.
Abstract:
 The notion of input to state stability (ISS) qualitatively describes stability of the mapping from initial states and inputs to internal states (and more generally outputs). This encyclopedia-style article entry gives a brief introduction to the definition of ISS and a discussion of equivalent characterizations. It is an update of the article in the 2015 edition, including additional citations to recent PDE work.

2. E.D. Sontag. Scale-invariance in biological sensing. In J. Baillieul and T. Samad, editors, Encyclopedia of Systems and Control, pages S1-. Springer-Verlag, 2020. [PDF] [doi:10.1007/978-1-4471-5102-9_100090-1] Keyword(s): adaptation, biological adaptation, perfect adaptation, fold-change detection.
Abstract:
 The phenomenon of fold-change detection, or scale-invariance, is exhibited by a variety of sensory systems, in both bacterial and eukaryotic signaling pathways. This encyclopedia-style article gives a brief introduction to the subject.

3. D.K. Agrawal, E.M. Dolan, N.E. Hernandez, K.M. Blacklock, S.D. Khare, and E.D. Sontag. Mathematical models of protease-based enzymatic biosensors. ACS Synthetic Biology, 9:198-208, 2020. [PDF] Keyword(s): synthetic biology, protease-based circuits, enzymatic circuits, systems biology, Boolean circuits.
Abstract:
 An important goal of synthetic biology is to build biosensors and circuits with well-defined input-output relationships that operate at speeds found in natural biological systems. However, for molecular computation, most commonly used genetic circuit elements typically involve several steps from input detection to output signal production: transcription, translation, and post-translational modifications. These multiple steps together require up to several hours to respond to a single stimulus, and this limits the overall speed and complexity of genetic circuits. To address this gap, molecular frameworks that rely exclusively on post-translational steps to realize reaction networks that can process inputs at a time scale of seconds to minutes have been proposed. Here, we build mathematical models of fast biosensors capable of producing Boolean logic functionality. We employ protease-based chemical and light-induced switches, investigate their operation, and provide selection guidelines for their use as on-off switches. As a proof of concept, we implement a rapamycin-induced switch in vitro and demonstrate that its response qualitatively agrees with the predictions from our models. We then use these switches as elementary blocks, developing models for biosensors that can perform OR and XOR Boolean logic computation while using reaction conditions as tuning parameters. We use sensitivity analysis to determine the time-dependent sensitivity of the output to proteolytic and protein-protein binding reaction parameters. These fast protease-based biosensors can be used to implement complex molecular circuits with a capability of processing multiple inputs controllably and algorithmically. Our framework for evaluating and optimizing circuit performance can be applied to other molecular logic circuits.

4. M.A. Al-Radhawi, D. Angeli, and E.D. Sontag. A computational framework for a Lyapunov-enabled analysis of biochemical reaction networks. PLoS Computational Biology, pp 16(2): e1007681, 2020. [PDF] Keyword(s): MAPK cascades, Lyapunov functions, stability, chemical networks, chemical rection networks, systems biology, RFM, ribosome flow model.
Abstract:
 This paper deals with the analysis of the dynamics of chemical reaction networks, developing a theoretical framework based only on graphical knowledge and applying regardless of the particular form of kinetics. It paper introduces a class of networks that are "structurally (mono) attractive", by which we mean that they are incapable of exhibiting multiple steady states, oscillation, or chaos by the virtue of their reaction graphs. These networks are characterized by the existence of a universal energy-like function which we call a Robust Lyapunov function (RLF). To find such functions, a finite set of rank-one linear systems is introduced, which form the extremals of a linear convex cone. The problem is then reduced to that of finding a common Lyapunov function for this set of extremals. Based on this characterization, a computational package, Lyapunov-Enabled Analysis of Reaction Networks (LEARN), is provided that constructs such functions or rules out their existence. An extensive study of biochemical networks demonstrates that LEARN offers a new unified framework. We study basic motifs, three-body binding, and transcriptional networks. We focus on cellular signalling networks including various post-translational modification cascades, phosphotransfer and phosphorelay networks, T-cell kinetic proofreading, ERK signaling, and the Ribosome Flow Model.

5. M. Ali Al-Radhawi, A.P. Tran, E. Ernst, T. Chen, C.A. Voigt, and E.D. Sontag. Distributed implementation of Boolean functions by transcriptional synthetic circuits. ACS Synthetic Biology, 9:2172-2187, 2020. [PDF] [doi:10.1021/acssynbio.0c00228] Keyword(s): synthetic biology, transcriptional networks, gene networks, boolean circuits, boolean gates, systems biology.
Abstract:
 Starting in the early 2000s, sophisticated technologies have been developed for the rational construction of synthetic genetic networks that implement specified logical functionalities. Despite impressive progress, however, the scaling necessary in order to achieve greater computational power has been hampered by many constraints, including repressor toxicity and the lack of large sets of mutually-orthogonal repressors. As a consequence, a typical circuit contains no more than roughly seven repressor-based gates per cell. A possible way around this scalability problem is to distribute the computation among multiple cell types, which communicate among themselves using diffusible small molecules (DSMs) and each of which implements a small sub-circuit. Examples of DSMs are those employed by quorum sensing systems in bacteria. This paper focuses on systematic ways to implement this distributed approach, in the context of the evaluation of arbitrary Boolean functions. The unique characteristics of genetic circuits and the properties of DSMs require the development of new Boolean synthesis methods, distinct from those classically used in electronic circuit design. In this work, we propose a fast algorithm to synthesize distributed realizations for any Boolean function, under constraints on the number of gates per cell and the number of orthogonal DSMs. The method is based on an exact synthesis algorithm to find the minimal circuit per cell, which in turn allows us to build an extensive database of Boolean functions up to a given number of inputs. For concreteness, we will specifically focus on circuits of up to 4 inputs, which might represent, for example, two chemical inducers and two light inputs at different frequencies. Our method shows that, with a constraint of no more than seven gates per cell, the use of a single DSM increases the total number of realizable circuits by at least 7.58-fold compared to centralized computation. Moreover, when allowing two DSM's, one can realize 99.995\% of all possible 4-input Boolean functions, still with at most 7 gates per cell. The methodology introduced here can be readily adapted to complement recent genetic circuit design automation software.

6. T. Chen, M. Ali Al-Radhawi, and E.D. Sontag. A mathematical model exhibiting the effect of DNA methylation on the stability boundary in cell-fate networks. Epigenetics, 15:1-22, 2020. Note: PMID: 32842865. [PDF] [doi:10.1080/15592294.2020.1805686] Keyword(s): methylation, differentiation, epigenetics, pluripotent cells, gene regulatory networks, bistability, bistability, systems biology.
Abstract:
 Cell-fate networks are traditionally studied within the framework of gene regulatory networks. This paradigm considers only interactions of genes through expressed transcription factors and does not incorporate chromatin modification processes. This paper introduces a mathematical model that seamlessly combines gene regulatory networks and DNA methylation, with the goal of quantitatively characterizing the contribution of epigenetic regulation to gene silencing. The Basin of Attraction percentage'' is introduced as a metric to quantify gene silencing abilities. As a case study, a computational and theoretical analysis is carried out for a model of the pluripotent stem cell circuit as well as a simplified self-activating gene model. The results confirm that the methodology quantitatively captures the key role that methylation plays in enhancing the stability of the silenced gene state.

7. J.L. Gevertz, J.M. Greene, C Hixahuary Sanchez Tapia, and E D Sontag. A novel COVID-19 epidemiological model with explicit susceptible and asymptomatic isolation compartments reveals unexpected consequences of timing social distancing. Journal of Theoretical Biology, 510:110539, 2020. [WWW] [PDF] Keyword(s): epidemiology, COVID-19, COVID, systems biology.
Abstract:
 Motivated by the current COVID-19 epidemic, this work introduces an epidemiological model in which separate compartments are used for susceptible and asymptomatic "socially distant" populations. Distancing directives are represented by rates of flow into these compartments, as well as by a reduction in contacts that lessens disease transmission. The dynamical behavior of this system is analyzed, under various different rate control strategies, and the sensitivity of the basic reproduction number to various parameters is studied. One of the striking features of this model is the existence of a critical implementation delay in issuing separation mandates: while a delay of about four weeks does not have an appreciable effect, issuing mandates after this critical time results in a far greater incidence of infection. In other words, there is a nontrivial but tight "window of opportunity" for commencing social distancing. Different relaxation strategies are also simulated, with surprising results. Periodic relaxation policies suggest a schedule which may significantly inhibit peak infective load, but that this schedule is very sensitive to parameter values and the schedule's frequency. Further, we considered the impact of steadily reducing social distancing measures over time. We find that a too-sudden reopening of society may negate the progress achieved under initial distancing guidelines, if not carefully designed.

8. J. M. Greene, C. Sanchez-Tapia, and E.D. Sontag. Mathematical details on a cancer resistance model. Frontiers in Bioengineering and Biotechnology, 8:501: 1-27, 2020. [PDF] [doi:10.3389/fbioe.2020.00501] Keyword(s): resistance, chemotherapy, phenotype, optimal control, singular controls, cancer, oncology, systems biology.
Abstract:
 One of the most important factors limiting the success of chemotherapy in cancer treatment is the phenomenon of drug resistance. We have recently introduced a framework for quantifying the effects of induced and non-induced resistance to cancer chemotherapy. In this work, we expound on the details relating to an optimal control problem outlined in our previous paper (Greene et al., 2018). The control structure is precisely characterized as a concatenation of bang-bang and path-constrained arcs via the Pontryagin Maximum Principle and differential Lie algebraic techniques. A structural identifiability analysis is also presented, demonstrating that patient-specific parameters may be measured and thus utilized in the design of optimal therapies prior to the commencement of therapy. For completeness, a detailed analysis of existence results is also included.

9. E.A. Hernandez-Vargas, G. Giordano, E. D. Sontag, J. G. Chase, H. Chang, and A. Astolfi. First special section on systems and control research efforts against COVID-19 and future pandemics. Annual Reviews in Control, 50:343-344, 2020. [WWW] [doi:https://doi.org/10.1016/j.arcontrol.2020.10.007] Keyword(s): SARS-CoV-2, COVID-19, Modelling, Control, Pandemics.

10. J. Miller, M.A. Al-Radhawi, and E.D. Sontag. Mediating ribosomal competition by splitting pools. IEEE Control Systems Letters, 5:1555-1560, 2020. [PDF] Keyword(s): systems biology, synthetic biology, ribosomes, RFM, ribosome flow model.
Abstract:
 Synthetic biology constructs often rely upon the introduction of "circuit" genes into host cells, in order to express novel proteins and thus endow the host with a desired behavior. The expression of these new genes "consumes" existing resources in the cell, such as ATP, RNA polymerase, amino acids, and ribosomes. Ribosomal competition among strands of mRNA may be described by a system of nonlinear ODEs called the Ribosomal Flow Model (RFM). The competition for resources between host and circuit genes can be ameliorated by splitting the ribosome pool by use of orthogonal ribosomes, where the circuit genes are exclusively translated by mutated ribosomes. In this work, the RFM system is extended to include orthogonal ribosome competition. This Orthogonal Ribosomal Flow Model (ORFM) is proven to be stable through the use of Robust Lyapunov Functions. The optimization problem of maximizing the weighted protein translation rate by adjusting allocation of ribosomal species is formulated and implemented. Note: publsihed Nov 2020, even though journal reprint says "Nov 2021".

11. E.D. Sontag. Bell-shaped dose response for a system with no IFFLs. bioRxiv, 2020. [PDF] Keyword(s): IFFL, feedforward loops, nonlinear systems, immunology.
Abstract:
 It is well known that the presence of an incoherent feedforward loop (IFFL) in a network may give rise to a steady state non-monotonic dose response. This note shows that the converse implication does not hold. It gives an example of a three-dimensional system that has no IFFLs, yet its dose response is bell-shaped. It also studies under what conditions the result is true for two-dimensional systems, in the process recovering, in far more generality, a result given in the T-cell activation literature.

12. A.P. Tran, M. Ali Al-Radhawi, I. Kareva, J. Wu, D.J. Waxman, and E.D. Sontag. Delicate balances in cancer chemotherapy: Modeling immune recruitment and emergence of systemic drug resistance. Frontiers in Immunology, 11:1376-, 2020. [PDF] [doi:10.3389/fimmu.2020.01376] Keyword(s): metronomic chemotherapy, cyclophosphamide, mathematical modeling, immune recruitment, cancer, resistance, oncology, immunology, systems biology.
Abstract:
 Metronomic chemotherapy can drastically enhance immunogenic tumor cell death. However, the responsible mechanisms are still incompletely understood. Here, we develop a mathematical model to elucidate the underlying complex interactions between tumor growth, immune system activation, and therapy-mediated immunogenic cell death. Our model is conceptually simple, yet it provides a surprisingly excellent fit to empirical data obtained from a GL261 mouse glioma model treated with cyclophosphamide on a metronomic schedule. The model includes terms representing immune recruitment as well as the emergence of drug resistance during prolonged metronomic treatments. Strikingly, a fixed set of parameters, not adjusted for individuals nor for drug schedule, excellently recapitulates experimental data across various drug regimens, including treatments administered at intervals ranging from 6 to 12 days. Additionally, the model predicts peak immune activation times, rediscovering experimental data that had not been used in parameter fitting or in model construction. The validated model was then used to make predictions about expected tumor-immune dynamics for novel drug administration schedules. Notably, the validated model suggests that immunostimulatory and immunosuppressive intermediates are responsible for the observed phenomena of resistance and immune cell recruitment, and thus for variation of responses with respect to different schedules of drug administration.

13. A.L. Williams, J.E. Fitzgerald, F. Ivich, E.D. Sontag, and M. Niedre. Short-term circulating tumor cell dynamics in mouse xenograft models and implications for liquid biopsy. Frontiers in Oncology, 10:2447-, 2020. [PDF] [doi:10.3389/fonc.2020.601085] Keyword(s): circulating tumor cells, liquid biopsy, cancer, oncology, multiple myeloma, systems biology.
Abstract:
 Circulating tumor cells (CTCs) are widely studied using liquid biopsy methods that analyze single, fractionally-small peripheral blood (PB) samples. However, little is known about fluctuations in CTC numbers that occur over short timescales in vivo, and how these may affect accurate enumeration from blood samples. Diffuse in vivo flow cytometry (DiFC) developed by the Niedre lab allows continuous, non-invasive counting of rare, green fluorescent protein expressing CTCs in large deeply-seated blood vessels in mice. Here, DiFC is used to study short-term changes in CTC numbers in multiple myeloma and Lewis lung carcinoma xenograft models. Both 35- to 50-minute data sets are analyzed, with intervals corresponding to approximately 1, 5, 10 and 20\% of the PB volume, as well as changes over 24-hour periods. For rare CTCs, the use of short DiFC intervals (corresponding to small PB samples) frequently resulted in no detections. For more abundant CTCs, CTC numbers frequently varied by an order of magnitude or more over the time-scales considered. This variability far exceeded that expected by Poisson statistics, and instead was consistent with rapidly changing mean numbers of CTCs in the PB. Because of these natural temporal changes, accurately enumerating CTCs from fractionally small blood samples is inherently problematic. The problem is likely to be compounded for multicellular CTC clusters or specific CTC subtypes. However, it is also shown that enumeration can be improved by averaging multiple samples, analysis of larger volumes, or development of new methods for enumeration of CTCs directly in vivo.

 2019
1. D.K. Agrawal, R. Marshall, V. Noireaux, and E.D. Sontag. In vitro implementation of robust gene regulation in a synthetic biomolecular integral controller. Nature Communications, 10:1-12, 2019. [PDF] Keyword(s): tracking, synthetic biology, integral feedback, TX/TL, systems biology, dynamical systems, adaptation, internal model principle, identifiability.
Abstract:
 Cells respond to biochemical and physical internal as well as external signals. These signals can be broadly classified into two categories: (a) actionable'' or reference'' inputs that should elicit appropriate biological or physical responses such as gene expression or motility, and (b) disturbances'' or perturbations'' that should be ignored or actively filtered-out. These disturbances might be exogenous, such as binding of nonspecific ligands, or endogenous, such as variations in enzyme concentrations or gene copy numbers. In this context, the term robustness describes the capability to produce appropriate responses to reference inputs while at the same time being insensitive to disturbances. These two objectives often conflict with each other and require delicate design trade-offs. Indeed, natural biological systems use complicated and still poorly understood control strategies in order to finely balance the goals of responsiveness and robustness. A better understanding of such natural strategies remains an important scientific goal in itself and will play a role in the construction of synthetic circuits for therapeutic and biosensing applications. A prototype problem in robustly responding to inputs is that of robust tracking'', defined by the requirement that some designated internal quantity (for example, the level of expression of a reporter protein) should faithfully follow an input signal while being insensitive to an appropriate class of perturbations. Control theory predicts that a certain type of motif, called integral feedback, will help achieve this goal, and this motif is, in fact, a necessary feature of any system that exhibits robust tracking. Indeed, integral feedback has always been a key component of electrical and mechanical control systems, at least since the 18th century when James Watt employed the centrifugal governor to regulate steam engines. Motivated by this knowledge, biological engineers have proposed various designs for biomolecular integral feedback control mechanisms. However, practical and quantitatively predictable implementations have proved challenging, in part due to the difficulty in obtaining accurate models of transcription, translation, and resource competition in living cells, and the stochasticity inherent in cellular reactions. These challenges prevent first-principles rational design and parameter optimization. In this work, we exploit the versatility of an Escherichia coli cell-free transcription-translation (TXTL) to accurately design, model and then build, a synthetic biomolecular integral controller that precisely controls the expression of a target gene. To our knowledge, this is the first design of a functioning gene network that achieves the goal of making gene expression track an externally imposed reference level, achieves this goal even in the presence of disturbances, and whose performance quantitatively agrees with mathematical predictions.

2. M. A. Al-Radhawi, D. Del Vecchio, and E. D. Sontag. Multi-modality in gene regulatory networks with slow gene binding. PLoS Computational Biology, 15:e1006784, 2019. [PDF] Keyword(s): multistability, gene networks, Markov Chains, Master Equation, cancer heterogeneity, phenotypic variation, nonlinear systems, stochastic systems, epigenetics, chemical master equations, systems biology.
Abstract:
 In biological processes such as embryonic development, hematopoietic cell differentiation, and the arising of tumor heterogeneity and consequent resistance to therapy, mechanisms of gene activation and deactivation may play a role in the emergence of phenotypically heterogeneous yet genetically identical (clonal) cellular populations. Mathematically, the variability in phenotypes in the absence of genetic variation can be modeled through the existence of multiple metastable attractors in nonlinear systems subject with stochastic switching, each one of them associated to an alternative epigenetic state. An important theoretical and practical question is that of estimating the number and location of these states, as well as their relative probabilities of occurrence. This paper focuses on a rigorous analytic characterization of multiple modes under slow promoter kinetics, which is a feature of epigenetic regulation. It characterizes the stationary distributions of Chemical Master Equations for gene regulatory networks as a mixture of Poisson distributions. As illustrations, the theory is used to tease out the role of cooperative binding in stochastic models in comparison to deterministic models, and applications are given to various model systems, such as toggle switches in isolation or in communicating populations and a trans-differentiation network.

3. J.M. Greene, J.L. Gevertz, and E. D. Sontag. A mathematical approach to distinguish spontaneous from induced evolution of drug resistance during cancer treatment. JCO Clinical Cancer Informatics, DOI: 10.1200/CCI.18.00087:1-20, 2019. [PDF] Keyword(s): cancer heterogeneity, phenotypic variation, nonlinear systems, epigenetics, oncology, cancer, systems biology.
Abstract:
 Resistance to chemotherapy is a major impediment to the successful treatment of cancer. Classically, resistance has been thought to arise primarily through random genetic mutations, after which mutated cells expand via Darwinian selection. However, recent experimental evidence suggests that the progression to resistance need not occur randomly, but instead may be induced by the therapeutic agent itself.This process of resistance induction can be a result of genetic changes, or can occur through epigenetic alterations that cause otherwise drug-sensitive cancer cells to undergo phenotype switching''. This relatively novel notion of resistance further complicates the already challenging task of designing treatment protocols that minimize the risk of evolving resistance. In an effort to better understand treatment resistance, we have developed a mathematical modeling framework that incorporates both random and drug-induced resistance. Our model demonstrates that the ability (or lack thereof) of a drug to induce resistance can result in qualitatively different responses to the same drug dose and delivery schedule. The importance of induced resistance in treatment response led us to ask if, in our model, one can determine the resistance induction rate of a drug for a given treatment protocol. Not only could we prove that the induction parameter in our model is theoretically identifiable, we have also proposed a possible in vitro experiment which could practically be used to determine a treatment's propensity to induce resistance.

4. M. Margaliot and E.D. Sontag. Revisiting totally positive differential systems: A tutorial and new results. Automatica, 101:1-14, 2019. [PDF] Keyword(s): tridiagonal systems, cooperative systems, monotone systems.
Abstract:
 A matrix is totally nonnegative (resp., totally positive) if all its minors are nonnegative (resp., positive). This paper draws connections between B. Schwarz's 1970 work on TN and TP matrices to Smillie's 1984 and Smith's 1991 work on stability of nonlinear tridiagonal cooperative systems, simplifying proofs in the later paper and suggesting new research questions.

5. E.V. Nikolaev, A. Zloza, and E.D. Sontag. Immunobiochemical reconstruction of influenza lung infection - melanoma skin cancer interactions. Frontiers in Immunology, 10:Article 4, 2019. [PDF] Keyword(s): oncology, cancer, infections, immunology, checkpoint inhibition, systems biology.
Abstract:
 Recent experimental results from the Zloza lab combined a mouse model of influenza A virus (IAV) infection (A/H1N1/PR8) and a highly aggressive model of infection-unrelated cancer, B16-F10 skin melanoma. This paper showed that acute influenza infection of the lung promotes distal melanoma growth in the dermis of the flank and leads to decreased host survival. Here, we proceed to ground the experimental observations in a mechanistic immunobiochemical model that incorporates the T cell receptor signaling pathway, various transcription factors, and a gene regulatory network (GRN). A core component of our model is a biochemical motif, which we call a Triple Incoherent Feed-Forward Loop (TIFFL), and which reflects known interactions between IRF4, Blimp-1, and Bcl-6. The different activity levels of the TIFFL components, as a function of the cognate antigen levels and the given inflammation context, manifest themselves in phenotypically distinct outcomes. Specifically, both the TIFFL reconstruction and quantitative estimates obtained from the model allowed us to formulate a hypothesis that it is the loss of the fundamental TIFFL-induced adaptation of the expression of PD-1 receptors on anti-melanoma CD8+ T cells that constitutes the essence of the previously unrecognized immunologic factor that promotes the experimentally observed distal tumor growth in the presence of acute non-ocogenic infection. We therefore hope that this work can further highlight the importance of adaptive mechanisms by which immune functions contribute to the balance between self and non-self immune tolerance, adaptive resistance, and the strength of TCR-induced activation, thus contributing to the understanding of a broader complexity of fundamental interactions between pathogens and tumors.

6. M. Sadeghi, M.A. Al-Radhawi, M. Margaliot, and E.D. Sontag. No switching policy is optimal for a positive linear system with a bottleneck entrance. IEEE Control Systems Letters, 3:889-894, 2019. Note: (Also in Proc. 2019 IEEE Conf. Decision and Control.). [PDF] Keyword(s): entrainment, switched systems, RFM, ribosome flow model, traffic systems, nonlinear systems, nonlinear control.
Abstract:
 We consider a nonlinear SISO system that is a cascade of a scalar "bottleneck entrance" with a stable positive linear system. In response to any periodic inflow, all solutions converge to a unique periodic solution with the same period. We study the problem of maximizing the averaged throughput via controlled switching. We compare two strategies: 1) switching between a high and low value, and 2 ~using a constant inflow equal to the prescribed mean value. We show that no possible switching policy can outperform a constant inflow rate, though it can approach it asymptotically. We describe several potential applications of this problem in traffic systems, ribosome flow models, and scheduling at security checks.

7. S. Wang, J.-R. Lin, E.D. Sontag, and P.K. Sorger. Inferring reaction network structure from single-cell, multiplex data, using toric systems theory. PLoS Computational Biology, 15:e1007311, 2019. [WWW] [PDF] Keyword(s): chemical reaction networks, stoichiometry, complex balancing, toric varieties, systems biology.
Abstract:
 The goal of many single-cell studies on eukaryotic cells is to gain insight into the biochemical reactions that control cell fate and state. This paper introduces the concept of effective stoichiometric space (ESS) to guide the reconstruction of biochemical networks from multiplexed, fixed time-point, single-cell data. In contrast to methods based solely on statistical models of data, the ESS method leverages the power of the geometric theory of toric varieties to begin unraveling the structure of chemical reaction networks (CRN). This application of toric theory enables a data-driven mapping of covariance relationships in single cell measurements into stoichiometric information, one in which each cell subpopulation has its associated ESS interpreted in terms of CRN theory. In the development of ESS we reframe certain aspects of the theory of CRN to better match data analysis. As an application of our approach we process cytomery- and image-based single-cell datasets and identify differences in cells treated with kinase inhibitors. Our approach is directly applicable to data acquired using readily accessible experimental methods such as Fluorescence Activated Cell Sorting (FACS) and multiplex immunofluorescence.

 2018
1. E.D. Sontag. Examples of computation of exact moment dynamics for chemical reaction networks. In R. Tempo, S. Yurkovich, and P. Misra, editors, Emerging Applications of Control and Systems Theory, volume 473 of Lecture Notes in Control and Inform. Sci., pages 295-312. Springer-Verlag, Berlin, 2018. [PDF] Keyword(s): chemical master equations, stochastic systems, moments, chemical reaction networks, incoherent feedforward loop, feedforward, IFFL, systems biology.
Abstract:
 The study of stochastic biomolecular networks is a key part of systems biology, as such networks play a central role in engineered synthetic biology constructs as well as in naturally occurring cells. This expository paper reviews in a unified way a pair of recent approaches to the finite computation of statistics for chemical reaction networks.

2. D. Del Vecchio, Y. Qian, R.M Murray, and E.D. Sontag. Future systems and control research in synthetic biology. Annual Reviews in Control, 45:5-17, 2018. [PDF] Keyword(s): synthetic biology, systems biology.
Abstract:
 This paper is a review of systems and control problems in synthetic biology, focusing on past accomplishments and open problems. It is partially a report on the workshop "The Compositionality Problem in Synthetic Biology: New Directions for Control Theory" held on June 26-27, 2017 at MIT, and organized by D. Del Vecchio, R. M. Murray, and E. D. Sontag

3. E.V. Nikolaev, S.J. Rahi, and E.D. Sontag. Chaos in simple periodically-forced biological models. Biophysical Journal, 114:1232-1240, 2018. [PDF] Keyword(s): chaos, entrainment, systems biology, periodic inputs, subharmonic responses, biochemical systems, forced oscillations.
Abstract:
 What complicated dynamics can arise in the simplest biochemical systems, in response to a periodic input? This paper discusses two models that commonly appear as components of larger sensing and signal transduction pathways in systems biology: a simple two-species negative feedback loop, and a prototype nonlinear integral feedback. These systems have globally attracting steady states when unforced, yet, when subject to a periodic excitation, subharmonic responses and strange attractors can arise via period-doubling cascades. These behaviors are similar to those exhibited by classical forced nonlinear oscillators such as those described by van der Pol or Duffing equations. The lack of entrainment to external oscillations, in even the simplest biochemical networks, represents a level of additional complexity in molecular biology.

4. T.H. Segall-Shapiro, E. D. Sontag, and C. A. Voigt. Engineered promoters enable constant gene expression at any copy number in bacteria. Nature Biotechnology, 36:352-358, 2018. [PDF] Keyword(s): synthetic biology, systems biology, genetic circuits, gene copy number, incoherent feedforward loop, feedforward, IFFL.
Abstract:
 This paper deals with the design of promoters that maintain constant levels of expression, whether they are carried at single copy in the genome or on high-copy plasmids. The design is based on an incoherent feedforward loop (iFFL) with a perfectly non-cooperative repression. The circuits are implemented in E. coli using Transcription Activator Like Effectors (TALEs). The resulting stabilized promoters generate near identical expression across different genome locations and plasmid backbones (pSC101, p15a, ColE1, pUC), and also provide robustness to strain mutations and growth media. Further, their strength is tunable and can be used to maintain constant ratios between proteins.

5. Y. Zarai, M. Margaliot, E.D. Sontag, and T. Tuller. Controllability analysis and control synthesis for the ribosome flow model. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 15:1351-1364, 2018. [PDF] Keyword(s): systems biology, ribosomes, controllability, RFM, ribosome flow model.
Abstract:
 The ribosomal density along the coding region of the mRNA molecule affects various fundamental intracellular phenomena including: protein production rates, organismal fitness, ribosomal drop off, and co-translational protein folding. Thus, regulating translation in order to obtain a desired ribosomal profile along the mRNA molecule is an important biological problem. This paper studies this problem formulated in the context of the ribosome flow model (RFM) in which one views the transition rates between site as controls.

 2017
1. M. Margaliot, E.D. Sontag, and T. Tuller. Checkable conditions for contraction after small transients in time and amplitude. In N. Petit, editor, Feedback Stabilization of Controlled Dynamical Systems - In Honor of Laurent Praly, volume 473 of Lecture Notes in Control and Inform. Sci., pages 279-305. Springer-Verlag, Berlin, 2017. [PDF] Keyword(s): contractions, contractive systems, stability.
Abstract:
 This is an expository paper, which compares in detail various alternative weak contraction ideas for nonlinear system stability.

2. S. Barish, M.F. Ochs, E.D. Sontag, and J.L. Gevertz. Evaluating optimal therapy robustness by virtual expansion of a sample population, with a case study in cancer immunotherapy. Proc Natl Acad Sci USA, 114:E6277-E6286, 2017. [WWW] [PDF] [doi:10.1073/pnas.1703355114] Keyword(s): cancer, oncolytic therapy, immunotherapy, optimal therapy, identifiability, systems biology.
Abstract:
 This paper proposes a technique that combines experimental data, mathematical modeling, and statistical analyses for identifying optimal treatment protocols that are robust with respect to individual variability. Experimental data from a small sample population is amplified using bootstrapping to obtain a large number of virtual populations that statistically match the expected heterogeneity. Alternative therapies chosen from among a set of clinically-realizable protocols are then compared and scored according to coverage. As proof of concept, the method is used to evaluate a treatment with oncolytic viruses and dendritic cell vaccines in a mouse model of melanoma. The analysis shows that while every scheduling variant of an experimentally-utilized treatment protocol is fragile (non-robust), there is an alternative region of dosing space (lower oncolytic virus dose, higher dendritic cell dose) for which a robust optimal protocol exists.

