Internal reports
  1. E.D. Sontag, D. Biswas, and N.J. Cowan. An observability result related to active sensing. Technical report, 2022. Note: ArXiv 2210.03848. [PDF] Keyword(s): nonlinear systems, observability, active sensing.
    For a general class of translationally invariant systems with a specific category of nonlinearity in the output, this paper presents necessary and sufficient conditions for global observability. Critically, this class of systems cannot be stabilized to an isolated equilibrium point by dynamic output feedback. These analyses may help explain the active sensing movements made by animals when they perform certain motor behaviors, despite the fact that these active sensing movements appear to run counter to the primary motor goals. The findings presented here establish that active sensing underlies the maintenance of observability for such biological systems, which are inherently nonlinear due to the presence of the high-pass sensor dynamics.

  1. T. Chen, M. A. Al-Radhawi, and E. D. Sontag. A mathematical model exhibiting the effect of DNA methylation on the stability boundary in cell-fate networks. Technical report, Cold Spring Harbor Laboratory, 2019. Note: BioRxiv preprint 10.1101/2019.12.19.883280. Keyword(s): Cell-fate networks, gene regulatory networks, DNA methylation, epigenetic regulation, pluripotent stem cell circuit.
    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.

  2. J.L. Gevertz, J.M. Greene, and E.D. Sontag. Validation of a mathematical model of cancer incorporating spontaneous and induced evolution to drug resistance. Technical report, Cold Spring Harbor Laboratory, 2019. Note: BioRxiv preprint 10.1101/2019.12.27.889444. Keyword(s): cancer heterogeneity, phenotypic variation, nonlinear systems, epigenetics, optimal control theory, oncology, cancer.
    This paper continues the study of a model which was introduced in earlier work by the authors to study spontaneous and induced evolution to drug resistance under chemotherapy. The model is fit to existing experimental data, and is then validated on additional data that had not been used when fitting. In addition, an optimal control problem is studied numerically.

  3. M. Margaliot and E.D. Sontag. Compact attractors of an antithetic integral feedback system have a simple structure. Technical report, bioRxiv 2019/868000v1, 2019. [PDF] Keyword(s): Poincare-Bendixson, k-cooperative dynamical systems, sign-regular matrices, synthetic biology, antithetic feedback.
    Since its introduction by Briat, Gupta and Khammash, the antithetic feedback controller design has attracted considerable attention in both theoretical and experimental systems biology. The case in which the plant is a two-dimensional linear system (making the closed-loop system a nonlinear four-dimensional system) has been analyzed in much detail. This system has a unique equilibrium but, depending on parameters, it may exhibit periodic orbits. This note shows that, for any parameter choices, every bounded trajectory satisfies a Poincare'-Bendixson property: the dynamics in the omega-limit set of any precompact solution is conjugate to the dynamics in a compact invariant subset of a two-dimensional Lipschitz dynamical system, thus precluding chaotic and other strange attractors.

  4. A. P. Tran, M. A. 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. Technical report, Cold Spring Harbor Laboratory, 2019. Note: BioRxiv 2019.12.12.874891. Keyword(s): chemotherapy, immunology, immune system, oncology, cancer, metronomic.
    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.

  1. J.M. Greene, C. Sanchez-Tapia, and E.D. Sontag. Mathematical details on a cancer resistance model. Technical report, bioRxiv 2018/475533, 2018. [PDF] Keyword(s): identifiability, drug resistance, chemotherapy, optimal control theory, singular controls, oncology, cancer.
    The primary factor 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, the control structure is precisely characterized as a concatenation of bang-bang and path-constrained arcs via the Pontryagin Maximum Principle and differential Lie techniques. A structural identfiability analysis is also presented, demonstrating that patient-specfic parameters may be measured and thus utilized in the design of optimal therapies prior to the commencement of therapy.

