Stochastic Bistability Research Articles

Stochastic Bistable Systems: Competing Hysteresis and Phase Coexistence

In this paper we describe the solution of a stochastic bistable system from a dynamical perspective. We show how a single framework with variable noise can explain hysteresis at zero temperature and two-state coexistence in the presence of noise. This feature is similar to the phase transition of thermodynamics. Our mathematical model for bistable systems also explains how the width of a hysteresis loop shrinks in the presence of noise, and how variation in initial conditions can take such systems to different final states.

Relatively slow stochastic gene-state switching in the presence of positive feedback significantly broadens the region of bimodality through stabilizing the uninduced phenotypic state.

Within an isogenic population, even in the same extracellular environment, individual cells can exhibit various phenotypic states. The exact role of stochastic gene-state switching regulating the transition among these phenotypic states in a single cell is not fully understood, especially in the presence of positive feedback. Recent high-precision single-cell measurements showed that, at least in bacteria, switching in gene states is slow relative to the typical rates of active transcription and translation. Hence using the lac operon as an archetype, in such a region of operon-state switching, we present a fluctuating-rate model for this classical gene regulation module, incorporating the more realistic operon-state switching mechanism that was recently elucidated. We found that the positive feedback mechanism induces bistability (referred to as deterministic bistability), and that the parameter range for its occurrence is significantly broadened by stochastic operon-state switching. We further show that in the absence of positive feedback, operon-state switching must be extremely slow to trigger bistability by itself. However, in the presence of positive feedback, which stabilizes the induced state, the relatively slow operon-state switching kinetics within the physiological region are sufficient to stabilize the uninduced state, together generating a broadened parameter region of bistability (referred to as stochastic bistability). We illustrate the opposite phenotype-transition rate dependence upon the operon-state switching rates in the two types of bistability, with the aid of a recently proposed rate formula for fluctuating-rate models. The rate formula also predicts a maximal transition rate in the intermediate region of operon-state switching, which is validated by numerical simulations in our model. Overall, our findings suggest a biological function of transcriptional “variations” among genetically identical cells, for the emergence of bistability and transition between phenotypic states.

On the Stochastic Sensitivity and Noise-Induced Transitions of a Kaldor-Type Business Cycle Model

In the paper, we consider a Kaldor-type model of the business cycle with external additive and internal parametric disturbances. We study analytically and numerically the probability properties of stochastically forced equilibria and limit cycles via stochastic sensitivity function technique. In particular, we discuss the effects of additive and parametric noises on the economic variables and we detect some stochastic bifurcations such as a P-bifurcation, i.e a phenomenon of noise-induced transition from monostability to bistability. This stochastic bistability causes a new trigger regime in economic dynamics.

Complete integrability of information processing by biochemical reactions.

Statistical mechanics provides an effective framework to investigate information processing in biochemical reactions. Within such framework far-reaching analogies are established among (anti-) cooperative collective behaviors in chemical kinetics, (anti-)ferromagnetic spin models in statistical mechanics and operational amplifiers/flip-flops in cybernetics. The underlying modeling – based on spin systems – has been proved to be accurate for a wide class of systems matching classical (e.g. Michaelis–Menten, Hill, Adair) scenarios in the infinite-size approximation. However, the current research in biochemical information processing has been focusing on systems involving a relatively small number of units, where this approximation is no longer valid. Here we show that the whole statistical mechanical description of reaction kinetics can be re-formulated via a mechanical analogy – based on completely integrable hydrodynamic-type systems of PDEs – which provides explicit finite-size solutions, matching recently investigated phenomena (e.g. noise-induced cooperativity, stochastic bi-stability, quorum sensing). The resulting picture, successfully tested against a broad spectrum of data, constitutes a neat rationale for a numerically effective and theoretically consistent description of collective behaviors in biochemical reactions.

