Stochastic Biochemical Networks Research Articles

Dynamic information transfer in stochastic biochemical networks

We develop numerical and analytical approaches to calculate mutual information between complete paths of two molecular components embedded into a larger reaction network. In particular, we focus on a continuous-time Markov chain formalism, frequently used to describe intracellular processes involving lowly abundant molecular species. Previously, we have shown how the path mutual information can be calculated for such systems when two molecular components interact directly with one another with no intermediate molecular components being present. In this paper, we generalize this approach to biochemical networks involving an arbitrary number of molecular components. We present an efficient Monte Carlo method as well as an analytical approximation to calculate the path mutual information and show how it can be decomposed into a pair of transfer entropies that capture the directed flow of information between two network components. We apply our methodology to study information transfer in a simple three-node feedforward network, as well as a more complex positive-feedback system that switches stochastically between two metastable modes.

Cellular Signaling beyond the Wiener-Kolmogorov Limit.

Accurate propagation of signals through stochastic biochemical networks involves significant expenditure of cellular resources. The same is true for regulatory mechanisms that suppress fluctuations in biomolecular populations. Wiener-Kolmogorov (WK) optimal noise filter theory, originally developed for engineering problems, has recently emerged as a valuable tool to estimate the maximum performance achievable in such biological systems for a given metabolic cost. However, WK theory has one assumption that potentially limits its applicability: it relies on a linear, continuum description of the reaction dynamics. Despite this, up to now no explicit test of the theory in nonlinear signaling systems with discrete molecular populations has ever seen performance beyond the WK bound. Here we report the first direct evidence of the bound being broken. To accomplish this, we develop a theoretical framework for multilevel signaling cascades, including the possibility of feedback interactions between input and output. In the absence of feedback, we introduce an analytical approach that allows us to calculate exact moments of the stationary distribution for a nonlinear system. With feedback, we rely on numerical solutions of the system's master equation. The results show WK violations in two common network motifs: a two-level signaling cascade and a negative feedback loop. However, the magnitude of the violation is biologically negligible, particularly in the parameter regime where signaling is most effective. The results demonstrate that while WK theory does not provide strict bounds, its predictions for performance limits are excellent approximations, even for nonlinear systems.

Structured Projection-Based Model Reduction With Application to Stochastic Biochemical Networks

The chemical master equation (CME) is well known to provide the highest resolution models of a biochemical reaction network. Unfortunately, even simulating the CME can be a challenging task. For this reason, simpler approximations to the CME have been proposed. In this paper, we focus on one such model, the linear noise approximation (LNA). Specifically, we consider implications of a recently proposed LNA time-scale separation method. We show that the reduced-order LNA converges to the full-order model in the mean square sense. Using this as motivation, we derive a network structure-preserving reduction algorithm based on structured projections. We discuss when these structured projections exist and we present convex optimization algorithms that describe how such projections can be computed. The algorithms are then applied to a linearized stochastic LNA model of the yeast glycolysis pathway.

Transient absolute robustness in stochastic biochemical networks.

Absolute robustness allows biochemical networks to sustain a consistent steady-state output in the face of protein concentration variability from cell to cell. This property is structural and can be determined from the topology of the network alone regardless of rate parameters. An important question regarding these systems is the effect of discrete biochemical noise in the dynamical behaviour. In this paper, a variable freezing technique is developed to show that under mild hypotheses the corresponding stochastic system has a transiently robust behaviour. Specifically, after finite time the distribution of the output approximates a Poisson distribution, centred around the deterministic mean. The approximation becomes increasingly accurate, and it holds for increasingly long finite times, as the total protein concentrations grow to infinity. In particular, the stochastic system retains a transient, absolutely robust behaviour corresponding to the deterministic case. This result contrasts with the long-term dynamics of the stochastic system, which eventually must undergo an extinction event that eliminates robustness and is completely different from the deterministic dynamics. The transiently robust behaviour may be sufficient to carry out many forms of robust signal transduction and cellular decision-making in cellular organisms.