  2. M. Sadeghi, M.A. Al-Radhawi, M. Margaliot, and E.D. Sontag. On the periodic gain of the Ribosome Flow Model. Technical report, bioRxiv 2018/507988, 2018. [PDF] Keyword(s): systems biology, biochemical networks, ribosomes, RFM, ribosome flow model.
    We consider a compartmental model for ribosome flow during RNA translation, the Ribosome Flow Model (RFM). This model includes a set of positive transition rates that control the flow from every site to the consecutive site. It has been shown that when these rates are time-varying and jointly T-periodic, the protein production rate converges to a unique T-periodic pattern. Here, we study a problem that can be roughly stated as: can periodic rates yield a higher average production rate than constant rates? We rigorously formulate this question and show via simulations, and rigorous analysis in one simple case, that the answer is no.

  1. E.D. Sontag. A remark on incoherent feedforward circuits as change detectors and feedback controllers. Technical report, arXiv:1602.00162, 2016. [PDF] Keyword(s): scale invariance, fold change detection, T cells, incoherent feedforward loops, immunology, incoherent feedforward loop, feedforward, IFFL.
    This note analyzes incoherent feedforward loops in signal processing and control. It studies the response properties of IFFL's to exponentially growing inputs, both for a standard version of the IFFL and for a variation in which the output variable has a positive self-feedback term. It also considers a negative feedback configuration, using such a device as a controller. It uncovers a somewhat surprising phenomenon in which stabilization is only possible in disconnected regions of parameter space, as the controlled system's growth rate is varied.

  2. E.D. Sontag. Examples of computation of exact moment dynamics for chemical reaction networks. Technical report, arXiv:1612.02393, 2016. [PDF] Keyword(s): systems biology, biochemical networks, stochastic systems, chemical master equation, chemical reaction networks, moments, molecular networks, complex-balanced networks.
    We review in a unified way results for two types of stochastic chemical reaction systems for which moments can be effectively computed: feedforward networks and complex-balanced networks.

  3. E.D. Sontag. Two-zone tumor tolerance can arise from a simple immunological feedforward motif that estimates tumor growth rates. Technical report, bioRxiv, 2016. [PDF] Keyword(s): scale invariance, fold change detection, T cells, incoherent feedforward loops, immunology, cancer.
    Preprint version of "A dynamical model of immune responses to antigen presentation predicts different regions of tumor or pathogen elimination", appeared in Cell Systems 2017. However, the journal version does not include Section 9 on degradation-based IFFL's from this preprint.

  1. E.D. Sontag. Incoherent feedforward motifs as immune change detectors. Technical report, bioRxiv, December 2015. [PDF] Keyword(s): scale invariance, fcd, fold change detection, T cells, incoherent feedforward loops, immunology, incoherent feedforward loop, feedforward, IFFL.
    We speculate that incoherent feedforward loops may be phenomenologically involved in self/nonself discrimination in immune-infection and immune-tumor interactions, acting as "change detectors". In turn, this may result in logarithmic sensing (Weber phenomenon) and even scale invariance (fold-change detection).

  1. J. Barton and E.D. Sontag. Remarks on the energy costs of insulators in enzymatic cascades. Technical report,, December 2014. [PDF] Keyword(s): retroactivity, systems biology, biochemical networks, futile cycles, singular perturbations, modularity.
    The connection between optimal biological function and energy use, measured for example by the rate of metabolite consumption, is a current topic of interest in the systems biology literature which has been explored in several different contexts. In [J. P. Barton and E. D. Sontag, Biophys. J. 104, 6 (2013)], we related the metabolic cost of enzymatic futile cycles with their capacity to act as insulators which facilitate modular interconnections in biochemical networks. There we analyzed a simple model system in which a signal molecule regulates the transcription of one or more target proteins by interacting with their promoters. In this note, we consider the case of a protein with an active and an inactive form, and whose activation is controlled by the signal molecule. As in the original case, higher rates of energy consumption are required for better insulator performance.