3. J. K. Kim and E.D. Sontag. Reduction of multiscale stochastic biochemical reaction networks using exact moment derivation. PLoS Computational Biology, 13:13(6): e1005571, 2017. [PDF] Keyword(s): systems biology, biochemical networks, stochastic systems, chemical master equation, chemical reaction networks, moments, molecular networks, complex-balanced networks.
Abstract:
 Biochemical reaction networks in cells frequently consist of reactions with disparate timescales. Stochastic simulations of such multiscale BRNs are prohibitively slow due to the high computational cost incurred in the simulations of fast reactions. One way to resolve this problem is to replace fast species by their stationary conditional expectation values conditioned on slow species. While various approximations schemes for this quasi-steady state approximation have been developed, they often lead to considerable errors. This paper considers two classes of multiscale BRNs which can be reduced by through an exact QSS rather than approximations. Specifically, we assume that fast species constitute either a feedforward network or a complex balanced network. Exact reductions for various examples are derived, and the computational advantages of this approach are illustrated through simulations.

4. M. Lang and E.D. Sontag. Zeros of nonlinear systems with input invariances. Automatica, 81:46-55, 2017. [PDF] Keyword(s): scale invariance, fold change detection, nonlinear systems, realization theory, internal model principle.
Abstract:
 This paper introduces two generalizations of systems invariant with respect to continuous sets of input transformations, that is, systems whose output dynamics remain invariant when applying a transformation to the input and simultaneously adjusting the initial conditions. These generalizations concern systems invariant with respect to time-dependent input transformations with exponentially increasing or decreasing strength'', and systems invariant with respect to transformations of the "nonlinear derivatives" of the input. Interestingly, these two generalizations of invariant systems encompass linear time-invariant (LTI) systems with real transfer function zeros of arbitrary multiplicity. Furthermore, the zero-dynamics of systems possessing our generalized invariances show properties analogous to those of LTI systems with transfer function zeros, generalizing concepts like pole-zero cancellation, the rejection of ramps by Hurwitz LTI systems with a zero at the origin with multiplicity two, and (to a certain extend) the superposition principle with respect to inputs zeroing the output.

5. F. Menolascina, R. Rusconi, V.I. Fernandez, S.P. Smriga, Z. Aminzare, E. D. Sontag, and R. Stocker. Logarithmic sensing in Bacillus subtilis aerotaxis. Nature Systems Biology and Applications, 3:16036-, 2017. [PDF] Keyword(s): adaptation, biological adaptation, perfect adaptation, Aerotaxis, chemotaxis, scale invariance, FCD, fold-change detection, B. subtilis, systems biology.
Abstract:
 Aerotaxis, the directed migration along oxygen gradients, allows many microorganisms to locate favorable oxygen concentrations. Despite oxygen's fundamental role for life, even key aspects of aerotaxis remain poorly understood. In Bacillus subtilis, for example, there is conflicting evidence of whether migration occurs to the maximal oxygen concentration available or to an optimal intermediate one, and how aerotaxis can be maintained over a broad range of conditions. Using precisely controlled oxygen gradients in a microfluidic device, spanning the full spectrum of conditions from quasi-anoxic to oxic (60nM-1mM), we resolved B. subtilis' oxygen preference conundrum'' by demonstrating consistent migration towards maximum oxygen concentrations. Surprisingly, the strength of aerotaxis was largely unchanged over three decades in oxygen concentration (131nM-196mM). We discovered that in this range B. subtilis responds to the logarithm of the oxygen concentration gradient, a log-sensing strategy that affords organisms high sensitivity over a wide range of conditions.

6. V. H. Nagaraj, J. M. Greene, A. M. Sengupta, and E.D. Sontag. Translation inhibition and resource balance in the TX-TL cell-free gene expression system. Synthetic Biology, 2:ysx005, 2017. [PDF] Keyword(s): tx/tl, cell-free systems, in vitro synthetic biology, synthetic biology, systems biology.
Abstract:
 Utilizing the synthetic transcription-translation (TX-TL) system, this paper studies the impact of nucleotide triphosphates (NTPs) and magnesium (Mg2+), on gene expression, in the context of the counterintuitive phenomenon of suppression of gene expression at high NTP concentration. Measuring translation rates for different Mg2+ and NTP concentrations, we observe a complex resource dependence. We demonstrate that translation is the rate-limiting process that is directly inhibited by high NTP concentrations. Additional Mg2+ can partially reverse this inhibition. In several experiments, we observe two maxima of the translation rate viewed as a function of both Mg2+ and NTP concentration, which can be explained in terms of an NTP-independent effect on the ribosome complex and an NTP- Mg2+ titration effect. The non-trivial compensatory effects of abundance of different vital resources signals the presence of complex regulatory mechanisms to achieve optimal gene expression.

7. S. J. Rahi, J. Larsch, K. Pecani, N. Mansouri, A. Y. Katsov, K. Tsaneva-Atanasova, E. D. Sontag, and F. R. Cross. Oscillatory stimuli differentiate adapting circuit topologies. Nature Methods, 14:1010-1016, 2017. [PDF] Keyword(s): biochemical networks, periodic behaviors, monotone systems, entrainment, oscillations, incoherent feedforward loop, feedforward, IFFL, systems biology.
Abstract:
 Elucidating the structure of biological intracellular networks from experimental data remains a major challenge. This paper studies two types of response signatures'' to identify specific circuit motifs, from the observed response to periodic inputs. In particular, the objective is to distinguish negative feedback loops (NFLs) from incoherent feedforward loops (IFFLs), which are two types of circuits capable of producing exact adaptation. The theory of monotone systems with inputs is used to show that period skipping'' (non-harmonic responses) is ruled out in IFFL's, and a notion called refractory period stabilization'' is also analyzed. The approach is then applied to identify a circuit dominating cell cycle timing in yeast, and to uncover a calcium-mediated NFL circuit in \emph{C.elegans} olfactory sensory neurons.

8. A. Rendall and E. D. Sontag. Multiple steady states and the form of response functions to antigen in a model for the initiation of T cell activation. Royal Society Open Science, 4:170821-, 2017. [PDF]
Abstract:
 This paper analizes a model for the initial stage of T cell activation. The state variables in the model are the concentrations of phosphorylation states of the T cell receptor complex and the phosphatase SHP-1 in the cell. It is shown that these quantities cannot approach zero, and that there is more than one positive steady state for certain values of the parameters; in addition, damped oscillations are possible. It is also shown that the chemical concentration which represents the degree of activation of the cell, represented by the maximally phosphorylated form of the T cell receptor complex, is in general a non-monotone function of the activating signal. In particular there are cases where there is a value of the dissociation constant of the ligand from the receptor which produces an optimal activation of the T cell. In this way the results of certain simulations in the literature have been confirmed rigorously and new features are discovered.

9. A. Silva, M. Silva, P. Sudalagunta, A. Distler, T. Jacobson, A. Collins, T. Nguyen, J. Song, D.T. Chen, Lu Chen, C. Cubitt, R. Baz, L. Perez, D. Rebatchouk, W. Dalton, J.M. Greene, R. Gatenby, R. Gillies, E.D. Sontag, M. Meads, and K. Shain. An ex vivo platform for the prediction of clinical response in multiple myeloma. Cancer Research, pp 10.1158/0008-5472.CAN-17-0502, 2017. [PDF] Keyword(s): cancer, multiple myeloma, personalized therapy.
Abstract:
 This paper describes a novel approach for characterization of chemosensitivity and prediction of clinical response in multiple myeloma. It relies upon a patient-specific computational model of clinical response, parameterized by a high-throughput ex vivo assay that quantifies sensitivity of primary MM cells to 31 agents or combinations, in a reconstruction of the tumor microenvironment. The mathematical model, which inherently accounts for intra-tumoral heterogeneity of drug sensitivity, combined with drug- and regimen-specific pharmacokinetics, produces patient-specific predictions of clinical response 5 days post-biopsy.

10. E.D. Sontag. A dynamical model of immune responses to antigen presentation predicts different regions of tumor or pathogen elimination. Cell Systems, 4:231-241, 2017. [PDF] Keyword(s): scale invariance, fold change detection, T cells, incoherent feedforward loops, immunology, cancer, internal model principle, incoherent feedforward loop, feedforward, IFFL, systems biology.
Abstract:
 Since the early 1990s, many authors have independently suggested that self/nonself recognition by the immune system might be modulated by the rates of change of antigen challenges. This paper introduces an extremely simple and purely conceptual mathematical model that allows dynamic discrimination of immune challenges. The main component of the model is a motif which is ubiquitous in systems biology, the incoherent feedforward loop, which endows the system with the capability to estimate exponential growth exponents, a prediction which is consistent with experimental work showing that exponentially increasing antigen stimulation is a determinant of immune reactivity. Combined with a bistable system and a simple feedback repression mechanism, an interesting phenomenon emerges as a tumor growth rate increases: elimination, tolerance (tumor growth), again elimination, and finally a second zone of tolerance (tumor escape). This prediction from our model is analogous to the two-zone tumor tolerance'' phenomenon experimentally validated since the mid 1970s. Moreover, we provide a plausible biological instantiation of our circuit using combinations of regulatory and effector T cells.

11. E.D. Sontag. Dynamic compensation, parameter identifiability, and equivariances. PLoS Computational Biology, 13:e1005447, 2017. Note: (Preprint was in bioRxiv https://doi.org/0.1101/095828, 2016). [WWW] [PDF] Keyword(s): fcd, fold-change detection, scale invariance, dynamic compensation, identifiability, observability, systems biology.
Abstract:
 A recent paper by Karin et al. introduced a mathematical notion called dynamical compensation (DC) of biological circuits. DC was shown to play an important role in glucose homeostasis as well as other key physiological regulatory mechanisms. Karin et al.\ went on to provide a sufficient condition to test whether a given system has the DC property. Here, we show how DC is a reformulation of a well-known concept in systems biology, statistics, and control theory -- that of parameter structural non-identifiability. Viewing DC as a parameter identification problem enables one to take advantage of powerful theoretical and computational tools to test a system for DC. We obtain as a special case the sufficient criterion discussed by Karin et al. We also draw connections to system equivalence and to the fold-change detection property.

12. Y. Vodovotz, A. Xia, E. Read, J. Bassaganya-Riera, D.A. Hafler, E.D. Sontag, J. Wang, J.S. Tsang, J.D. Day, S. Kleinstein, A.J. Butte, M.C. Altman, R. Hammond, C. Benoist, and S.C. Sealfon. Solving Immunology?. Trends in Immunology, 38:116-127, 2017. [PDF] Keyword(s): Immunology.
Abstract:
 Emergent responses of the immune system result from the integration of molecular and cellular networks over time and across multiple organs. High-content and high-throughput analysis technologies, concomitantly with data-driven and mechanistic modeling, hold promise for the systematic interrogation of these complex pathways. However, connecting genetic variation and molecular mechanisms to individual phenotypes and health outcomes has proven elusive. Gaps remain in data, and disagreements persist about the value of mechanistic modeling for immunology. This paper presents perspectives that emerged from the National Institute of Allergy and Infectious Disease (NIAID) workshop Complex Systems Science, Modeling and Immunity' and subsequent discussions regarding the potential synergy of high-throughput data acquisition, data-driven modeling, and mechanistic modeling to define new mechanisms of immunological disease and to accelerate the translation of these insights into therapies.

13. L. Yang, E.M. Dolan, S.K. Tan, T. Lin, E.D. Sontag, and S.D. Khare. Computation-guided design of a stimulus-responsive multi-enzyme supramolecular assembly, systems biology. ChemBioChem, 18:2000-2006, 2017. [PDF]
Abstract:
 This paper reports on the construction of a phosphorylation- and optically-responsive supramolecular complex of metabolic pathway enzymes for the biodegradation of an environmental pollutant. Fusing of enzymes led to an increase in pathway efficiency, and illustrates the possibility of spatio-temporal control over formation and functioning of a wide variety of synthetic biotransformations.

 2016
1. Z. Aminzare and E.D. Sontag. Some remarks on spatial uniformity of solutions of reaction-diffusion PDEs. Nonlinear Analysis, 147:125-144, 2016. [PDF] Keyword(s): contractions, contractive systems, matrix measures, logarithmic norms, synchronization, consensus, reaction-diffusion PDEs, partial differential equations.
Abstract:
 This paper presents a condition which guarantees spatial uniformity for the asymptotic behavior of the solutions of a reaction diffusion partial differential equation (PDE) with Neumann boundary conditions in one dimension, using the Jacobian matrix of the reaction term and the first Dirichlet eigenvalue of the Laplacian operator on the given spatial domain. The estimates are based on logarithmic norms in non-Hilbert spaces, which allow, in particular for a class of examples of interest in biology, tighter estimates than other previously proposed methods.

2. J.A. Ascensao, P. Datta, B. Hancioglu, E.D. Sontag, M.L. Gennaro, and O.A. Igoshin. Non-monotonic response dynamics of glyoxylate shunt genes in Mycobacterium tuberculosis. PLoS Computational Biology, 12:e1004741, 2016. [PDF] Keyword(s): cell signaling, monotone systems, monotone systems, systems biology.
Abstract:
 Understanding how dynamical responses of biological networks are constrained by underlying network topology is one of the fundamental goals of systems biology. Here we employ monotone systems theory to formulate a theorem stating necessary conditions for non-monotonic time-response of a biochemical network to a monotonic stimulus. We apply this theorem to analyze the non-monotonic dynamics of the sigmaB-regulated glyoxylate shunt gene expression in Mycobacterium tuberculosis cells exposed to hypoxia. We first demonstrate that the known network structure is inconsistent with observed dynamics. To resolve this inconsistency we employ the formulated theorem, modeling simulations and optimization along with follow-up dynamic experimental measurements. We show a requirement for post-translational modulation of sigmaB activity in order to reconcile the network dynamics with its topology. The results of this analysis make testable experimental predictions and demonstrate wider applicability of the developed methodology to a wide class of biological systems.

3. M. Margaliot, E.D. Sontag, and T. Tuller. Contraction after small transients. Automatica, 67:178-184, 2016. [PDF] Keyword(s): entrainment, nonlinear systems, stability, contractions, contractive systems, systems biology.
Abstract:
 Contraction theory is a powerful tool for proving asymptotic properties of nonlinear dynamical systems including convergence to an attractor and entrainment to a periodic excitation. We introduce three new forms of generalized contraction (GC) that are motivated by allowing contraction to take place after small transients in time and/or amplitude. These forms of GC are useful for several reasons. First, allowing small transients does not destroy the asymptotic properties provided by standard contraction. Second, in some cases as we change the parameters in a contractive system it becomes a GC just before it looses contractivity. In this respect, GC is the analogue of marginal stability in Lyapunov stability theory. We provide checkable sufficient conditions for GC, and demonstrate their usefulness using several models from systems biology that are not contractive, with respect to any norm, yet are GC.

4. E.V. Nikolaev and E.D. Sontag. Quorum-sensing synchronization of synthetic toggle switches: A design based on monotone dynamical systems theory. PLoS Computational Biology, 12:e1004881, 2016. [PDF] Keyword(s): quorum sensing, toggle switches, monotone systems, systems biology.
Abstract:
 Synthetic constructs in biotechnology, bio-computing, and proposed gene therapy interventions are often based on plasmids or transfected circuits which implement some form of on-off (toggle or flip-flop) switch. For example, the expression of a protein used for therapeutic purposes might be triggered by the recognition of a specific combination of inducers (e.g., antigens), and memory of this event should be maintained across a cell population until a specific stimulus commands a coordinated shut-off. The robustness of such a design is hampered by molecular (intrinsic) or environmental (extrinsic) noise, which may lead to spontaneous changes of state in a subset of the population and is reflected in the bimodality of protein expression, as measured for example using flow cytometry. In this context, a majority-vote correction circuit, which brings deviant cells back into the required state, is highly desirable. To address this concrete challenge, we have developed a new theoretical design for quorum-sensing (QS) synthetic toggles. QS provides a way for cells to broadcast their states to the population as a whole so as to facilitate consensus. Our design is endowed with strong theoretical guarantees, based on monotone dynamical systems theory, of global stability and no oscillations, and which leads to robust consensus states.

5. A. Raveh, M. Margaliot, E.D. Sontag, and T. Tuller. A model for competition for ribosomes in the cell. Proc. Royal Society Interface, 13:2015.1062, 2016. [PDF] Keyword(s): resource competition, ribosomes, entrainment, nonlinear systems, stability, contractions, contractive systems, systems biology, RFM, ribosome flow model.
Abstract:
 We develop and analyze a general model for large-scale simultaneous mRNA translation and competition for ribosomes. Such models are especially important when dealing with highly expressed genes, as these consume more resources. For our model, we prove that the compound system always converges to a steady-state and that it always entrains or phase locks to periodically time-varying transition rates in any of the mRNA molecules. We use this model to explore the interactions between the various mRNA molecules and ribosomes at steady-state. We show that increasing the length of an mRNA molecule decreases the production rate of all the mRNAs. Increasing any of the codon translation rates in a specific mRNA molecule yields a local effect: an increase in the translation rate of this mRNA, and also a global effect: the translation rates in the other mRNA molecules all increase or all decrease. These results suggest that the effect of codon decoding rates of endogenous and heterologous mRNAs on protein production might be more complicated than previously thought.

 2015
1. E.D. Sontag. Input-to-State Stability. In J. Baillieul and T. Samad, editors, Encyclopedia of Systems and Control. Springer-Verlag, 2015. [PDF] Keyword(s): input to state stability, integral input to state stability, iISS, ISS, input to output stability.
Abstract:
 The notion of input to state stability (ISS) qualitatively describes stability of the mapping from initial states and inputs to internal states (and more generally outputs). This entry focuses on the definition of ISS and a discussion of equivalent characterizations.

2. P. Bastiaens, M. R. Birtwistle, N. Bluthgen, F. J. Bruggeman, K.-H. Cho, C. Cosentino, A. de la Fuente, J. B. Hoek, A. Kiyatkin, S. Klamt, W. Kolch, S. Legewie, P. Mendes, T. Naka, T. Santra, E.D. Sontag, H. V. Westerhoff, and B. N. Kholodenko. Silence on the relevant literature and errors in implementation. Nature Biotech, 33:336-339, 2015. [PDF] Keyword(s): reverse engineering, systems biology, systems identification.
Abstract:
 This letter discusses a paper in the same journal which reported a method for reconstructing network topologies. Here we show that the method is a variant of a previously published method, modular response analysis. We also demonstrate that the implementation of the algorithm in that paper using statistical similarity measures as a proxy for global network responses to perturbations is erroneous and its performance is overestimated.

3. T. Kang, R. Moore, Y. Li, E.D. Sontag, and L. Bleris. Discriminating direct and indirect connectivities in biological networks. Proc Natl Acad Sci USA, 112:12893-12898, 2015. [PDF] Keyword(s): modular response analysis, stochastic systems, reverse engineering, gene networks, synthetic biology, feedforward, systems biology.
Abstract:
 Reverse engineering of biological pathways involves an iterative process between experiments, data processing, and theoretical analysis. In this work, we engineer synthetic circuits, subject them to perturbations, and then infer network connections using a combination of nonparametric single-cell data resampling and modular response analysis. Intriguingly, we discover that recovered weights of specific network edges undergo divergent shifts under differential perturbations, and that the particular behavior is markedly different between different topologies. Investigating topological changes under differential perturbations may address the longstanding problem of discriminating direct and indirect connectivities in biological networks.

4. M. Skataric, E.V. Nikolaev, and E.D. Sontag. A fundamental limitation to fold-change detection by biological systems with multiple time scales. IET Systems Biology, 9:1-15, 2015. [PDF] Keyword(s): adaptation, biological adaptation, perfect adaptation, singular perturbations, scale invariance, systems biology, transient behavior, symmetries, fcd, fold-change detection, incoherent feedforward loop, feedforward, IFFL.
Abstract:
 The phenomenon of fold-change detection, or scale invariance, is exhibited by a variety of sensory systems, in both bacterial and eukaryotic signaling pathways. It has been often remarked in the systems biology literature that certain systems whose output variables respond at a faster time scale than internal components give rise to an approximate scale-invariant behavior, allowing approximate fold-change detection in stimuli. This paper establishes a fundamental limitation of such a mechanism, showing that there is a minimal fold-change detection error that cannot be overcome, no matter how large the separation of time scales is. To illustrate this theoretically predicted limitation, we discuss two common biomolecular network motifs, an incoherent feedforward loop and a feedback system, as well as a published model of the chemotaxis signaling pathway of Dictyostelium discoideum.

5. E.D. Sontag and A. Singh. Exact moment dynamics for feedforward nonlinear chemical reaction networks. IEEE Life Sciences Letters, 1:26-29, 2015. [PDF] Keyword(s): systems biology, biochemical networks, stochastic systems, chemical master equation, chemical reaction networks.
Abstract:
 Chemical systems are inherently stochastic, as reactions depend on random (thermal) motion. This motivates the study of stochastic models, and specifically the Chemical Master Equation (CME), a discrete-space continuous-time Markov process that describes stochastic chemical kinetics. Exact studies using the CME are difficult, and several moment closure tools related to "mass fluctuation kinetics" and "fluctuation-dissipation" formulas can be used to obtain approximations of moments. This paper, in contrast, introduces a class of nonlinear chemical reaction networks for which exact computation is possible, by means of finite-dimensional linear differential equations. This class allows second and higher order reactions, but only under special assumptions on structure and/or conservation laws.

6. M. Marcondes de Freitas and E.D. Sontag. A small-gain theorem for random dynamical systems with inputs and outputs. SIAM J. Control and Optimization, 53:2657-2695, 2015. [PDF] Keyword(s): random dynamical systems, monotone systems, small-gain theorem, stochastic systems.
Abstract:
 A formalism for the study of random dynamical systems with inputs and outputs (RDSIO) is introduced. An axiomatic framework and basic properties of RDSIO are developed, and a theorem is shown that guarantees the stability of interconnected systems.

 2014
1. Z. Aminzare, Y. Shafi, M. Arcak, and E.D. Sontag. Guaranteeing spatial uniformity in reaction-diffusion systems using weighted $L_2$-norm contractions. In V. Kulkarni, G.-B. Stan, and K. Raman, editors, A Systems Theoretic Approach to Systems and Synthetic Biology I: Models and System Characterizations, pages 73-101. Springer-Verlag, 2014. [PDF] Keyword(s): contractions, contractive systems, Turing instabilities, diffusion, partial differential equations, synchronization.
Abstract:
 This paper gives conditions that guarantee spatial uniformity of the solutions of reaction-diffusion partial differential equations, stated in terms of the Jacobian matrix and Neumann eigenvalues of elliptic operators on the given spatial domain, and similar conditions for diffusively-coupled networks of ordinary differential equations. Also derived are numerical tests making use of linear matrix inequalities that are useful in certifying these conditions.

2. Z. Aminzare and E.D. Sontag. Synchronization of diffusively-connected nonlinear systems: results based on contractions with respect to general norms. IEEE Transactions on Network Science and Engineering, 1(2):91-106, 2014. [PDF] Keyword(s): matrix measures, logarithmic norms, synchronization, consensus, contractions, contractive systems.
Abstract:
 Contraction theory provides an elegant way to analyze the behavior of certain nonlinear dynamical systems. In this paper, we discuss the application of contraction to synchronization of diffusively interconnected components described by nonlinear differential equations. We provide estimates of convergence of the difference in states between components, in the cases of line, complete, and star graphs, and Cartesian products of such graphs. We base our approach on contraction theory, using matrix measures derived from norms that are not induced by inner products. Such norms are the most appropriate in many applications, but proofs cannot rely upon Lyapunov-like linear matrix inequalities, and different techniques, such as the use of the Perron-Frobenious Theorem in the cases of L1 or L-infinity norms, must be introduced.

3. D. Angeli, G.A. Enciso, and E.D. Sontag. A small-gain result for orthant-monotone systems under mixed feedback. Systems and Control Letters, 68:9-19, 2014. [PDF] Keyword(s): small-gain theorem, monotone systems.
Abstract:
 This paper introduces a small-gain result for interconnected orthant-monotone systems for which no matching condition is required between the partial orders in input and output spaces. Previous results assumed that the partial orders adopted would be induced by positivity cones in input and output spaces and that such positivity cones should fulfill a compatibility rule: namely either be coincident or be opposite. Those two configurations correspond to positive feedback or negative feedback cases. We relax those results by allowing arbitrary orthant orders.

4. M. Margaliot, E.D. Sontag, and T. Tuller. Entrainment to periodic initiation and transition rates in a computational model for gene translation. PLoS ONE, 9(5):e96039, 2014. [WWW] [PDF] [doi:10.1371/journal.pone.0096039] Keyword(s): ribosomes, entrainment, nonlinear systems, stability, contractions, contractive systems, systems biology, RFM, ribosome flow model.
Abstract:
 A recent biological study has demonstrated that the gene expression pattern entrains to a periodically varying abundance of tRNA molecules. This motivates developing mathematical tools for analyzing entrainment of translation elongation to intra-cellular signals such as tRNAs levels and other factors affecting translation. We consider a recent deterministic mathematical model for translation called the Ribosome Flow Model (RFM). We analyze this model under the assumption that the elongation rate of the tRNA genes and/or the initiation rate are periodic functions with a common period T. We show that the protein synthesis pattern indeed converges to a unique periodic trajectory with period T. The analysis is based on introducing a novel property of dynamical systems, called contraction after a short transient (CAST), that may be of independent interest. We provide a sufficient condition for CAST and use it to prove that the RFM is CAST, and that this implies entrainment. Our results support the conjecture that periodic oscillations in tRNA levels and other factors related to the translation process can induce periodic oscillations in protein levels, and suggest a new approach for engineering genes to obtain a desired, periodic, synthesis rate.

5. S. Prabakaran, J. Gunawardena, and E.D. Sontag. Paradoxical results in perturbation-based signaling network reconstruction. Biophysical Journal, 106:2720-2728, 2014. [PDF] Keyword(s): MAPK cascades, systems biology, biochemical networks, gene and protein networks, reverse engineering, systems identification, retroactivity.
Abstract:
 This paper describes a potential pitfall of perturbation-based approaches to network inference It is shows experimentally, and then explained mathematically, how even in the simplest signaling systems, perturbation methods may lead to paradoxical conclusions: for any given pair of two components X and Y, and depending upon the specific intervention on Y, either an activation or a repression of X could be inferred. The experiments are performed in an in vitro minimal system, thus isolating the effect and showing that it cannot be explained by feedbacks due to unknown intermediates; this system utilizes proteins from a pathway in mammalian (and other eukaryotic) cells that play a central role in proliferation, gene expression, differentiation, mitosis, cell survival, and apoptosis and is a perturbation target of contemporary therapies for various types of cancers. The results show that the simplistic view of intracellular signaling networks being made up of activation and repression links is seriously misleading, and call for a fundamental rethinking of signaling network analysis and inference methods.

6. T.H. Segall-Shapiro, A.J. Meyer, A.D. Ellington, E.D. Sontag, and C.A. Voigt. A resource allocator' for transcription based on a highly fragmented T7 RNA polymerase. Molecular Systems Biology, 10:742-, 2014. [WWW] [PDF] Keyword(s): systems biology, synthetic biology, gene expression.
Abstract:
 A transcriptional system is built based on a 'resource allocator' that sets a core RNAP concentration, which is then shared by multiple sigma fragments, which provide specificity. Adjusting the concentration of the core sets the maximum transcriptional capacity available to a synthetic system.

7. E.D. Sontag. A technique for determining the signs of sensitivities of steady states in chemical reaction networks. IET Systems Biology, 8:251-267, 2014. [PDF] Keyword(s): sensitivity, retroactivity, biomolecular networks, systems biology, stoichiometry, biochemical networks, systems biology.
Abstract:
 This paper studies the direction of change of steady states to parameter perturbations in chemical reaction networks, and, in particular, to changes in conserved quantities. Theoretical considerations lead to the formulation of a computational procedure that provides a set of possible signs of such sensitivities. The procedure is purely algebraic and combinatorial, only using information on stoichiometry, and is independent of the values of kinetic constants. Two examples of important intracellular signal transduction models are worked out as an illustration. In these examples, the set of signs found is minimal, but there is no general guarantee that the set found will always be minimal in other examples. The paper also briefly discusses the relationship of the sign problem to the question of uniqueness of steady states in stoichiometry classes.

 2013
1. D. Angeli and E.D. Sontag. Behavior of responses of monotone and sign-definite systems. In K. Hüper and Jochen Trumpf, editors, Mathematical System Theory - Festschrift in Honor of Uwe Helmke on the Occasion of his Sixtieth Birthday, pages 51-64. CreateSpace, 2013. [PDF] Keyword(s): monotone systems, reverse engineering, systems biology.
Abstract:
 This paper study systems with sign-definite interactions between variables, providing a sufficient condition to characterize the possible transitions between intervals of increasing and decreasing behavior. It also provides a discussion illustrating how our approach can help identify interactions in models, using information from time series of observations.

2. M. Marcondes de Freitas and E.D. Sontag. Random dynamical systems with inputs. In C. Pötzsche and P. Kloeden, editors, Nonautonomous Dynamical Systems in the Life Sciences, Lecture Notes in Mathematics vol. 2102, pages 41-87. Springer-Verlag, 2013. [PDF] Keyword(s): random dynamical systems, monotone systems.
Abstract:
 This work introduces a notion of random dynamical systems with inputs, providing several basic definitions and results on equilibria and convergence. It also presents a "converging input to converging state" result, a concept that plays a key role in the analysis of stability of feedback interconnections, for monotone systems.

3. Z. Aminzare and E.D. Sontag. Logarithmic Lipschitz norms and diffusion-induced instability. Nonlinear Analysis: Theory, Methods & Applications, 83:31-49, 2013. [PDF] Keyword(s): contractions, contractive systems, matrix measures, logarithmic norms, Turing instabilities, diffusion, partial differential equations, synchronization.
Abstract:
 This paper proves that ordinary differential equation systems that are contractive with respect to $L^p$ norms remain so when diffusion is added. Thus, diffusive instabilities, in the sense of the Turing phenomenon, cannot arise for such systems, and in fact any two solutions converge exponentially to each other. The key tools are semi-inner products and logarithmic Lipschitz constants in Banach spaces. An example from biochemistry is discussed, which shows the necessity of considering non-Hilbert spaces. An analogous result for graph-defined interconnections of systems defined by ordinary differential equations is given as well.

4. J. Barton and E.D. Sontag. The energy costs of insulators in biochemical networks. Biophysical Journal, 104:1390-1380, 2013. [PDF] Keyword(s): biochemical networks, futile cycles, enzymatic cycles, cell signaling, retroactivity, modularity, systems biology.
Abstract:
 Complex networks of biochemical reactions, such as intracellular protein signaling pathways and genetic networks, are often conceptualized in terms of modules,'' semi-independent collections of components that perform a well-defined function and which may be incorporated in multiple pathways. However, due to sequestration of molecular messengers during interactions and other effects, collectively referred to as retroactivity, real biochemical systems do not exhibit perfect modularity. Biochemical signaling pathways can be insulated from impedance and competition effects, which inhibit modularity, through enzymatic futile cycles'' which consume energy, typically in the form of ATP. We hypothesize that better insulation necessarily requires higher energy consumption. We test this hypothesis through a combined theoretical and computational analysis of a simplified physical model of covalent cycles, using two innovative measures of insulation, as well as a new way to characterize optimal insulation through the balancing of these two measures in a Pareto sense. Our results indicate that indeed better insulation requires more energy. While insulation may facilitate evolution by enabling a modular plug and play'' interconnection architecture, allowing for the creation of new behaviors by adding targets to existing pathways, our work suggests that this potential benefit must be balanced against the metabolic costs of insulation necessarily incurred in not affecting the behavior of existing processes.