Precise parameter synthesis for stochastic biochemical systems

We consider the problem of synthesising rate parameters for stochastic biochemical networks so that a given time-bounded CSL property is guaranteed to hold, or, in the case of quantitative properties, the probability of satisfying the property is maximised or minimised. Our method is based on extending CSL model checking and standard uniformisation to parametric models, in order to compute safe bounds on the satisfaction probability of the property. We develop synthesis algorithms that yield answers that are precise to within an arbitrarily small tolerance value. The algorithms combine the computation of probability bounds with the refinement and sampling of the parameter space. Our methods are precise and efficient, and improve on existing approximate techniques that employ discretisation and refinement. We evaluate the usefulness of the methods by synthesising rates for three biologically motivated case studies: infection control for a SIR epidemic model; reliability analysis of molecular computation by a DNA walker; and bistability in the gene regulation of the mammalian cell cycle.

The Spatial Chemical Langevin Equation and Reaction Diffusion Master Equations: moments and qualitative solutions.

BackgroundIt has been established that stochastic effects play an important role in spatio-temporal biochemical networks. A popular method of representing such stochastic systems is the Reaction Diffusion Master Equation (RDME). However, simulating sample paths from the RDME can be computationally expensive, particularly at large populations. Here we investigate an uncommon, but much faster alternative: the Spatial Chemical Langevin Equation (SCLE).MethodsWe investigate moment equations and correlation functions analytically, then we compare sample paths and moments of the SCLE to the RDME and associated deterministic solutions. Sample paths are generated computationally by the Next Subvolume method (RDME) and the Euler-Maruyama method (SCLE), while a deterministic solution is obtained with an Euler method. We consider the Gray-Scott model, a well-known pattern generating system, and a predator–prey system with spatially inhomogeneous parameters as sample applications.ResultsFor linear reaction networks, it is well known that the first order moments of all three approaches match, that the RDME and SCLE match to the second moment, and that all approaches diverge at third order moments. For non-linear reaction networks, differential equations governing moments do not form a closed system, but a general moment equation can be compared term wise. All approaches match at the leading order, and the RDME and SCLE match at the second leading order. As expected, the SCLE captures many dynamics of the RDME where deterministic methods fail to represent them. However, areas of the parameter space in the Gray-Scott model exist where either the SCLE and RDME give qualitatively different predictions, or the RDME predicts patterns, while the SCLE does not.ConclusionsThe SCLE provides a fast alternative to existing methods for simulation of spatial stochastic biochemical networks, capturing many aspects of dynamics represented by the RDME. This becomes very useful in search of quantitative parameters yielding desired qualitative solutions. However, there exist parameter sets where both the qualitative and quantitative behaviour of the SCLE can differ when compared to the RDME, so care should be taken in its use for applications demanding greater accuracy.Electronic supplementary materialThe online version of this article (doi:10.1186/s12976-015-0001-6) contains supplementary material, which is available to authorized users.

A master equation and moment approach for biochemical systems with creation-time-dependent bimolecular rate functions.

Noise and stochasticity are fundamental to biology and derive from the very nature of biochemical reactions where thermal motion of molecules translates into randomness in the sequence and timing of reactions. This randomness leads to cell-to-cell variability even in clonal populations. Stochastic biochemical networks have been traditionally modeled as continuous-time discrete-state Markov processes whose probability density functions evolve according to a chemical master equation (CME). In diffusion reaction systems on membranes, the Markov formalism, which assumes constant reaction propensities is not directly appropriate. This is because the instantaneous propensity for a diffusion reaction to occur depends on the creation times of the molecules involved. In this work, we develop a chemical master equation for systems of this type. While this new CME is computationally intractable, we make rational dimensional reductions to form an approximate equation, whose moments are also derived and are shown to yield efficient, accurate results. This new framework forms a more general approach than the Markov CME and expands upon the realm of possible stochastic biochemical systems that can be efficiently modeled.

Unveiling the hidden structure of complex stochastic biochemical networks

Complex Markov models are widely used and powerful predictive tools to analyze stochastic biochemical processes. However, when the network of states is unknown, it is necessary to extract information from the data to partially build the network and estimate the values of the rates. The short-time behavior of the first-passage time distributions between two states in linear chains has been shown recently to behave as a power of time with an exponent equal to the number of intermediate states. For a general Markov model we derive the complete Taylor expansion of the first-passage time distribution between two arbitrary states. By combining algebraic methods and graph theory approaches it is shown that the first term of the Taylor expansion is determined by the shortest path from the initial state to the final state. When this path is unique, we prove that the coefficient of the first term can be written in terms of the product of the transition rates along the path. It is argued that the application of our results to first-return times may be used to estimate the dependence of rates on external parameters in experimentally measured time distributions.

