Piecewise-linear Models of Genetic Regulatory Networks: Equilibria and their Stability

Efficient parameter inference in networked dynamical systems via steady states: A surrogate objective function approach integrating mean-field and nonlinear least squares

In networked dynamical systems, inferring governing parameters is crucial for predicting nodal dynamics, such as gene expression levels, species abundance, or population density. While many parameter estimation techniques rely on time-series data, particularly systems that converge over extreme time ranges, only noisy steady-state data is available, requiring a new approach to infer dynamical parameters from noisy observations of steady states. However, the traditional optimization process is computationally demanding, requiring repeated simulation of coupled ordinary differential equations. To overcome these limitations, we introduce a surrogate objective function that leverages decoupled equations to compute steady states, significantly reducing computational complexity. Furthermore, by optimizing the surrogate objective function, we obtain steady states that more accurately approximate the ground truth than noisy observations and predict future equilibria when topology changes. We empirically demonstrate the effectiveness of the proposed method across ecological, gene regulatory, and epidemic networks. Our approach provides an efficient and effective way to estimate parameters from steady-state data and has the potential to improve predictions in networked dynamical systems.

Cycle dynamics and synchronization in a coupled network of peripheral circadian clocks

The intercellular interactions between peripheral circadian clocks, located in tissues and organs other than the suprachiasmatic nuclei of the hypothalamus, are still very poorly understood. We propose a theoretical and computational study of the coupling between two or more clocks, using a calibrated, reduced model of the circadian clock to describe some synchronization properties between peripheral cellular clocks. Based on a piecewise linearization of the dynamics of the mutual CLOCK:BMAL1/PER:CRY inactivation term, we suggest a segmentation of the circadian cycle into six stages, to help analyse different types of synchronization between two clocks, including single stage duration, total period and maximal amplitudes. Finally, our model reproduces some recent experimental results on the effects of different regimes of time-restricted feeding in liver circadian clocks of mice.

On convergence for hybrid models of gene regulatory networks under polytopic uncertainties: a Lyapunov approach

Hybrid models of genetic regulatory networks allow for a simpler analysis with respect to fully detailed quantitative models, still maintaining the main dynamical features of interest. In this paper we consider a piecewise affine model of a genetic regulatory network, in which the parameters describing the production function are affected by polytopic uncertainties. In the first part of the paper, after recalling how the problem of finding a Lyapunov function is solved in the nominal case, we present the considered polytopic uncertain system and then, after describing how to deal with sliding mode solutions, we prove a result of existence of a parameter dependent Lyapunov function subject to the solution of a feasibility linear matrix inequalities problem. In the second part of the paper, based on the previously described Lyapunov function, we are able to determine a set of domains where the system is guaranteed to converge, with the exception of a zero measure set of times, independently from the uncertainty realization. Finally a three nodes network example shows the validity of the results.

Qualitative Modeling, Analysis and Control of Synthetic Regulatory Circuits

Qualitative modeling approaches are promising and still underexploited tools for the analysis and design of synthetic circuits. They can make predictions of circuit behavior in the absence of precise, quantitative information. Moreover, they provide direct insight into the relation between the feedback structure and the dynamical properties of a network. We review qualitative modeling approaches by focusing on two specific formalisms, Boolean networks and piecewise-linear differential equations, and illustrate their application by means of three well-known synthetic circuits. We describe various methods for the analysis of state transition graphs, discrete representations of the network dynamics that are generated in both modeling frameworks. We also briefly present the problem of controlling synthetic circuits, an emerging topic that could profit from the capacity of qualitative modeling approaches to rapidly scan a space of design alternatives.

Topology-induced dynamics in a network of synthetic oscillators with piecewise affine approximation

In synthetic biology approaches, minimal systems are used to reproduce complex molecular mechanisms that appear in the core functioning of multi-cellular organisms. In this paper, we study a piecewise affine model of a synthetic two-gene oscillator and prove existence and stability of a periodic solution for all parameters in a given region. Motivated by the synchronization of circadian clocks in a cluster of cells, we next consider a network of N identical oscillators under diffusive coupling to investigate the effect of the topology of interactions in the network's dynamics. Our results show that both all-to-all and one-to-all coupling topologies may introduce new stable steady states in addition to the expected periodic orbit. Both topologies admit an upper bound on the coupling parameter that prevents the generation of new steady states. However, this upper bound is independent of the number of oscillators in the network and less conservative for the one-to-all topology.