5. A.O. Hamadeh, B.P. Ingalls, and E.D. Sontag. Transient dynamic phenotypes as criteria for model discrimination: fold-change detection in Rhodobacter sphaeroides chemotaxis. Proc. Royal Society Interface, 10:20120935, 2013. [PDF] Keyword(s): adaptation, biological adaptation, perfect adaptation, scale invariance, systems biology, transient behavior, symmetries, fcd, fold-change detection, chemotaxis.
Abstract:
 The chemotaxis pathway of the bacterium Rhodobacter sphaeroides has many similarities to that of Escherichia coli. It exhibits robust adaptation and has several homologues of the latter's chemotaxis proteins. Recent theoretical results have correctly predicted that, in response to a scaling of its ligand input signal, Escherichia coli exhibits the same output behavior, a property known as fold-change detection (FCD). In light of recent experimental results suggesting that R. sphaeroides may also show FCD, we present theoretical assumptions on the R. sphaeroides chemosensory dynamics that can be shown to yield FCD behavior. Furthermore, it is shown that these assumptions make FCD a property of this system that is robust to structural and parametric variations in the chemotaxis pathway, in agreement with experimental results. We construct and examine models of the full chemotaxis pathway that satisfy these assumptions and reproduce experimental time-series data from earlier studies. We then propose experiments in which models satisfying our theoretical assumptions predict robust FCD behavior where earlier models do not. In this way, we illustrate how transient dynamic phenotypes such as FCD can be used for the purposes of discriminating between models that reproduce the same experimental time-series data.

6. T. Kang, J.T. White, Z. Xie, Y. Benenson, E.D. Sontag, and L. Bleris. Reverse engineering validation using a benchmark synthetic gene circuit in human cells. ACS Synthetic Biology, 2:255-262, 2013. [PDF] Keyword(s): reverse engineering, systems biology, synthetic biology.
Abstract:
 This work introduces an experimental platform customized for the development and verification of reverse engineering and pathway characterization algorithms in mammalian cells. Specifically, we stably integrate a synthetic gene network in human kidney cells and use it as a benchmark for validating reverse engineering methodologies. The network, which is orthogonal to endogenous cellular signaling, contains a small set of regulatory interactions that can be used to quantify the reconstruction performance. By performing successive perturbations to each modular component of the network and comparing protein and RNA measurements, we study the conditions under which we can reliably reconstruct the causal relationships of the integrated synthetic network.

7. L. Liu, G. Duclos, B. Sun, J. Lee, A. Wu, Y. Kam, E.D. Sontag, H.A. Stone, J.C. Sturm, R.A. Gatenby, and R.H. Austin. Minimization of thermodynamic costs in cancer cell invasion. Proc Natl Acad Sci USA, 110:1686-1691, 2013. [PDF] Keyword(s): chemotaxis, cancer, metastasis.
Abstract:
 This paper shows that metastatic breast cancer cells cooperatively invade a 3D collagen matrix while following a glucose gradient. The front cell leadership is dynamic, and invading cells act in a cooperative manner by exchanging leaders in the invading front.

8. G. Russo, M. di Bernardo, and E.D. Sontag. A contraction approach to the hierarchical analysis and design of networked systems. IEEE Transactions Autom. Control, 58:1328-1331, 2013. [PDF] Keyword(s): contractions, contractive systems, matrix measures, logarithmic norms, synchronization, systems biology.
Abstract:
 This paper studies networks of components, and shows that a contraction property on the interconnection matrix, coupled with contractivity of the individual component subsystems, suffices to insure contractivity of the overall system.

9. V. Shimoga, J.T. White, Y. Li, E.D. Sontag, and L. Bleris. Synthetic mammalian transgene negative autoregulation. Molecular Systems Biology, 9:670-, 2013. [PDF] Keyword(s): systems biology, synthetic biology, gene expression.
Abstract:
 Using synthetic circuits stably integrated in human kidney cells, we study the effect of negative feedback regulation on cell-wide (extrinsic) and gene-specific (intrinsic) sources of uncertainty. We develop a theoretical approach to extract the two noise components from experiments and show that negative feedback reduces extrinsic noise while marginally increasing intrinsic noise, resulting to significant total noise reduction. We compare the results to simple negative regulation, where a constitutively transcribed transcription factor represses a reporter protein. We observe that the control architecture also reduces the extrinsic noise but results in substantially higher intrinsic fluctuations. We conclude that negative feedback is the most efficient way to mitigate the effects of extrinsic fluctuations by a sole regulatory wiring.

10. A. White, B. Lees, H.-L. Kao, G. Cipriani, E. Munarriz, A. Paaby, K. Erickson, S. Guzman, K. Rattanakorn, E.D. Sontag, D. Geiger, K. Gunsalus, and F. Piano. DevStaR: A novel algorithm for quantitative phenotyping of C. elegans development. IEEE Transactions on Medical Imaging, 32:1791-1803, 2013. [PDF]

 2012
1. M. Miller, M. Hafner, E.D. Sontag, N. Davidsohn, S. Subramanian, P. E. M. Purnick, D. Lauffenburger, and R. Weiss. Modular design of artificial tissue homeostasis: robust control through synthetic cellular heterogeneity. PLoS Computational Biology, 8:e1002579-, 2012. [PDF] Keyword(s): systems biology, homeostasis, stem cells, synthetic biology.
Abstract:
 Synthetic biology efforts have largely focused on small engineered gene networks, yet understanding how to integrate multiple synthetic modules and interface them with endogenous pathways remains a challenge. Here we present the design, system integration, and analysis of several large scale synthetic gene circuits for artificial tissue homeostasis. Diabetes therapy represents a possible application for engineered homeostasis, where genetically programmed stem cells maintain a steady population of beta-cells despite continuous turnover. We develop a new iterative process that incorporates modular design principles with hierarchical performance optimization targeted for environments with uncertainty and incomplete information. We employ theoretical analysis and computational simulations of multicellular reaction/diffusion models to design and understand system behavior, and find that certain features often associated with robustness (e.g., multicellular synchronization and noise attenuation) are actually detrimental for tissue homeostasis. We overcome these problems by engineering a new class of genetic modules for 'synthetic cellular heterogeneity' that function to generate beneficial population diversity. We design two such modules (an asynchronous genetic oscillator and a signaling throttle mechanism), demonstrate their capacity for enhancing robust control, and provide guidance for experimental implementation with various computational techniques. We found that designing modules for synthetic heterogeneity can be complex, and in general requires a framework for non-linear and multifactorial analysis. Consequently, we adapt a 'phenotypic sensitivity analysis' method to determine how functional module behaviors combine to achieve optimal system performance. We ultimately combine this analysis with Bayesian network inference to extract critical, causal relationships between a module's biochemical rate-constants, its high level functional behavior in isolation, and its impact on overall system performance once integrated.

2. A. Rufino Ferreira, M. Arcak, and E.D. Sontag. Stability certification of large scale stochastic systems using dissipativity of subsystems. Automatica, 48:2956-2964, 2012. [PDF] Keyword(s): stochastic systems, passivity, noise-to-state stability, ISS, input to state stability.
Abstract:
 This paper deals with the stability of interconnections of nonlinear stochastic systems, using concepts of passivity and noise-to-state stability.

3. M. Skataric and E.D. Sontag. A characterization of scale invariant responses in enzymatic networks. PLoS Computational Biology, 8:e1002748, 2012. [PDF] Keyword(s): adaptation, biological adaptation, perfect adaptation, scale invariance, systems biology, transient behavior, symmetries, fcd, fold-change detection.
Abstract:
 This paper studies a recently discovered remarkable feature that was shown in many adapting systems: scale invariance, which means that the initial, transient behavior stays approximately the same when the background signal level is scaled. Not every adapting system is scale-invariant: we investigate under which conditions a broadly used model of biochemical enzymatic networks will show scale invariant behavior. For all 3-node enzymatic networks, we performed a wide computational study to find candidates for scale invariance, among 16,038 possible topologies. This effort led us to discover a new necessary and sufficient mechanism that explains the behavior of all 3-node enzyme networks that have this property, which we calluniform linearizations with fast output''. We also apply our theoretical results to a concrete biological example of order six, a model of the response of the chemotaxis signaling pathway of Dictyostelium discoideum to changes in chemoeffector cyclic adenosine monophosphate (cAMP).

4. K. Wood, S. Nishida, E.D. Sontag, and P. Cluzel. Mechanism-independent method for predicting response to multiple drug exposure in bacteria. Proc Natl Acad Sci USA, 109:12254-12259, 2012. [PDF] Keyword(s): systems biology, drug interactions.
Abstract:
 Drugs are commonly used in combinations larger than two for treating bacterial infections. It is generally impossible to infer directly from the effects of individual drugs the net effect of a multi-drug combination. This paper describes an empirically derived mechanism-independent method for predicting the microbial growth response to combinations of more than two drugs, experimentally tested on both gram-negative (Escherichia coli) and grampositive (Staphylococcus aureus) bacteria. The method shows that for a wide range of drugs, the bacterial responses to drug pairs are sufficient to infer the effects of larger drug combinations, and provides a simple formula for the prediction.

 2011
1. G. Craciun, C. Pantea, and E.D. Sontag. Graph-theoretic analysis of multistability and monotonicity for biochemical reaction networks. In H. Koeppl, G. Setti, M. di Bernardo, and D. Densmore, editors, Design and Analysis of Biomolecular Circuits, pages 63-72. Springer-Verlag, 2011. [PDF] Keyword(s): biochemical networks, monotone systems.
Abstract:
 This is a short expository article describing how the species-reaction graph (SR graph) can be used to analyze both multistability and monotonicity of biochemical networks.

2. E.D. Sontag. Input to State Stability. In W. S. Levine, editor, The Control Systems Handbook: Control System Advanced Methods, Second Edition., pages 45.1-45.21 (1034-1054). CRC Press, Boca Raton, 2011. [PDF] Keyword(s): input to state stability, integral input to state stability, iISS, ISS, input to output stability.
Abstract:
 An encyclopedia-type article on foundations of ISS.

3. E.D. Sontag. Modularity, retroactivity, and structural identification. In H. Koeppl, G. Setti, M. di Bernardo, and D. Densmore, editors, Design and Analysis of Biomolecular Circuits, pages 183-202. Springer-Verlag, 2011. [PDF] Keyword(s): modularity, retroactivity, identification.
Abstract:
 Many reverse-engineering techniques in systems biology rely upon data on steady-state (or dynamic) perturbations --obtained from siRNA, gene knock-down or overexpression, kinase and phosphatase inhibitors, or other interventions-- in order to understand the interactions between different modules'' in a network. This paper first reviews one such popular such technique, introduced by the author and collaborators, and focuses on why conclusions drawn from its use may be misleading due to retroactivity'' (impedance or load) effects. A theoretical result characterizing stoichiometric-induced steady-state retroactivity effects is given for a class of biochemical networks.

4. E.D. Sontag. Stability and feedback stabilization. In Robert Meyers, editor, Mathematics of Complexity and Dynamical Systems, pages 1639-1652. Springer-Verlag, Berlin, 2011. [PDF] Keyword(s): stability, nonlinear control, feedback stabilization.
Abstract:
 The problem of stabilization of equilibria is one of the central issues in control. In addition to its intrinsic interest, it represents a first step towards the solution of more complicated problems, such as the stabilization of periodic orbits or general invariant sets, or the attainment of other control objectives, such as tracking, disturbance rejection, or output feedback, all of which may be interpreted as requiring the stabilization of some quantity (typically, some sort of error'' signal). A very special case, when there are no inputs, is that of stability. This short and informal article provides an introduction to the subject.

5. A.R. Teel, T.T. Georgiou, L. Praly, and E.D. Sontag. Input-Output Stability. In W. S. Levine, editor, The Control Systems Handbook: Control System Advanced Methods, Second Edition., pages 44.1-44.23 (1011-1033). CRC Press, Boca Raton, 2011. [PDF]
Abstract:
 An encyclopedia-type article on foundations of input/output stability.

6. R. Albert, B. DasGupta, R. Hegde, G.S. Sivanathan, A. Gitter, G. Gürsoy, P. Paul, and E.D. Sontag. A new computationally efficient measure of topological redundancy of biological and social networks. Physical Review E, 84:036117, 2011. [PDF]
Abstract:
 In this paper, we introduce a topological redundancy measure for labeled directed networks that is formal, computationally efficient and applicable to a variety of directed networks such as cellular signaling, metabolic and social interaction networks. We demonstrate the computational efficiency of our measure by computing its value and statistical significance on a number of biological and social networks with up to several thousands of nodes and edges. Our results suggest a number of interesting observations: (1) social networks are more redundant that their biological counterparts, (2) transcriptional networks are less redundant than signaling networks, (3) the topological redundancy of the C. elegans metabolic network is largely due to its inclusion of currency metabolites, and (4) the redundancy of signaling networks is highly (negatively) correlated with monotonicity of their dynamics.

7. D. Angeli, P. de Leenheer, and E.D. Sontag. Persistence results for chemical reaction networks with time-dependent kinetics and no global conservation laws. SIAM Journal on Applied Mathematics, 71:128-146, 2011. [PDF] Keyword(s): biochemical networks, fluxes, Petri nets, persistence, biochemical networks with inputs.
Abstract:
 New checkable criteria for persistence of chemical reaction networks are proposed, which extend and complement existing ones. The new results allow the consideration of reaction rates which are time-varying, thus incorporating the effects of external signals, and also relax the assumption of existence of global conservation laws, thus allowing for inflows (production) and outflows (degradation). For time-invariant networks parameter-dependent conditions for persistence of certain classes of networks are provided. As an illustration, two networks arising in the systems biology literature are analyzed, namely a hypoxia and an apoptosis network.

8. L. Bleris, Z. Xie, D. Glass, A. Adadey, E.D. Sontag, and Y. Benenson. Synthetic incoherent feed-forward circuits show adaptation to the amount of their genetic template. Molecular Systems Biology, 7:519-, 2011. [PDF] Keyword(s): adaptation, feedforward loops, systems biology, synthetic biology, incoherent feedforward loop, feedforward, IFFL.
Abstract:
 Natural and synthetic biological networks must function reliably in the face of fluctuating stoichiometry of their molecular components. These fluctuations are caused in part by changes in relative expression efficiency and the DNA template amount of the network-coding genes. Gene product levels could potentially be decoupled from these changes via built-in adaptation mechanisms, thereby boosting network reliability. Here we show that a mechanism based on an incoherent feed-forward motif enables adaptive gene expression in mammalian cells. We modeled, synthesized, and tested transcriptional and post-transcriptional incoherent loops and found that in all cases the gene product adapts to changes in DNA template abundance. We also observed that the post-transcriptional form results in superior adaptation behavior, higher absolute expression levels, and lower intrinsic fluctuations. Our results support a previously-hypothesized endogenous role in gene dosage compensation for such motifs and suggest that their incorporation in synthetic networks will improve their robustness and reliability.

9. S.N. Dashkovskiy, D.V. Efimov, and E.D. Sontag. Ustoichivost' ot vhoda k sostoyaniu i smezhnie svoystva sistem (In Russian, Input to state stability and allied system properties). Avtomatika i Telemekhanika (Automation and Remote Control), 72(8):1579-1614, 2011. [PDF]

10. A.C. Jiang, A. C. Ventura, E. D. Sontag, S. D. Merajver, A. J. Ninfa, and D. Del Vecchio. Load-induced modulation of signal transduction networks. Science Signaling, 4, issue 194:ra67, 2011. [PDF] Keyword(s): systems biology, biochemical networks, synthetic biology, futile cycles, singular perturbations, modularity.
Abstract:
 Biological signal transduction networks are commonly viewed as circuits that pass along in the process amplifying signals, enhancing sensitivity, or performing other signal-processing to transcriptional and other components. Here, we report on a "reverse-causality" phenomenon, which we call load-induced modulation. Through a combination of analytical and experimental tools, we discovered that signaling was modulated, in a surprising way, by downstream targets that receive the signal and, in doing so, apply what in physics is called a load. Specifically, we found that non-intuitive changes in response dynamics occurred for a covalent modification cycle when load was present. Loading altered the response time of a system, depending on whether the activity of one of the enzymes was maximal and the other was operating at its minimal rate or whether both enzymes were operating at submaximal rates. These two conditions, which we call "limit regime" and "intermediate regime," were associated with increased or decreased response times, respectively. The bandwidth, the range of frequency in which the system can process information, decreased in the presence of load, suggesting that downstream targets participate in establishing a balance between noise-filtering capabilities and a its ability to process high-frequency stimulation. Nodes in a signaling network are not independent relay devices, but rather are modulated by their downstream targets

11. O. Shoval, U. Alon, and E.D. Sontag. Symmetry invariance for adapting biological systems. SIAM Journal on Applied Dynamical Systems, 10:857-886, 2011. Note: (See here for a small typo: http://www.sontaglab.org/FTPDIR/shoval.alon.sontag.erratum.pdf). [PDF] Keyword(s): adaptation, biological adaptation, perfect adaptation, adaptation, feedforward loops, integral feedback, scale invariance, systems biology, transient behavior, symmetries, fcd, fold-change detection, incoherent feedforward loop, feedforward, IFFL.
Abstract:
 Often, the ultimate goal of regulation is to maintain a narrow range of concentration levels of vital quantities (homeostasis, adaptation) while at the same time appropriately reacting to changes in the environment (signal detection or sensitivity). Much theoretical, modeling, and analysis effort has been devoted to the understanding of these questions, traditionally in the context of steady-state responses to constant or step-changing stimuli. In this paper, we present a new theorem that provides a necessary and sufficient characterization of invariance of transient responses to symmetries in inputs. A particular example of this property, scale invariance (a.k.a. "fold change detection"), appears to be exhibited by biological sensory systems ranging from bacterial chemotaxis pathways to signal transduction mechanisms in eukaryotes. The new characterization amounts to the solvability of an associated partial differential equation. It is framed in terms of a notion which considerably extends equivariant actions of compact Lie groups. For several simple system motifs that are recurrent in biology, the solvability criterion may be checked explicitly.

 2010
1. R. Albert, B. Dasgupta, and E.D. Sontag. Inference of signal transduction networks from double causal evidence. In David Fenyö, editor, Computational Biology, Methods in Molecular Biology vol. 673, pages 239-251. Springer, 2010. [PDF] Keyword(s): systems biology, biochemical networks, algorithms, signal transduction networks, graph algorithms.
Abstract:
 We present a novel computational method, and related software, to synthesize signal transduction networks from single and double causal evidence.

2. B. Dasgupta, P. Vera-Licona, and E.D. Sontag. Reverse engineering of molecular networks from a common combinatorial approach. In M. Elloumi and A.Y. Zomaya, editors, Algorithms in computational molecular biology: Techniques, Approaches and Applications, pages 941-954. Wiley, Hoboken, 2010. [PDF] Keyword(s): reverse engineering, systems biology.

3. E.D. Sontag. Contractive systems with inputs. In Jan Willems, Shinji Hara, Yoshito Ohta, and Hisaya Fujioka, editors, Perspectives in Mathematical System Theory, Control, and Signal Processing, pages 217-228. Springer-verlag, 2010. [PDF] Keyword(s): contractions, contractive systems, consensus, synchronization.
Abstract:
 Contraction theory provides an elegant way of analyzing the behaviors of systems subject to external inputs. Under sometimes easy to check hypotheses, systems can be shown to have the incremental stability property that all trajectories converge to a unique solution. This property is especially interesting when forcing functions are periodic (globally attracting limit cycles result), as well as in the context of establishing synchronization results. The present paper provides a self-contained introduction to some basic results, with a focus on contractions with respect to non-Euclidean metrics.

4. D. Angeli, P. de Leenheer, and E.D. Sontag. Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates. J. Mathematical Biology, 61:581-616, 2010. [PDF] Keyword(s): MAPK cascades, biochemical networks, fluxes, monotone systems, reaction cordinates, Petri nets, persistence, futile cycles.
Abstract:
 This paper derives new results for certain classes of chemical reaction networks, linking structural to dynamical properties. In particular, it investigates their monotonicity and convergence without making assumptions on the form of the kinetics (e.g., mass-action) of the dynamical equations involved, and relying only on stoichiometric constraints. The key idea is to find an alternative representation under which the resulting system is monotone. As a simple example, the paper shows that a phosphorylation/dephosphorylation process, which is involved in many signaling cascades, has a global stability property.

5. G. Russo, M. di Bernardo, and E.D. Sontag. Global entrainment of transcriptional systems to periodic inputs. PLoS Computational Biology, 6:e1000739, 2010. [PDF] Keyword(s): contractive systems, contractions, systems biology, biochemical networks, gene and protein networks.
Abstract:
 This paper addresses the problem of giving conditions for transcriptional systems to be globally entrained to external periodic inputs. By using contraction theory, a powerful tool from dynamical systems theory, it is shown that certain systems driven by external periodic signals have the property that all solutions converge to fixed limit cycles. General results are proved, and the properties are verified in the specific case of some models of transcriptional systems.

6. L. Scardovi, M. Arcak, and E.D. Sontag. Synchronization of interconnected systems with applications to biochemical networks: an input-output approach. IEEE Transactions Autom. Control, 55:1367-1379, 2010. [PDF]
Abstract:
 This paper provides synchronization conditions for networks of nonlinear systems, where each component of the network itself consists of subsystems represented as operators in the extended L2 space. The synchronization conditions are provided by combining the input-output properties of the subsystems with information about the structure of network. The paper also explores results for state-space models as well as biochemical applications. The work is motivated by cellular networks where signaling occurs both internally, through interactions of species, and externally, through intercellular signaling.

7. O. Shoval, L. Goentoro, Y. Hart, A. Mayo, E.D. Sontag, and U. Alon. Fold change detection and scalar symmetry of sensory input fields. Proc Natl Acad Sci USA, 107:15995-16000, 2010. [PDF] Keyword(s): adaptation, biological adaptation, perfect adaptation, adaptation, feedforward loops, integral feedback, scale invariance, systems biology, transient behavior, symmetries, fcd, fold-change detection, incoherent feedforward loop, feedforward, IFFL.
Abstract:
 Certain cellular sensory systems display fold-change detection (FCD): a response whose entire shape, including amplitude and duration, depends only on fold-changes in input, and not on absolute changes. Thus, a step change in input from, say, level 1 to 2, gives precisely the same dynamical output as a step from level 2 to 4, since the steps have the same fold-change. We ask what is the benefit of FCD, and show that FCD is necessary and sufficient for sensory search to be independent of multiplying the input-field by a scalar. Thus the FCD search pattern depends only on the spatial profile of the input, and not on its amplitude. Such scalar symmetry occurs in a wide range of sensory inputs, such as source strength multiplying diffusing/convecting chemical fields sensed in chemotaxis, ambient light multiplying the contrast field in vision, and protein concentrations multiplying the output in cellular signaling-systems.Furthermore, we demonstrate that FCD entails two features found across sensory systems, exact adaptation and Weber's law, but that these two features are not sufficient for FCD. Finally, we present a wide class of mechanisms that have FCD, including certain non-linear feedback and feedforward loops.. We find that bacterial chemotaxis displays feedback within the present class, and hence is expected to show FCD. This can explain experiments in which chemotaxis searches are insensitive to attractant source levels. This study thus suggests a connection between properties of biological sensory systems and scalar symmetry stemming from physical properties of their input-fields.

8. E.D. Sontag. Remarks on Feedforward Circuits, Adaptation, and Pulse Memory. IET Systems Biology, 4:39-51, 2010. [PDF] Keyword(s): adaptation, feedforward loops, integral feedback, systems biology, transient behavior, incoherent feedforward loop, feedforward, IFFL.
Abstract:
 This note studies feedforward circuits as models for perfect adaptation to step signals in biological systems. A global convergence theorem is proved in a general framework, which includes examples from the literature as particular cases. A notable aspect of these circuits is that they do not adapt to pulse signals, because they display a memory phenomenon. Estimates are given of the magnitude of this effect.

9. E.D. Sontag. Rudolf E. Kalman and his students. Control Systems Magazine, 30:87-103, 2010. [PDF]
Abstract:
 An edited set of articles about Rudolf Kalman's legacy through his Ph.D. students.

10. E.D. Sontag and D. Zeilberger. A symbolic computation approach to a problem involving multivariate Poisson distributions. Advances in Applied Mathematics, 44:359-377, 2010. Note: There are a few typos in the published version. Please see this file for corrections: https://drive.google.com/file/d/0BzWFHczJF2INUlEtVkFJOUJiUFU/view. [PDF] Keyword(s): probability theory, stochastic systems, systems biology, biochemical networks, chemical master equation.
Abstract:
 Multivariate Poisson random variables subject to linear integer constraints arise in several application areas, such as queuing and biomolecular networks. This note shows how to compute conditional statistics in this context, by employing WZ Theory and associated algorithms. A symbolic computation package has been developed and is made freely available. A discussion of motivating biomolecular problems is also provided.

11. L. Wang, P. de Leenheer, and E.D. Sontag. Conditions for global stability of monotone tridiagonal systems with negative feedback. Systems and Control Letters, 59:138-130, 2010. [PDF] Keyword(s): systems biology, monotone systems, tridiagonal systems, global stability.
Abstract:
 This paper studies monotone tridiagonal systems with negative feedback. These systems possess the Poincar{\'e}-Bendixson property, which implies that, if orbits are bounded, if there is a unique steady state and this unique equilibrium is asymptotically stable, and if one can rule out periodic orbits, then the steady state is globally asymptotically stable. Different approaches are discussed to rule out period orbits. One is based on direct linearization, while the other uses the theory of second additive compound matrices. Among the examples that will illustrate our main theoretical results is the classical Goldbeter model of circadian rhythms.

 2009
1. D. Angeli and E.D. Sontag. Graphs and the Dynamics of Biochemical Networks. In B.P. Ingalls and P. Iglesias, editors, Control Theory in Systems Biology, pages 125-142. MIT Press, 2009.
Abstract:
 This is an expository paper about graph-theoretical properties of biochemical networks, discussing two approaches, one based on bipartite graphs and Petri net concepts, and another based on decompositions into order-preserving subsystems. Other papers on this website contain basically the same material.

2. D. Del Vecchio and E.D. Sontag. Synthetic Biology: A Systems Engineering Perspective. In B.P. Ingalls and P. Iglesias, editors, Control Theory in Systems Biology, pages 101-123. MIT Press, 2009.
Abstract:
 This is an expository paper about certain aspects of Synthetic Biology, including a discussion of the issue of modularity (load effects from downstream components).

3. D. Angeli, M.W. Hirsch, and E.D. Sontag. Attractors in coherent systems of differential equations. J. of Differential Equations, 246:3058-3076, 2009. [PDF] Keyword(s): monotone systems, positive feedback systems.
Abstract:
 Attractors of cooperative dynamical systems are particularly simple; for example, a nontrivial periodic orbit cannot be an attractor. This paper provides characterizations of attractors for the wider class of systems defined by the property that all directed feedback loops are positive. Several new results for cooperative systems are obtained in the process.

4. D. Angeli, P. de Leenheer, and E.D. Sontag. Chemical networks with inflows and outflows: A positive linear differential inclusions approach. Biotechnology Progress, 25:632-642, 2009. [PDF] Keyword(s): biochemical networks, fluxes, differential inclusions, positive systems, Petri nets, persistence, switched systems.
Abstract:
 Certain mass-action kinetics models of biochemical reaction networks, although described by nonlinear differential equations, may be partially viewed as state-dependent linear time-varying systems, which in turn may be modeled by convex compact valued positive linear differential inclusions. A result is provided on asymptotic stability of such inclusions, and applied to biochemical reaction networks with inflows and outflows. Included is also a characterization of exponential stability of general homogeneous switched systems

5. M. Chaves, A. M. Sengupta, and E.D. Sontag. Geometry and topology of parameter space: investigating measures of robustness in regulatory networks. J. of Mathematical Biology, 59:315-358, 2009. [PDF] Keyword(s): identifiability, robust, robustness, geometry.
Abstract:
 The concept of robustness of regulatory networks has been closely related to the nature of the interactions among genes, and the capability of pattern maintenance or reproducibility. Defining this robustness property is a challenging task, but mathematical models have often associated it to the volume of the space of admissible parameters. Not only the volume of the space but also its topology and geometry contain information on essential aspects of the network, including feasible pathways, switching between two parallel pathways or distinct/disconnected active regions of parameters. A method is presented here to characterize the space of admissible parameters, by writing it as a semi-algebraic set, and then theoretically analyzing its topology and geometry, as well as volume. This method provides a more objective and complete measure of the robustness of a developmental module. As a detailed case study, the segment polarity gene network is analyzed.

6. A. Dayarian, M. Chaves, E.D. Sontag, and A. M. Sengupta. Shape, Size and Robustness: Feasible Regions in the Parameter Space of Biochemical Networks. PLoS Computational Biology, 5:e10000256, 2009. [PDF] Keyword(s): identifiability, robust, robustness, geometry.
Abstract:
 The concept of robustness of regulatory networks has received much attention in the last decade. One measure of robustness has been associated with the volume of the feasible region, namely, the region in the parameter space in which the system is functional. In recent work, we emphasized that topology and geometry matter, as well as volume. In this paper, and using the segment polarity gene network to illustrate our approach, we show that random walks in parameter space and how they exit the feasible region provide a rich perspective on the different modes of failure of a model. In particular, for the segment polarity network, we found that, between two alternative ways of activating Wingless, one is more robust. Our method provides a more complete measure of robustness to parameter variation. As a general modeling strategy, our approach is an interesting alternative to Boolean representation of biochemical networks.

7. D. Del Vecchio and E.D. Sontag. Engineering Principles in Bio-Molecular Systems: From Retroactivity to Modularity. European Journal of Control, 15:389-397, 2009. Note: Preliminary version appeared as paper MoB2.2 in Proceedings of the European Control Conference 2009, August 23-26, 2009, Budapest. [PDF] Keyword(s): systems biology, biochemical networks, synthetic biology, futile cycles, singular perturbations, modularity.

8. T. Riley, X. Yu, E.D. Sontag, and A. Levine. The P53HMM algorithm: using novel profile Hidden Markov Models to detect p53-responsive genes. BMC Bioinformatics, 10:111, 2009. [PDF] [doi:10.1186/1471-2105-10-111] Keyword(s): Hidden Markov Models, p53, transcription factors.
Abstract:
 A novel computational method (called p53HMM) is presented that utilizes Profile Hidden Markov Models (PHMM's) to estimate the relative binding affinities of putative p53 response elements (RE's), both p53 single-sites and cluster-sites. These models incorporate a novel Correlated Baum Welch'' training algorithm that provides increased predictive power by exploiting the redundancy of information found in the repeated, palindromic p53-binding motif. The predictive accuracy of these new models are compared against other predictive models, including position specic score matrices (PSSM's, or weight matrices). Finally, we provide experimental evidence that verifies a predicted p53-target site that regu- lates the CHMP4C gene. The P53HMM algorithm is available on-line from http://tools.csb.ias.edu.

