Transcription-based circadian mechanism controls the duration of molecular clock states in response to signaling inputs.

The molecular oscillator of the mammalian circadian clock consists in a dynamical network of genes and proteins whose main regulatory mechanisms occur at the transcriptional level. From a dynamical point of view, the mechanisms leading to an oscillatory solution with an orderly protein peak expression and a clear day/night phase distinction remain unclear. Our goal is to identify the essential interactions needed to generate phase opposition between the activating CLOCK:BMAL1 and the repressing PER:CRY complexes and to better distinguish these two main clock molecular phases relating to rest/activity and fast/feeding cycles. To do this, we develop a transcription-based mathematical model centered on linear combinations of the clock controlled elements (CCEs): E-box, R-box and D-box. Each CCE is responsive to activators and repressors. After model calibration with single-cell data, we explore entrainment and period tuning via interplay with metabolism. Variation of the PER degradation rate γp, relating to the tau mutation, results in asymmetric changes in the duration of the different clock molecular phases. Time spent at the state of high PER/PER:CRY decreases with γp, while time spent at the state of high BMAL1 and CRY1, both proteins with activity in promoting insulin sensitivity, remains constant. This result suggests a possible mechanism behind the altered metabolism of tau mutation animals. Furthermore, we expose the clock system to two regulatory inputs, one relating to the fast/feeding cycle and the other to the light-dependent synchronization signaling. We observe the phase difference between these signals to also affect the relative duration of molecular clock states. Simulated circadian misalignment, known to correlate with insulin resistance, leads to decreased duration of BMAL1 expression. Our results reveal a possible mechanism for clock-controlled metabolic homeostasis, whereby the circadian clock controls the relative duration of different molecular (and metabolic) states in response to signaling inputs.

Analysis of a genetic-metabolic oscillator with piecewise linear models.

Interactions between gene regulatory networks and metabolism produce a diversity of dynamics, including multistability and oscillations. Here, we characterize a regulatory mechanism that drives the emergence of periodic oscillations in metabolic networks subject to genetic feedback regulation by pathway intermediates. We employ a qualitative formalism based on piecewise linear models to systematically analyze the behavior of gene-regulated metabolic pathways. For a pathway with two metabolites and three enzymes, we prove the existence of two co-existing oscillatory behaviors: damped oscillations towards a fixed point or sustained oscillations along a periodic orbit. We show that this mechanism closely resembles the "metabolator", a genetic-metabolic circuit engineered to produce autonomous oscillations in vivo.

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.

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.

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.

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.

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.

Two-level modeling approach to identify the regulatory dynamics capturing drug response heterogeneity in single-cells.

Single-cell multimodal technologies reveal the scales of cellular heterogeneity impairing cancer treatment, yet cell response dynamics remain largely underused to decipher the mechanisms of drug resistance they take part in. As the phenotypic heterogeneity of a clonal cell population informs on the capacity of each single-cell to recapitulate the whole range of observed behaviors, we developed a modeling approach utilizing single-cell response data to identify regulatory reactions driving population heterogeneity in drug response. Dynamic data of hundreds of HeLa cells treated with TNF-related apoptosis-inducing ligand (TRAIL) were used to characterize the fate-determining kinetic parameters of an apoptosis receptor reaction model. Selected reactions sets were augmented to incorporate a mechanism that leads to the separation of the opposing response phenotypes. Using a positive feedback loop motif to identify the reaction set, we show that caspase-8 is able to encapsulate high levels of heterogeneity by introducing a response delay and amplifying the initial differences arising from natural protein expression variability. Our approach enables the identification of fate-determining reactions that drive the population response heterogeneity, providing regulatory targets to curb the cell dynamics of drug resistance.

Shape, size, and robustness: feasible regions in the parameter space of biochemical networks.

The concept of robustness of regulatory networks has received much attention in the last decade. One measure of robustness has been associated with the volume of the feasible region, namely, the region in the parameter space in which the system is functional. In this paper, we show that, in addition to volume, the geometry of this region has important consequences for the robustness and the fragility of a network. We develop an approximation within which we could algebraically specify the feasible region. We analyze the segment polarity gene network to illustrate our approach. The study of random walks in the parameter space and how they exit the feasible region provide us with a rich perspective on the different modes of failure of this network model. In particular, we found that, between two alternative ways of activating Wingless, one is more robust than the other. Our method provides a more complete measure of robustness to parameter variation. As a general modeling strategy, our approach is an interesting alternative to Boolean representation of biochemical networks.

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.

Geometry and topology of parameter space: investigating measures of robustness in regulatory networks.

The concept of robustness of regulatory networks has been closely related to the nature of the interactions among genes, and the capability of pattern maintenance or reproducibility. Defining this robustness property is a challenging task, but mathematical models have often associated it to the volume of the space of admissible parameters. Not only the volume of the space but also its topology and geometry contain information on essential aspects of the network, including feasible pathways, switching between two parallel pathways or distinct/disconnected active regions of parameters. A method is presented here to characterize the space of admissible parameters, by writing it as a semi-algebraic set, and then theoretically analyzing its topology and geometry, as well as volume. This method provides a more objective and complete measure of the robustness of a developmental module. As a detailed case study, the segment polarity gene network is analyzed.

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.

Analysis Tools for Interconnected Boolean Networks With Biological Applications.

Boolean networks with asynchronous updates are a class of logical models particularly well adapted to describe the dynamics of biological networks with uncertain measures. The state space of these models can be described by an asynchronous state transition graph, which represents all the possible exits from every single state, and gives a global image of all the possible trajectories of the system. In addition, the asynchronous state transition graph can be associated with an absorbing Markov chain, further providing a semi-quantitative framework where it becomes possible to compute probabilities for the different trajectories. For large networks, however, such direct analyses become computationally untractable, given the exponential dimension of the graph. Exploiting the general modularity of biological systems, we have introduced the novel concept of asymptotic graph, computed as an interconnection of several asynchronous transition graphs and recovering all asymptotic behaviors of a large interconnected system from the behavior of its smaller modules. From a modeling point of view, the interconnection of networks is very useful to address for instance the interplay between known biological modules and to test different hypotheses on the nature of their mutual regulatory links. This paper develops two new features of this general methodology: a quantitative dimension is added to the asymptotic graph, through the computation of relative probabilities for each final attractor and a companion cross-graph is introduced to complement the method on a theoretical point of view.

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.

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.

Robustness and fragility of Boolean models for genetic regulatory 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. We propose a new modeling 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 time-scales between pre- and post-translational processes and a minimal prepattern ensure convergence to the wild-type expression pattern regardless of fluctuations.

Steady-states of receptor-ligand dynamics: a theoretical framework.

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, which is based upon the rich and deep formalism of irreducible biochemical networks. When represented in this manner, the mass action kinetics of biochemical processes can be clearly seen in terms of their component biochemical interactions, their kinetic rate constants, and the stoichiometry for the system. A minimal parametrization is provided for models for two- or multi-state receptor interaction with ligand, and an "affinity quotient" is introduced, which allows an elegant classification of ligands into agonists, neutral agonists, and inverse agonists.