Two-fold singularities in nonsmooth dynamics-Higher dimensional analogs

When a system of ordinary differential equations is discontinuous along some threshold, its flow may become tangent to that threshold from one side or the other, creating a fold singularity, or from both sides simultaneously, creating a two-fold singularity. The classic two-fold exhibits intricate local dynamics and accumulating sequences of local bifurcations and is by now rather well understood, but it is just the simplest of an infinite hierarchy of two-folds and multi-folds in higher dimensions. These arise when a system is discontinuous along multiple intersecting thresholds, and the induced sliding flows on those thresholds become tangent to their intersections. We show here, surprisingly, that these higher dimensional analogs of the two-fold reduce to the equations of the classic two-fold, providing the first step into their study and a new tool to understand higher dimensional systems with discontinuities.

A Robustness Analysis of Dynamic Boolean Models of Cellular Circuits

With ever growing amounts of omics data, the next challenge in biological research is the interpretation of these data to gain mechanistic insights about cellular function. Dynamic models of cellular circuits that capture the activity levels of proteins and other molecules over time offer great expressive power by allowing the simulation of the effects of specific internal or external perturbations on the workings of the cell. However, the study of such models is at its infancy and no large-scale analysis of the robustness of real models to changing conditions has been conducted to date. Here we provide a computational framework to study the robustness of such models using a combination of stochastic simulations and integer linear programming techniques. We apply our framework to a large collection of cellular circuits and benchmark the results against randomized models. We find that the steady states of real circuits tend to be more robust in multiple aspects compared with their randomized counterparts.

Inferring parameters of prey switching in a 1 predator-2 prey plankton system with a linear preference tradeoff

We construct two ordinary-differential-equation models of a predator feeding adaptively on two prey types, and we evaluate the models' ability to fit data on freshwater plankton. We model the predator's switch from one prey to the other in two different ways: (i) smooth switching using a hyperbolic tangent function; and (ii) by incorporating a parameter that changes abruptly across the switching boundary as a system variable that is coupled to the population dynamics. We conduct linear stability analyses, use approximate Bayesian computation (ABC) combined with a population Monte Carlo (PMC) method to fit model parameters, and compare model results quantitatively to data for ciliate predators and their two algal prey groups collected from Lake Constance on the German-Swiss-Austrian border. We show that the two models fit the data well when the smooth transition is steep, supporting the simplifying assumption of a discontinuous prey-switching behavior for this scenario. We thus conclude that prey switching is a possible mechanistic explanation for the observed ciliate-algae dynamics in Lake Constance in spring, but that these data cannot distinguish between the details of prey switching that are encoded in these different models.

Global dynamics for switching systems and their extensions by linear differential equations

Switching systems use piecewise constant nonlinearities to model gene regulatory networks. This choice provides advantages in the analysis of behavior and allows the global description of dynamics in terms of Morse graphs associated to nodes of a parameter graph. The parameter graph captures spatial characteristics of a decomposition of parameter space into domains with identical Morse graphs. However, there are many cellular processes that do not exhibit threshold-like behavior and thus are not well described by a switching system. We consider a class of extensions of switching systems formed by a mixture of switching interactions and chains of variables governed by linear differential equations. We show that the parameter graphs associated to the switching system and any of its extensions are identical. For each parameter graph node, there is an order-preserving map from the Morse graph of the switching system to the Morse graph of any of its extensions. We provide counterexamples that show why possible stronger relationships between the Morse graphs are not valid.

Applying differential dynamic logic to reconfigurable biological networks

Qualitative and quantitative modeling frameworks are widely used for analysis of biological regulatory networks, the former giving a preliminary overview of the system's global dynamics and the latter providing more detailed solutions. Another approach is to model biological regulatory networks as hybrid systems, i.e., systems which can display both continuous and discrete dynamic behaviors. Actually, the development of synthetic biology has shown that this is a suitable way to think about biological systems, which can often be constructed as networks with discrete controllers, and present hybrid behaviors. In this paper we discuss this approach as a special case of the reconfigurability paradigm, well studied in Computer Science (CS). In CS there are well developed computational tools to reason about hybrid systems. We argue that it is worth applying such tools in a biological context. One interesting tool is differential dynamic logic (dL), which has recently been developed by Platzer and applied to many case-studies. In this paper we discuss some simple examples of biological regulatory networks to illustrate how dL can be used as an alternative, or also as a complement to methods already used.

