Publications about 'systems biology' |
Books and proceedings |
Articles in journal or book chapters |
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. |
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. |
The emergence of and transitions between distinct phenotypes in isogenic cells can be attributed to the intricate interplay of epigenetic marks, external signals, and gene regulatory elements. These elements include chromatin remodelers, histone modifiers, transcription factors, and regulatory RNAs. Mathematical models known as Gene Regulatory Networks (GRNs) are an increasingly important tool to unravel the workings of such complex networks. In such models, epigenetic factors are usually proposed to act on the chromatin regions directly involved in the expression of relevant genes. However, it has been well-established that these factors operate globally and compete with each other for targets genome-wide. Therefore, a perturbation of the activity of a regulator can redistribute epigenetic marks across the genome and modulate the levels of competing regulators. In this paper, we propose a conceptual and mathematical modeling framework that incorporates both local and global competition effects between antagonistic epigenetic regulators in addition to local transcription factors, and show the counter-intuitive consequences of such interactions. We apply our approach to recent experimental findings on the Epithelial-Mesenchymal Transition (EMT). We show that it can explain the puzzling experimental data as well provide new verifiable predictions. |
Differentiation within multicellular organisms is a complex process that helps to establish spatial patterning and tissue formation within the body. Often, the differentiation of cells is governed by morphogens and intercellular signaling molecules that guide the fate of each cell, frequently using toggle-like regulatory components. Synthetic biologists have long sought to recapitulate patterned differentiation with engineered cellular communities and various methods for differentiating bacteria have been invented. Here, we couple a synthetic co-repressive toggle switch with intercellular signaling pathways to create a "quorum-sensing toggle." We show that this circuit not only exhibits population-wide bistability in a well-mixed liquid environment, but also generates patterns of differentiation in colonies grown on agar containing an externally supplied morphogen. |
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. |
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. |
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. |
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. |
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. |
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. |
Correspondence regarding circulating tumor cell detection |
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. |
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. This 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. |
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. |
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. |
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. |
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. |
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". |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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 |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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 call``uniform 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). |
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. |
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. |
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. |
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. |
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 |
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. |
We present a novel computational method, and related software, to synthesize signal transduction networks from single and double causal evidence. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
See abstract and pdf for ``Monotone and near-monotone biochemical networks''. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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". |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
errata for Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction |
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. |
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. |
Conference articles |
Conference version of paper published in IEEE Control Systems Letters, 2020 |
Integral feedback can help achieve robust tracking independently of external disturbances. Motivated by this knowledge, biological engineers have proposed various designs of biomolecular integral feedback controllers to regulate biological processes. In this paper, we theoretically analyze the operation of a particular synthetic biomolecular integral controller, which we have recently proposed and implemented experimentally. Using a combination of methods, ranging from linearized analysis to sum-of-squares (SOS) Lyapunov functions, we demonstrate that, when the controller is operated in closed-loop, it is capable of providing integral corrections to the concentration of an output species in such a manner that the output tracks a reference signal linearly over a large dynamic range. We investigate the output dependency on the reaction parameters through sensitivity analysis, and quantify performance using control theory metrics to characterize response properties, thus providing clear selection guidelines for practical applications. We then demonstrate the stable operation of the closed-loop control system by constructing quartic Lyapunov functions using SOS optimization techniques, and establish global stability for a unique equilibrium. Our analysis suggests that by incorporating effective molecular sequestration, a biomolecular closed-loop integral controller that is capable of robustly regulating gene expression is feasible. |
Cellular reprogramming is traditionally accomplished through an open loop control approach, wherein key transcription factors are injected in cells to steer a gene regulatory network toward a pluripotent state. Recently, a closed loop feedback control strategy was proposed in order to achieve more accurate control. Previous analyses of the controller were based on deterministic models, ignoring the substantial stochasticity in these networks, Here we analyze the Chemical Master Equation for reaction models with and without the feedback controller. We computationally and analytically investigate the performance of the controller in biologically relevant parameter regimes where stochastic effects dictate system dynamics. Our results indicate that the feedback control approach still ensures reprogramming even when analyzed using a stochastic model. |
In the mathematical modeling of cell differentiation, it is common to think of internal states of cells (quanfitied by activation levels of certain genes) as determining different cell types. We study here the "PU.1/GATA-1 circuit" that controls the development of mature blood cells from hematopoietic stem cells (HSCs). We introduce a rigorous chemical reaction network model of the PU.1/GATA-1 circuit, which incorporates current biological knowledge and find that the resulting ODE model of these biomolecular reactions is incapable of exhibiting multistability, contradicting the fact that differentiation networks have, by definition, alternative stable steady states. When considering instead the stochastic version of this chemical network, we analytically construct the stationary distribution, and are able to show that this distribution is indeed capable of admitting a multiplicity of modes. Finally, we study how a judicious choice of system parameters serves to bias the probabilities towards different stationary states. We remark that certain changes in system parameters can be physically implemented by a biological feedback mechanism; tuning this feedback gives extra degrees of freedom that allow one to assign higher likelihood to some cell types over others. |