9. E.D. Sontag, Y. Wang, and A. Megretski. Input classes for identification of bilinear systems. IEEE Transactions Autom. Control, 54:195-207, 2009. Note: Also arXiv math.OC/0610633, 20 Oct 2006, and short version in ACC'07.[PDF] Keyword(s): realization theory, observability, identifiability, bilinear systems.
Abstract:
 This paper asks what classes of input signals are sufficient in order to completely identify the input/output behavior of generic bilinear systems. The main results are that step inputs are not sufficient, nor are single pulses, but the family of all pulses (of a fixed amplitude but varying widths) do suffice for identification.

10. A.M. Weinstein and E.D. Sontag. Modeling proximal tubule cell homeostasis: Tracking changes in luminal flow. Bulletin of Mathematical Biology, 71:1285-1322, 2009. [PDF]
Abstract:
 During normal kidney function, there are are routinely wide swings in proximal tubule fluid flow and proportional changes in Na+ reabsorption across tubule epithelial cells. This "glomerulotubular balance" occurs in the absence of any substantial change in cell volume, and is thus a challenge to coordinate luminal membrane solute entry with peritubular membrane solute exit. In this work, linear optimal control theory is applied to generate a configuration of regulated transporters that could achieve this result. A previously developed model of rat proximal tubule epithelium is linearized about a physiologic reference condition; the approximate linear system is recast as a dynamical system; and a Riccati equation is solved to yield optimal linear feedback that stabilizes Na+ flux, cell volume, and cell pH. This optimal feedback control is largely consigned to three physiologic variables, cell volume, cell electrical potential, and lateral intercellular hydrostatic pressure. Transport modulation by cell volume stabilizes cell volume; transport modulation by electrical potential or interspace pressure act to stabilize Na+ flux and cell pH. This feedback control is utilized in a tracking problem, in which reabsorptive Na+ flux varies over a factor of two. The resulting control parameters consist of two terms, an autonomous term and a feedback term, and both terms include transporters on both luminal and peritubular cell membranes. Overall, the increase in Na+ flux is achieved with upregulation of luminal Na+/H+ exchange and Na+-glucose cotransport, with increased peritubular Na+-3HCO_3- and K+-Cl- cotransport, and with increased Na+,K+-ATPase activity. The configuration of activated transporters emerges as testable hypothesis of the molecular basis for glomerulotubular balance. It is suggested that the autonomous control component at each cell membrane could represent the cytoskeletal effects of luminal flow.

 2008
1. M. Arcak and E.D. Sontag. Passivity-based Stability of Interconnection Structures. In V. Blondel, S. Boyd, and H. Kimura, editors, Recent Advances in Learning and Control, volume Volume 371, pages 195-204. Springer-Verlag, NY, 2008. [PDF] [doi:10.1007/978-1-84800-155-8_14] Keyword(s): passive systems, secant condition, biochemical networks.
Abstract:
 In this expository paper, we provide a streamlined version of the key lemma on stability of interconnections due to Vidyasagar and Moylan and Hill, and then show how it its hypotheses may be verified for network structures of great interest in biology.

2. R. Albert, B. Dasgupta, R. Dondi, and E.D. Sontag. Inferring (biological) signal transduction networks via transitive reductions of directed graphs. Algorithmica, 51:129-159, 2008. [PDF] [doi:10.1007/s00453-007-9055-0] Keyword(s): systems biology, biochemical networks, algorithms, signal transduction networks, graph algorithms.
Abstract:
 The transitive reduction problem is that of inferring a sparsest possible biological signal transduction network consistent with a set of experimental observations, with a goal to minimize false positive inferences even if risking false negatives. This paper provides computational complexity results as well as approximation algorithms with guaranteed performance.

3. D. Angeli and E.D. Sontag. Oscillations in I/O monotone systems. IEEE Transactions on Circuits and Systems, Special Issue on Systems Biology, 55:166-176, 2008. Note: Preprint version in arXiv q-bio.QM/0701018, 14 Jan 2007. [PDF] Keyword(s): monotone systems, hopf bifurcations, circadian rhythms, tridiagonal systems, nonlinear dynamics, systems biology, biochemical networks, oscillations, periodic behavior.
Abstract:
 In this note, we show how certain properties of Goldbeter's 1995 model for circadian oscillations can be proved mathematically, using techniques from the recently developed theory of monotone systems with inputs and outputs. The theory establishes global asymptotic stability, and in particular no oscillations, if the rate of transcription is somewhat smaller than that assumed by Goldbeter, based on the application of a tight small gain condition. This stability persists even under arbitrary delays in the feedback loop. On the other hand, when the condition is violated a Poincare'-Bendixson result allows to conclude existence of oscillations, for sufficiently high delays.

4. D. Angeli and E.D. Sontag. Translation-invariant monotone systems, and a global convergence result for enzymatic futile cycles. Nonlinear Analysis Series B: Real World Applications, 9:128-140, 2008. [PDF] [doi:10.1016/j.nonrwa.2006.09.006] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 Strongly monotone systems of ordinary differential equations which have a certain translation-invariance property are shown to have the property that all projected solutions converge to a unique equilibrium. This result may be seen as a dual of a well-known theorem of Mierczynski for systems that satisfy a conservation law. As an application, it is shown that enzymatic futile cycles have a global convergence property.

5. M. Arcak and E.D. Sontag. A passivity-based stability criterion for a class of interconnected systems and applications to biochemical reaction networks. Mathematical Biosciences and Engineering, 5:1-19, 2008. Note: Also, preprint: arxiv0705.3188v1 [q-bio], May 2007. [PDF] Keyword(s): MAPK cascades, systems biology, biochemical networks, cyclic feedback systems, secant condition, nonlinear stability, dynamical systems.
Abstract:
 This paper presents a stability test for a class of interconnected nonlinear systems motivated by biochemical reaction networks. One of the main results determines global asymptotic stability of the network from the diagonal stability of a "dissipativity matrix" which incorporates information about the passivity properties of the subsystems, the interconnection structure of the network, and the signs of the interconnection terms. This stability test encompasses the "secant criterion" for cyclic networks presented in our previous paper, and extends it to a general interconnection structure represented by a graph. A second main result allows one to accommodate state products. This extension makes the new stability criterion applicable to a broader class of models, even in the case of cyclic systems. The new stability test is illustrated on a mitogen activated protein kinase (MAPK) cascade model, and on a branched interconnection structure motivated by metabolic networks. Finally, another result addresses the robustness of stability in the presence of diffusion terms in a compartmental system made out of identical systems.

6. D. Del Vecchio, A.J. Ninfa, and E.D. Sontag. Modular Cell Biology: Retroactivity and Insulation. Molecular Systems Biology, 4:161, 2008. [PDF] Keyword(s): retroactivity, systems biology, biochemical networks, synthetic biology, futile cycles, singular perturbations, modularity.
Abstract:
 Modularity plays a fundamental role in the prediction of the behavior of a system from the behavior of its components, guaranteeing that the properties of individual components do not change upon interconnection. Just as electrical, hydraulic, and other physical systems often do not display modularity, nor do many biochemical systems, and specifically, genetic networks. Here, we study the effect of interconnections on the input/output dynamic characteristics of transcriptional components, focusing on a property, which we call "retroactivity," that plays a role analogous to non-zero output impedance in electrical systems. In transcriptional networks, retroactivity is large when the amount of transcription factor is comparable to, or smaller than, the amount of promoter binding sites, or when the affinity of such binding sites is high. In order to attenuate the effect of retroactivity, we propose a feedback mechanism inspired by the design of amplifiers in electronics. We introduce, in particular, a mechanism based on a phosphorylation/dephosphorylation cycle. This mechanism enjoys a remarkable insulation property, due to the fast time scales of the phosphorylation and dephosphorylation reactions. Such a mechanism, when viewed as a signal transduction system, has thus an inherent capacity to provide insulation and hence to increase the modularity of the system in which it is placed.

7. G.A. Enciso and E.D. Sontag. Monotone bifurcation graphs. Journal of Biological Dynamics, 2:121-139, 2008. [PDF]
Abstract:
 This paper generalizes the approach to bistability based on the existence of characteristics for open-loop monotone systems to the case when characteristics do not exist. A set-valued version is provided, instead.

8. M.R. Jovanovic, M. Arcak, and E.D. Sontag. A passivity-based approach to stability of spatially distributed systems with a cyclic interconnection structure. IEEE Transactions on Circuits and Systems, Special Issue on Systems Biology, 55:75-86, 2008. Note: Preprint: also arXiv math.OC/0701622, 22 January 2007.[PDF] Keyword(s): MAPK cascades, systems biology, biochemical networks, nonlinear stability, nonlinear dynamics, diffusion, secant condition, cyclic feedback systems.
Abstract:
 A class of distributed systems with a cyclic interconnection structure is considered. These systems arise in several biochemical applications and they can undergo diffusion driven instability which leads to a formation of spatially heterogeneous patterns. In this paper, a class of cyclic systems in which addition of diffusion does not have a destabilizing effect is identified. For these systems global stability results hold if the "secant" criterion is satisfied. In the linear case, it is shown that the secant condition is necessary and sufficient for the existence of a decoupled quadratic Lyapunov function, which extends a recent diagonal stability result to partial differential equations. For reaction-diffusion equations with nondecreasing coupling nonlinearities global asymptotic stability of the origin is established. All of the derived results remain true for both linear and nonlinear positive diffusion terms. Similar results are shown for compartmental systems.

9. S. Kachalo, R. Zhang, E.D. Sontag, R. Albert, and B. Dasgupta. NET-SYNTHESIS: A software for synthesis, inference and simplification of signal transduction networks. Bioinformatics, 24:293 - 295, 2008. [PDF] Keyword(s): systems biology, biochemical networks, algorithms, signal transduction networks, graph algorithms.
Abstract:
 This paper presents a software tool for inference and simplification of signal transduction networks. The method relies on the representation of observed indirect causal relationships as network paths, using techniques from combinatorial optimization to find the sparsest graph consistent with all experimental observations. We illustrate the biological usability of our software by applying it to a previously published signal transduction network and by using it to synthesize and simplify a novel network corresponding to activation-induced cell death in large granular lymphocyte leukemia.

10. A. Maayan, R. Iyengar, and E.D. Sontag. Intracellular Regulatory Networks are close to Monotone Systems. IET Systems Biology, 2:103-112, 2008. [PDF]
Abstract:
 We find that three intracellular regulatory networks contain far more positive "sign-consistent" feedback and feed-forward loops than negative loops. Negative inconsistent loops can be more easily removed from real regulatory network topologies compared to removing negative loops from shuffled networks. The abundance of positive feed-forward loops and feedback loops in real networks emerges from the presence of hubs that are enriched with either negative or positive links, and from the non-uniform connectivity distribution. Boolean dynamics applied to the signaling network further support the stability of its topology. These observations suggest that the "close-to-monotone" structure of intracellular regulatory networks may contribute to the dynamical stability observed in cellular behavior.

11. T. Riley, E.D. Sontag, P. Chen, and A. Levine. The transcriptional regulation of human p53-regulated genes. Nature Reviews Molecular Cell Biology, 9:402-412, 2008. [PDF] Keyword(s): Hidden Markov Models, p53, transcription.
Abstract:
 The p53 protein regulates the transcription of many different genes in response to a wide variety of stress signals. Following DNA damage, p53 regulates key processes, including DNA repair, cell-cycle arrest, senescence and apoptosis, in order to suppress cancer. This Analysis article provides an overview of the current knowledge of p53-regulated genes in these pathways and others, and the mechanisms of their regulation. In addition, we present the most comprehensive list so far of human p53-regulated genes and their experimentally validated, functional binding sites that confer p53 regulation.

12. E.D. Sontag. Network reconstruction based on steady-state data. Essays in Biochemistry, 45:161-176, 2008. [PDF] Keyword(s): MAPK cascades, systems biology, biochemical networks, gene and protein networks, reverse engineering, systems identification.
Abstract:
 The reverse engineering problem'' in systems biology is that of unraveling of the web of interactions among the components of protein and gene regulatory networks, so as to map out the direct or local interactions among components. These direct interactions capture the topology of the functional network. An intrinsic difficulty in capturing these direct interactions, at least in intact cells, is that any perturbation to a particular gene or signaling component may rapidly propagate throughout the network, thus causing global changes which cannot be easily distinguished from direct effects. Thus, a major goal in reverse engineering is to use these observed global responses - such as steady-state changes in concentrations of active proteins, mRNA levels, or transcription rates - in order to infer the local interactions between individual nodes. One approach to solving this global-to-local problem is the Modular Response Analysis'' (MRA) method proposed in work of the author with Kholodenko et. al. (PNAS, 2002) and further elaborated in other papers. The basic method deals only with steady-state data. However, recently, quasi-steady state MRA has been used by Santos et. al. (Nature Cell Biology, 2007) for quantifying positive and negative feedback effects in the Raf/Mek/Erk MAPK network in rat adrenal pheochromocytoma (PC-12) cells. This paper presents an overview of the MRA technique, as well as a generalization of the algorithm to that quasi-steady state case.

13. E.D. Sontag, A. Veliz-Cuba, R. Laubenbacher, and A.S. Jarrah. The effect of negative feedback loops on the dynamics of Boolean networks. Biophysical Journal, 95:518-526, 2008. [PDF] Keyword(s): monotone systems, positive feedback systems, Boolean networks, limit cycles.
Abstract:
 Feedback loops play an important role in determining the dynamics of biological networks. In order to study the role of negative feedback loops, this paper introduces the notion of "distance to positive feedback (PF-distance)" which in essence captures the number of "independent" negative feedback loops in the network, a property inherent in the network topology. Through a computational study using Boolean networks it is shown that PF-distance has a strong influence on network dynamics and correlates very well with the number and length of limit cycles in the phase space of the network. To be precise, it is shown that, as the number of independent negative feedback loops increases, the number (length) of limit cycles tends to decrease (increase). These conclusions are consistent with the fact that certain natural biological networks exhibit generally regular behavior and have fewer negative feedback loops than randomized networks with the same numbers of nodes and connectivity.

14. L. Wang and E.D. Sontag. On the number of steady states in a multiple futile cycle. Journal of Mathematical Biology, 57:29-52, 2008. [PDF] Keyword(s): singular perturbations, futile cycles, MAPK cascades, systems biology, biochemical networks, multistability.
Abstract:
 This note studies the number of positive steady states in biomolecular reactions consisting of activation/deactivation futile cycles, such as those arising from phosphorylations and dephosphorylations at each level of a MAPK cascade. It is shown that: (1) for some parameter ranges, there are at least n+1 (if n is even) or n (if n is odd) steady states; (2) there never are more than 2n-1 steady states (so, for n=2, there are no more than 3 steady states); (3) for parameters near the standard Michaelis-Menten quasi-steady state conditions, there are at most n+1 steady states; and (4) for parameters far from the standard Michaelis-Menten quasi-steady state conditions, there is at most one steady state.

15. L. Wang and E.D. Sontag. Singularly perturbed monotone systems and an application to double phosphorylation cycles. J. Nonlinear Science, 18:527-550, 2008. [PDF] Keyword(s): singular perturbations, futile cycles, MAPK cascades, systems biology, biochemical networks, nonlinear stability, nonlinear dynamics, multistability, monotone systems.
Abstract:
 The theory of monotone dynamical systems has been found very useful in the modeling of some gene, protein, and signaling networks. In monotone systems, every net feedback loop is positive. On the other hand, negative feedback loops are important features of many systems, since they are required for adaptation and precision. This paper shows that, provided that these negative loops act at a comparatively fast time scale, the main dynamical property of (strongly) monotone systems, convergence to steady states, is still valid. An application is worked out to a double-phosphorylation "futile cycle" motif which plays a central role in eukaryotic cell signaling The workis heavily based on Fenichel-Jones geometric singular perturbation theory.

 2007
1. R. Albert, B. DasGupta, R. Dondi, S. Kachalo, E.D. Sontag, A. Zelikovsky, and K. Westbrooks. A novel method for signal transduction network inference from indirect experimental evidence. In R. Giancarlo and S. Hannenhalli, editors, 7th Workshop on Algorithms in Bioinformatics (WABI), volume 14, pages 407-419. Springer-Verlag, Berlin, 2007. Note: Conference version of journal paper with same title. Keyword(s): systems biology, biochemical networks, algorithms, signal transduction networks, graph algorithms.

2. D. Angeli, P. De Leenheer, and E.D. Sontag. A Petri net approach to persistence analysis in chemical reaction networks. In I. Queinnec, S. Tarbouriech, G. Garcia, and S-I. Niculescu, editors, Biology and Control Theory: Current Challenges (Lecture Notes in Control and Information Sciences Volume 357), pages 181-216. Springer-Verlag, Berlin, 2007. Note: See abstract for A Petri net approach to the study of persistence in chemical reaction networks.[PDF]

3. E.D. Sontag. Input to state stability: Basic concepts and results. In P. Nistri and G. Stefani, editors, Nonlinear and Optimal Control Theory, pages 163-220. Springer-Verlag, Berlin, 2007. [PDF] Keyword(s): input to state stability, stability, input to state stability, nonlinear systems, detectability, nonlinear regulation.
Abstract:
 This expository presentation, prepared for a summer course, addresses the precise formulation of questions of robustness with respect to disturbances, using the paradigm of input to state stability. It provides an intuitive and informal presentation of the main concepts.

4. E.D. Sontag. Monotone and near-monotone systems. In I. Queinnec, S. Tarbouriech, G. Garcia, and S-I. Niculescu, editors, Biology and Control Theory: Current Challenges (Lecture Notes in Control and Information Sciences Volume 357), pages 79-122. Springer-Verlag, Berlin, 2007. Note: Conference version of Monotone and near-monotone biochemical networks,'' basically the same paper.Keyword(s): systems biology, biochemical networks, monotone systems, Ising spin models, nonlinear stability, dynamical systems, consistent graphs, gene networks.
Abstract:
 See abstract and pdf for Monotone and near-monotone biochemical networks''.

5. E.D. Sontag. Stability and Feedback Stabilization. In Robert Meyers, editor, Encyclopedia of Complexity and Systems Science. Springer-Verlag, Berlin, 2007. Keyword(s): stability, nonlinear control, feedback stabilization.
Abstract:
 The problem of stabilization of equilibria is one of the central issues in control. In addition to its intrinsic interest, it represents a first step towards the solution of more complicated problems, such as the stabilization of periodic orbits or general invariant sets, or the attainment of other control objectives, such as tracking, disturbance rejection, or output feedback, all of which may be interpreted as requiring the stabilization of some quantity (typically, some sort of error'' signal). A very special case, when there are no inputs, is that of stability. This short and informal article provides an introduction to the subject.

6. E.D. Sontag and Y. Wang. Uniformly Universal Inputs. In Alessandro Astolfi, editor, Analysis and Design of Nonlinear Control Systems, volume 224, pages 9-24. Springer-Verlag, London, 2007. [PDF] Keyword(s): observability, identification.
Abstract:
 A result is presented showing the existence of inputs universal for observability, uniformly with respect to the class of all continuous-time analytic systems. This represents an ultimate generalization of a 1977 theorem, for bilinear systems, due to Alberto Isidori and Osvaldo Grasselli.

7. R. Albert, B. DasGupta, R. Dondi, S. Kachalo, E.D. Sontag, A. Zelikovsky, and K. Westbrooks. A novel method for signal transduction network inference from indirect experimental evidence. Journal of Computational Biology, 14:927-949, 2007. [PDF] Keyword(s): systems biology, biochemical networks, algorithms, signal transduction networks, graph algorithms.
Abstract:
 This paper introduces a new method of combined synthesis and inference of biological signal transduction networks. The main idea lies in representing observed causal relationships as network paths, and using techniques from combinatorial optimization to find the sparsest graph consistent with all experimental observations. The paper formalizes the approach, studies its computational complexity, proves new results for exact and approximate solutions of the computationally hard transitive reduction substep of the approach, validates the biological applicability by applying it to a previously published signal transduction network by Li et al., and shows that the algorithm for the transitive reduction substep performs well on graphs with a structure similar to those observed in transcriptional regulatory and signal transduction networks.

8. D. Angeli, P. de Leenheer, and E.D. Sontag. A Petri net approach to the study of persistence in chemical reaction networks. Mathematical Biosciences, 210:598-618, 2007. Note: Please look at the paper A Petri net approach to persistence analysis in chemical reaction networks'' for additional results, not included in the journal paper due to lack of space. See also the preprint: arXiv q-bio.MN/068019v2, 10 Aug 2006. [PDF] Keyword(s): Petri nets, systems biology, biochemical networks, nonlinear stability, dynamical systems, futile cycles.
Abstract:
 Persistency is the property, for differential equations in Rn, that solutions starting in the positive orthant do not approach the boundary. For chemical reactions and population models, this translates into the non-extinction property: provided that every species is present at the start of the reaction, no species will tend to be eliminated in the course of the reaction. This paper provides checkable conditions for persistence of chemical species in reaction networks, using concepts and tools from Petri net theory, and verifies these conditions on various systems which arise in the modeling of cell signaling pathways.

9. P. Berman, B. Dasgupta, and E.D. Sontag. Algorithmic issues in reverse engineering of protein and gene networks via the modular response analysis method. Annals of the NY Academy of Sciences, 1115:132-141, 2007. [PDF] Keyword(s): systems biology, biochemical networks, gene and protein networks, reverse engineering, systems identification, graph algorithms.
Abstract:
 This paper studies a computational problem motivated by the modular response analysis method for reverse engineering of protein and gene networks. This set-cover problem is hard to solve exactly for large networks, but efficient approximation algorithms are given and their complexity is analyzed.

10. P. Berman, B. Dasgupta, and E.D. Sontag. Randomized approximation algorithms for set multicover problems with applications to reverse engineering of protein and gene networks. Discrete Applied Mathematics Special Series on Computational Molecular Biology, 155:733-749, 2007. [PDF] Keyword(s): systems biology, biochemical networks, gene and protein networks, systems identification, reverse engineering.
Abstract:
 This paper investigates computational complexity aspects of a combinatorial problem that arises in the reverse engineering of protein and gene networks, showing relations to an appropriate set multicover problem with large "coverage" factor, and providing a non-trivial analysis of a simple randomized polynomial-time approximation algorithm for the problem.

11. B. DasGupta, G.A. Enciso, E.D. Sontag, and Y. Zhang. Algorithmic and complexity aspects of decompositions of biological networks into monotone subsystems. BioSystems, 90:161-178, 2007. [PDF] Keyword(s): monotone systems, systems biology, biochemical networks.
Abstract:
 A useful approach to the mathematical analysis of large-scale biological networks is based upon their decompositions into monotone dynamical systems. This paper deals with two computational problems associated to finding decompositions which are optimal in an appropriate sense. In graph-theoretic language, the problems can be recast in terms of maximal sign-consistent subgraphs. The theoretical results include polynomial-time approximation algorithms as well as constant-ratio inapproximability results. One of the algorithms, which has a worst-case guarantee of 87.9% from optimality, is based on the semidefinite programming relaxation approach of Goemans-Williamson. The algorithm was implemented and tested on a Drosophila segmentation network and an Epidermal Growth Factor Receptor pathway model.

12. T. Gedeon and E.D. Sontag. Oscillations in multi-stable monotone systems with slowly varying feedback. J. of Differential Equations, 239:273-295, 2007. [PDF] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 This paper gives a theorem showing that a slow feedback adaptation, acting entirely analogously to the role of negative feedback for ordinary relaxation oscillations, leads to periodic orbits for bistable monotone systems. The proof is based upon a combination of i/o monotone systems theory and Conley Index theory.

13. W. Maass, P. Joshi, and E.D. Sontag. Computational aspects of feedback in neural circuits. PLoS Computational Biology, 3:e165 1-20, 2007. [PDF] Keyword(s): machine learning, neural networks, feedback linearization, computation by cortical microcircuits, fading memory.
Abstract:
 It had previously been shown that generic cortical microcircuit models can perform complex real-time computations on continuous input streams, provided that these computations can be carried out with a rapidly fading memory. We investigate in this article the computational capability of such circuits in the more realistic case where not only readout neurons, but in addition a few neurons within the circuit have been trained for specific tasks. This is essentially equivalent to the case where the output of trained readout neurons is fed back into the circuit. We show that this new model overcomes the limitation of a rapidly fading memory. In fact, we prove that in the idealized case without noise it can carry out any conceivable digital or analog computation on time-varying inputs. But even with noise the resulting computational model can perform a large class of biologically relevant real-time computations that require a non-fading memory.

14. E.D. Sontag. Monotone and near-monotone biochemical networks. Systems and Synthetic Biology, 1:59-87, 2007. [PDF] [doi:10.1007/s11693-007-9005-9] Keyword(s): systems biology, biochemical networks, monotone systems, Ising spin models, nonlinear stability, dynamical systems, consistent graphs, gene networks.
Abstract:
 This paper provides an expository introduction to monotone and near-monotone biochemical network structures. Monotone systems respond in a predictable fashion to perturbations, and have very robust dynamical characteristics. This makes them reliable components of more complex networks, and suggests that natural biological systems may have evolved to be, if not monotone, at least close to monotone. In addition, interconnections of monotone systems may be fruitfully analyzed using tools from control theory.

15. P. de Leenheer, D. Angeli, and E.D. Sontag. Monotone chemical reaction networks. J. Math Chemistry, 41:295-314, 2007. [PDF] [doi:10.1007/s10910-006-9075-z] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 We analyze certain chemical reaction networks and show that every solution converges to some steady state. The reaction kinetics are assumed to be monotone but otherwise arbitrary. When diffusion effects are taken into account, the conclusions remain unchanged. The main tools used in our analysis come from the theory of monotone dynamical systems. We review some of the features of this theory and provide a self-contained proof of a particular attractivity result which is used in proving our main result.

 2006
1. B. Dasgupta, P. Berman, and E.D. Sontag. Computational complexities of combinatorial problems with applications to reverse engineering of biological networks. In D. Liu and F-Y. Wan, editors, Advances in Computational Intelligence: Theory & Applications, pages 303-316. World Scientific, Hackensack, 2006. Keyword(s): systems biology, biochemical networks, gene and protein networks, reverse engineering, systems identification, theory of computing and complexity.

2. B. Dasgupta, G.A. Enciso, E.D. Sontag, and Y. Zhang. Algorithmic and complexity results for decompositions of biological networks into monotone subsystems. In C. Àlvarez and M. Serna, editors, Lecture Notes in Computer Science: Experimental Algorithms: 5th International Workshop, WEA 2006, pages 253-264. Springer-Verlag, 2006. Note: (Cala Galdana, Menorca, Spain, May 24-27, 2006). Keyword(s): systems biology, biochemical networks, monotone systems, theory of computing and complexity.

3. W. Maass, P. Joshi, and E.D. Sontag. Principles of real-time computing with feedback applied to cortical microcircuit models. In Advances in Neural Information Processing Systems 18. MIT Press, Cambridge, 2006. [PDF] Keyword(s): neural networks.
Abstract:
 The network topology of neurons in the brain exhibits an abundance of feedback connections, but the computational function of these feedback connections is largely unknown. We present a computational theory that characterizes the gain in computational power achieved through feedback in dynamical systems with fading memory. It implies that many such systems acquire through feedback universal computational capabilities for analog computing with a non-fading memory. In particular, we show that feedback enables such systems to process time-varying input streams in diverse ways according to rules that are implemented through internal states of the dynamical system. In contrast to previous attractor-based computational models for neural networks, these flexible internal states are high-dimensional attractors of the circuit dynamics, that still allow the circuit state to absorb new information from online input streams. In this way one arrives at novel models for working memory, integration of evidence, and reward expectation in cortical circuits. We show that they are applicable to circuits of conductance-based Hodgkin-Huxley (HH) neurons with high levels of noise that reflect experimental data on invivo conditions.

4. M. Arcak and E.D. Sontag. Diagonal stability of a class of cyclic systems and its connection with the secant criterion. Automatica, 42:1531-1537, 2006. [PDF] Keyword(s): passive systems, systems biology, biochemical networks, cyclic feedback systems, secant condition, nonlinear stability, dynamical systems.
Abstract:
 This paper considers a class of systems with a cyclic structure that arises, among other examples, in dynamic models for certain biochemical reactions. We first show that a criterion for local stability, derived earlier in the literature, is in fact a necessary and sufficient condition for diagonal stability of the corresponding class of matrices. We then revisit a recent generalization of this criterion to output strictly passive systems, and recover the same stability condition using our diagonal stability result as a tool for constructing a Lyapunov function. Using this procedure for Lyapunov construction we exhibit classes of cyclic systems with sector nonlinearities and characterize their global stability properties.

5. M. Chaves and E.D. Sontag. Exact computation of amplification for a class of nonlinear systems arising from cellular signaling pathways. Automatica, 42:1987-1992, 2006. [PDF] Keyword(s): MAPK cascades, systems biology, biochemical networks, nonlinear stability, dynamical systems.
Abstract:
 A commonly employed measure of the signal amplification properties of an input/output system is its induced L2 norm, sometimes also known as H-infinity gain. In general, however, it is extremely difficult to compute the numerical value for this norm, or even to check that it is finite, unless the system being studied is linear. This paper describes a class of systems for which it is possible to reduce this computation to that of finding the norm of an associated linear system. In contrast to linearization approaches, a precise value, not an estimate, is obtained for the full nonlinear model. The class of systems that we study arose from the modeling of certain biological intracellular signaling cascades, but the results should be of wider applicability.

6. M. Chaves, E.D. Sontag, and R. Albert. Methods of robustness analysis for Boolean models of gene control networks. IET Systems Biology, 153:154-167, 2006. [PDF] Keyword(s): systems biology, biochemical networks, boolean systems, identifiability, robust, robustness, geometry, Boolean, segment polarity network, gene and protein networks, hybrid systems.
Abstract:
 As a discrete approach to genetic regulatory networks, Boolean models provide an essential qualitative description of the structure of interactions among genes and proteins. Boolean models generally assume only two possible states (expressed or not expressed) for each gene or protein in the network as well as a high level of synchronization among the various regulatory processes. In this paper, we discuss and compare two possible methods of adapting qualitative models to incorporate the continuous-time character of regulatory networks. The first method consists of introducing asynchronous updates in the Boolean model. In the second method, we adopt the approach introduced by L. Glass to obtain a set of piecewise linear differential equations which continuously describe the states of each gene or protein in the network. We apply both methods to a particular example: a Boolean model of the segment polarity gene network of Drosophila melanogaster. We analyze the dynamics of the model, and provide a theoretical characterization of the model's gene pattern prediction as a function of the timescales of the various processes.

7. B. DasGupta, J.P. Hespanha, J. Riehl, and E.D. Sontag. Honey-pot constrained searching with local sensory information. Nonlinear Analysis, 65:1773-1793, 2006. [PDF] Keyword(s): search problems, algorithms, computational complexity.
Abstract:
 This paper investigates the problem of searching for a hidden target in a bounded region of the plane by an autonomous robot which is only able to use limited local sensory information. It proposes an aggregation-based approach to solve this problem, in which the continuous search space is partitioned into a finite collection of regions on which we define a discrete search problem and a solution to the original problem is obtained through a refinement procedure that lifts the discrete path into a continuous one. The resulting solution is in general not optimal but one can construct bounds to gauge the cost penalty incurred. The discrete version is formalized and an optimization problem is stated as a reward-collecting' bounded-length path problem. NP-completeness and efficient approximation algorithms for various cases of this problem are discussed.