Stochastic Bistability and Switching in Virus/Immune Cell Networks

In recent years, theoretical modeling of biochemical reaction networks has shed light on the connection between network features (i.e., structure and dynamics) and biological function. At the same time, the pervasiveness of stochastic fluctuations in cellular processes has been amply demonstrated. These fluctuations can fundamentally influence dynamics--for example, we found that network models of virus evolution coupled to cellular immune responses have stochastic steady-state distributions that are qualitatively different from those predicted by deterministic equations. I will discuss how the features of the network that give rise to this disparity are related to those underlying stochastic bistabilities encountered in gene regulation and cell signaling. I will also discuss recent advances in numerical methods for inferring critical dynamical mechanisms in complex networks, which have the potential to inform strategies for perturbing cell dynamics.

Biological Noise Abatement: Coordinating the Responses of Autonomous Bacteria in a Synthetic Biofilm to a Fluctuating Environment Using a Stochastic Bistable Switch

Noise is inherent to single cell behavior. Its origins can be traced to the stochasticity associated with a few copies of genes and low concentrations of protein and ligands. We have studied the mechanisms by which the response of noisy elements can be entrained for biological signal processing. To elicit predictable biological function, we have engineered a gene environment that incorporates a gene regulatory network with the stringently controlled microenvironment found in a synthetic biofilm. The regulatory network leverages the positive feedback found in quorum-sensing regulatory components of the lux operon, which is used to coordinate cellular responses to environmental fluctuations. Accumulation of the Lux receptor in cells, resulting from autoregulation, confers a rapid response and enhanced sensitivity to the quorum-sensing molecule that is retained after cell division as epigenetic memory. The memory of the system channels stochastic noise into a coordinated response among quorum-sensing signal receivers in a synthetic biofilm in which the noise diminishes with repeated exposure to noisy transmitters on the input of a signaling cascade integrated into the same biofilm. Thus, gene expression in the receivers, which are autonomous and do not communicate with each other, is synchronized to fluctuations in the environment.

Modeling Stochastic Dynamics in Biochemical Systems with Feedback Using Maximum Caliber

Complex feedback systems are ubiquitous in biology. Modeling such systems with mass action laws or master equations requires information rarely measured directly. Thus rates and reaction topologies are often treated as adjustable parameters. Here we present a general stochastic modeling method for small chemical and biochemical systems with emphasis on feedback systems. The method, Maximum Caliber (MaxCal), is more parsimonious than others in constructing dynamical models requiring fewer model assumptions and parameters to capture the effects of feedback. MaxCal is the dynamical analogue of Maximum Entropy. It uses average rate quantities and correlations obtained from short experimental trajectories to construct dynamical models. We illustrate the method on the bistable genetic toggle switch. To test our method, we generate synthetic data from an underlying stochastic model. MaxCal reliably infers the statistics of the stochastic bistability and other full dynamical distributions of the simulated data, without having to invoke complex reaction schemes. The method should be broadly applicable to other systems.

Cellular Biology in Terms of Stochastic Nonlinear Biochemical Dynamics: Emergent Properties, Isogenetic Variations and Chemical System Inheritability

Based on a stochastic, nonlinear, open biochemical reaction system perspective, we present an analytical theory for cellular biochemical processes. The chemical master equation (CME) approach provides a unifying mathematical framework for cellular modeling. We apply this theory to both self-regulating gene networks and phosphorylation-dephosphorylation signaling modules with feedbacks. Two types of bistability are illustrated in mesoscopic biochemical systems: one that has a macroscopic, deterministic counterpart and another that does not. In certain cases, the latter stochastic bistability is shown to be a “ghost” of the extinction phenomenon. We argue the thermal fluctuations inherent in molecular processes do not disappear in mesoscopic cell-sized nonlinear systems; rather they manifest themselves as isogenetic variations on a different time scale. Isogenetic biochemical variations in terms of the stochastic attractors can have extremely long lifetime. Transitions among discrete stochastic attractors spend most of the time in “waiting”, exhibit punctuated equilibria. It can be naturally passed to “daughter cells” via a simple growth and division process. The CME system follows a set of nonequilibrium thermodynamic laws that include non-increasing free energy F(t) with external energy drive Q hk ≥0, and total entropy production rate e p =−dF/dt+Q hk ≥0. In the thermodynamic limit, with a system’s size being infinitely large, the nonlinear bistability in the CME exhibits many of the characteristics of macroscopic equilibrium phase transition.