This is a tutorial paper on control-theoretic methods for the analysis of biological systems. |
Combining in-vivo experiments with system identification methods, we determine a simple model of aerotaxis in B. subtilis, and we subsequently employ this model in order to compute the sequence of oxygen gradients needed in order to achieve set-point regulation with respect to a signal tracking the center of mass of the bacterial population. We then successfully validate both the model and the control scheme, by showing that in-vivo positioning control can be achieved via the application of the precomputed inputs in-vivo in an open-loop configuration. |
Applying Modular Response Analysis to a synthetic gene circuit, which was introduced in a recent paper by the authors, leads to the inference of a nontrivial "ghost" regulation edge which was not explicitly engineered into the network and which is, in fact, not immediately apparent from experimental measurements. One may thus hypothesize that this ghost regulatory effect is due to competition for resources. A mathematical model is proposed, and analyzed in closed form, that lends validation to this hypothesis. |
This conference paper (a) summarizes material from "A fundamental limitation to fold-change detection by biological systems with multiple time scales" (IET Systems Biology 2014) and presents additional remarks regarding (b) expansion techniques to compute FCD error and (c) stochastic adaptation and FCD |
This paper studies model-based estimation methods of a rate of a nonhomogeneous Poisson processes that describes events arising from modeling biological phenomena in which discrete events are measured. We describe an approach based on observers and Kalman filters as well as preliminary simulation results, and compare these to other methods (not model-based) in the literature. The problem is motivated by the question of identification of internal states from neural spikes and bacterial tumbling behavior. |
Recent experimental work has shown that transient E. coli chemotactic response is unchanged by a scaling of its ligand input signal (fold change detection, or FCD), and this is in agreement with earlier mathematical predictions. However, this prediction was based on certain particular assumptions on the structure of the chemotaxis pathway. In this work, we begin by showing that behavior similar to FCD can be obtained under weaker conditions on the system structure. Namely, we show that under relaxed conditions, a scaling of the chemotaxis system's inputs leads to a time scaling of the output response. We propose that this may be a contributing factor to the robustness of the experimentally observed FCD. We further show that FCD is a special case of this time scaling behavior for which the time scaling factor is unity. We then proceed to extend the conditions for output time scaling to more general adapting systems, and demonstrate this time scaling behavior on a published model of the chemotaxis pathway of the bacterium Rhodobacter sphaeroides. This work therefore provides examples of how robust biological behavior can arise from simple yet realistic conditions on the underlying system structure. |
This is a conference version of ``A characterization of scale invariant responses in enzymatic networks. |
This paper studies invariance with respect to symmetries in sensory fields, a particular case of which, scale invariance, has recently been found in certain eukaryotic as well as bacterial cell signaling systems. We describe a necessary and sufficient characterization of symmetry invariance in terms of equivariant transformations, show how this characterization helps find all possible symmetries in standard models of biological adaptation, and discuss symmetry-invariant searches. |
Preliminary conference version of ''A contraction approach to the hierarchical analysis and design of networked systems''. |
See abstract and link to pdf in entry for Journal paper. |
See abstract and link to pdf in entry for Journal paper. |
Conference version of paper "Conditions for global stability of monotone tridiagonal systems with negative feedback" |
This tutorial paper presents an introduction to systems and synthetic molecular biology. It provides an introduction to basic biological concepts, and describes some of the techniques as well as challenges in the analysis and design of biomolecular networks. |
For distributed systems with a cyclic interconnection structure, a global stability result is shown to hold if the secant criterion is satisfied. |
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. |
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 structure (e.g., mass-action kinetics) of the dynamical equations involved, and relying only on stoichiometric constraints. The key idea is to find a suitable set of coordinates under which the resulting system is cooperative. As a simple example, the paper shows that a phosphorylation/dephosphorylation process, which is involved in many signaling cascades, has a global stability property. |
This paper deals with global convergence to equilibria, and in particular Hirsch's generic convergence theorem for strongly monotone systems, for singular perturbations of monotone systems. |
This paper deals with global convergence to equilibria, and in particular Hirsch's generic convergence theorem for strongly monotone systems, for singular perturbations of monotone systems. |
We show how certain properties of Goldbeter's original 1995 model for circadian oscillations can be proved mathematically. We establish global asymptotic stability, and in particular no oscillations, if the rate of transcription is somewhat smaller than that assumed by Goldbeter, but, on the other hand, this stability persists even under arbitrary delays in the feedback loop. We are mainly interested in illustrating certain mathematical techniques, including the use of theorems concerning tridiagonal cooperative systems and the recently developed theory of monotone systems with inputs and outputs. |
Monotone systems are dynamical systems for which the flow preserves a partial order. Some applications will be briefly reviewed in this paper. Much of the appeal of the class of monotone systems stems from the fact that roughly, most solutions converge to the set of equilibria. However, this usually requires a stronger monotonicity property which is not always satisfied or easy to check in applications. Following work of J.F. Jiang, we show that monotonicity is enough to conclude global attractivity if there is a unique equilibrium and if the state space satisfies a particular condition. The proof given here is self-contained and does not require the use of any of the results from the theory of monotone systems. We will illustrate it on a class of chemical reaction networks with monotone, but otherwise arbitrary, reaction kinetics. |
Internal reports |
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. |
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. |
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. |
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. |
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. |
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