8. G.A. Enciso, H.L. Smith, and E.D. Sontag. Non-monotone systems decomposable into monotone systems with negative feedback. J. of Differential Equations, 224:205-227, 2006. [PDF] Keyword(s): nonlinear stability, dynamical systems, monotone systems.
Abstract:
 Motivated by the theory of monotone i/o systems, this paper shows that certain finite and infinite dimensional semi-dynamical systems with negative feedback can be decomposed into a monotone open loop system with inputs and a decreasing output function. The original system is reconstituted by plugging the output into the input. By embedding the system into a larger symmetric monotone system, this paper obtains finer information on the asymptotic behavior of solutions, including existence of positively invariant sets and global convergence. An important new result is the extension of the "small gain theorem" of monotone i/o theory to reaction-diffusion partial differential equations: adding diffusion preserves the global attraction of the ODE equilibrium.

9. G.A. Enciso and E.D. Sontag. Global attractivity, I/O monotone small-gain theorems, and biological delay systems. Discrete Contin. Dyn. Syst., 14(3):549-578, 2006. [PDF] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 This paper further develops a method, originally introduced in a paper by Angeli and Sontag, for proving global attractivity of steady states in certain classes of dynamical systems. In this aproach, one views the given system as a negative feedback loop of a monotone controlled system. An auxiliary discrete system, whose global attractivity implies that of the original system, plays a key role in the theory, which is presented in a general Banach space setting. Applications are given to delay systems, as well as to systems with multiple inputs and outputs, and the question of expressing a given system in the required negative feedback form is addressed.

10. M. Malisoff, M. Krichman, and E.D. Sontag. Global stabilization for systems evolving on manifolds. Journal of Dynamical and Control Systems, 12:161-184, 2006. [PDF] Keyword(s): nonlinear stability, nonlinear control, feedback stabilization.
Abstract:
 This paper shows that any globally asymptotically controllable system on any smooth manifold can be globally stabilized by a state feedback. Since discontinuous feedbacks are allowed, solutions are understood in the sample and hold'' sense introduced by Clarke-Ledyaev-Sontag-Subbotin (CLSS). This work generalizes the CLSS Theorem, which is the special case of our result for systems on Euclidean space. We apply our result to the input-to-state stabilization of systems on manifolds relative to actuator errors, under small observation noise.

11. E.P. Ryan and E.D. Sontag. Well-defined steady-state response does not imply CICS. Systems and Control Letters, 55:707-710, 2006. [PDF] [doi:10.1016/j.sysconle.2006.02.001] Keyword(s): nonlinear stability, dynamical systems.
Abstract:
 Systems for which each constant input gives rise to a unique globally attracting equilibrium are considered. A counterexample is provided to show that inputs which are only asymptotically constant may not result in states converging to equilibria (failure of the converging-input converging state, or CICS'' property).

12. E.D. Sontag. Passivity gains and the secant condition'' for stability. Systems Control Lett., 55(3):177-183, 2006. [PDF] Keyword(s): cyclic feedback systems, systems biology, biochemical networks, nonlinear stability, dynamical systems, passive systems, secant condition, biochemical networks.
Abstract:
 A generalization of the classical secant condition for the stability of cascades of scalar linear systems is provided for passive systems. The key is the introduction of a quantity that combines gain and phase information for each system in the cascade. For linear one-dimensional systems, the known result is recovered exactly.

13. E.D. Sontag and Y. Wang. A cooperative system which does not satisfy the limit set dichotomy. J. of Differential Equations, 224:373-384, 2006. [PDF] Keyword(s): dynamical systems, monotone systems.
Abstract:
 The fundamental property of strongly monotone systems, and strongly cooperative systems in particular, is the limit set dichotomy due to Hirsch: if x < y, then either Omega(x) < Omega (y), or Omega(x) = Omega(y) and both sets consist of equilibria. We provide here a counterexample showing that this property need not hold for (non-strongly) cooperative systems.

14. P. de Leenheer, D. Angeli, and E.D. Sontag. Crowding effects promote coexistence in the chemostat. Journal of Mathematical Analysis and Applications, 319:48-60, 2006. [PDF] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 We provide an almost-global stability result for a particular chemostat model, in which crowding effects are taken into consideration. The model can be rewritten as a negative feedback interconnection of two monotone i/o systems with well-defined characteristics, which allows the use of a small-gain theorem for feedback interconnections of monotone systems. This leads to a sufficient condition for almost-global stability, and we show that coexistence occurs in this model if the crowding effects are large enough.

15. P. de Leenheer, S.A. Levin, E.D. Sontag, and C.A. Klausmeier. Global stability in a chemostat with multiple nutrients. J. Mathematical Biology, 52:419-438, 2006. [PDF] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 We study a single species in a chemostat, limited by two nutrients, and separate nutrient uptake from growth. For a broad class of uptake and growth functions it is proved that a nontrivial equilibrium may exist. Moreover, if it exists it is unique and globally stable, generalizing a previous result by Legovic and Cruzado.

16. N.A.W. van Riel and E.D. Sontag. Parameter estimation in models combining signal transduction and metabolic pathways: The dependent input approach. IET Systems Biology, 153:263-274, 2006. [PDF] Keyword(s): systems biology, biochemical networks, parameter identification.
Abstract:
 Biological complexity and limited quantitative measurements impose severe challenges to standard engineering methodologies for systems identification. This paper presents an approach, justified by the theory of universal inputs for distinguishability, based on replacing unmodeled dynamics by fictitious dependent inputs'. The approach is particularly useful in validation experiments, because it allows one to fit model parameters to experimental data generated by a reference (wild-type) organism and then testing this model on data generated by a variation (mutant), so long as the mutations only affect the unmodeled dynamics that produce the dependent inputs. As a case study, this paper addresses the pathways that control the nitrogen uptake fluxes in baker's yeast Saccharomyces cerevisiae enabling it to optimally respond to changes in nitrogen availability. Well-defined perturbation experiments were performed on cells growing in steady-state. Time-series data of extracellular and intracellular metabolites were obtained, as well as mRNA levels. A nonlinear model was proposed, and shown to be structurally identifiable given input/output data. The identified model correctly predicted the responses of different yeast strains and different perturbations.

 2005
1. M. Andrec, B.N. Kholodenko, R.M. Levy, and E.D. Sontag. Inference of signaling and gene regulatory networks by steady-state perturbation experiments: structure and accuracy. J. Theoret. Biol., 232(3):427-441, 2005. Note: Supplementary materials are here: http://sontaglab.org/FTPDIR/andrec-kholodenko-levy-sontag-JTB04-supplementary.pdf. [PDF] Keyword(s): systems biology, biochemical networks, gene and protein networks, systems identification, reverse engineering.
Abstract:
 One of the fundamental problems of cell biology is the understanding of complex regulatory networks. Such networks are ubiquitous in cells, and knowledge of their properties is essential for the understanding of cellular behavior. This paper studies the effect of experimental uncertainty on the accuracy of the inferred structure of the networks determined using the method in "Untangling the wires: a novel strategy to trace functional interactions in signaling and gene networks".

2. M. Chaves, R. Albert, and E.D. Sontag. Robustness and fragility of Boolean models for genetic regulatory networks. J. Theoret. Biol., 235(3):431-449, 2005. [PDF] Keyword(s): systems biology, biochemical networks, boolean systems, gene and protein networks.
Abstract:
 Interactions between genes and gene products give rise to complex circuits that enable cells to process information and respond to external signals. Theoretical studies often describe these interactions using continuous, stochastic, or logical approaches. Here we propose a framework for gene regulatory networks that combines the intuitive appeal of a qualitative description of gene states with a high flexibility in incorporating stochasticity in the duration of cellular processes. We apply our methods to the regulatory network of the segment polarity genes, thus gaining novel insights into the development of gene expression patterns. For example, we show that very short synthesis and decay times can perturb the wild type pattern. On the other hand, separation of timescales between pre- and post-translational processes and a minimal prepattern ensure convergence to the wild type expression pattern regardless of fluctuations.

3. G.A. Enciso and E.D. Sontag. Monotone systems under positive feedback: multistability and a reduction theorem. Systems Control Lett., 54(2):159-168, 2005. [PDF] Keyword(s): multistability, systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 For feedback loops involving single input, single output monotone systems with well-defined I/O characteristics, a previous paper provided an approach to determining the location and stability of steady states. A result on global convergence for multistable systems followed as a consequence of the technique. The present paper extends the approach to multiple inputs and outputs. A key idea is the introduction of a reduced system which preserves local stability properties. New results characterizing strong monotonicity of feedback loops involving cascades are also presented.

4. J.P. Hespanha, D. Liberzon, D. Angeli, and E.D. Sontag. Nonlinear norm-observability notions and stability of switched systems. IEEE Trans. Automat. Control, 50(2):154-168, 2005. [PDF] Keyword(s): observability, input to state stability, observability, invariance principle.
Abstract:
 This paper proposes several definitions of observability for nonlinear systems and explores relationships among them. These observability properties involve the existence of a bound on the norm of the state in terms of the norms of the output and the input on some time interval. A Lyapunov-like sufficient condition for observability is also obtained. As an application, we prove several variants of LaSalle's stability theorem for switched nonlinear systems. These results are demonstrated to be useful for control design in the presence of switching as well as for developing stability results of Popov type for switched feedback systems.

5. J. L. Mancilla-Aguilar, R. Garcìa, E.D. Sontag, and Y. Wang. On the representation of switched systems with inputs by perturbed control systems. Nonlinear Anal., 60(6):1111-1150, 2005. [PDF]
Abstract:
 This paper provides representations of switched systems described by controlled differential inclusions, in terms of perturbed control systems. The control systems have dynamics given by differential equations, and their inputs consist of the original controls together with disturbances that evolve in compact sets; their sets of maximal trajectories contain, as a dense subset, the set of maximal trajectories of the original system. Several applications to control theory, dealing with properties of stability with respect to inputs and of detectability, are derived as a consequence of the representation theorem.

6. J. L. Mancilla-Aguilar, R. Garcìa, E.D. Sontag, and Y. Wang. Uniform stability properties of switched systems with switchings governed by digraphs. Nonlinear Anal., 63(3):472-490, 2005. [PDF]
Abstract:
 This paper develops characterizations of various uniform stability properties of switched systems described by differential inclusions, and whose switchings are governed by a digraph. These characterizations are given in terms of stability properties of the system with restricted switchings and also in terms of Lyapunov functions.

7. E.D. Sontag. Molecular systems biology and control. Eur. J. Control, 11(4-5):396-435, 2005. [PDF] Keyword(s): cell biology, systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems, molecular biology, systems biology, cellular signaling.
Abstract:
 This paper, prepared for a tutorial at the 2005 IEEE Conference on Decision and Control, presents an introduction to molecular systems biology and some associated problems in control theory. It provides an introduction to basic biological concepts, describes several questions in dynamics and control that arise in the field, and argues that new theoretical problems arise naturally in this context. A final section focuses on the combined use of graph-theoretic, qualitative knowledge about monotone building-blocks and steady-state step responses for components.

8. P. de Leenheer, D. Angeli, and E.D. Sontag. On predator-prey systems and small-gain theorems. Math. Biosci. Eng., 2(1):25-42, 2005. [PDF] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 This paper deals with an almost global attractivity result for Lotka-Volterra systems with predator-prey interactions. These systems can be written as (negative) feedback systems. The subsystems of the feedback loop are monotone control systems, possessing particular input-output properties. We use a small-gain theorem, adapted to a context of systems with multiple equilibrium points to obtain the desired almost global attractivity result. It provides sufficient conditions to rule out oscillatory or more complicated behavior which is often observed in predator-prey systems.

 2004
1. D. Angeli and E.D. Sontag. Interconnections of monotone systems with steady-state characteristics. In Optimal control, stabilization and nonsmooth analysis, volume 301 of Lecture Notes in Control and Inform. Sci., pages 135-154. Springer, Berlin, 2004. [PDF] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 One of the key ideas in control theory is that of viewing a complex dynamical system as an interconnection of simpler subsystems, thus deriving conclusions regarding the complete system from properties of its building blocks. Following this paradigm, and motivated by questions in molecular biology modeling, the authors have recently developed an approach based on components which are monotone systems with respect to partial orders in state and signal spaces. This paper presents a brief exposition of recent results, with an emphasis on small gain theorems for negative feedback, and the emergence of multi-stability and associated hysteresis effects under positive feedback.

2. M. Malisoff and E.D. Sontag. Asymptotic controllability and input-to-state stabilization: the effect of actuator errors. In Optimal control, stabilization and nonsmooth analysis, volume 301 of Lecture Notes in Control and Inform. Sci., pages 155-171. Springer, Berlin, 2004. [PDF] Keyword(s): input to state stability, control-Lyapunov functions, nonlinear control, feedback stabilization, ISS.
Abstract:
 We discuss several issues related to the stabilizability of nonlinear systems. First, for continuously stabilizable systems, we review constructions of feedbacks that render the system input-to-state stable with respect to actuator errors. Then, we discuss a recent paper which provides a new feedback design that makes globally asymptotically controllable systems input-to-state stable to actuator errors and small observation noise. We illustrate our constructions using the nonholonomic integrator, and discuss a related feedback design for systems with disturbances.

3. D. Angeli, J. E. Ferrell, and E.D. Sontag. Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems.. Proc Natl Acad Sci USA, 101(7):1822-1827, 2004. Note: A revision of Suppl. Fig. 7(b) is here: http://sontaglab.org/FTPDIR/nullclines-f-g-REV.jpg; and typos can be found here: http://sontaglab.org/FTPDIR/angeli-ferrell-sontag-pnas04-errata.txt. [WWW] [PDF] [doi:10.1073/pnas.0308265100] Keyword(s): MAPK cascades, multistability, systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 Multistability is an important recurring theme in cell signaling, of particular relevance to biological systems that switch between discrete states, generate oscillatory responses, or "remember" transitory stimuli. Standard mathematical methods allow the detection of bistability in some very simple feedback systems (systems with one or two proteins or genes that either activate each other or inhibit each other), but realistic depictions of signal transduction networks are invariably much more complex than this. Here we show that for a class of feedback systems of arbitrary order, the stability properties of the system can be deduced mathematically from how the system behaves when feedback is blocked. Provided that this "open loop," feedback-blocked system is monotone and possesses a sigmoidal characteristic, the system is guaranteed to be bistable for some range of feedback strengths. We present a simple graphical method for deducing the stability behavior and bifurcation diagrams for such systems, and illustrate the method with two examples taken from recent experimental studies of bistable systems: a two-variable Cdc2/Wee1 system and a more complicated five-variable MAPK cascade.

4. D. Angeli, B.P. Ingalls, E.D. Sontag, and Y. Wang. Separation principles for input-output and integral-input-to-state stability. SIAM J. Control Optim., 43(1):256-276, 2004. [PDF] [doi:http://dx.doi.org/10.1137/S0363012902419047] Keyword(s): input to state stability, integral input to state stability, iISS, ISS, input to output stability.
Abstract:
 We present new characterizations of input-output-to-state stability. This is a notion of detectability formulated in the ISS framework. Equivalent properties are presented in terms of asymptotic estimates of the state trajectories based on the magnitudes of the external input and output signals. These results provide a set of "separation principles" for input-output-to-state stability , characterizations of the property in terms of weaker stability notions. When applied to the closely related notion of integral ISS, these characterizations yield analogous results.

5. D. Angeli, B.P. Ingalls, E.D. Sontag, and Y. Wang. Uniform global asymptotic stability of differential inclusions. J. Dynam. Control Systems, 10(3):391-412, 2004. [PDF] [doi:http://dx.doi.org/10.1023/B:JODS.0000034437.54937.7f] Keyword(s): differential inclusions.
Abstract:
 The stability of differential inclusions defined by locally Lipschitz compact valued maps is addressed. It is shown that if such a differential inclusion is globally asymptotically stable, then in fact it is uniformly globally asymptotically stable (with respect to initial states in compacts). This statement is trivial for differential equations, but here we provide the extension to compact (not necessarily convex) valued differential inclusions. The main result is presented in a context which is useful for control-theoretic applications: a differential inclusion with two outputs is considered, and the result applies to the property of global error detectability.

6. D. Angeli and E.D. Sontag. Multi-stability in monotone input/output systems. Systems Control Lett., 51(3-4):185-202, 2004. [PDF] Keyword(s): multistability, systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 This paper studies the emergence of multi-stability and hysteresis in those systems that arise, under positive feedback, from monotone systems with well-defined steady-state responses. Such feedback configurations appear routinely in several fields of application, and especially in biology. The results are stated in terms of directly checkable conditions which do not involve explicit knowledge of basins of attractions of each equilibria.

7. D. Angeli, P. de Leenheer, and E.D. Sontag. A small-gain theorem for almost global convergence of monotone systems. Systems Control Lett., 52(5):407-414, 2004. [PDF] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 A small-gain theorem is presented for almost global stability of monotone control systems which are open-loop almost globally stable, when constant inputs are applied. The theorem assumes "negative feedback" interconnections. This typically destroys the monotonicity of the original flow and potentially destabilizes the resulting closed-loop system.

8. M. Chaves, R.J. Dinerstein, and E.D. Sontag. Optimal length and signal amplification in weakly activated signal transduction cascades. J. Physical Chemistry, 108:15311-15320, 2004. [PDF] Keyword(s): systems biology, biochemical networks, dynamical systems.
Abstract:
 Weakly activated signaling cascades can be modeled as linear systems. The input-to-output transfer function and the internal gain of a linear system, provide natural measures for the propagation of the input signal down the cascade and for the characterization of the final outcome. The most efficient design of a cascade for generating sharp signals, is obtained by choosing all the off rates equal, and a "universal" finite optimal length.

9. M. Chaves, E.D. Sontag, and R. J. Dinerstein. Steady-states of receptor-ligand dynamics: A theoretical framework. J. Theoret. Biol., 227(3):413-428, 2004. [PDF] Keyword(s): zero-deficiency networks, systems biology, biochemical networks, receptor-ligand models, dynamical systems.
Abstract:
 This paper studies aspects of the dynamics of a conventional mechanism of ligand-receptor interactions, with a focus on the stability and location of steady-states. A theoretical framework is developed, and, as an application, a minimal parametrization is provided for models for two- or multi-state receptor interaction with ligand. In addition, an "affinity quotient" is introduced, which allows an elegant classification of ligands into agonists, neutral agonists, and inverse agonists.

10. G.A. Enciso and E.D. Sontag. On the stability of a model of testosterone dynamics. J. Math. Biol., 49(6):627-634, 2004. [PDF] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 We prove the global asymptotic stability of a well-known delayed negative-feedback model of testosterone dynamics, which has been proposed as a model of oscillatory behavior. We establish stability (and hence the impossibility of oscillations) even in the presence of delays of arbitrary length.

11. P. Kuusela, D. Ocone, and E.D. Sontag. Learning Complexity Dimensions for a Continuous-Time Control System. SIAM J. Control Optim., 43(3):872-898, 2004. [PDF] [doi:http://dx.doi.org/10.1137/S0363012901384302] Keyword(s): machine learning, theory of computing and complexity, VC dimension, neural networks.
Abstract:
 This paper takes a computational learning theory approach to a problem of linear systems identification. It is assumed that input signals have only a finite number k of frequency components, and systems to be identified have dimension no greater than n. The main result establishes that the sample complexity needed for identification scales polynomially with n and logarithmically with k.

12. M. Malisoff, L. Rifford, and E.D. Sontag. Global Asymptotic Controllability Implies Input-to-State Stabilization. SIAM J. Control Optim., 42(6):2221-2238, 2004. [PDF] [doi:http://dx.doi.org/10.1137/S0363012903422333] Keyword(s): input to state stability, control-Lyapunov functions, nonlinear control, feedback stabilization.
Abstract:
 The main problem addressed in this paper is the design of feedbacks for globally asymptotically controllable (GAC) control affine systems that render the closed loop systems input to state stable with respect to actuator errors. Extensions for fully nonlinear GAC systems with actuator errors are also discussed. Our controllers have the property that they tolerate small observation noise as well.

13. E.D. Sontag. Some new directions in control theory inspired by systems biology. IET Systems Biology, 1:9-18, 2004. [PDF] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems, cellular signaling.
Abstract:
 This paper, addressed primarily to engineers and mathematicians with an interest in control theory, argues that entirely new theoretical problems arise naturally when addressing questions in the field of systems biology. Examples from the author's recent work are used to illustrate this point.

14. E.D. Sontag, A. Kiyatkin, and B.N. Kholodenko. Inferring dynamic architecture of cellular networks using time series of gene expression, protein and metabolite data. Bioinformatics, 20(12):1877-1886, 2004. Note: Supplementary materials are here: http://sontaglab.org/FTPDIR/sontag-kiyatkin-kholodenko-informatics04-supplement.pdf. [PDF] [doi:http://dx.doi.org/10.1093/bioinformatics/bth173] Keyword(s): systems biology, biochemical networks, systems identification, gene and protein networks, reverse engineering.
Abstract:
 High-throughput technologies have facilitated the acquisition of large genomics and proteomics data sets. However, these data provide snapshots of cellular behavior, rather than help us reveal causal relations. Here, we propose how these technologies can be utilized to infer the topology and strengths of connections among genes, proteins, and metabolites by monitoring time-dependent responses of cellular networks to experimental interventions. We show that all connections leading to a given network node, e.g., to a particular gene, can be deduced from responses to perturbations none of which directly influences that node, e.g., using strains with knock-outs to other genes. To infer all interactions from stationary data, each node should be perturbed separately or in combination with other nodes. Monitoring time series provides richer information and does not require perturbations to all nodes.

 2003
1. P. de Leenheer, D. Angeli, and E.D. Sontag. A feedback perspective for chemostat models with crowding effects. In Positive systems (Rome, 2003), volume 294 of Lecture Notes in Control and Inform. Sci., pages 167-174. Springer, Berlin, 2003. Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.

2. P. de Leenheer, D. Angeli, and E.D. Sontag. Small-gain theorems for predator-prey systems. In Positive systems (Rome, 2003), volume 294 of Lecture Notes in Control and Inform. Sci., pages 191-198. Springer, Berlin, 2003. Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.

3. D. Angeli and E.D. Sontag. Monotone control systems. IEEE Trans. Automat. Control, 48(10):1684-1698, 2003. Note: Errata are here: http://sontaglab.org/FTPDIR/angeli-sontag-monotone-TAC03-typos.txt. [PDF] Keyword(s): MAPK cascades, systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 Monotone systems constitute one of the most important classes of dynamical systems used in mathematical biology modeling. The objective of this paper is to extend the notion of monotonicity to systems with inputs and outputs, a necessary first step in trying to understand interconnections, especially including feedback loops, built up out of monotone components. Basic definitions and theorems are provided, as well as an application to the study of a model of one of the cell's most important subsystems.

4. D. Angeli, E.D. Sontag, and Y. Wang. Input-to-state stability with respect to inputs and their derivatives. Internat. J. Robust Nonlinear Control, 13(11):1035-1056, 2003. [PDF] Keyword(s): input to state stability, ISS, input to state stability, ISS.
Abstract:
 A new notion of input-to-state stability involving infinity norms of input derivatives up to a finite order k is introduced and characterized. An example shows that this notion of stability is indeed weaker than the usual ISS. Applications to the study of global asymptotic stability of cascaded nonlinear systems are discussed.

5. M. Chyba, N. E. Leonard, and E.D. Sontag. Singular trajectories in multi-input time-optimal problems: Application to controlled mechanical systems. Journal of Dynamical and Control Systems, 9(1):103-129, 2003. [PDF] [doi:http://dx.doi.org/10.1023/A:1022159318457] Keyword(s): optimal control.
Abstract:
 This paper addresses the time-optimal control problem for a class of control systems which includes controlled mechanical systems with possible dissipation terms. The Lie algebras associated with such mechanical systems enjoy certain special properties. These properties are explored and are used in conjunction with the Pontryagin maximum principle to determine the structure of singular extremals and, in particular, time-optimal trajectories. The theory is illustrated with an application to a time-optimal problem for a class of underwater vehicles.

6. B.P. Ingalls, E.D. Sontag, and Y. Wang. An infinite-time relaxation theorem for differential inclusions. Proc. Amer. Math. Soc., 131(2):487-499, 2003. [PDF]
Abstract:
 The fundamental relaxation result for Lipschitz differential inclusions is the Filippov-Wazewski Relaxation Theorem, which provides approximations of trajectories of a relaxed inclusion on finite intervals. A complementary result is presented, which provides approximations on infinite intervals, but does not guarantee that the approximation and the reference trajectory satisfy the same initial condition.

7. L. Moreau and E.D. Sontag. Balancing at the border of instability. Phys. Rev. E (3), 68(2):020901, 4, 2003. [PDF] Keyword(s): bifurcations, adaptive control.
Abstract:
 Some biological systems operate at the critical point between stability and instability and this requires a fine-tuning of parameters. We bring together two examples from the literature that illustrate this: neural integration in the nervous system and hair cell oscillations in the auditory system. In both examples the question arises as to how the required fine-tuning may be achieved and maintained in a robust and reliable way. We study this question using tools from nonlinear and adaptive control theory. We illustrate our approach on a simple model which captures some of the essential features of neural integration. As a result, we propose a large class of feedback adaptation rules that may be responsible for the experimentally observed robustness of neural integration. We mention extensions of our approach to the case of hair cell oscillations in the ear.

8. L. Moreau, E.D. Sontag, and M. Arcak. Feedback tuning of bifurcations. Systems Control Lett., 50(3):229-239, 2003. [PDF] Keyword(s): bifurcations, adaptive control.
Abstract:
 This paper studies a feedback regulation problem that arises in at least two different biological applications. The feedback regulation problem under consideration may be interpreted as an adaptive control problem for tuning bifurcation parameters, and it has not been studied in the control literature. The goal of the paper is to formulate this problem and to present some preliminary results.

9. J. R. Pomerening, E.D. Sontag, and J. E. Ferrell. Building a cell cycle oscillator: hysteresis and bistability in the activation of Cdc2. Nature Cell Biology, 5(4):346-351, 2003. Note: Supplementary materials 2-4 are here: http://sontaglab.org/FTPDIR/pomerening-sontag-ferrell-additional.pdf. [WWW] [PDF] [doi:10.1038/ncb954] Keyword(s): systems biology, biochemical networks, oscillations, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 In the early embryonic cell cycle, Cdc2-cyclin B functions like an autonomous oscillator, at whose core is a negative feedback loop: cyclins accumulate and produce active mitotic Cdc2-cyclin B Cdc2 activates the anaphase-promoting complex (APC); the APC then promotes cyclin degradation and resets Cdc2 to its inactive, interphase state. Cdc2 regulation also involves positive feedback4, with active Cdc2-cyclin B stimulating its activator Cdc25 and inactivating its inhibitors Wee1 and Myt1. Under the correct circumstances, these positive feedback loops could function as a bistable trigger for mitosis, and oscillators with bistable triggers may be particularly relevant to biological applications such as cell cycle regulation. This paper examined whether Cdc2 activation is bistable, confirming that the response of Cdc2 to non-degradable cyclin B is temporally abrupt and switchlike, as would be expected if Cdc2 activation were bistable. It is also shown that Cdc2 activation exhibits hysteresis, a property of bistable systems with particular relevance to biochemical oscillators. These findings help establish the basic systems-level logic of the mitotic oscillator.

10. E.D. Sontag. A remark on the converging-input converging-state property. IEEE Trans. Automat. Control, 48(2):313-314, 2003. [PDF]
Abstract:
 Suppose that an equilibrium is asymptotically stable when external inputs vanish. Then, every bounded trajectory which corresponds to a control which approaches zero and which lies in the domain of attraction of the unforced system, must also converge to the equilibrium. This "well-known" but hard-to-cite fact is proved and slightly generalized here.

11. E.D. Sontag. Adaptation and regulation with signal detection implies internal model. Systems Control Lett., 50(2):119-126, 2003. [PDF] Keyword(s): biological adaptation, internal model principle.
Abstract:
 This note provides a simple result showing, under suitable technical assumptions, that if a system S adapts to a class of external signals U, then S must necessarily contain a subsystem which is capable of generating all the signals in U. It is not assumed that regulation is robust, nor is there a prior requirement for the system to be partitioned into separate plant and controller components. Instead, a "signal detection" capability is imposed. These weaker assumptions make the result better applicable to cellular phenomena such as the adaptation of E-coli chemotactic tumbling rate to constant concentrations.

12. E.D. Sontag and M. Krichman. An example of a GAS system which can be destabilized by an integrable perturbation. IEEE Trans. Automat. Control, 48(6):1046-1049, 2003. [PDF] Keyword(s): nonlinear stability.
Abstract:
 A construction is given of a globally asymptotically stable time-invariant system which can be destabilized by some integrable perturbation. Besides its intrinsic interest, this serves to provide counterexamples to an open question regarding Lyapunov functions.

 2002
1. M. Arcak, D. Angeli, and E.D. Sontag. A unifying integral ISS framework for stability of nonlinear cascades. SIAM J. Control Optim., 40(6):1888-1904, 2002. [PDF] [doi:http://dx.doi.org/10.1137/S0363012901387987] Keyword(s): input to state stability, integral input to state stability, iISS, ISS.
Abstract:
 We analyze nonlinear cascades in which the driven subsystem is integral ISS, and characterize the admissible integral ISS gains for stability. This characterization makes use of the convergence speed of the driving subsystem, and allows a larger class of gain functions when the convergence is faster. We show that our integral ISS gain characterization unifies different approaches in the literature which restrict the nonlinear growth of the driven subsystem and the convergence speed of the driving subsystem.

2. M. Chaves and E.D. Sontag. State-Estimators for chemical reaction networks of Feinberg-Horn-Jackson zero deficiency type. European J. Control, 8:343-359, 2002. [PDF] Keyword(s): observability, zero-deficiency networks, systems biology, biochemical networks, observers, nonlinear stability, dynamical systems.
Abstract:
 This paper provides a necessary and sufficient condition for detectability, and an explicit construction of observers when this condition is satisfied, for chemical reaction networks of the Feinberg-Horn-Jackson zero deficiency type.

3. B.N. Kholodenko, A. Kiyatkin, F.J. Bruggeman, E.D. Sontag, H.V. Westerhoff, and J. Hoek. Untangling the wires: a novel strategy to trace functional interactions in signaling and gene networks. Proceedings of the National Academy of Sciences USA, 99:12841-12846, 2002. [PDF] Keyword(s): MAPK cascades, systems biology, biochemical networks, reverse engineering, gene and protein networks, protein networks, gene networks, systems identification.
Abstract:
 Emerging technologies have enabled the acquisition of large genomics and proteomics data sets. This paper proposes a novel quantitative method for determining functional interactions in cellular signaling and gene networks. It can be used to explore cell systems at a mechanistic level, or applied within a modular framework, which dramatically decreases the number of variables to be assayed. The topology and strength of network connections are retrieved from experimentally measured network responses to successive perturbations of all modules. In addition, the method can reveal functional interactions even when the components of the system are not all known, in which case some connections retrieved by the analysis will not be direct but correspond to the interaction routes through unidentified elements. The method is tested and illustrated using computer-generated responses of a modeled MAPK cascade and gene network.