Design of a bistable switch to control cellular uptake

Bistable switches are widely used in synthetic biology to trigger cellular functions in response to environmental signals. All bistable switches developed so far, however, control the expression of target genes without access to other layers of the cellular machinery. Here, we propose a bistable switch to control the rate at which cells take up a metabolite from the environment. An uptake switch provides a new interface to command metabolic activity from the extracellular space and has great potential as a building block in more complex circuits that coordinate pathway activity across cell cultures, allocate metabolic tasks among different strains or require cell-to-cell communication with metabolic signals. Inspired by uptake systems found in nature, we propose to couple metabolite import and utilization with a genetic circuit under feedback regulation. Using mathematical models and analysis, we determined the circuit architectures that produce bistability and obtained their design space for bistability in terms of experimentally tuneable parameters. We found an activation-repression architecture to be the most robust switch because it displays bistability for the largest range of design parameters and requires little fine-tuning of the promoters' response curves. Our analytic results are based on on-off approximations of promoter activity and are in excellent qualitative agreement with simulations of more realistic models. With further analysis and simulation, we established conditions to maximize the parameter design space and to produce bimodal phenotypes via hysteresis and cell-to-cell variability. Our results highlight how mathematical analysis can drive the discovery of new circuits for synthetic biology, as the proposed circuit has all the hallmarks of a toggle switch and stands as a promising design to control metabolic phenotypes across cell cultures.

Convergence Properties of Posttranslationally Modified Protein-Protein Switching Networks with Fast Decay Rates

A significant conceptual difficulty in the use of switching systems to model regulatory networks is the presence of so-called "black walls," co-dimension 1 regions of phase space with a vector field pointing inward on both sides of the hyperplane. Black walls result from the existence of direct negative self-regulation in the system. One biologically inspired way of removing black walls is the introduction of intermediate variables that mediate the negative self-regulation. In this paper, we study such a perturbation. We replace a switching system with a higher-dimensional switching system with rapidly decaying intermediate proteins, and compare the dynamics between the two systems. We find that the while the individual solutions of the original system can be approximated for a finite time by solutions of a sufficiently close perturbed system, there are always solutions that are not well approximated for any fixed perturbation. We also study a particular example, where global basins of attraction of the perturbed system have a strikingly different form than those of the original system. We perform this analysis using techniques that are adapted to dealing with non-smooth systems.

A Model of Yeast Cell-Cycle Regulation Based on a Standard Component Modeling Strategy for Protein Regulatory Networks

To understand the molecular mechanisms that regulate cell cycle progression in eukaryotes, a variety of mathematical modeling approaches have been employed, ranging from Boolean networks and differential equations to stochastic simulations. Each approach has its own characteristic strengths and weaknesses. In this paper, we propose a “standard component” modeling strategy that combines advantageous features of Boolean networks, differential equations and stochastic simulations in a framework that acknowledges the typical sorts of reactions found in protein regulatory networks. Applying this strategy to a comprehensive mechanism of the budding yeast cell cycle, we illustrate the potential value of standard component modeling. The deterministic version of our model reproduces the phenotypic properties of wild-type cells and of 125 mutant strains. The stochastic version of our model reproduces the cell-to-cell variability of wild-type cells and the partial viability of the CLB2-dbΔ clb5Δ mutant strain. Our simulations show that mathematical modeling with “standard components” can capture in quantitative detail many essential properties of cell cycle control in budding yeast.

Time-Delayed Models of Gene Regulatory Networks

We discuss different mathematical models of gene regulatory networks as relevant to the onset and development of cancer. After discussion of alternative modelling approaches, we use a paradigmatic two-gene network to focus on the role played by time delays in the dynamics of gene regulatory networks. We contrast the dynamics of the reduced model arising in the limit of fast mRNA dynamics with that of the full model. The review concludes with the discussion of some open problems.

A Modelling Framework for Gene Regulatory Networks Including Transcription and Translation

Qualitative models of gene regulatory networks have generally considered transcription factors to regulate directly the expression of other transcription factors, without any intermediate variables. In fact, gene expression always involves transcription, which produces mRNA molecules, followed by translation, which produces protein molecules, which can then act as transcription factors for other genes (in some cases after post-transcriptional modifications). Suppressing these multiple steps implicitly assumes that the qualitative behaviour does not depend on them. Here we explore a class of expanded models that explicitly includes both transcription and translation, keeping track of both mRNA and protein concentrations. We mainly deal with regulation functions that are steep sigmoids or step functions, as is often done in protein-only models. We find that flow cannot be constrained to switching domains, though there can still be asymptotic approach to singular stationary points (fixed points in the vicinity of switching thresholds). This avoids the thorny issue of singular flow, but leads to somewhat more complicated possibilities for flow between threshold crossings. In the infinitely fast limit of either mRNA or protein rates, we find that solutions converge uniformly to solutions of the corresponding protein-only model on arbitrary finite time intervals. This leaves open the possibility that the limit system (with one type of variable infinitely fast) may have different asymptotic behaviour, and indeed, we find an example in which stability of a fixed point in the protein-only model is lost in the expanded model. Our results thus show that including mRNA as a variable may change the behaviour of solutions.