The Chemical Master Equation Approach to Nonequilibrium Steady-State of Open Biochemical Systems: Linear Single-Molecule Enzyme Kinetics and Nonlinear Biochemical Reaction Networks

We develop the stochastic, chemical master equation as a unifying approach to the dynamics of biochemical reaction systems in a mesoscopic volume under a living environment. A living environment provides a continuous chemical energy input that sustains the reaction system in a nonequilibrium steady state with concentration fluctuations. We discuss the linear, unimolecular single-molecule enzyme kinetics, phosphorylation-dephosphorylation cycle (PdPC) with bistability, and network exhibiting oscillations. Emphasis is paid to the comparison between the stochastic dynamics and the prediction based on the traditional approach based on the Law of Mass Action. We introduce the difference between nonlinear bistability and stochastic bistability, the latter has no deterministic counterpart. For systems with nonlinear bistability, there are three different time scales: (a) individual biochemical reactions, (b) nonlinear network dynamics approaching to attractors, and (c) cellular evolution. For mesoscopic systems with size of a living cell, dynamics in (a) and (c) are stochastic while that with (b) is dominantly deterministic. Both (b) and (c) are emergent properties of a dynamic biochemical network; We suggest that the (c) is most relevant to major cellular biochemical processes such as epi-genetic regulation, apoptosis, and cancer immunoediting. The cellular evolution proceeds with transitions among the attractors of (b) in a “punctuated equilibrium” manner.

Stochastic Bistability and Bifurcation in a Mesoscopic Signaling System with Autocatalytic Kinase

Bistability is a nonlinear phenomenon widely observed in nature including in biochemical reaction networks. Deterministic chemical kinetics studied in the past has shown that bistability occurs in systems with strong (cubic) nonlinearity. For certain mesoscopic, weakly nonlinear (quadratic) biochemical reaction systems in a small volume, however, stochasticity can induce bistability and bifurcation that have no macroscopic counterpart. We report the simplest yet known reactions involving driven phosphorylation-dephosphorylation cycle kinetics with autocatalytic kinase. We show that the noise-induced phenomenon is correlated with free energy dissipation and thus conforms with the open-chemical system theory. A previous reported noise-induced bistability in futile cycles is found to have originated from the kinase synchronization in a bistable system with slow transitions, as reported here.

Stochastic Modeling of Gene Positive Autoregulation Networks Involving Signal Molecules

Positive autoregulation in gene regulation networks has been shown in the past to exhibit stochastic behavior, including stochastic bistability, in which an initially uniform cell population develops into two distinct subpopulations. However, positive autoregulation is often mediated by signal molecules, which have not been considered in prior stochastic analysis of these networks. Here we propose both a full model of such a network that includes a signal molecule, and a simplified model in which the signal molecules have been eliminated through the use of two simplifications. The simplified model is amenable to direct mathematical analysis that shows that stochastic bistability is possible. We use stochastic Petri networks for simulating both types of models. The simulation results show that 1), the stochastic behavior of the two models is similar; and 2), that the analytical steady-state distribution of the simplified model matches well the transient results at times equal to that of a cell generation. A discussion of the simplifications we used in the context of the results indicates the importance of the signal molecule number as a factor determining the presence of bistability. This is further supported from a deterministic steady-state analysis of the full model that is shown to be a useful indicator of potential stochastic bistability. We use the regulation of SdiA in Escherichia coli as an example, due to the importance of this protein and of the signal molecule, a bacterial autoinducer, that is involved. However, the use of kinetic parameter values representing typical cellular activities make the conclusions applicable to other signal-mediated positive autoregulation networks as well.