4. M. Krichman and E.D. Sontag. Characterizations of detectability notions in terms of discontinuous dissipation functions. Internat. J. Control, 75(12):882-900, 2002. [PDF] Keyword(s): input to state stability, detectability, input to output stability, detectability.
Abstract:
 We consider a new Lyapunov-type characterization of detectability for nonlinear systems without controls, in terms of lower-semicontinuous (not necessarily smooth, or even continuous) dissipation functions, and prove its equivalence to the GASMO (global asymptotic stability modulo outputs) and UOSS (uniform output-to-state stability) properties studied in previous work. The result is then extended to provide a construction of a discontinuous dissipation function characterization of the IOSS (input-to-state stability) property for systems with controls. This paper complements a recent result on smooth Lyapunov characterizations of IOSS. The utility of non-smooth Lyapunov characterizations is illustrated by application to a well-known transistor network example.

5. D. Liberzon, A. S. Morse, and E.D. Sontag. Output-input stability and minimum-phase nonlinear systems. IEEE Trans. Automat. Control, 47(3):422-436, 2002. [PDF] Keyword(s): input to state stability, detectability, minimum-phase systems, ISS, nonlinear control, minimum phase, adaptive control.
Abstract:
 This paper introduces and studies a new definition of the minimum-phase property for general smooth nonlinear control systems. The definition does not rely on a particular choice of coordinates in which the system takes a normal form or on the computation of zero dynamics. In the spirit of the input-to-state stability'' philosophy, it requires the state and the input of the system to be bounded by a suitable function of the output and derivatives of the output, modulo a decaying term depending on initial conditions. The class of minimum-phase systems thus defined includes all affine systems in global normal form whose internal dynamics are input-to-state stable and also all left-invertible linear systems whose transmission zeros have negative real parts. As an application, we explain how the new concept enables one to develop a natural extension to nonlinear systems of a basic result from linear adaptive control.

6. D. Liberzon, E.D. Sontag, and Y. Wang. Universal construction of feedback laws achieving ISS and integral-ISS disturbance attenuation. Systems Control Lett., 46(2):111-127, 2002. Note: Errata here: http://sontaglab.org/FTPDIR/iiss-clf-errata.pdf. [PDF] Keyword(s): input to state stability, integral input to state stability, ISS, iISS, nonlinear control, feedback stabilization.
Abstract:
 We study nonlinear systems with both control and disturbance inputs. The main problem addressed in the paper is design of state feedback control laws that render the closed-loop system integral-input-to-state stable (iISS) with respect to the disturbances. We introduce an appropriate concept of control Lyapunov function (iISS-CLF), whose existence leads to an explicit construction of such a control law. The same method applies to the problem of input-to-state stabilization. Converse results and techniques for generating iISS-CLFs are also discussed.

7. E.D. Sontag. Asymptotic amplitudes and Cauchy gains: A small-gain principle and an application to inhibitory biological feedback. Systems Control Lett., 47(2):167-179, 2002. [PDF] Keyword(s): MAPK cascades, cyclic feedback systems, small-gain.
Abstract:
 The notions of asymptotic amplitude for signals, and Cauchy gain for input/output systems, and an associated small-gain principle, are introduced. These concepts allow the consideration of systems with multiple, and possibly feedback-dependent, steady states. A Lyapunov-like characterization allows the computation of gains for state-space systems, and the formulation of sufficient conditions insuring the lack of oscillations and chaotic behaviors in a wide variety of cascades and feedback loops. An application in biology (MAPK signaling) is worked out in detail.

8. E.D. Sontag. Correction to: Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction'' [IEEE Trans. Automat. Control 46 (2001), no. 7, 1028--1047; MR1842137 (2002e:92006)]. IEEE Trans. Automat. Control, 47(4):705, 2002. [PDF] Keyword(s): zero-deficiency networks, systems biology, biochemical networks, nonlinear stability, dynamical systems.
Abstract:
 errata for Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction

9. E.D. Sontag. For differential equations with r parameters, 2r+1 experiments are enough for identification. J. Nonlinear Sci., 12(6):553-583, 2002. [PDF] Keyword(s): identifiability, observability, systems biology, biochemical networks, parameter identification.
Abstract:
 Given a set of differential equations whose description involves unknown parameters, such as reaction constants in chemical kinetics, and supposing that one may at any time measure the values of some of the variables and possibly apply external inputs to help excite the system, how many experiments are sufficient in order to obtain all the information that is potentially available about the parameters? This paper shows that the best possible answer (assuming exact measurements) is 2r+1 experiments, where r is the number of parameters.

10. E.D. Sontag and B.P. Ingalls. A small-gain theorem with applications to input/output systems, incremental stability, detectability, and interconnections. J. Franklin Inst., 339(2):211-229, 2002. [PDF] Keyword(s): input to state stability, ISS, Small-Gain Theorem.
Abstract:
 A general ISS-type small-gain result is presented. It specializes to a small-gain theorem for ISS operators, and it also recovers the classical statement for ISS systems in state-space form. In addition, we highlight applications to incrementally stable systems, detectable systems, and to interconnections of stable systems.

 2001
1. A. C. Antoulas, E. D. Sontag, and Y. Yamamoto. Controllability and Observability, pages 264-281. John Wiley & Sons, Inc., 2001. [WWW] [PDF] [doi:10.1002/047134608X.W1006] Keyword(s): reachability, controllability, observability, Lie algebra accessibility.

2. E.D. Sontag. The ISS philosophy as a unifying framework for stability-like behavior. In Nonlinear control in the year 2000, Vol. 2 (Paris), volume 259 of Lecture Notes in Control and Inform. Sci., pages 443-467. Springer, London, 2001. [PDF] Keyword(s): input to state stability, integral input to state stability, iISS, ISS, input to output stability.
Abstract:
 (This is an expository paper prepared for a plenary talk given at the Second Nonlinear Control Network Workshop, Paris, June 9, 2000.) The input to state stability (ISS) paradigm is motivated as a generalization of classical linear systems concepts under coordinate changes. A summary is provided of the main theoretical results concerning ISS and related notions of input/output stability and detectability. A bibliography is also included, listing extensions, applications, and other current work.

3. B. DasGupta and E.D. Sontag. A polynomial-time algorithm for checking equivalence under certain semiring congruences motivated by the state-space isomorphism problem for hybrid systems. Theor. Comput. Sci., 262(1-2):161-189, 2001. [PDF] [doi:http://dx.doi.org/10.1016/S0304-3975(00)00188-2] Keyword(s): hybrid systems, computational complexity.
Abstract:
 The area of hybrid systems concerns issues of modeling, computation, and control for systems which combine discrete and continuous components. The subclass of piecewise linear (PL) systems provides one systematic approach to discrete-time hybrid systems, naturally blending switching mechanisms with classical linear components. PL systems model arbitrary interconnections of finite automata and linear systems. Tools from automata theory, logic, and related areas of computer science and finite mathematics are used in the study of PL systems, in conjunction with linear algebra techniques, all in the context of a "PL algebra" formalism. PL systems are of interest as controllers as well as identification models. Basic questions for any class of systems are those of equivalence, and, in particular, if state spaces are equivalent under a change of variables. This paper studies this state-space equivalence problem for PL systems. The problem was known to be decidable, but its computational complexity was potentially exponential; here it is shown to be solvable in polynomial-time.

4. W. Desch, H. Logemann, E. P. Ryan, and E.D. Sontag. Meagre functions and asymptotic behaviour of dynamical systems. Nonlinear Anal., 44(8, Ser. A: Theory Methods):1087-1109, 2001. [PDF] [doi:http://dx.doi.org/10.1016/S0362-546X(99)00323-5] Keyword(s): invariance principle.
Abstract:
 A measurable function x from a subset J of R into a metric space X is said to be C-meagre if C is non-empty subset of X and, for every closed subset K of X disjoint from C, the preimage of K under x has finite Lebesgue measure. This concept of meagreness, applied to trajectories, is shown to provide a unifying framework which facilitates a variety of characterizations, extensions or generalizations of diverse facts pertaining to asymptotic behaviour of dynamical systems.

5. M. Krichman, E.D. Sontag, and Y. Wang. Input-output-to-state stability. SIAM J. Control Optim., 39(6):1874-1928, 2001. [PDF] [doi:http://dx.doi.org/10.1137/S0363012999365352] Keyword(s): input to state stability.
Abstract:
 This work explores Lyapunov characterizations of the input-output-to-state stability (IOSS) property for nonlinear systems. The notion of IOSS is a natural generalization of the standard zero-detectability property used in the linear case. The main contribution of this work is to establish a complete equivalence between the input-output-to-state stability property and the existence of a certain type of smooth Lyapunov function. As corollaries, one shows the existence of "norm-estimators", and obtains characterizations of nonlinear detectability in terms of relative stability and of finite-energy estimates.

6. E.D. Sontag. Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction. IEEE Trans. Automat. Control, 46(7):1028-1047, 2001. [PDF] Keyword(s): zero-deficiency networks, systems biology, biochemical networks, nonlinear stability, dynamical systems, kinetic proofreading, T cells, immunology.
Abstract:
 This paper deals with the theory of structure, stability, robustness, and stabilization for an appealing class of nonlinear systems which arises in the analysis of chemical networks. The results given here extend, but are also heavily based upon, certain previous work by Feinberg, Horn, and Jackson, of which a self-contained and streamlined exposition is included. The theoretical conclusions are illustrated through an application to the kinetic proofreading model proposed by McKeithan for T-cell receptor signal transduction.

 2000
1. L. Grüne, E.D. Sontag, and F.R. Wirth. On equivalence of exponential and asymptotic stability under changes of variables. In International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), pages 850-852. World Sci. Publishing, River Edge, NJ, 2000. Keyword(s): input to state stability.

2. D. Angeli, E.D. Sontag, and Y. Wang. A characterization of integral input-to-state stability. IEEE Trans. Automat. Control, 45(6):1082-1097, 2000. [PDF] Keyword(s): input to state stability, integral input to state stability, iISS, ISS.
Abstract:
 Just as input to state stability (ISS) generalizes the idea of finite gains with respect to supremum norms, the new notion of integral input to state stability (IISS) generalizes the concept of finite gain when using an integral norm on inputs. In this paper, we obtain a necessary and sufficient characterization of the IISS property, expressed in terms of dissipation inequalities.

3. D. Angeli, E.D. Sontag, and Y. Wang. Further equivalences and semiglobal versions of integral input to state stability. Dynamics and Control, 10(2):127-149, 2000. [PDF] [doi:http://dx.doi.org/10.1023/A:1008356223747] Keyword(s): input to state stability, integral input to state stability, iISS, ISS.
Abstract:
 This paper continues the study of the integral input-to-state stability (IISS) property. It is shown that the IISS property is equivalent to one which arises from the consideration of mixed norms on states and inputs, as well as to the superposition of a bounded energy bounded state'' requirement and the global asymptotic stability of the unforced system. A semiglobal version of IISS is shown to imply the global version, though a counterexample shows that the analogous fact fails for input to state stability (ISS). The results in this note complete the basic theoretical picture regarding IISS and ISS.

4. X. Bao, Z. Lin, and E.D. Sontag. Finite gain stabilization of discrete-time linear systems subject to actuator saturation. Automatica, 36(2):269-277, 2000. [PDF] Keyword(s): discrete-time, saturation, input-to-state stability, stabilization, ISS, bounded inputs.
Abstract:
 It is shown that, for neutrally stable discrete-time linear systems subject to actuator saturation, finite gain lp stabilization can be achieved by linear output feedback, for all p>1. An explicit construction of the corresponding feedback laws is given. The feedback laws constructed also result in a closed-loop system that is globally asymptotically stable, and in an input-to-state estimate.

5. W. Maass and E.D. Sontag. Neural Systems as Nonlinear Filters. Neural Comput., 12(8):1743-1772, 2000. [PDF] [doi:http://dx.doi.org/10.1162/089976600300015123] Keyword(s): neural networks, Volterra series.
Abstract:
 We analyze computations on temporal patterns and spatio-temporal patterns in formal network models whose temporal dynamics arises from empirically established quantitative models for short term dynamics at biological synapses. We give a complete characterization of all linear and nonlinear filters that can be approximated by such dynamic network models: it is the class of all filters that can be approximated by Volterra series. This characterization is shown to be rather stable with regard to changes in the model. For example it is shown that synaptic facilitation and one layer of neurons suffices for approximating arbitrary filters from this class. Our results provide a new complexity hierarchy for all filters that are approximable by Volterra series, which appears to be closer related to the actual cost of implementing such filters in neural hardware than preceding complexity measures. Our results also provide a new parameterization for approximations to such filters in terms of parameters that are arguable related to those that are tunable in biological neural systems.

6. M. Malisoff and E.D. Sontag. Universal formulas for feedback stabilization with respect to Minkowski balls. Systems Control Lett., 40(4):247-260, 2000. [PDF] Keyword(s): nonlinear control, feedback stabilization, saturation, control-Lyapunov functions, bounded inputs.
Abstract:
 This note provides explicit algebraic stabilizing formulas for clf's when controls are restricted to certain Minkowski balls in Euclidean space. Feedbacks of this kind are known to exist by a theorem of Artstein, but the proof of Artstein's theorem is nonconstructive. The formulas are obtained from a general feedback stabilization technique and are used to construct approximation solutions to some stabilization problems.

7. L. Rosier and E.D. Sontag. Remarks regarding the gap between continuous, Lipschitz, and differentiable storage functions for dissipation inequalities appearing in H infinity control. Systems Control Lett., 41(4):237-249, 2000. [PDF] Keyword(s): viscosity solutions, H-infinity control.
Abstract:
 This paper deals with the regularity of solutions of the Hamilton-Jacobi Inequality which arises in H-infinity control. It shows by explicit counterexamples that there are gaps between existence of continuous and locally Lipschitz (positive definite and proper) solutions, and between Lipschitz and continuously differentiable ones. On the other hand, it is shown that it is always possible to smooth-out solutions, provided that an infinitesimal increase in gain is allowed.

8. E.D. Sontag and Y. Wang. Lyapunov characterizations of input to output stability. SIAM J. Control Optim., 39(1):226-249, 2000. [PDF] [doi:http://dx.doi.org/10.1137/S0363012999350213] Keyword(s): input to state stability.
Abstract:
 This paper presents necessary and sufficient characterizations of several notions of input to output stability. Similar Lyapunov characterizations have been found to play a key role in the analysis of the input to state stability property, and the results given here extend their validity to the case when the output, but not necessarily the entire internal state, is being regulated.

 1999
1. E.D. Sontag. Control-Lyapunov functions. In Open problems in mathematical systems and control theory, Comm. Control Engrg. Ser., pages 211-216. Springer, London, 1999. Keyword(s): control-Lyapunov functions.

2. E.D. Sontag. Nonlinear feedback stabilization revisited. In Dynamical systems, control, coding, computer vision (Padova, 1998), volume 25 of Progr. Systems Control Theory, pages 223-262. Birkhäuser, Basel, 1999. Note: This is a short conference proceedings paper. Please consult the full version Stability and stabilization: discontinuities and the effect of disturbances.

3. E.D. Sontag. Stability and stabilization: discontinuities and the effect of disturbances. In Nonlinear analysis, differential equations and control (Montreal, QC, 1998), volume 528 of NATO Sci. Ser. C Math. Phys. Sci., pages 551-598. Kluwer Acad. Publ., Dordrecht, 1999. [PDF] Keyword(s): feedback stabilization, nonlinear control, input to state stability.
Abstract:
 In this expository paper, we deal with several questions related to stability and stabilization of nonlinear finite-dimensional continuous-time systems. We review the basic problem of feedback stabilization, placing an emphasis upon relatively new areas of research which concern stability with respect to "noise" (such as errors introduced by actuators or sensors). The table of contents is as follows: Review of Stability and Asymptotic Controllability, The Problem of Stabilization, Obstructions to Continuous Stabilization, Control-Lyapunov Functions and Artstein's Theorem, Discontinuous Feedback, Nonsmooth CLF's, Insensitivity to Small Measurement and Actuator Errors, Effect of Large Disturbances: Input-to-State Stability, Comments on Notions Related to ISS.

4. F. Albertini and E.D. Sontag. Continuous control-Lyapunov functions for asymptotically controllable time-varying systems. Internat. J. Control, 72(18):1630-1641, 1999. [PDF] Keyword(s): control-Lyapunov functions.
Abstract:
 This paper shows that, for time varying systems, global asymptotic controllability to a given closed subset of the state space is equivalent to the existence of a continuous control-Lyapunov function with respect to the set.

5. D. Angeli and E.D. Sontag. Forward completeness, unboundedness observability, and their Lyapunov characterizations. Systems Control Lett., 38(4-5):209-217, 1999. [PDF] Keyword(s): observability, input to state stability, dynamical systems.
Abstract:
 A finite-dimensional continuous-time system is forward complete if solutions exist globally, for positive time. This paper shows that forward completeness can be characterized in a necessary and sufficient manner by means of smooth scalar growth inequalities. Moreover, a version of this fact is also proved for systems with inputs, and a generalization is also provided for systems with outputs and a notion (unboundedness observability) of relative completeness. We apply these results to obtain a bound on reachable states in terms of energy-like estimates of inputs.

6. L. Grüne, E.D. Sontag, and F.R. Wirth. Asymptotic stability equals exponential stability, and ISS equals finite energy gain---if you twist your eyes. Systems Control Lett., 38(2):127-134, 1999. [PDF] Keyword(s): input to state stability, ISS.
Abstract:
 This paper shows that uniformly global asymptotic stability for a family of ordinary differential equations is equivalent to uniformly global exponential stability under a suitable nonlinear change of variables. The same is shown respectively for input-to-state stability, input-to-state exponential stability, and the property of finite square-norm gain ("nonlinear H-infty"). The results are shown for systems of any dimension not equal to 4 or 5.

7. Y.S. Ledyaev and E.D. Sontag. A Lyapunov characterization of robust stabilization. Nonlinear Anal., 37(7, Ser. A: Theory Methods):813-840, 1999. [PDF] Keyword(s): nonlinear control, feedback stabilization.
Abstract:
 One of the fundamental facts in control theory (Artstein's theorem) is the equivalence, for systems affine in controls, between continuous feedback stabilizability to an equilibrium and the existence of smooth control Lyapunov functions. This equivalence breaks down for general nonlinear systems, not affine in controls. One of the main results in this paper establishes that the existence of smooth Lyapunov functions implies the existence of (in general, discontinuous) feedback stabilizers which are insensitive to small errors in state measurements. Conversely, it is shown that the existence of such stabilizers in turn implies the existence of smooth control Lyapunov functions. Moreover, it is established that, for general nonlinear control systems under persistently acting disturbances, the existence of smooth Lyapunov functions is equivalent to the existence of (possibly) discontinuous) feedback stabilizers which are robust with respect to small measurement errors and small additive external disturbances.

8. W. Maass and E.D. Sontag. Analog neural nets with Gaussian or other common noise distributions cannot recognize arbitrary regular languages. Neural Comput., 11(3):771-782, 1999. [PDF] [doi:http://dx.doi.org/10.1162/089976699300016656] Keyword(s): machine learning, neural networks.
Abstract:
 We consider recurrent analog neural nets where the output of each gate is subject to Gaussian noise, or any other common noise distribution that is nonzero on a large set. We show that many regular languages cannot be recognized by networks of this type, and we give a precise characterization of those languages which can be recognized. This result implies severe constraints on possibilities for constructing recurrent analog neural nets that are robust against realistic types of analog noise. On the other hand we present a method for constructing feedforward analog neural nets that are robust with regard to analog noise of this type.

9. D. Nesic, A.R. Teel, and E.D. Sontag. Formulas relating KL stability estimates of discrete-time and sampled-data nonlinear systems. Systems Control Lett., 38(1):49-60, 1999. [PDF] Keyword(s): input to state stability, sampled-data systems, discrete-time systems, sampling, ISS.
Abstract:
 We provide an explicit KL stability or input-to-state stability (ISS) estimate for a sampled-data nonlinear system in terms of the KL estimate for the corresponding discrete-time system and a K function describing inter-sample growth. It is quite obvious that a uniform inter-sample growth condition, plus an ISS property for the exact discrete-time model of a closed-loop system, implies uniform ISS of the sampled-data nonlinear system; our results serve to quantify these facts by means of comparison functions. Our results can be used as an alternative to prove and extend results of Aeyels et al and extend some results by Chen et al to a class of nonlinear systems. Finally, the formulas we establish can be used as a tool for some other problems which we indicate.

10. E.D. Sontag. Clocks and insensitivity to small measurement errors. ESAIM Control Optim. Calc. Var., 4:537-557, 1999. [PDF] Keyword(s): nonlinear control, feedback stabilization, hybrid systems, discontinuous feedback, measurement noise.
Abstract:
 This paper provides a precise result which shows that insensitivity to small measurement errors in closed-loop stabilization can be attained provided that the feedback controller ignores observations during small time intervals.

11. E.D. Sontag and Y. Qiao. Further results on controllability of recurrent neural networks. Systems Control Lett., 36(2):121-129, 1999. [PDF] Keyword(s): machine learning, controllability, recurrent neural networks, neural networks.
Abstract:
 This paper studies controllability properties of recurrent neural networks. The new contributions are: (1) an extension of the result in "Complete controllability of continuous-time recurrent neural networks" to a slightly different model, where inputs appear in an affine form, (2) a formulation and proof of a necessary and sufficient condition, in terms of local-local controllability, and (3) a complete analysis of the 2-dimensional case for which the hypotheses made in previous work do not apply.

12. E.D. Sontag and Y. Wang. Notions of input to output stability. Systems Control Lett., 38(4-5):235-248, 1999. [PDF] Keyword(s): input to state stability, ISS, input to output stability.
Abstract:
 This paper deals with several related notions of output stability with respect to inputs (which may be thought of as disturbances). The main such notion is called input to output stability (IOS), and it reduces to input to state stability (ISS) when the output equals the complete state. For systems with no inputs, IOS provides a generalization of the classical concept of partial stability. Several variants, which formalize in different manners the transient behavior, are introduced. The main results provide a comparison among these notions

 1998
1. E.D. Sontag. A general approach to path planning for systems without drift. In J. Baillieul, S. S. Sastry, and H.J. Sussmann, editors, Essays on mathematical robotics (Minneapolis, MN, 1993), volume 104 of IMA Vol. Math. Appl., pages 151-168. Springer, New York, 1998. [PDF] Keyword(s): path-planning, systems without drift, nonlinear control, controllability.
Abstract:
 This paper proposes a generally applicable technique for the control of analytic systems with no drift. The method is based on the generation of "nonsingular loops" that allow linearized controllability. One can then implement Newton and/or gradient searches in the search for a control. A general convergence theorem is proved.

2. E.D. Sontag. Automata and neural networks. In The handbook of brain theory and neural networks, pages 119-122. MIT Press, Cambridge, MA, USA, 1998. [PDF] Keyword(s): neural networks.

3. E.D. Sontag. VC dimension of neural networks. In C.M. Bishop, editor, Neural Networks and Machine Learning, pages 69-95. Springer, Berlin, 1998. [PDF] Keyword(s): machine learning, VC dimension, learning, neural networks, shattering.
Abstract:
 The Vapnik-Chervonenkis (VC) dimension is an integer which helps to characterize distribution-independent learning of binary concepts from positive and negative samples. This paper, based on lectures delivered at the Isaac Newton Institute in August of 1997, presents a brief introduction, establishes various elementary results, and discusses how to estimate the VC dimension in several examples of interest in neural network theory. (It does not address the learning and estimation-theoretic applications of VC dimension, and the applications to uniform convergence theorems for empirical probabilities, for which many suitable references are available.)

4. P. Koiran and E.D. Sontag. Vapnik-Chervonenkis dimension of recurrent neural networks. Discrete Appl. Math., 86(1):63-79, 1998. [PDF] [doi:http://dx.doi.org/10.1016/S0166-218X(98)00014-6] Keyword(s): machine learning, neural networks, recurrent neural networks.
Abstract:
 This paper provides lower and upper bounds for the VC dimension of recurrent networks. Several types of activation functions are discussed, including threshold, polynomial, piecewise-polynomial and sigmoidal functions. The bounds depend on two independent parameters: the number w of weights in the network, and the length k of the input sequence. Ignoring multiplicative constants, the main results say roughly the following: 1. For architectures whose activation is any fixed nonlinear polynomial, the VC dimension is proportional to wk. 2. For architectures whose activation is any fixed piecewise polynomial, the VC dimension is between wk and w**2k. 3. For architectures with threshold activations, the VC dimension is between wlog(k/w) and the smallest of wklog(wk) and w**2+wlog(wk). 4. For the standard sigmoid tanh(x), the VC dimension is between wk and w**4 k**2.

5. D. Nesic and E.D. Sontag. Input-to-state stabilization of linear systems with positive outputs. Systems Control Lett., 35(4):245-255, 1998. [PDF] Keyword(s): input to state stability, ISS, stabilization.
Abstract:
 This paper considers the problem of stabilization of linear systems for which only the magnitudes of outputs are measured. It is shown that, if a system is controllable and observable, then one can find a stabilizing controller, which is robust with respect to observation noise (in the ISS sense).

6. E.D. Sontag. A learning result for continuous-time recurrent neural networks. Systems Control Lett., 34(3):151-158, 1998. [PDF] [doi:http://dx.doi.org/10.1016/S0167-6911(98)00006-1] Keyword(s): machine learning, neural networks, VC dimension, recurrent neural networks.
Abstract:
 The following learning problem is considered, for continuous-time recurrent neural networks having sigmoidal activation functions. Given a black box'' representing an unknown system, measurements of output derivatives are collected, for a set of randomly generated inputs, and a network is used to approximate the observed behavior. It is shown that the number of inputs needed for reliable generalization (the sample complexity of the learning problem) is upper bounded by an expression that grows polynomially with the dimension of the network and logarithmically with the number of output derivatives being matched.

7. E.D. Sontag. Comments on integral variants of ISS. Systems Control Lett., 34(1-2):93-100, 1998. [PDF] [doi:http://dx.doi.org/10.1016/S0167-6911(98)00003-6] Keyword(s): input to state stability, integral input to state stability, iISS, ISS.
Abstract:
 This note discusses two integral variants of the input-to-state stability (ISS) property, which represent nonlinear generalizations of L2 stability, in much the same way that ISS generalizes L-infinity stability. Both variants are equivalent to ISS for linear systems. For general nonlinear systems, it is shown that one of the new properties is strictly weaker than ISS, while the other one is equivalent to it. For bilinear systems, a complete characterization is provided of the weaker property. An interesting fact about functions of type KL is proved as well.

8. E.D. Sontag and F.R. Wirth. Remarks on universal nonsingular controls for discrete-time systems. Systems Control Lett., 33(2):81-88, 1998. [PDF] [doi:http://dx.doi.org/10.1016/S0167-6911(97)00117-5] Keyword(s): discrete time, controllability.
Abstract:
 For analytic discrete-time systems, it is shown that uniform forward accessibility implies the generic existence of universal nonsingular control sequences. A particular application is given by considering forward accessible systems on compact manifolds. For general systems, it is proved that the complement of the set of universal sequences of infinite length is of the first category. For classes of systems satisfying a descending chain condition, and in particular for systems defined by polynomial dynamics, forward accessibility implies uniform forward accessibility.

 1997
1. P. Koiran and E.D. Sontag. Vapnik-Chervonenkis dimension of recurrent neural networks. In Computational learning theory (Jerusalem, 1997), volume 1208 of Lecture Notes in Comput. Sci., pages 223-237. Springer-Verlag, London, UK, 1997. Keyword(s): machine learning, neural networks, VC dimension, recurrent neural networks.

2. Y.S. Ledyaev and E.D. Sontag. A notion of discontinuous feedback. In Control using logic-based switching (Block Island, RI, 1995), volume 222 of Lecture Notes in Control and Inform. Sci., pages 97-103. Springer, London, 1997.

3. E.D. Sontag. Recurrent neural networks: Some systems-theoretic aspects. In M. Karny, K. Warwick, and V. Kurkova, editors, Dealing with Complexity: a Neural Network Approach, pages 1-12. Springer-Verlag, London, 1997. [PDF] Keyword(s): machine learning, neural networks, recurrent neural networks, learning, VC dimension.
Abstract:
 This paper provides an exposition of some recent results regarding system-theoretic aspects of continuous-time recurrent (dynamic) neural networks with sigmoidal activation functions. The class of systems is introduced and discussed, and a result is cited regarding their universal approximation properties. Known characterizations of controllability, observability, and parameter identifiability are reviewed, as well as a result on minimality. Facts regarding the computational power of recurrent nets are also mentioned.

4. F. H. Clarke, Y.S. Ledyaev, E.D. Sontag, and A.I. Subbotin. Asymptotic controllability implies feedback stabilization. IEEE Trans. Automat. Control, 42(10):1394-1407, 1997. [PDF]
Abstract:
 It is shown that every asymptotically controllable system can be stabilized by means of some (discontinuous) feedback law. One of the contributions of the paper is in defining precisely the meaning of stabilization when the feedback rule is not continuous. The main ingredients in our construction are: (a) the notion of control-Lyapunov function, (b) methods of nonsmooth analysis, and (c) techniques from positional differential games.

5. M. J. Donahue, L. Gurvits, C. Darken, and E.D. Sontag. Rates of convex approximation in non-Hilbert spaces. Constr. Approx., 13(2):187-220, 1997. [PDF] Keyword(s): machine learning, neural networks, optimization, approximation theory.
Abstract:
 This paper deals with sparse approximations by means of convex combinations of elements from a predetermined "basis" subset S of a function space. Specifically, the focus is on the rate at which the lowest achievable error can be reduced as larger subsets of S are allowed when constructing an approximant. The new results extend those given for Hilbert spaces by Jones and Barron, including in particular a computationally attractive incremental approximation scheme. Bounds are derived for broad classes of Banach spaces. The techniques used borrow from results regarding moduli of smoothness in functional analysis as well as from the theory of stochastic processes on function spaces.

6. P. Koiran and E.D. Sontag. Neural networks with quadratic VC dimension. J. Comput. System Sci., 54(1, part 2):190-198, 1997. Note: (1st Annual Dagstuhl Seminar on Neural Computing, 1994). [PDF] [doi:http://dx.doi.org/10.1006/jcss.1997.1479] Keyword(s): machine learning, neural networks, VC dimension.
Abstract:
 This paper shows that neural networks which use continuous activation functions have VC dimension at least as large as the square of the number of weights w. This result settles the open question of whether whether the well-known O(w log w) bound, known for hard-threshold nets, also held for more general sigmoidal nets. Implications for the number of samples needed for valid generalization are discussed.

7. R. Koplon and E.D. Sontag. Using Fourier-neural recurrent networks to fit sequential input/output data. Neurocomputing, 15:225-248, 1997. [PDF] Keyword(s): machine learning, neural networks, recurrent neural networks.
Abstract:
 This paper suggests the use of Fourier-type activation functions in fully recurrent neural networks. The main theoretical advantage is that, in principle, the problem of recovering internal coefficients from input/output data is solvable in closed form.