Piecewise linear and Boolean models of chemical reaction networks

Models of biochemical networks are frequently complex and high-dimensional. Reduction methods that preserve important dynamical properties are therefore essential for their study. Interactions in biochemical networks are frequently modeled using Hill functions ([Formula: see text]). Reduced ODEs and Boolean approximations of such model networks have been studied extensively when the exponent [Formula: see text] is large. However, while the case of small constant [Formula: see text] appears in practice, it is not well understood. We provide a mathematical analysis of this limit and show that a reduction to a set of piecewise linear ODEs and Boolean networks can be mathematically justified. The piecewise linear systems have closed-form solutions that closely track those of the fully nonlinear model. The simpler, Boolean network can be used to study the qualitative behavior of the original system. We justify the reduction using geometric singular perturbation theory and compact convergence, and illustrate the results in network models of a toggle switch and an oscillator.

Links between topology of the transition graph and limit cycles in a two-dimensional piecewise affine biological model

A class of piecewise affine differential (PWA) models, initially proposed by Glass and Kauffman (in J Theor Biol 39:103-129, 1973), has been widely used for the modelling and the analysis of biological switch-like systems, such as genetic or neural networks. Its mathematical tractability facilitates the qualitative analysis of dynamical behaviors, in particular periodic phenomena which are of prime importance in biology. Notably, a discrete qualitative description of the dynamics, called the transition graph, can be directly associated to this class of PWA systems. Here we present a study of periodic behaviours (i.e. limit cycles) in a class of two-dimensional piecewise affine biological models. Using concavity and continuity properties of Poincaré maps, we derive structural principles linking the topology of the transition graph to the existence, number and stability of limit cycles. These results notably extend previous works on the investigation of structural principles to the case of unequal and regulated decay rates for the 2-dimensional case. Some numerical examples corresponding to minimal models of biological oscillators are treated to illustrate the use of these structural principles.

Hierarchy of models: from qualitative to quantitative analysis of circadian rhythms in cyanobacteria

A hierarchy of models, ranging from high to lower levels of abstraction, is proposed to construct "minimal" but predictive and explanatory models of biological systems. Three hierarchical levels will be considered: Boolean networks, piecewise affine differential (PWA) equations, and a class of continuous, ordinary, differential equations' models derived from the PWA model. This hierarchy provides different levels of approximation of the biological system and, crucially, allows the use of theoretical tools to more exactly analyze and understand the mechanisms of the system. The Kai ABC oscillator, which is at the core of the cyanobacterial circadian rhythm, is analyzed as a case study, showing how several fundamental properties-order of oscillations, synchronization when mixing oscillating samples, structural robustness, and entrainment by external cues-can be obtained from basic mechanisms.

Introduction to focus issue: quantitative approaches to genetic networks

All cells of living organisms contain similar genetic instructions encoded in the organism's DNA. In any particular cell, the control of the expression of each different gene is regulated, in part, by binding of molecular complexes to specific regions of the DNA. The molecular complexes are composed of protein molecules, called transcription factors, combined with various other molecules such as hormones and drugs. Since transcription factors are coded by genes, cellular function is partially determined by genetic networks. Recent research is making large strides to understand both the structure and the function of these networks. Further, the emerging discipline of synthetic biology is engineering novel gene circuits with specific dynamic properties to advance both basic science and potential practical applications. Although there is not yet a universally accepted mathematical framework for studying the properties of genetic networks, the strong analogies between the activation and inhibition of gene expression and electric circuits suggest frameworks based on logical switching circuits. This focus issue provides a selection of papers reflecting current research directions in the quantitative analysis of genetic networks. The work extends from molecular models for the binding of proteins, to realistic detailed models of cellular metabolism. Between these extremes are simplified models in which genetic dynamics are modeled using classical methods of systems engineering, Boolean switching networks, differential equations that are continuous analogues of Boolean switching networks, and differential equations in which control is based on power law functions. The mathematical techniques are applied to study: (i) naturally occurring gene networks in living organisms including: cyanobacteria, Mycoplasma genitalium, fruit flies, immune cells in mammals; (ii) synthetic gene circuits in Escherichia coli and yeast; and (iii) electronic circuits modeling genetic networks using field-programmable gate arrays. Mathematical analyses will be essential for understanding naturally occurring genetic networks in diverse organisms and for providing a foundation for the improved development of synthetic genetic networks.