Stem cell proliferation and differentiation and stochastic bistability in gene expression

The process of proliferation and differentiation of stem cells is inherently stochastic in the sense that the outcome of cell division is characterized by probabilities that depend on the intracellular properties, extracellular medium, and cell-cell communication. Despite four decades of intensive studies, the understanding of the physics behind this stochasticity is still limited, both in details and conceptually. Here, we suggest a simple scheme showing that the stochastic behavior of a single stem cell may be related to (i) the existence of a short stage of decision whether it will proliferate or differentiate and (ii) control of this stage by stochastic bistability in gene expression or, more specifically, by transcriptional “bursts.” Our Monte Carlo simulations indicate that our proposed scheme may operate if the number of mRNA (or protein) molecules generated during the high-reactive periods of gene expression is below or about 50. The stochastic-burst window in the space of kinetic parameters is found to increase with decreasing the mRNA and/or regulatory-protein numbers and increasing the number of regulatory sites. For mRNA production with three regulatory sites, for example, the mRNA degradation rate constant may change in the range ±10%.

A Model for Perceptual Averaging and Stochastic Bistable Behavior and the Role of Voluntary Control

We combine population coding, winner-take-all competition, and differentiated inhibitory feedback to model the process by which information from different, continuously variable signals is integrated for perceptual awareness. We focus on "slant rivalry," where binocular disparity is in conflict with monocular perspective in specifying surface slant. Using a robust single parameter set, our model successfully replicates three key experimental results: (1) transition from signal averaging to bistability with increasing signal conflict, (2) change in perceptual reversal rates as a function of signal conflict, and (3) a shift in the distribution of percept durations through voluntary control exertion. Voluntary control is implemented through the use of a single top-down bias input. The transition from signal averaging to bistability arises as a natural consequence of combining population coding and wide receptive fields, common to higher cortical areas. The model architecture does not contain any assumption that would limit it to this particular example of stimulus rivalry. An emergent physiological interpretation is that differentiated inhibitory feedback may play an important role for increasing percept stability without reducing sensitivity to large stimulus changes, which for bistable conditions leads to increased alternation rate as a function of signal conflict.

Stochasticity in Transcriptional Regulation: Origins, Consequences, and Mathematical Representations

Transcriptional regulation is an inherently noisy process. The origins of this stochastic behavior can be traced to the random transitions among the discrete chemical states of operators that control the transcription rate and to finite number fluctuations in the biochemical reactions for the synthesis and degradation of transcripts. We develop stochastic models to which these random reactions are intrinsic and a series of simpler models derived explicitly from the first as approximations in different parameter regimes. This innate stochasticity can have both a quantitative and qualitative impact on the behavior of gene-regulatory networks. We introduce a natural generalization of deterministic bifurcations for classification of stochastic systems and show that simple noisy genetic switches have rich bifurcation structures; among them, bifurcations driven solely by changing the rate of operator fluctuations even as the underlying deterministic system remains unchanged. We find stochastic bistability where the deterministic equations predict monostability and vice-versa. We derive and solve equations for the mean waiting times for spontaneous transitions between quasistable states in these switches.

Mean Switching Frequency Locking in Stochastic Bistable Systems Driven by a Periodic Force.

The nonlinear response of noisy bistable systems driven by a strong amplitude-periodic force is investigated by physical experiment. The new phenomenon of locking of the mean switching frequency between states of a bistable system is found. It is shown that there is an interval of noise intensities in which the mean switching frequency remains constant and coincides with the frequency of the external periodic force. The region on the parameter plane "noise intensity---amplitude of periodic excitation" which corresponds to this phenomenon is similar to the synchronization (phase locking) region (Arnold's tongue) in classical oscillatory systems.

Path integral solutions for non-Markovian processes

For a nonlinear stochastic flow driven by Markovian or non-Markovian colored noise ζ(t) we present the path integral solution for the single-event probabilityp(x,t). The solution has the structure of a complex-valued double path integral. Explicit formulas for the action functional, i.e., the non-Markovian Onsager-Machlup functional, are derived for the case that ζ(t) is characterized by a stationary Gaussian process. Moreover, we derive explicit results for (generalized) Poissonian colored shot noise ζ(t). The use of the path integral solution is elucidated by a weak noise analysis of the WKB-type. As a simple application, we consider stochastic bistability driven by colored noise with an extremely long correlation time.