8. E.D. Sontag. Shattering all sets of k points in general position' requires (k-1)/2 parameters. Neural Comput., 9(2):337-348, 1997. [PDF] Keyword(s): machine learning, neural networks, VC dimension.
Abstract:
 For classes of concepts defined by certain classes of analytic functions depending on k parameters, there are nonempty open sets of samples of length 2k+2 which cannot be shattered. A slighly weaker result is also proved for piecewise-analytic functions. The special case of neural networks is discussed.

9. E.D. Sontag and H.J. Sussmann. Complete controllability of continuous-time recurrent neural networks. Systems Control Lett., 30(4):177-183, 1997. [PDF] [doi:http://dx.doi.org/10.1016/S0167-6911(97)00002-9] Keyword(s): machine learning, neural networks, recurrent neural networks.
Abstract:
 This paper presents a characterization of controllability for the class of control systems commonly called (continuous-time) recurrent neural networks. The characterization involves a simple condition on the input matrix, and is proved when the activation function is the hyperbolic tangent.

10. E.D. Sontag and Y. Wang. Output-to-state stability and detectability of nonlinear systems. Systems Control Lett., 29(5):279-290, 1997. [PDF] [doi:http://dx.doi.org/10.1016/S0167-6911(97)90013-X] Keyword(s): input to state stability, integral input to state stability, iISS, ISS, detectability, output to state stability, detectability, input to state stability.
Abstract:
 The notion of input-to-state stability (ISS) has proved to be useful in nonlinear systems analysis. This paper discusses a dual notion, output-to-state stability (OSS). A characterization is provided in terms of a dissipation inequality involving storage (Lyapunov) functions. Combining ISS and OSS there results the notion of input/output-to-state stability (IOSS), which is also studied and related to the notion of detectability, the existence of observers, and output injection.

11. Y. Yang, E.D. Sontag, and H.J. Sussmann. Global stabilization of linear discrete-time systems with bounded feedback. Systems Control Lett., 30(5):273-281, 1997. [PDF] [doi:http://dx.doi.org/10.1016/S0167-6911(97)00021-2] Keyword(s): discrete-time, saturation, bounded inputs.
Abstract:
 This paper deals with the problem of global stabilization of linear discrete time systems by means of bounded feedback laws. The main result proved is an analog of one proved for the continuous time case by the authors, and shows that such stabilization is possible if and only if the system is stabilizable with arbitrary controls and the transition matrix has spectral radius less or equal to one. The proof provides in principle an algorithm for the construction of such feedback laws, which can be implemented either as cascades or as parallel connections (single hidden layer neural networks'') of simple saturation functions.

 1996
1. E.D. Sontag. Interconnected automata and linear systems: a theoretical framework in discrete-time. In R. Alur, T.A. Henzinger, and E.D. Sontag, editors, Proceedings of the DIMACS/SYCON workshop on Hybrid systems III : verification and control, pages 436-448. Springer-Verlag New York, Inc., Secaucus, NJ, USA, 1996. [PDF] Keyword(s): hybrid systems.
Abstract:
 This paper summarizes the definitions and several of the main results of an approach to hybrid systems, which combines finite automata and linear systems, developed by the author in the early 1980s. Some related more recent results are briefly mentioned as well.

2. E.D. Sontag and H.J. Sussmann. General classes of control-Lyapunov functions. In Stability theory (Ascona, 1995), volume 121 of Internat. Ser. Numer. Math., pages 87-96. Birkhäuser, Basel, 1996. [PDF] Keyword(s): control-Lyapunov functions.
Abstract:
 Shorter and more expository version of "Nonsmooth control-Lyapunov functions"

3. B. DasGupta and E.D. Sontag. Sample complexity for learning recurrent perceptron mappings. IEEE Trans. Inform. Theory, 42(5):1479-1487, 1996. [PDF] Keyword(s): machine learning, neural networks, VC dimension, recurrent neural networks.
Abstract:
 Recurrent perceptron classifiers generalize the usual perceptron model. They correspond to linear transformations of input vectors obtained by means of "autoregressive moving-average schemes", or infinite impulse response filters, and allow taking into account those correlations and dependences among input coordinates which arise from linear digital filtering. This paper provides tight bounds on sample complexity associated to the fitting of such models to experimental data. The results are expressed in the context of the theory of probably approximately correct (PAC) learning.

4. Y. Lin, E.D. Sontag, and Y. Wang. A smooth converse Lyapunov theorem for robust stability. SIAM J. Control Optim., 34(1):124-160, 1996. [PDF] [doi:http://dx.doi.org/10.1137/S0363012993259981] Keyword(s): input to state stability.
Abstract:
 This paper presents a Converse Lyapunov Function Theorem motivated by robust control analysis and design. Our result is based upon, but generalizes, various aspects of well-known classical theorems. In a unified and natural manner, it (1) allows arbitrary bounded time-varying parameters in the system description, (2) deals with global asymptotic stability, (3) results in smooth (infinitely differentiable) Lyapunov functions, and (4) applies to stability with respect to not necessarily compact invariant sets.

5. W. Liu, Y. Chitour, and E.D. Sontag. On finite-gain stabilizability of linear systems subject to input saturation. SIAM J. Control Optim., 34(4):1190-1219, 1996. [PDF] [doi:http://dx.doi.org/10.1137/S0363012994263469] Keyword(s): saturation, bounded inputs.
Abstract:
 This paper deals with (global) finite-gain input/output stabilization of linear systems with saturated controls. For neutrally stable systems, it is shown that the linear feedback law suggested by the passivity approach indeed provides stability, with respect to every Lp-norm. Explicit bounds on closed-loop gains are obtained, and they are related to the norms for the respective systems without saturation. These results do not extend to the class of systems for which the state matrix has eigenvalues on the imaginary axis with nonsimple (size >1) Jordan blocks, contradicting what may be expected from the fact that such systems are globally asymptotically stabilizable in the state-space sense; this is shown in particular for the double integrator.

6. E.D. Sontag. Critical points for least-squares problems involving certain analytic functions, with applications to sigmoidal nets. Adv. Comput. Math., 5(2-3):245-268, 1996. [PDF] Keyword(s): machine learning, subanalytic sets, semianalytic sets, analytic geometry, critical points, approximation theory, neural networks.
Abstract:
 This paper deals with nonlinear least-squares problems involving the fitting to data of parameterized analytic functions. For generic regression data, a general result establishes the countability, and under stronger assumptions finiteness, of the set of functions giving rise to critical points of the quadratic loss function. In the special case of what are usually called "single-hidden layer neural networks", which are built upon the standard sigmoidal activation tanh(x) or equivalently 1/(1+exp(-x)), a rough upper bound for this cardinality is provided as well.

7. E.D. Sontag and Y. Wang. New characterizations of input-to-state stability. IEEE Trans. Automat. Control, 41(9):1283-1294, 1996. [PDF] Keyword(s): input to state stability, ISS.
Abstract:
 We present new characterizations of the Input to State Stability property. As a consequence of these results, we show the equivalence between the ISS property and several (apparent) variations proposed in the literature.

 1995
1. E.D. Sontag. State-space and i/o stability for nonlinear systems. In Feedback control, nonlinear systems, and complexity (Montreal, PQ, 1994), volume 202 of Lecture Notes in Control and Inform. Sci., pages 215-235. Springer, London, 1995. Note: (Expository paper, placed online per request. The paper Input to state stability: Basic concepts and results'' is far more up to date and should be downloaded instead of this one!). [PDF] Keyword(s): input to state stability.

2. A.R. Teel, T.T. Georgiou, L. Praly, and E.D. Sontag. Input-Output Stability. In W. S. Levine, editor, The Control Handbook, pages 895-908. CRC Press, Boca Raton, 1995. [PDF]
Abstract:
 An encyclopedia-type article on foundations of input/output stability.

3. Y. Chitour, W. Liu, and E.D. Sontag. On the continuity and incremental-gain properties of certain saturated linear feedback loops. Internat. J. Robust Nonlinear Control, 5(5):413-440, 1995. [PDF] Keyword(s): saturation, bounded inputs, incremental gains.
Abstract:
 This paper discusses various continuity and incremental-gain properties for neutrally stable linear systems under linear feedback subject to actuator saturation. The results complement our previous ones, which applied to the same class of problems and provided finite-gain stability.

4. M. A. Dahleh, E.D. Sontag, D. N. C. Tse, and J. N. Tsitsiklis. Worst-case identification of nonlinear fading memory systems. Automatica, 31(3):503-508, 1995. [PDF] [doi:http://dx.doi.org/10.1016/0005-1098(94)00131-2] Keyword(s): information-based complexity, fading-memory systems, stability, system identification, structured uncertainty.
Abstract:
 We consider the problem of characterizing possible supply functions for a given dissipative nonlinear system, and provide a result that allows some freedom in the modification of such functions.

5. B. DasGupta, H.T. Siegelmann, and E.D. Sontag. On the complexity of training neural networks with continuous activation functions. IEEE Trans. Neural Networks, 6:1490-1504, 1995. [PDF] Keyword(s): machine learning, neural networks, analog computing, theory of computing, neural networks, computational complexity, machine learning.
Abstract:
 Blum and Rivest showed that any possible neural net learning algorithm based on fixed architectures faces severe computational barriers. This paper extends their NP-completeness result, which applied only to nets based on hard threshold activations, to nets that employ a particular continuous activation. In view of neural network practice, this is a more relevant result to understanding the limitations of backpropagation and related techniques.

6. Y. Lin and E.D. Sontag. Control-Lyapunov universal formulas for restricted inputs. Control Theory Adv. Tech., 10(4, part 5):1981-2004, 1995. [PDF] Keyword(s): control-Lyapunov functions, saturation, bounded inputs.
Abstract:
 We deal with the question of obtaining explicit feedback control laws that stabilize a nonlinear system, under the assumption that a "control Lyapunov function" is known. In previous work, the case of unbounded controls was considered. Here we obtain results for bounded and/or positive controls. We also provide some simple preliminary remarks regarding a set stability version of the problem and a version for systems subject to disturbances.

7. Y. Lin, E.D. Sontag, and Y. Wang. Input to state stabilizability for parametrized families of systems. Internat. J. Robust Nonlinear Control, 5(3):187-205, 1995. [PDF] Keyword(s): ISS, stabilization.
Abstract:
 This paper studies various stability issues for parameterized families of systems, including problems of stabilization with respect to sets. The study of such families is motivated by robust control applications. A Lyapunov-theoretic necessary and sufficient characterization is obtained for a natural notion of robust uniform set stability; this characterization allows replacing ad hoc conditions found in the literature by more conceptual stability notions. We then use these techniques to establish a result linking state space stability to input to state'' (bounded-input bounded-state) stability. In addition, the preservation of stabilizability under certain types of cascade interconnections is analyzed.

8. H. T. Siegelmann and E.D. Sontag. On the computational power of neural nets. J. Comput. System Sci., 50(1):132-150, 1995. [PDF] [doi:http://dx.doi.org/10.1006/jcss.1995.1013] Keyword(s): machine learning, neural networks, recurrent neural networks, machine learning, analog computing, theory of computing, neural networks, computational complexity, super-Turing computation.
Abstract:
 This paper deals with finite size networks which consist of interconnections of synchronously evolving processors. Each processor updates its state by applying a "sigmoidal" function to a rational-coefficient linear combination of the previous states of all units. We prove that one may simulate all Turing Machines by such nets. In particular, one can simulate any multi-stack Turing Machine in real time, and there is a net made up of 886 processors which computes a universal partial-recursive function. Products (high order nets) are not required, contrary to what had been stated in the literature. Non-deterministic Turing Machines can be simulated by non-deterministic rational nets, also in real time. The simulation result has many consequences regarding the decidability, or more generally the complexity, of questions about recursive nets.

9. E.D. Sontag. Control of systems without drift via generic loops. IEEE Trans. Automat. Control, 40(7):1210-1219, 1995. [PDF] Keyword(s): stabilization, non-holonomic systems, path-planning, systems without drift, nonlinear control, controllability.
Abstract:
 This paper proposes a simple numerical technique for the steering of arbitrary analytic systems with no drift. It is based on the generation of "nonsingular loops" which allow linearized controllability along suitable trajetories. Once such loops are available, it is possible to employ standard Newton or steepest descent methods, as classically done in numerical control. The theoretical justification of the approach relies on recent results establishing the genericity of nonsingular controls, as well as a simple convergence lemma.

10. E.D. Sontag. On the input-to-state stability property. European J. Control, 1:24-36, 1995. [PDF] Keyword(s): input to state stability, ISS.
Abstract:
 The "input to state stability" (ISS) property provides a natural framework in which to formulate notions of stability with respect to input perturbations. In this expository paper, we review various equivalent definitions expressed in stability, Lyapunov-theoretic, and dissipation terms. We sketch some applications to the stabilization of cascades of systems and of linear systems subject to control saturation.

11. E.D. Sontag and A.R. Teel. Changing supply functions in input/state stable systems. IEEE Trans. Automat. Control, 40(8):1476-1478, 1995. [PDF] Keyword(s): input to state stability, ISS, input to state stability, Lyapunov functions.
Abstract:
 We consider the problem of characterizing possible supply functions for a given dissipative nonlinear system, and provide a result that allows some freedom in the modification of such functions.

12. E.D. Sontag and Y. Wang. On characterizations of the input-to-state stability property. Systems Control Lett., 24(5):351-359, 1995. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(94)00050-6] Keyword(s): input to state stability, ISS.
Abstract:
 We show that the well-known Lyapunov sufficient condition for input-to-state stability is also necessary, settling positively an open question raised by several authors during the past few years. Additional characterizations of the ISS property, including one in terms of nonlinear stability margins, are also provided.

13. Y. Wang and E.D. Sontag. Orders of input/output differential equations and state-space dimensions. SIAM J. Control Optim., 33(4):1102-1126, 1995. [PDF] [doi:http://dx.doi.org/10.1137/S0363012993246828] Keyword(s): identifiability, observability, realization theory.
Abstract:
 This paper deals with the orders of input/output equations satisfied by nonlinear systems. Such equations represent differential (or difference, in the discrete-time case) relations between high-order derivatives (or shifts, respectively) of input and output signals. It is shown that, under analyticity assumptions, there cannot exist equations of order less than the minimal dimension of any observable realization; this generalizes the known situation in the classical linear case. The results depend on new facts, themselves of considerable interest in control theory, regarding universal inputs for observability in the discrete case, and observation spaces in both the discrete and continuous cases. Included in the paper is also a new and simple self-contained proof of Sussmann's universal input theorem for continuous-time analytic systems.

 1994
1. B. DasGupta, H.T. Siegelmann, and E.D. Sontag. On the Intractability of Loading Neural Networks. In V. P. Roychowdhury, Siu K. Y., and Orlitsky A., editors, Theoretical Advances in Neural Computation and Learning, pages 357-389. Kluwer Academic Publishers, 1994. [PDF] Keyword(s): analog computing, neural networks, computational complexity, machine learning.

2. W. Maass, G. Schnitger, and E.D. Sontag. A comparison of the computational power of sigmoid and Boolean threshold circuits. In V. P. Roychowdhury, Siu K. Y., and Orlitsky A., editors, Theoretical Advances in Neural Computation and Learning, pages 127-151. Kluwer Academic Publishers, 1994. [PDF] Keyword(s): machine learning, neural networks, boolean systems.
Abstract:
 We examine the power of constant depth circuits with sigmoid threshold gates for computing boolean functions. It is shown that, for depth 2, constant size circuits of this type are strictly more powerful than constant size boolean threshold circuits (i.e. circuits with linear threshold gates). On the other hand it turns out that, for any constant depth d, polynomial size sigmoid threshold circuits with polynomially bounded weights compute exactly the same boolean functions as the corresponding circuits with linear threshold gates.

3. F. Albertini and E.D. Sontag. Further results on controllability properties of discrete-time nonlinear systems. Dynam. Control, 4(3):235-253, 1994. [PDF] [doi:http://dx.doi.org/10.1007/BF01985073] Keyword(s): discrete-time, nonlinear control.
Abstract:
 Controllability questions for discrete-time nonlinear systems are addressed in this paper. In particular, we continue the search for conditions under which the group-like notion of transitivity implies the stronger and semigroup-like property of forward accessibility. We show that this implication holds, pointwise, for states which have a weak Poisson stability property, and globally, if there exists a global "attractor" for the system.

4. F. Albertini and E.D. Sontag. State observability in recurrent neural networks. Systems Control Lett., 22(4):235-244, 1994. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(94)90054-X] Keyword(s): machine learning, neural networks, recurrent neural networks, observability, identifiability.
Abstract:
 This paper concerns recurrent networks x'=s(Ax+Bu), y=Cx, where s is a sigmoid, in both discrete time and continuous time. Our main result is that observability can be characterized, if one assumes certain conditions on the nonlinearity and on the system, in a manner very analogous to that of the linear case. Recall that for the latter, observability is equivalent to the requirement that there not be any nontrivial A-invariant subspace included in the kernel of C. We show that the result generalizes in a natural manner, except that one now needs to restrict attention to certain special "coordinate" subspaces.

5. R. Koplon, E.D. Sontag, and M. L. J. Hautus. Observability of linear systems with saturated outputs. Linear Algebra Appl., 205/206:909-936, 1994. [PDF] Keyword(s): observability, saturation, bounded inputs.
Abstract:
 In this paper, we present necessary and sufficient conditions for observability of the class of output-saturated systems. These are linear systems whose output passes through a saturation function before it can be measured.

6. H. T. Siegelmann and E.D. Sontag. Analog computation via neural networks. Theoret. Comput. Sci., 131(2):331-360, 1994. [PDF] [doi:http://dx.doi.org/10.1016/0304-3975(94)90178-3] Keyword(s): analog computing, neural networks, computational complexity, super-Turing computation, recurrent neural networks, neural networks, computational complexity.
Abstract:
 We consider recurrent networks with real-valued weights. If allowed exponential time for computation, they turn out to have unbounded power. However, under polynomial-time constraints there are limits on their capabilities, though being more powerful than Turing Machines. Moreover, there is a precise correspondence between nets and standard non-uniform circuits with equivalent resources, and as a consequence one has lower bound constraints on what they can compute. We note that these networks are not likely to solve polynomially NP-hard problems, as the equality "P=NP" in our model implies the almost complete collapse of the standard polynomial hierarchy. We show that a large class of different networks and dynamical system models have no more computational power than this neural (first-order) model with real weights. The results suggest the following Church-like Thesis of Time-bounded Analog Computing: "Any reasonable analog computer will have no more power (up to polynomial time) than first-order recurrent networks."

7. H.J. Sussmann, E.D. Sontag, and Y. Yang. A general result on the stabilization of linear systems using bounded controls. IEEE Trans. Automat. Control, 39(12):2411-2425, 1994. [PDF] Keyword(s): saturation, neural networks, global stability, nonlinear stability, bounded inputs.
Abstract:
 We present two constructions of controllers that globally stabilize linear systems subject to control saturation. We allow essentially arbitrary saturation functions. The only conditions imposed on the system are the obvious necessary ones, namely that no eigenvalues of the uncontrolled system have positive real part and that the standard stabilizability rank condition hold. One of the constructions is in terms of a "neural-network type" one-hidden layer architecture, while the other one is in terms of cascades of linear maps and saturations.

 1993
1. F. Albertini, E.D. Sontag, and V. Maillot. Uniqueness of weights for neural networks. In R. Mammone, editor, Artificial Neural Networks for Speech and Vision, pages 115-125. Chapman and Hall, London, 1993. [PDF] Keyword(s): machine learning, neural networks, recurrent neural networks.
Abstract:
 In this short expository survey, we sketch various known facts about uniqueness of weights in neural networks, including results about recurrent nets, and we provide a new and elementary complex-variable proof of a uniqueness result that applies in the single hidden layer case.

2. E.D. Sontag. Neural networks for control. In H. L. Trentelman and J. C. Willems, editors, Essays on control: perspectives in the theory and its applications (Groningen, 1993), volume 14 of Progr. Systems Control Theory, pages 339-380. Birkhäuser Boston, Boston, MA, 1993. Note: A longer version (tech report with more details) is here: http://sontaglab.org/FTPDIR/neural-nets-siemens.pdf. [PDF] Keyword(s): neural networks, recurrent neural networks, machine learning, neural networks.
Abstract:
 This paper has an expository introduction to two related topics: (a) Some mathematical results regarding "neural networks", and (b) so-called "neurocontrol" and "learning control" (each part can be read independently of the other). It was prepared for a short course given at the 1993 European Control Conference.

3. E.D. Sontag and H.J. Sussmann. Time-optimal control of manipulators (reprint of 1986 IEEE Int Conf on Robotics and Automation paper. In M.W. Spong, F.L. Lewis, and C.T. Abdallah, editors, Robot Control, pages 266-271. IEEE Press, New York, 1993. Keyword(s): robotics, optimal control.

4. F. Albertini and E.D. Sontag. Discrete-time transitivity and accessibility: analytic systems. SIAM J. Control Optim., 31(6):1599-1622, 1993. [PDF] [doi:http://dx.doi.org/10.1137/0331075]
Abstract:
 A basic open question for discrete-time nonlinear systems is that of determining when, in analogy with the classical continuous-time "positive form of Chow's Lemma", accessibility follows from transitivity of a natural group action. This paper studies the problem, and establishes the desired implication for analytic systems in several cases: (i) compact state space, (ii) under a Poisson stability condition, and (iii) in a generic sense. In addition, the paper studies accessibility properties of the "control sets" recently introduced in the context of dynamical systems studies. Finally, various examples and counterexamples are provided relating the various Lie algebras introduced in past work.

5. F. Albertini and E.D. Sontag. For neural networks, function determines form. Neural Networks, 6(7):975-990, 1993. [PDF] Keyword(s): machine learning, neural networks, identifiability, recurrent neural networks, realization theory, observability, neural networks.
Abstract:
 This paper shows that the weights of continuous-time feedback neural networks x'=s(Ax+Bu), y=Cx (where s is a sigmoid) are uniquely identifiable from input/output measurements. Under very weak genericity assumptions, the following is true: Assume given two nets, whose neurons all have the same nonlinear activation function s; if the two nets have equal behaviors as "black boxes" then necessarily they must have the same number of neurons and -except at most for sign reversals at each node- the same weights. Moreover, even if the activations are not a priori known to coincide, they are shown to be also essentially determined from the external measurements.

6. R. Koplon and E.D. Sontag. Linear systems with sign-observations. SIAM J. Control Optim., 31(5):1245-1266, 1993. [PDF] [doi:http://dx.doi.org/10.1137/0331059] Keyword(s): observability.
Abstract:
 This paper deals with systems that are obtained from linear time-invariant continuous- or discrete-time devices followed by a function that just provides the sign of each output. Such systems appear naturally in the study of quantized observations as well as in signal processing and neural network theory. Results are given on observability, minimal realizations, and other system-theoretic concepts. Certain major differences exist with the linear case, and other results generalize in a surprisingly straightforward manner.

 1992
1. E.D. Sontag. Feedback stabilization using two-hidden-layer nets. IEEE Trans. Neural Networks, 3:981-990, 1992. [PDF] Keyword(s): machine learning, neural networks, feedback stabilization.
Abstract:
 This paper compares the representational capabilities of one hidden layer and two hidden layer nets consisting of feedforward interconnections of linear threshold units. It is remarked that for certain problems two hidden layers are required, contrary to what might be in principle expected from the known approximation theorems. The differences are not based on numerical accuracy or number of units needed, nor on capabilities for feature extraction, but rather on a much more basic classification into "direct" and "inverse" problems. The former correspond to the approximation of continuous functions, while the latter are concerned with approximating one-sided inverses of continuous functions - and are often encountered in the context of inverse kinematics determination or in control questions. A general result is given showing that nonlinear control systems can be stabilized using two hidden layers, but not in general using just one.

2. E.D. Sontag. Feedforward nets for interpolation and classification. J. Comput. System Sci., 45(1):20-48, 1992. [PDF] [doi:http://dx.doi.org/10.1016/0022-0000(92)90039-L] Keyword(s): machine learning, neural networks, VC dimension, boolean systems.
Abstract:
 This paper deals with single-hidden-layer feedforward nets, studying various aspects of classification power and interpolation capability. In particular, a worst-case analysis shows that direct input to output connections in threshold nets double the recognition but not the interpolation power, while using sigmoids rather than thresholds allows doubling both. For other measures of classification, including the Vapnik-Chervonenkis dimension, the effect of direct connections or sigmoidal activations is studied in the special case of two-dimensional inputs.

3. E.D. Sontag. Universal nonsingular controls. Systems Control Lett., 19(3):221-224, 1992. Note: Erratum appeared in SCL 20(1993), p. 77, can be found in same file.[PDF] [doi:http://dx.doi.org/10.1016/0167-6911(92)90116-A] Keyword(s): controllability.
Abstract:
 For analytic systems satisfying the strong accessibility rank condition, generic inputs produce trajectories along which the linearized system is controllable. Applications to the steering of systems without drift are briefly mentioned.

4. Y. Wang and E.D. Sontag. Algebraic differential equations and rational control systems. SIAM J. Control Optim., 30(5):1126-1149, 1992. [PDF] Keyword(s): identifiability, observability, realization theory, input/output system representations.
Abstract:
 It is shown that realizability of an input/output operators by a finite-dimensional continuous-time rational control system is equivalent to the existence of a high-order algebraic differential equation satisfied by the corresponding input/output pairs ("behavior"). This generalizes, to nonlinear systems, the classical equivalence between autoregressive representations and finite dimensional linear realizability.

5. Y. Wang and E.D. Sontag. Generating series and nonlinear systems: analytic aspects, local realizability, and i/o representations. Forum Math., 4(3):299-322, 1992. [PDF] Keyword(s): identifiability, observability, realization theory, input/output system representations.
Abstract:
 This paper studies fundamental analytic properties of generating series for nonlinear control systems, and of the operators they define. It then applies the results obtained to the extension of facts, which relate realizability and algebraic input/output equations, to local realizability and analytic equations.

 1991
1. F. Albertini and E.D. Sontag. Transitivity and forward accessibility of discrete-time nonlinear systems. In Analysis of controlled dynamical systems (Lyon, 1990), volume 8 of Progr. Systems Control Theory, pages 21-34. Birkhäuser Boston, Boston, MA, 1991.

2. E.D. Sontag. Capabilities and training of feedforward nets. In Neural networks (New Brunswick, NJ, 1990), pages 303-321. Academic Press, Boston, MA, 1991. [PDF] Keyword(s): machine learning, machine learning, neural networks.
Abstract:
 This paper surveys recent work by the author on learning and representational capabilities of feedforward nets. The learning results show that, among two possible variants of the so-called backpropagation training method for sigmoidal nets, both of which variants are used in practice, one is a better generalization of the older perceptron training algorithm than the other. The representation results show that nets consisting of sigmoidal neurons have at least twice the representational capabilities of nets that use classical threshold neurons, at least when this increase is quantified in terms of classification power. On the other hand, threshold nets are shown to be more useful when approximating implicit functions, as illustrated with an application to a typical control problem.

3. E.D. Sontag. Input/output and state-space stability. In New trends in systems theory (Genoa, 1990), volume 7 of Progr. Systems Control Theory, pages 684-691. Birkhäuser Boston, Boston, MA, 1991. [PDF] Keyword(s): input to state stability, input to state stability.
Abstract:
 This conference paper reviews various results relating state-space (Lyapunov) stabilization and exponential stabilization to several notions of input/output or bounded-input bounded-output stabilization. It also provides generalizations of some of these results to systems with saturating controls. Some of these latter results were not included in journal papers.

4. E.D. Sontag. Kalman's controllability rank condition: from linear to nonlinear. In Mathematical system theory, pages 453-462. Springer, Berlin, 1991. [PDF] Keyword(s): controllability.
Abstract:
 The notion of controllability was identified by Kalman as one of the central properties determining system behavior. His simple rank condition is ubiquitous in linear systems analysis. This article presents an elementary and expository overview of the generalizations of this test to a condition for testing accessibility of discrete and continuous time nonlinear systems.

5. Y. Lin and E.D. Sontag. A universal formula for stabilization with bounded controls. Systems Control Lett., 16(6):393-397, 1991. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(91)90111-Q] Keyword(s): saturation, bounded inputs.
Abstract:
 We provide a formula for a stabilizing feedback law using a bounded control, under the assumption that an appropriate control-Lyapunov function is known. Such a feedback, smooth away from the origin and continuous everywhere, is known to exist via Artstein's Theorem. As in the unbounded-control case treated in a previous note, we provide an explicit and universal'' formula given by an algebraic function of Lie derivatives. In particular, we extend to the bounded case the result that the feedback can be chosen analytic if the Lyapunov function and the vector fields defining the system are analytic.

6. H. T. Siegelmann and E.D. Sontag. Turing computability with neural nets. Appl. Math. Lett., 4(6):77-80, 1991. [PDF] Keyword(s): machine learning, neural networks, computational complexity, recurrent neural networks.
Abstract:
 This paper shows the existence of a finite neural network, made up of sigmoidal neurons, which simulates a universal Turing machine. It is composed of less than 100,000 synchronously evolving processors, interconnected linearly. High-order connections are not required. (Note: this paper was placed here by special request. The results in this paper have been by now improved considerably: see the JCSS pape which among other aspects provides a polynomial time simulation. This paper, based on a unary encoding, results in an exponential slowdown).

7. E.D. Sontag and H.J. Sussmann. Back propagation separates where perceptrons do. Neural Networks, 4(2):243-249, 1991. [PDF] [doi:http://dx.doi.org/10.1016/0893-6080(91)90008-S] Keyword(s): machine learning, neural networks.
Abstract:
 Feedforward nets with sigmoidal activation functions are often designed by minimizing a cost criterion. It has been pointed out before that this technique may be outperformed by the classical perceptron learning rule, at least on some problems. In this paper, we show that no such pathologies can arise if the error criterion is of a threshold LMS type, i.e., is zero for values beyond'' the desired target values. More precisely, we show that if the data are linearly separable, and one considers nets with no hidden neurons, then an error function as above cannot have any local minima that are not global. In addition, the proof gives the following stronger result, under the stated hypotheses: the continuous gradient adjustment procedure is such that from any initial weight configuration a separating set of weights is obtained in finite time. This is a precise analogue of the Perceptron Learning Theorem. The results are then compared with the more classical pattern recognition problem of threshold LMS with linear activations, where no spurious local minima exist even for nonseparable data: here it is shown that even if using the threshold criterion, such bad local minima may occur, if the data are not separable and sigmoids are used. keywords = { neural networks , feedforward neural nets },

 1990
1. E.D. Sontag. Constant McMillan degree and the continuous stabilization of families of transfer matrices. In Control of uncertain systems (Bremen, 1989), volume 6 of Progr. Systems Control Theory, pages 289-295. Birkhäuser Boston, Boston, MA, 1990. [PDF] Keyword(s): systems over rings.