Sensitive dependence on initial conditions in gene networks

Active regulation in gene networks poses mathematical challenges that have led to conflicting approaches to analysis. Competing regulation that keeps concentrations of some transcription factors at or near threshold values leads to so-called singular dynamics when steeply sigmoidal interactions are approximated by step functions. An extension, due to Artstein and coauthors, of the classical singular perturbation approach was suggested as an appropriate way to handle the complex situation where non-trivial dynamics, such as a limit cycle, of fast variables occur in switching domains. This non-trivial behaviour can occur when a gene regulates multiple other genes at the same threshold. Here, it is shown that it is possible for nonuniqueness to arise in such a system in the case of limiting step-function interactions. This nonuniqueness is reminiscent of but not identical to the nonuniqueness of Filippov solutions. More realistic gene network models have sigmoidal interactions, however, and in the example considered here, it is shown numerically that the corresponding phenomenon in smooth systems is a sensitivity to initial conditions that leads in the limit to densely interwoven basins of attraction of different fixed point attractors.

A gene regulatory motif that generates oscillatory or multiway switch outputs

The pattern of gene expression in a developing tissue determines the spatial organization of cell type generation. We previously defined regulatory interactions between a set of transcription factors that specify the pattern of gene expression in progenitors of different neuronal subtypes of the vertebrate neural tube. These transcription factors form a circuit that acts as a multistate switch, patterning the tissue in response to a gradient of Sonic Hedgehog. Here, by simplifying aspects of the regulatory interactions, we found that the topology of the circuit allows either switch-like or oscillatory behaviour depending on parameter values. The qualitative dynamics appear to be controlled by a simpler sub-circuit, which we term the AC–DC motif. We argue that its topology provides a natural way to implement a multistate gene expression switch and we show that the circuit is readily extendable to produce more distinct stripes of gene expression. Our analysis also suggests that AC–DC motifs could be deployed in tissues patterned by oscillatory mechanisms, thus blurring the distinction between pattern-formation mechanisms relying on temporal oscillations or graded signals. Furthermore, during evolution, mechanisms of gradient interpretation might have arisen from oscillatory circuits, or vice versa.

Probabilistic approach for predicting periodic orbits in piecewise affine differential models

Piecewise affine models provide a qualitative description of the dynamics of a system, and are often used to study genetic regulatory networks. The state space of a piecewise affine system is partitioned into hyperrectangles, which can be represented as nodes in a directed graph, so that the system's trajectories follow a path in a transition graph. This paper proposes and compares two definitions of probability of transition between two nodes A and B of the graph, based on the volume of the initial conditions on the hyperrectangle A whose trajectories cross to B. The parameters of the system can thus be compared to the observed transitions between two hyperrectangles. This property may become useful to identify sets of parameters for which the system yields a desired periodic orbit with a high probability, or to predict the most likely periodic orbit given a set of parameters, as illustrated by a gene regulatory system composed of two intertwined negative loops.

A self-organized model for cell-differentiation based on variations of molecular decay rates

Systemic properties of living cells are the result of molecular dynamics governed by so-called genetic regulatory networks (GRN). These networks capture all possible features of cells and are responsible for the immense levels of adaptation characteristic to living systems. At any point in time only small subsets of these networks are active. Any active subset of the GRN leads to the expression of particular sets of molecules (expression modes). The subsets of active networks change over time, leading to the observed complex dynamics of expression patterns. Understanding of these dynamics becomes increasingly important in systems biology and medicine. While the importance of transcription rates and catalytic interactions has been widely recognized in modeling genetic regulatory systems, the understanding of the role of degradation of biochemical agents (mRNA, protein) in regulatory dynamics remains limited. Recent experimental data suggests that there exists a functional relation between mRNA and protein decay rates and expression modes. In this paper we propose a model for the dynamics of successions of sequences of active subnetworks of the GRN. The model is able to reproduce key characteristics of molecular dynamics, including homeostasis, multi-stability, periodic dynamics, alternating activity, differentiability, and self-organized critical dynamics. Moreover the model allows to naturally understand the mechanism behind the relation between decay rates and expression modes. The model explains recent experimental observations that decay-rates (or turnovers) vary between differentiated tissue-classes at a general systemic level and highlights the role of intracellular decay rate control mechanisms in cell differentiation.