  1. Z. Aminzare and E. D. Sontag. Remarks on a population-level model of chemotaxis: advection-diffusion approximation and simulations. Technical report, arxiv:1302.2605, 2013. [PDF]
    This note works out an advection-diffusion approximation to the density of a population of E. coli bacteria undergoing chemotaxis in a one-dimensional space. Simulations show the high quality of predictions under a shallow-gradient regime.

  2. E.D. Sontag. A remark about polynomials with specified local minima and no other critical points. Technical report, arxiv 1302.0759, 2013. [PDF]
    The following observation must surely be "well-known", but it seems worth giving a simple and quite explicit proof. Take any finite subset X of Rn, n>1. Then, there is a polynomial function P:Rn -> R which has local minima on the set X, and has no other critical points. Applied to the negative gradient flow of P, this implies that there is a polynomial vector field with asymptotically stable equilibria on X and no other equilibria. Some trajectories of this vector field are not pre-compact; a complementary observation says that, again for arbitrary X, one can find a vector field with asymptotically stable equilibria on X, no other equilibria except saddles, and all omega-limit sets consisting of singletons.

  1. J. Barton and E.D. Sontag. The energy costs of biological insulators. Technical report,, October 2012. Keyword(s): retroactivity, systems biology, biochemical networks, futile cycles, singular perturbations, modularity.
    Biochemical signaling pathways can be insulated from impedance and competition effects through enzymatic "futile cycles" which consume energy, typically in the form of ATP. We hypothesize that better insulation necessarily requires higher energy consumption, and provide evidence, through the computational analysis of a simplified physical model, to support this hypothesis.

  2. M. Marcondes de Freitas and E.D. Sontag. Remarks on random dynamical systems with inputs and outputs and a small-gain theorem for monotone RDS. Technical report,, July 2012. Keyword(s): random dynamical systems, monotone systems.

  1. E.D. Sontag. Remarks on invariance of population distributions for systems with equivariant internal dynamics. Technical report, arxiv.1108.3245, August 2011. [PDF] Keyword(s): adaptation, biological adaptation, perfect adaptation, scale invariance, systems biology, transient behavior, symmetries, fcd, fold-change detection, jump Markov processes.

  1. E.D. Sontag. An observation regarding systems which converge to steady states for all constant inputs, yet become chaotic with periodic inputs. Technical report, arxiv 0906.2166, 2009. [PDF]

  1. E.D. Sontag and F.R. Wirth. Remarks on universal nonsingular controls for discrete-time systems. Technical report 381, Institute for Dynamical Systems, University of Bremen, 1996.

  1. F. Albertini and E.D. Sontag. Some connections between chaotic dynamical systems and control systems. Technical report SYCON-90-13, Rutgers Center for Systems and Control, 1990.

  1. E.D. Sontag. Sigmoids distinguish more efficiently than Heavisides. Technical report SYCON-89-12, Rutgers Center for Systems and Control, 1989. Keyword(s): machine learning, neural networks.

  1. E.D. Sontag. Integrability of certain distributions associated to actions on manifolds and an introduction to Lie-algebraic control. Technical report SYCON-88-04, Rutgers Center for Systems and Control, 1988.

  2. E.D. Sontag. Some remarks on the backpropagation algorithm for neural net learning. Technical report SYCON-88-02, Rutgers Center for Systems and Control, 1988. [PDF] Keyword(s): machine learning, neural networks.
    This is a very old informal report that discusses the study of local minima of quadratic loss functions for fitting errors in sigmoidal neural net learning. It also includes several remarks concerning the growth of weights during gradient descent. There is nothing very interesting here - far better knowledge is now available - but the report was placed here by request.

  1. E.D. Sontag and H.J. Sussmann. Optimization algorithms for image restoration and segmentation. Technical report 34, Rutgers Center for Computer Aids for Industrial Productivity, 1987.

  1. E.D. Sontag and D.E. Stevenson. Remarks on multi-server, multi-priority queuing models related to MVS job scheduling. Technical report TM-81-45281-1, Bell Telephone Labs., 1981.



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