Bistable stochastic biochemical networks: large chemical networks and systems with many molecules

In this paper we continue the program started in Hwang and Velazquez (J Math Chem, to appear). We describe some chemical systems exhibiting bistable behavior with reaction constants of order one, but where bistability is due to the presence of a large number of chemical species or a large number of molecules of some of the species. We derive generalizations of the classical Kramers’ formula that gives the switching times for some particular systems exhibiting a large number of species.

Time-ordered product expansions for computational stochastic system biology

The time-ordered product framework of quantum field theory can also be used to understand salient phenomena in stochastic biochemical networks. It is used here to derive Gillespie’s stochastic simulation algorithm (SSA) for chemical reaction networks; consequently, the SSA can be interpreted in terms of Feynman diagrams. It is also used here to derive other, more general simulation and parameter-learning algorithms including simulation algorithms for networks of stochastic reaction-like processes operating on parameterized objects, and also hybrid stochastic reaction/differential equation models in which systems of ordinary differential equations evolve the parameters of objects that can also undergo stochastic reactions. Thus, the time-ordered product expansion can be used systematically to derive simulation and parameter-fitting algorithms for stochastic systems.

Bistable stochastic biochemical networks: highly specific systems with few chemicals

In this paper we describe a class of stochastic biochemical systems exhibiting bistable behavior, in the sense that the invariant measure associated to the system is concentrated near two different classes of states of the system. We develop methods that allow us to describe the behavior of the invariant measure in some suitable asymptotic limits, as well as the switching times for the transitions between the states close to each of the states with high probability. Due to the discrete character of the problem, switching times cannot be computed using the classical Kramers’ formula, and alternative methods are required.

Efficient calculation of steady state probability distribution for stochastic biochemical reaction network

The Steady State (SS) probability distribution is an important quantity needed to characterize the steady state behavior of many stochastic biochemical networks. In this paper, we propose an efficient and accurate approach to calculating an approximate SS probability distribution from solution of the Chemical Master Equation (CME) under the assumption of the existence of a unique deterministic SS of the system. To find the approximate solution to the CME, a truncated state-space representation is used to reduce the state-space of the system and translate it to a finite dimension. The subsequent ill-posed eigenvalue problem of a linear system for the finite state-space can be converted to a well-posed system of linear equations and solved. The proposed strategy yields efficient and accurate estimation of noise in stochastic biochemical systems. To demonstrate the approach, we applied the method to characterize the noise behavior of a set of biochemical networks of ligand-receptor interactions for Bone Morphogenetic Protein (BMP) signaling. We found that recruitment of type II receptors during the receptor oligomerization by itself doesn't not tend to lower noise in receptor signaling, but regulation by a secreted co-factor may provide a substantial improvement in signaling relative to noise. The steady state probability approximation method shortened the time necessary to calculate the probability distributions compared to earlier approaches, such as Gillespie's Stochastic Simulation Algorithm (SSA) while maintaining high accuracy.

A data-integrated method for analyzing stochastic biochemical networks.

Variability and fluctuations among genetically identical cells under uniform experimental conditions stem from the stochastic nature of biochemical reactions. Understanding network function for endogenous biological systems or designing robust synthetic genetic circuits requires accounting for and analyzing this variability. Stochasticity in biological networks is usually represented using a continuous-time discrete-state Markov formalism, where the chemical master equation (CME) and its kinetic Monte Carlo equivalent, the stochastic simulation algorithm (SSA), are used. These two representations are computationally intractable for many realistic biological problems. Fitting parameters in the context of these stochastic models is particularly challenging and has not been accomplished for any but very simple systems. In this work, we propose that moment equations derived from the CME, when treated appropriately in terms of higher order moment contributions, represent a computationally efficient framework for estimating the kinetic rate constants of stochastic network models and subsequent analysis of their dynamics. To do so, we present a practical data-derived moment closure method for these equations. In contrast to previous work, this method does not rely on any assumptions about the shape of the stochastic distributions or a functional relationship among their moments. We use this method to analyze a stochastic model of a biological oscillator and demonstrate its accuracy through excellent agreement with CME/SSA calculations. By coupling this moment-closure method with a parameter search procedure, we further demonstrate how a model's kinetic parameters can be iteratively determined in order to fit measured distribution data.