An ensemble approach for inferring semi-quantitative regulatory dynamics for the differentiation of mouse embryonic stem cells using prior knowledge

The process of differentiation of embryonic stem cells (ESCs) is currently becoming the focus of many systems biologists not only due to mechanistic interest but also since it is expected to play an increasingly important role in regenerative medicine, in particular with the advert to induced pluripotent stem cells. These ESCs give rise to the formation of the three germ layers and therefore to the formation of all tissues and organs. Here, we present a computational method for inferring regulatory interactions between the genes involved in ESC differentiation based on time resolved microarray profiles. Fully quantitative methods are commonly unavailable on such large-scale data; on the other hand, purely qualitative methods may fail to capture some of the more detailed regulations. Our method combines the beneficial aspects of qualitative and quantitative (ODE-based) modeling approaches searching for quantitative interaction coefficients in a discrete and qualitative state space. We further optimize on an ensemble of networks to detect essential properties and compare networks with respect to robustness. Applied to a toy model our method is able to reconstruct the original network and outperforms an entire discrete boolean approach. In particular, we show that including prior knowledge leads to more accurate results. Applied to data from differentiating mouse ESCs reveals new regulatory interactions, in particular we confirm the activation of Foxh1 through Oct4, mediating Nodal signaling.

Genetic network analyzer: a tool for the qualitative modeling and simulation of bacterial regulatory networks

Genetic Network Analyzer (GNA) is a tool for the qualitative modeling and simulation of gene regulatory networks, based on so-called piecewise-linear differential equation models. We describe the use of this tool in the context of the modeling of bacterial regulatory networks, notably the network of global regulators controlling the adaptation of Escherichia coli to carbon starvation conditions. We show how the modeler, by means of GNA, can define a regulatory network, build a model of the network, determine the steady states of the system, perform a qualitative simulation of the network dynamics, and analyze the simulation results using model-checking tools. The example illustrates the interest of qualitative approaches for the analysis of the dynamics of bacterial regulatory networks.

Multistability and oscillations in genetic control of metabolism

Genetic control of enzyme activity drives metabolic adaptations to environmental changes, and therefore the feedback interaction between gene expression and metabolism is essential to cell fitness. In this paper we develop a new formalism to detect the equilibrium regimes of an unbranched metabolic network under transcriptional feedback from one metabolite. Our results indicate that one-to-all transcriptional feedback can induce a wide range of metabolic phenotypes, including mono-, multistability and oscillatory behavior. The analysis is based on the use of switch-like models for transcriptional control and the exploitation of the time scale separation between metabolic and genetic dynamics. For any combination of activation and repression feedback loops, we derive conditions for the emergence of a specific phenotype in terms of genetic parameters such as enzyme expression rates and regulatory thresholds. We find that metabolic oscillations can emerge under uniform thresholds and, in the case of operon-controlled networks, the analysis reveals how nutrient-induced bistability and oscillations can emerge as a consequence of the transcriptional feedback.

A theoretical exploration of birhythmicity in the p53-Mdm2 network

Experimental observations performed in the p53-Mdm2 network, one of the key protein modules involved in the control of proliferation of abnormal cells in mammals, revealed the existence of two frequencies of oscillations of p53 and Mdm2 in irradiated cells depending on the irradiation dose. These observations raised the question of the existence of birhythmicity, i.e. the coexistence of two oscillatory regimes for the same external conditions, in the p53-Mdm2 network which would be at the origin of these two distinct frequencies. A theoretical answer has been recently suggested by Ouattara, Abou-Jaoudé and Kaufman who proposed a 3-dimensional differential model showing birhythmicity to reproduce the two frequencies experimentally observed. The aim of this work is to analyze the mechanisms at the origin of the birhythmic behavior through a theoretical analysis of this differential model. To do so, we reduced this model, in a first step, into a 3-dimensional piecewise linear differential model where the Hill functions have been approximated by step functions, and, in a second step, into a 2-dimensional piecewise linear differential model by setting one autonomous variable as a constant in each domain of the phase space. We find that two features related to the phase space structure of the system are at the origin of the birhythmic behavior: the existence of two embedded cycles in the transition graph of the reduced models; the presence of a bypass in the orbit of the large amplitude oscillatory regime of low frequency. Based on this analysis, an experimental strategy is proposed to test the existence of birhythmicity in the p53-Mdm2 network. From a methodological point of view, this approach greatly facilitates the computational analysis of complex oscillatory behavior and could represent a valuable tool to explore mathematical models of biological rhythms showing sufficiently steep nonlinearities.

On the Gause predator-prey model with a refuge: a fresh look at the history

This article re-analyses a prey-predator model with a refuge introduced by one of the founders of population ecology Gause and his co-workers to explain discrepancies between their observations and predictions of the Lotka-Volterra prey-predator model. They replaced the linear functional response used by Lotka and Volterra by a saturating functional response with a discontinuity at a critical prey density. At concentrations below this critical density prey were effectively in a refuge while at a higher densities they were available to predators. Thus, their functional response was of the Holling type III. They analyzed this model and predicted existence of a limit cycle in predator-prey dynamics. In this article I show that their model is ill posed, because trajectories are not well defined. Using the Filippov method, I define and analyze solutions of the Gause model. I show that depending on parameter values, there are three possibilities: (1) trajectories converge to a limit cycle, as predicted by Gause, (2) trajectories converge to an equilibrium, or (3) the prey population escapes predator control and grows to infinity.