2. E.D. Sontag. Feedback stabilization of nonlinear systems. In Robust control of linear systems and nonlinear control (Amsterdam, 1989), volume 4 of Progr. Systems Control Theory, pages 61-81. Birkhäuser Boston, Boston, MA, 1990. [PDF]
Abstract:
 This paper surveys some well-known facts as well as some recent developments on the topic of stabilization of nonlinear systems. (NOTE: figures are not included in file; they were pasted-in.)

3. E.D. Sontag. Integrability of certain distributions associated with actions on manifolds and applications to control problems. In Nonlinear controllability and optimal control, volume 133 of Monogr. Textbooks Pure Appl. Math., pages 81-131. Dekker, New York, 1990. [PDF] Keyword(s): controllability.
Abstract:
 Results are given on the integrability of certain distributions which arise from smoothly parametrized families of diffeomorphisms acting on manifolds. Applications to control problems and in particular to the problem of sampling are discussed. Pages 42-50 apply the results to the control of continuous time systems; this is an exposition of some of the basic results of the Lie algebraic accessibility theory.

4. E.D. Sontag and Y. Wang. Input/output equations and realizability. In Realization and modelling in system theory (Amsterdam, 1989), volume 3 of Progr. Systems Control Theory, pages 125-132. Birkhäuser Boston, Boston, MA, 1990. [PDF] Keyword(s): identifiability, observability, realization theory.

5. B. Jakubczyk and E.D. Sontag. Controllability of nonlinear discrete-time systems: a Lie-algebraic approach. SIAM J. Control Optim., 28(1):1-33, 1990. [PDF] [doi:http://dx.doi.org/10.1137/0328001] Keyword(s): discrete-time.
Abstract:
 This paper presents a geometric study of controllability for discrete-time nonlinear systems. Various accessibility properties are characterized in terms of Lie algebras of vector fields. Some of the results obtained are parallel to analogous ones in continuous-time, but in many respects the theory is substantially different and many new phenomena appear.

6. E.D. Sontag. Further facts about input to state stabilization. IEEE Trans. Automat. Control, 35(4):473-476, 1990. [PDF] Keyword(s): input to state stability, ISS, stabilization.
Abstract:
 Previous results about input to state stabilizability are shown to hold even for systems which are not linear in controls, provided that a more general type of feedback be allowed. Applications to certain stabilization problems and coprime factorizations, as well as comparisons to other results on input to state stability, are also briefly discussed.d local minima may occur, if the data are not separable and sigmoids are used.

7. E.D. Sontag and Y. Wang. Pole shifting for families of linear systems depending on at most three parameters. Linear Algebra Appl., 137/138:3-38, 1990. [PDF] Keyword(s): systems over rings, systems over rings.
Abstract:
 We prove that for any family of n-dimensional controllable linear systems, continuously parameterized by up to three parameters, and for any continuous selection of n eigenvalues (in complex conjugate pairs), there is some dynamic controller of dimension 3n which is itself continuously parameterized and for which the closed-loop eigenvalues are these same eigenvalues, each counted 4 times. An analogous result holds also for smooth parameterizations.

 1989
1. B. Jakubczyk and E.D. Sontag. Nonlinear discrete-time systems. Accessibility conditions. In Modern optimal control, volume 119 of Lecture Notes in Pure and Appl. Math., pages 173-185. Dekker, New York, 1989. [PDF]

2. A. Arapostathis, B. Jakubczyk, H.-G. Lee, S. I. Marcus, and E.D. Sontag. The effect of sampling on linear equivalence and feedback linearization. Systems Control Lett., 13(5):373-381, 1989. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(89)90103-5] Keyword(s): discrete-time, sampled-data systems, discrete-time systems, sampling.
Abstract:
 We investigate the effect of sampling on linearization for continuous time systems. It is shown that the discretized system is linearizable by state coordinate change for an open set of sampling times if and only if the continuous time system is linearizable by state coordinate change. Also, it is shown that linearizability via digital feedback imposes highly nongeneric constraints on the structure of the plant, even if this is known to be linearizable with continuous-time feedback.

3. E.D. Sontag. A universal'' construction of Artstein's theorem on nonlinear stabilization. Systems Control Lett., 13(2):117-123, 1989. [PDF] Keyword(s): control-Lyapunov functions, stabilization.
Abstract:
 This note presents an explicit proof of the theorem - due to Artstein - which states that the existence of a smooth control-Lyapunov function implies smooth stabilizability. Moreover, the result is extended to the real-analytic and rational cases as well. The proof uses a "universal" formula given by an algebraic function of Lie derivatives; this formula originates in the solution of a simple Riccati equation.

4. E.D. Sontag. Sigmoids distinguish more efficiently than Heavisides. Neural Computation, 1:470-472, 1989. [PDF] Keyword(s): machine learning, neural networks, boolean systems.
Abstract:
 Every dichotomy on a 2k-point set in Rn can be implemented by a neural net with a single hidden layer containing k sigmoidal neurons. If the neurons were of a hardlimiter (Heaviside) type, 2k-1 would be in general needed.

5. E.D. Sontag. Smooth stabilization implies coprime factorization. IEEE Trans. Automat. Control, 34(4):435-443, 1989. [PDF] Keyword(s): input to state stability, ISS, input to state stability.
Abstract:
 This paper shows that coprime right factorizations exist for the input to state mapping of a continuous time nonlinear system provided that the smooth feedback stabilization problem be solvable for this system. In particular, it follows that feedback linearizable systems admit such factorizations. In order to establish the result a Lyapunov-theoretic definition is proposed for bounded input bounded output stability. The main technical fact proved relates the notion of stabilizability studied in the state space nonlinear control literature to a notion of stability under bounded control perturbations analogous to those studied in operator theoretic approaches to systems; it states that smooth stabilization implies smooth input-to-state stabilization. (Note: This is the original ISS paper, but the ISS results have been much improved in later papers. The material on coprime factorizations is still of interest, but the 89 CDC paper has some improvements and should be read too.)

6. E.D. Sontag and H.J. Sussmann. Backpropagation can give rise to spurious local minima even for networks without hidden layers. Complex Systems, 3(1):91-106, 1989. [PDF] Keyword(s): machine learning, neural networks.
Abstract:
 We give an example of a neural net without hidden layers and with a sigmoid transfer function, together with a training set of binary vectors, for which the sum of the squared errors, regarded as a function of the weights, has a local minimum which is not a global minimum. The example consists of a set of 125 training instances, with four weights and a threshold to be learnt. We do not know if substantially smaller binary examples exist.

7. E.D. Sontag and H.J. Sussmann. Further comments on the stabilizability of the angular velocity of a rigid body. Systems Control Lett., 12(3):213-217, 1989. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(89)90052-2] Keyword(s): satellite control, feedback stabilization.
Abstract:
 We prove that the angular velocity equations can be smoothly stabilized with a single torque controller for bodies having an axis of symmetry. This complements a recent result of Aeyels and Szafranski.

8. E.D. Sontag and Y. Yamamoto. On the existence of approximately coprime factorizations for retarded systems. Systems Control Lett., 13(1):53-58, 1989. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(89)90020-0] Keyword(s): delay-differential systems.
Abstract:
 This note establishes a result linking algebraically coprime factorizations of transfer matrices of delay systems to approximately coprime factorizations in the sense of distributions. The latter have been employed by the second author in the study of function-space controllability for such systems.

9. Y. Wang and E.D. Sontag. On two definitions of observation spaces. Systems Control Lett., 13(4):279-289, 1989. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(89)90116-3] Keyword(s): observability, identifiability, observables, observation space, realization theory.
Abstract:
 This paper establishes the equality of the observation spaces defined by means of piecewise constant controls with those defined in terms of differentiable controls.

 1988
1. E.D. Sontag. A Chow property for sampled bilinear systems. In C.I. Byrnes, C.F. Martin, and R. Saeks, editors, Analysis and Control of Nonlinear Systems, pages 205-211. North Holland, Amsterdam, 1988. [PDF] Keyword(s): discrete-time, bilinear systems.
Abstract:
 This paper studies accessibility (weak controllability) of bilinear systems under constant sampling rates. It is shown that the property is preserved provided that the sampling period satisfies a condition related to the eigenvalues of the autonomous dynamics matrix. This condition generalizes the classical Kalman-Ho-Narendra criterion which is well known in the linear case, and which, for observability, results in the classical Nyquist theorem.

2. E.D. Sontag. An explicit construction of the equilinearization controller. In C.I. Byrnes, C.F. Martin, and R. Saek, editors, Analysis and Control of Nonlinear Systems, pages 483-492. North Holland, Amsterdam, 1988. [PDF]
Abstract:
 This paper provides further results about the equilinearization method of control design recently introduced by the author. A simplified derivation of the controller is provided, as well as a theorem on local stabilization along reference trajectories.

3. E.D. Sontag. Bilinear realizability is equivalent to existence of a singular affine differential I/O equation. Systems Control Lett., 11(3):181-187, 1988. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(88)90057-6] Keyword(s): identification, identifiability, observability, observation space.
Abstract:
 For continuous time analytic input/output maps, the existence of a singular differential equation relating derivatives of controls and outputs is shown to be equivalent to bilinear realizability. A similar result holds for the problem of immersion into bilinear systems. The proof is very analogous to that of the corresponding, and previously known, result for discrete time.

4. E.D. Sontag. Controllability is harder to decide than accessibility. SIAM J. Control Optim., 26(5):1106-1118, 1988. [PDF] [doi:http://dx.doi.org/10.1137/0326061] Keyword(s): computational complexity, controllability, computational complexity.
Abstract:
 The present article compares the difficulties of deciding controllability and accessibility. These are standard properties of control systems, but complete algebraic characterizations of controllability have proved elusive. We show in particular that for subsystems of bilinear systems, accessibility can be decided in polynomial time, but controllability is NP-hard.

5. E.D. Sontag. Finite-dimensional open-loop control generators for nonlinear systems. Internat. J. Control, 47(2):537-556, 1988. [PDF]
Abstract:
 This paper concerns itself with the existence of open-loop control generators for nonlinear (continuous-time) systems. The main result is that, under relatively mild assumptions on the original system, and for each fixed compact subset of the state space, there always exists one such generator. This is a new system with the property that the controls it produces are sufficiently rich to preserve complete controllability along nonsingular trajectories. General results are also given on the continuity and differentiability of the input to state mapping for various p-norms on controls, as well as a comparison of various nonlinear controllability notions.

 1987
1. E.D. Sontag. Reachability, observability, and realization of a class of discrete-time nonlinear systems. In Encycl. of Systems and Control, pages 3288-3293. Pergamon Press, 1987. Keyword(s): observability.

2. E.D. Sontag. A remark on bilinear systems and moduli spaces of instantons. Systems Control Lett., 9(5):361-367, 1987. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(87)90064-8] Keyword(s): bilinear systems, moduli spaces, instantons.
Abstract:
 Explicit equations are given for the moduli space of framed instantons as a quasi-affine variety, based on the representation theory of noncommutative power series, or equivalently, the minimal realization theory of bilinear systems.

3. E.D. Sontag. Controllability and linearized regulation. IEEE Trans. Automat. Control, 32(10):877-888, 1987. [PDF]
Abstract:
 A nonlinear controllable plant, under mild technical conditions, admits a precompensator with the following property: along control trajectories joining pairs of states, the composite system (precompensator plus plant) is, up to first order, isomorphic to a parallel connection of integrators.

4. E.D. Sontag. Review of Multidimensional Systems Theory. Linear Alg. and Applications, 87:273-278, 1987. [PDF]
Abstract:
 A book review which also provides a quick introduction to questions of stability and positivity of multivariable polynomials for 2D and spatially-distributed systems.

 1986
1. E.D. Sontag. Orbit theorems and sampling. In Algebraic and geometric methods in nonlinear control theory, volume 29 of Math. Appl., pages 441-483. Reidel, Dordrecht, 1986. [PDF]
Abstract:
 This paper proposes a notion of smooth action on a manifold, and establishes a general integrability result for certain associated distributions. As corollaries, various classical and new results on manifold structures of orbits are established, and the main theorem on preservation of transitivity under sampling is shown to be a simple consequence.

2. M. L. J. Hautus and E.D. Sontag. New results on pole-shifting for parametrized families of systems. J. Pure Appl. Algebra, 40(3):229-244, 1986. [PDF] Keyword(s): systems over rings.
Abstract:
 New results are given on the pole-shifting problem for commutative rings, and these are then applied to conclude that rings of continuous, smooth, or real-analytic functions on a manifold X are PA rings if and only if X is one-dimensional.

3. E.D. Sontag. An eigenvalue condition for sample weak controllability of bilinear systems. Systems Control Lett., 7(4):313-315, 1986. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(86)90045-9] Keyword(s): discrete-time.
Abstract:
 Weak controllability of bilinear systems is preserved under sampling provided that the sampling period satisfies a condition related to the eigenvalues of the autonomous dynamics matrix. This condition generalizes the classical Kalman-Ho-Narendra criterion which is well known in the linear case.

4. E.D. Sontag. Comments on: `Some results on pole-placement and reachability'' [Systems Control Lett. 6 (1986), no. 5, 325--328; MR0821927 (87c:93032)] by P. K. Sharma. Systems Control Lett., 8(1):79-83, 1986. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(86)90034-4]
Abstract:
 We present various comments on a question about systems over rings posed in a recent note by Sharma, proving that a ring R is pole-assignable if and only if, for every reachable system (F,G), G contains a rank-one summand of the state space. We also provide a generalization to deal with dynamic feedback.

5. E.D. Sontag. Continuous stabilizers and high-gain feedback. IMA Journal of Mathematical Control and Information, 3:237-253, 1986. [PDF] Keyword(s): adaptive control, systems over rings.
Abstract:
 A controller is shown to exist, universal for the family of all systems of fixed dimension n, and m controls, which stabilizes those systems that are stabilizable, if certain gains are large enough. The controller parameters are continuous, in fact polynomial, functions of the entries of the plant. As a consequence, a result is proved on polynomial stabilization of families of systems.

 1985
1. E.D. Sontag. An introduction to the stabilization problem for parametrized families of linear systems. In Linear algebra and its role in systems theory (Brunswick, Maine, 1984), volume 47 of Contemp. Math., pages 369-400. Amer. Math. Soc., Providence, RI, 1985. [PDF] Keyword(s): systems over rings.
Abstract:
 This paper provides an introduction to definitions and known facts relating to the stabilization of parametrized families of linear systems using static and dynamic controllers. New results are given in the rational and polynomial cases.

2. B.W. Dickinson and E.D. Sontag. Dynamic realizations of sufficient sequences. IEEE Trans. Inform. Theory, 31(5):670-676, 1985. [PDF] Keyword(s): realization theory, statistics, innovations, sufficient statistics.
Abstract:
 Let Ul, U2, ... be a sequence of observed random variables and (T1(U1),T2(Ul,U2),...) be a corresponding sequence of sufficient statistics (a sufficient sequence). Under certain regularity conditions, the sufficient sequence defines the input/output map of a time-varying, discrete-time nonlinear system. This system provides a recursive way of updating the sufficient statistic as new observations are made. Conditions are provided assuring that such a system evolves in a state space of minimal dimension. Several examples are provided to illustrate how this notion of dimensional minimality is related to other properties of sufficient sequences. The results can be used to verify the form of the minimum dimension (discrete-time) nonlinear filter associated with the autoregressive parameter estimation problem.

3. E.D. Sontag. Real addition and the polynomial hierarchy. Inform. Process. Lett., 20(3):115-120, 1985. [PDF]
Abstract:
 The k-th alternation level of the theory of real numbers under addition and order is log-complete for the k-th level of the polynomial hierarchy.

 1984
1. E.D. Sontag. An approximation theorem in nonlinear sampling. In Mathematical theory of networks and systems (Beer Sheva, 1983), volume 58 of Lecture Notes in Control and Inform. Sci., pages 806-812. Springer, London, 1984. [PDF]
Abstract:
 We continue here our investigation into the preservation of structural properties under the sampling of nonlinear systems. The main new result is that, under minimal hypothesis, a controllable system always satisfies a strong type of approximate sampled controllability.

2. C.A. Schwartz, B.W. Dickinson, and E.D. Sontag. Characterizing innovations realizations for random processes. Stochastics, 11(3-4):159-172, 1984. [PDF] Keyword(s): statistics, innovations, sufficient statistics.
Abstract:
 In this paper we are concerned with the theory of second order (linear) innovations for discrete random processes. We show that of existence of a finite dimensional linear filter realizing the mapping from a discrete random process to its innovations is equivalent to a certain semiiseparable structure of the covariance sequence of the process. We also show that existence of a finite dimensional realization (linear or nonlinear) of the mapping from a process to its innovations implies that the process have this serniseparable covariance sequence property. In particular, for a stationary random process, the spectral density function must be rational.

3. E.D. Sontag. A concept of local observability. Systems Control Lett., 5(1):41-47, 1984. [PDF] Keyword(s): observability.
Abstract:
 A notion of local observability, which is natural in the context of nonlinear input/output regulation. is introduced. A simple characterization is provided, a comparison is made with other local nonlinear observability definitions. and its behavior under constant-rate sampling is analyzed.

4. E.D. Sontag. An algebraic approach to bounded controllability of linear systems. Internat. J. Control, 39(1):181-188, 1984. [PDF] Keyword(s): saturation, bounded inputs.
Abstract:
 In this note we present an algebraic approach to the proof that a linear system with matrices (A,B) is null-controllable using bounded inputs iff it is null-controllable (with unbounded inputs) and all eigenvalues of A have nonpositive real parts (continuous time) or magnitude not greater than one (discrete time). We also give the analogous results for the asymptotic case. Finally, we give an interpretation of these results in the context of local nonlinear controllability.

5. E.D. Sontag. Parametric stabilization is easy. Systems Control Lett., 4(4):181-188, 1984. [PDF] Keyword(s): systems over rings.
Abstract:
 A polynomially parametrized family of continuous-time controllable linear systems is always stabilizable by polynomially parametrized feedback. (Note: appendix had a MACSYMA computation. I cannot find the source file for that. Please look at journal if interested, but this is not very important. Also, two figures involving root loci are not in the web version.)

 1983
1. E.D. Sontag. Remarks on the preservation of various controllability properties under sampling. In Mathematical tools and models for control, systems analysis and signal processing, Vol. 3 (Toulouse/Paris, 1981/1982), Travaux Rech. Coop. Programme 567, pages 623-637. CNRS, Paris, 1983. [PDF]
Abstract:
 This note studies the preservation of controllability (and other properties) under sampling of a nonlinear system. More detailed results are obtained in the cases of analytic systems and of systems with finite dimensional Lie algebras.

2. R.T. Bumby and E.D. Sontag. Stabilization of polynomially parametrized families of linear systems. The single-input case. Systems Control Lett., 3(5):251-254, 1983. [PDF] Keyword(s): systems over rings.
Abstract:
 Given a continuous-time family of finite dimensional single input linear systems, parametrized polynomially, such that each of the systems in the family is controllable, there exists a polynomially parametrized control law making each of the systems in the family stable.

3. E.D. Sontag. A Lyapunov-like characterization of asymptotic controllability. SIAM J. Control Optim., 21(3):462-471, 1983. [PDF] Keyword(s): control-Lyapunov functions.
Abstract:
 It is shown that a control system in Rn is asymptotically controllable to the origin if and only if there exists a positive definite continuous functional of the states whose derivative can be made negative by appropriate choices of controls.

 1982
1. E.D. Sontag. A characterization of asymptotic controllability. In A. Bednarek and L. Cesari, editors, Dynamical Systems II, pages 645-648. Academic Press, NY, 1982. [PDF] Keyword(s): control-Lyapunov functions.
Abstract:
 This paper was a conference version of the SIAM paper that introduced the idea of control-Lyapunov functions for arbitrary nonlinear systems. (The journal paper was submitted in 1981 but only published in 1983.)

2. E.D. Sontag. Abstract regulation of nonlinear systems: stabilization. In Feedback control of linear and nonlinear systems (Bielefeld/Rome, 1981), volume 39 of Lecture Notes in Control and Inform. Sci., pages 227-243. Springer, Berlin, 1982.

3. E.D. Sontag. Linear systems over commutative rings: a (partial) updated survey. In Control science and technology for the progress of society, Vol. 1 (Kyoto, 1981), pages 325-330. IFAC, Laxenburg, 1982. Keyword(s): systems over rings.

4. P.P. Khargonekar and E.D. Sontag. On the relation between stable matrix fraction factorizations and regulable realizations of linear systems over rings. IEEE Trans. Automat. Control, 27(3):627-638, 1982. [PDF] Keyword(s): systems over rings.
Abstract:
 Various types of transfer matrix factorizations are of interest when designing regulators for generalized types of linear systems (delay differential. 2-D, and families of systems). This paper studies the existence of stable and of stable proper factorizations, in the context of the thery of systems over rings. Factorability is related to stabilizability and detectability properties of realizations of the transfer matrix. The original formulas for coprime factorizations (which are valid, in particular, over the field of reals) were given in this paper.

5. E.D. Sontag. Remarks on piecewise-linear algebra. Pacific J. Math., 98(1):183-201, 1982. [PDF] Keyword(s): hybrid systems, piecewise linear systems.
Abstract:
 Algebraic study of functions defined by piecewise linear (generally discontinuous) equations. File obtained by scanning a reprint.

 1981
1. R.T. Bumby, E.D. Sontag, H.J. Sussmann, and W. Vasconcelos. Remarks on the pole-shifting problem over rings. J. Pure Appl. Algebra, 20(2):113-127, 1981. [PDF] Keyword(s): systems over rings, systems over rings.
Abstract:
 Problems that appear in trying to extend linear control results to systems over rings R have attracted considerable attention lately. This interest has been due mainly to applications-oriented motivations (in particular, dealing with delay-differential equations), and partly to a purely algebraic interest. Given a square n-matrix F and an n-row matrix G. pole-shifting problems consist in obtaining more or less arbitrary characteristic polynomials for F+GK, for suitable ("feedback") matrices K. A review of known facts is given, various partial results are proved, and the case n=2 is studied in some detail.

2. E.D. Sontag. Conditions for abstract nonlinear regulation. Inform. and Control, 51(2):105-127, 1981. [PDF] Keyword(s): feedback stabilization.
Abstract:
 A paper that introduces a separation principle for general finite dimensional analytic continuous-time systems, proving the equivalence between existence of an output regulator (which is an abstract dynamical system) and certain "0-detectability" and asymptotic controllability assumptions.

3. E.D. Sontag. Nonlinear regulation: the piecewise linear approach. IEEE Trans. Automat. Control, 26(2):346-358, 1981. [PDF] Keyword(s): hybrid systems.
Abstract:
 Development of an approach to nonlinear control based on mixtures of linear systems and finite automata. File obtained by scanning.

 1980
1. M. L. J. Hautus and E.D. Sontag. An approach to detectability and observers. In Algebraic and geometric methods in linear systems theory (AMS-NASA-NATO Summer Sem., Harvard Univ., Cambridge, Mass., 1979), volume 18 of Lectures in Appl. Math., pages 99-135. Amer. Math. Soc., Providence, R.I., 1980. [PDF] Keyword(s): observability.
Abstract:
 This paper proposes an approach to the problem of establishing the existence of observers for deterministic dynamical systems. This approach differs from the standard one based on Luenberger observers in that the observation error is not required to be Markovian given the past input and output data. A general abstract result is given, which special- izes to new results for parametrized families of linear systems, delay systems and other classes of systems. Related problems of feedback control and regulation are also studied.

2. E.D. Sontag. On quasireachable realizations of a polynomial response. In Systems analysis (Conf., Bordeaux, 1978), volume 75 of Astérisque, pages 207-217. Soc. Math. France, Paris, 1980.

3. E.D. Sontag. On generalized inverses of polynomial and other matrices. IEEE Trans. Automat. Control, 25(3):514-517, 1980. [PDF]
Abstract:
 Necessary and sufficient conditions are given for a matrix over a ring to admit a Moore-Penrose generalized inverse in a weak sense. (Attached is also a Math Review with additional comments on strong inverses.)

4. E.D. Sontag. On the length of inputs necessary in order to identify a deterministic linear system. IEEE Trans. Automat. Control, 25(1):120-121, 1980. [PDF]
Abstract:
 The family of m-input, n-dimensional linear systems can be globally Identified with a generic input sequence of length 2mn. This bound is the best possible. A best bound is proved also for a corresponding local identification problem.

 1979
1. Y. Rouchaleau and E.D. Sontag. On the existence of minimal realizations of linear dynamical systems over Noetherian integral domains. J. Comput. System Sci., 18(1):65-75, 1979. [PDF] Keyword(s): systems over rings.
Abstract:
 This paper studies the problem of obtaining minimal realizations of linear input/output maps defined over rings. In particular, it is shown that, contrary to the case of systems over fields, it is in general impossible to obtain realizations whose dimiension equals the rank of the Hankel matrix. A characterization is given of those (Noetherian) rings over which realizations of such dimensions can he always obtained, and the result is applied to delay-differential systems.

2. E.D. Sontag. On finitary linear systems. Kybernetika (Prague), 15(5):349-358, 1979. [PDF] Keyword(s): systems over rings.
Abstract:
 An abstract operator approach is introduced, permitting a unified study of discrete- and continuous-time linear control systems. As an application, an algorithm is given for deciding if a linear system can be built from any fixed set of linear components. Finally, a criterion is given for reachability of the abstract systems introduced, giving thus a unified proof of known reachability results for discrete-time, continuous-time, and delay-differential systems.

3. E.D. Sontag. On the observability of polynomial systems. I. Finite-time problems. SIAM J. Control Optim., 17(1):139-151, 1979. [PDF] Keyword(s): observability, observability, polynomial systems.
Abstract:
 Different notions of observability are compared for systems defined by polynomial difference equations. The main result states that, for systems having the standard property of (multiple-experiment initial-state) observability, the response to a generic input sequence is sufficient for final-state determination. Some remarks are made on results for nonpolynomial and/or continuous-time systems. An identifiability result is derived from the above.

4. E.D. Sontag. Realization theory of discrete-time nonlinear systems. I. The bounded case. IEEE Trans. Circuits and Systems, 26(5):342-356, 1979. [PDF]
Abstract:
 A state-space realization theory is presented for a wide class of discrete time input/output behaviors. Although In many ways restricted, this class does include as particular cases those treated in the literature (linear, multilinear, internally bilinear, homogeneous), as well ss certain nonanalytic nonlinearities. The theory is conceptually simple, and matrix-theoretic algorithms are straightforward. Finite-realizability of these behaviors by state-affine systems is shown to be equivalent both to the existence of high-order input/output equadons and to realizability by more general types of systems.

 1978
1. W. Dicks and E.D. Sontag. Sylvester domains. J. Pure Appl. Algebra, 13(3):243-275, 1978. [PDF]
Abstract:
 The inner rank of an m x n matrix A over a ring is defined as the least integer r such that A can be expressed as the product of an m x r and an r x n matrix. For example, over a (skew) field this concept coincides with the usual notion of rank. This notion is studied in this paper, and is related to Sylvester's law of nullity and work by P.M. Cohn.

2. E.D. Sontag. On first-order equations for multidimensional filters. IEEE Trans. Acoustics, Speech, and Signal Processing, 26:480-482, 1978. [PDF]
Abstract:
 A construction is given to obtain first-order equation representations of a multidimensional filter, whose dimension is of the order of the degree of the transfer function.

3. E.D. Sontag. On split realizations of response maps over rings. Information and Control, 37(1):23-33, 1978. [PDF] Keyword(s): systems over rings.
Abstract:
 This paper deals with observability properties of realizations of linear response maps defined over commutative rings. A characterization is given for those maps which admit realizations which are simultaneously reachable and observable in a strong sense. Applications are given to delay-differential systems.

 1977
1. E.D. Sontag. On the internal realization of nonlinear behaviors. In A. Bednarek and L. Cesari, editors, Dynamical Systems, pages 93-497. Academic Press, New York, 1977.

2. E.D. Sontag. The lattice of minimal realizations of response maps over rings. Math. Systems Theory, 11(2):169-175, 1977. [PDF] Keyword(s): systems over rings.
Abstract:
 A lattice characterization is given for the class of minimal-rank realizations of a linear response map defined over a (commutative) Noetherian integral domain. As a corollary, it is proved that there are only finitely many nonisomorphic minimal-rank realizations of a response map over the integers, while for delay -differential systems these are classified by a lattice of subspaces of a finite-dimensional real vector space.

3. E.D. Sontag and Y. Rouchaleau. Sur les anneaux de Fatou forts. C. R. Acad. Sci. Paris Sér. A-B, 284(5):A331-A333, 1977. [PDF] Keyword(s): systems over rings.
Abstract:
 It is well known that principal rings are strong Fatou rings. We construct here a more general type of strong Fatou rings. We also prove that the monoid of divisor classes of a noetherian strong Fatou ring contains only the zero element, and that the dimension of such a ring is at most two.

 1976
1. E.D. Sontag. Linear systems over commutative rings: A survey. Ricerche di Automatica, 7:1-34, 1976. [PDF] Keyword(s): systems over rings.
Abstract:
 An elementary presentation is given of some of the main motivations and known results on linear systems over rings, including questions of realization and control. The analogies and differences with the more standard case of systems over fields are emphasized throughout.

2. E.D. Sontag. On finitely accessible and finitely observable rings. J. Pure Appl. Algebra, 8(1):97-104, 1976. [PDF] Keyword(s): systems over rings, observability, noncommutative rings.
Abstract:
 Two classes of rings which occur in linear system theory are introduced and compared. Characterizations of one of them are given in terms, of integral extensions (every finite extension of R is integral) and Cayley--Hamilton type matrix condition. A comparison is made in the case of no zero-divisors with Ore domains.

3. E.D. Sontag and Y. Rouchaleau. On discrete-time polynomial systems. Nonlinear Anal., 1(1):55-64, 1976. [PDF] Keyword(s): identifiability, observability, polynomial systems, realization theory, discrete-time.
Abstract:
 Considered here are a type of discrete-time systems which have algebraic constraints on their state set and for which the state transitions are given by (arbitrary) polynomial functions of the inputs and state variables. The paper studies reachability in bounded time, the problem of deciding whether two systems have the same external behavior by applying finitely many inputs, the fact that finitely many inputs (which can be chosen quite arbitrarily) are sufficient to separate those states of a system which are distinguishable, and introduces the subject of realization theory for this class of systems.

 1975
1. E.D. Sontag. On linear systems and noncommutative rings. Math. Systems Theory, 9(4):327-344, 1975. [PDF] Keyword(s): systems over rings.
Abstract:
 This paper studies some problems appearing in the extension of the theory of linear dynamical systems to the case in which parameters are taken from noncommutative rings. Purely algebraic statements of some of the problems are also obtained. Through systems defined by operator rings, the theory of linear systems over rings may be applied to other areas of automata and control theory; several such applications are outlined.

2. E.D. Sontag. On some questions of rationality and decidability. J. Comput. System Sci., 11(3):375-381, 1975. [PDF] Keyword(s): theory of computing and complexity.
Abstract:
 Some results are given in the theory of rational power series over a broad class of semirings. In particular, it is shown that for unambiguous sets the notion of rationality is independent of the semiring over which representations are defined. The undecidability of the rationality of probabilistic word functions is also established.

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