Model reduction using piecewise-linear approximations preserves dynamic properties of the carbon starvation response in Escherichia coli

The adaptation of the bacterium Escherichia coli to carbon starvation is controlled by a large network of biochemical reactions involving genes, mRNAs, proteins, and signalling molecules. The dynamics of these networks is difficult to analyze, notably due to a lack of quantitative information on parameter values. To overcome these limitations, model reduction approaches based on quasi-steady-state (QSS) and piecewise-linear (PL) approximations have been proposed, resulting in models that are easier to handle mathematically and computationally. These approximations are not supposed to affect the capability of the model to account for essential dynamical properties of the system, but the validity of this assumption has not been systematically tested. In this paper, we carry out such a study by evaluating a large and complex PL model of the carbon starvation response in E. coli using an ensemble approach. The results show that, in comparison with conventional nonlinear models, the PL approximations generally preserve the dynamics of the carbon starvation response network, although with some deviations concerning notably the quantitative precision of the model predictions. This encourages the application of PL models to the qualitative analysis of bacterial regulatory networks, in situations where the reference time scale is that of protein synthesis and degradation.

Design of regulation and dynamics in simple biochemical pathways

Complex regulation of biochemical pathways in a cell is brought about by the interaction of simpler regulatory structures. Among the basic regulatory designs, feedback inhibition of gene expression is the most common motif in gene regulation and a ubiquitous control structure found in nature. In this work, we have studied a common structural feature (delayed feedback) in gene organisation and shown, both theoretically and experimentally, its subtle but important functional role in gene expression kinetics in a negatively auto-regulated system. Using simple deterministic and stochastic models with varying levels of realism, we present detailed theoretical representations of negatively auto-regulated transcriptional circuits with increasing delays in the establishment of feedback of repression. The models of the circuits with and without delay are studied analytically as well as numerically for variation of parameters and delay lengths. The positive invariance, boundedness of the solutions, local and global asymptotic stability of both the systems around the unique positive steady state are studied analytically. Existence of transient temporal dynamics is shown mathematically. Comparison of the two types of model circuits shows that even though the long-term dynamics is stable and not affected by delays in repression, there is interesting variation in the transient dynamical features with increasing delays. Theoretical predictions are validated through experimentally constructed gene circuits of similar designs. This combined theoretical and experimental study helps delineate the opposing effects of delay-induced instability, and the stability-enhancing property of negative feedback in the pathway behaviour, and gives rationale for the abundance of similar designs in real biochemical pathways.

Linear control theory for gene network modeling

Systems biology is an interdisciplinary field that aims at understanding complex interactions in cells. Here we demonstrate that linear control theory can provide valuable insight and practical tools for the characterization of complex biological networks. We provide the foundation for such analyses through the study of several case studies including cascade and parallel forms, feedback and feedforward loops. We reproduce experimental results and provide rational analysis of the observed behavior. We demonstrate that methods such as the transfer function (frequency domain) and linear state-space (time domain) can be used to predict reliably the properties and transient behavior of complex network topologies and point to specific design strategies for synthetic networks.

Comparing Boolean and piecewise affine differential models for genetic networks

Multi-level discrete models of genetic networks, or the more general piecewise affine differential models, provide qualitative information on the dynamics of the system, based on a small number of parameters (such as synthesis and degradation rates). Boolean models also provide qualitative information, but are based simply on the structure of interconnections. To explore the relationship between the two formalisms, a piecewise affine differential model and a Boolean model are compared, for the carbon starvation response network in E. coli. The asymptotic dynamics of both models are shown to be quite similar. This study suggests new tools for analysis and reduction of biological networks.

Catastrophic sliding bifurcations and onset of oscillations in a superconducting resonator

This paper presents a general analysis and a concrete example of the catastrophic case of a discontinuity-induced bifurcation in so-called Filippov nonsmooth dynamical systems. Such systems are characterized by discontinuous jumps in the right-hand sides of differential equations across a phase space boundary and are often used as physical models of stick-slip motion and relay control. Sliding bifurcations of periodic orbits have recently been shown to underlie the onset of complex dynamics including chaos. In contrast to previously analyzed cases, in this work a periodic orbit is assumed to graze the boundary of a repelling sliding region, resulting in its abrupt destruction without any precursive change in its stability or period. Necessary conditions for the occurrence of such catastrophic grazing-sliding bifurcations are derived. The analysis is illustrated in a piecewise-smooth model of a stripline resonator, where it can account for the abrupt onset of self-modulating current fluctuations. The resonator device is based around a ring of NbN containing a microbridge bottleneck, whose switching between normal and super conducting states can be modeled as discontinuous, and whose fast temperature versus slow current fluctuations are modeled by a slow-fast timescale separation in the dynamics. By approximating the slow component as Filippov sliding, explicit conditions are derived for catastrophic grazing-sliding bifurcations, which can be traced out as parameters vary. The results are shown to agree well with simulations of the slow-fast model and to offer a simple explanation of one of the key features of this experimental device.

Periodic solutions of piecewise affine gene network models with non uniform decay rates: the case of a negative feedback loop

This paper concerns periodic solutions of a class of equations that model gene regulatory networks. Unlike the vast majority of previous studies, it is not assumed that all decay rates are identical. To handle this more general situation, we rely on monotonicity properties of these systems. Under an alternative assumption, it is shown that a classical fixed point theorem for monotone, concave operators can be applied to these systems. The required assumption is expressed in geometrical terms as an alignment condition on so-called focal points. As an application, we show the existence and uniqueness of a stable periodic orbit for negative feedback loop systems in dimension 3 or more, and of a unique stable equilibrium point in dimension 2. This extends a theorem of Snoussi, which showed the existence of these orbits only.

Pattern formation by dynamically interacting network motifs

Systematic validation of pattern formation mechanisms revealed by molecular studies of development is essentially impossible without mathematical models. Models can provide a compact summary of a large number of experiments that led to mechanism formulation and guide future studies of pattern formation. Here, we realize this program by analyzing a mathematical model of epithelial patterning by the highly conserved EGFR and BMP signaling pathways in Drosophila oogenesis. The model accounts for the dynamic interaction of the feedforward and feedback network motifs that control the expression of Broad, a zinc finger transcription factor expressed in the cells that form the upper part of the respiratory eggshell appendages. Based on the combination of computational analysis and genetic experiments, we show that the model accounts for the key features of wild-type pattern formation, correctly predicts patterning defects in multiple mutants, and guides the identification of additional regulatory links in a complex pattern formation mechanism.

Studying the effect of cell division on expression patterns of the segment polarity genes

The segment polarity gene family, and its gene regulatory network, is at the basis of Drosophila embryonic development. The network's capacity for generating and robustly maintaining a specific gene expression pattern has been investigated through mathematical modelling. The models have provided several useful insights by suggesting essential network links, or uncovering the importance of the relative time scales of different biological processes in the formation of the segment polarity genes' expression patterns. But the developmental pattern formation process raises many other questions. Two of these questions are analysed here: the dependence of the signalling protein sloppy paired on the segment polarity genes and the effect of cell division on the segment polarity genes' expression patterns. This study suggests that cell division increases the robustness of the segment polarity network with respect to perturbations in biological processes.

Search for steady states of piecewise-linear differential equation models of genetic regulatory networks

Analysis of the attractors of a genetic regulatory network gives a good indication of the possible functional modes of the system. In this paper we are concerned with the problem of finding all steady states of genetic regulatory networks described by piecewise-linear differential equation (PLDE) models. We show that the problem is NP-hard and translate it into a propositional satisfiability (SAT) problem. This allows the use of existing, efficient SAT solvers and has enabled the development of a steady state search module of the computer tool Genetic Network Analyzer (GNA). The practical use of this module is demonstrated by means of the analysis of a number of relatively small bacterial regulatory networks as well as randomly generated networks of several hundreds of genes.

Periodicity in piecewise-linear switching networks with delay

Gene regulatory networks and neural networks can be modeled by piecewise-linear switching systems of differential equations, known as Glass networks. These biological networks exhibit delays in regulatory activity, for example, transcription, translation and spatial transport in gene networks, and transmission delays in neural networks. Such delays may be significant in determining their dynamical behavior. Here Glass networks with a discrete delay are introduced and analyzed. Fixed points away from thresholds are straightforward to identify, even in the presence of delays, so the focus of this work is on cyclic patterns of switching. Under a condition that ensures an unambiguous pattern of switching, it is shown by means of a fractional linear mapping that delayed Glass networks have a periodic orbit for all positive finite delays. Furthermore, an algorithm is presented to locate the periodic orbit for a given cycle, to determine whether the periodic orbit is locally asymptotically stable, and to check if it is unique. In addition, the complete dynamics of the two-dimensional delayed Glass network is provided: if there is a cycle of four orthants, then there exists a unique globally stable limit cycle; whereas if there is a black wall, then across the wall there exists a unique limit cycle that is globally stable with respect to the associated orthants. This behavior is in contrast to the non-delayed case, in which spiralling approach to fixed points on threshold boundaries can occur.