Cholesky-Langevin Model for Biochemical Networks with Accurate Account of Stochastic Effects of Couplings between Molecular Species

The dynamics of stochastic biochemical networks are often modeled using Langevin equations as an approximation to the underlying Chemical Master Equation. However, the coupling effects between different molecular species are neglected in current Langevin models. These coupling effects are important when intrinsic stochasticity due to the small-copy number nature and the discrete nature of the reactions system is non-negligible. Here we develop a generalized Langevin approach, called the Cholesky-Langevin Model (CLM), to incorporate rigorously both the fluctuations of chemical species and the stochasticity of their pairwise-coupling effects. We exploit the property of positive semi-definiteness of the stochastic covariance structure of a biochemical network of arbitrary stoichiometry. By comparing the calculated probabilistic landscapes using CLM with that from the conventional Langevin Model and the exact landscape obtained by directly solving the Chemical Master Equation using a method recently developed in (Cao and Liang, 2008), we show through several examples that the CLM model significantly improves the accuracy in approximating the underlying Chemical Master Equation. This study shows the importance of the coupling effects between molecular species in biochemical networks, and points out an effective approach for their rigorous account. We also discuss the biological implications of our approach. (Cao and Liang, BMC Systems Biology 2008, 2:30)

A rigorous framework for multiscale simulation of stochastic cellular networks

Noise and stochasticity are fundamental to biology and derive from the very nature of biochemical reactions where thermal motion of molecules translates into randomness in the sequence and timing of reactions. This randomness leads to cell-cell variability even in clonal populations. Stochastic biochemical networks are modeled as continuous time discrete state Markov processes whose probability density functions evolve according to a chemical master equation (CME). The CME is not solvable but for the simplest cases, and one has to resort to kinetic Monte Carlo techniques to simulate the stochastic trajectories of the biochemical network under study. A commonly used such algorithm is the stochastic simulation algorithm (SSA). Because it tracks every biochemical reaction that occurs in a given system, the SSA presents computational difficulties especially when there is a vast disparity in the timescales of the reactions or in the number of molecules involved in these reactions. This is common in cellular networks, and many approximation algorithms have evolved to alleviate the computational burdens of the SSA. Here, we present a rigorously derived modified CME framework based on the partition of a biochemically reacting system into restricted and unrestricted reactions. Although this modified CME decomposition is as analytically difficult as the original CME, it can be naturally used to generate a hierarchy of approximations at different levels of accuracy. Most importantly, some previously derived algorithms are demonstrated to be limiting cases of our formulation. We apply our methods to biologically relevant test systems to demonstrate their accuracy and efficiency.

Adiabatic coarse-graining and simulations of stochastic biochemical networks.

We propose a universal approach for analysis and fast simulations of stiff stochastic biochemical networks, which rests on elimination of fast chemical species without a loss of information about mesoscopic, non-Poissonian fluctuations of the slow ones. Our approach is similar to the Born-Oppenheimer approximation in quantum mechanics and follows from the stochastic path integral representation of the cumulant generating function of reaction events. In applications with a small number of chemical reactions, it produces analytical expressions for cumulants of chemical fluxes between the slow variables. This allows for a low-dimensional, interpretable representation and can be used for high-accuracy, low-complexity coarse-grained numerical simulations. As an example, we derive the coarse-grained description for a chain of biochemical reactions and show that the coarse-grained and the microscopic simulations agree, but the former is 3 orders of magnitude faster.

Dynamic Simulations of Single-Molecule Enzyme Networks

Along with the growth of technologies allowing accurate visualization of biochemical reactions to the scale of individual molecules has arisen an appreciation of the role of statistical fluctuations in intracellular biochemistry. The stochastic nature of metabolism can no longer be ignored. It can be probed empirically, and theoretical studies have established its importance. Traditional methods for modeling stochastic biochemistry are derived from an elegant and physically satisfying theory developed by Gillespie. However, although Gillespie's algorithm and its derivatives efficiently model small-scale systems, complex networks are harder to manage on easily available computer systems. Here we present a novel method of simulating stochastic biochemical networks using discrete events simulation techniques borrowed from manufacturing production systems. The method is very general and can be mapped to an arbitrarily complex network. As an illustration, we apply the technique to the glucose phosphorylation steps of the Embden-Meyerhof-Parnas pathway in E. coli . We show that a deterministic version of the discrete event simulation reproduces the behavior of an analogous deterministic differential equation model. The stochastic version of the same model predicts that catastrophic bottlenecks in the system are more likely than one would expect from deterministic theory.

Robust Engineered Circuit Design Principles for Stochastic Biochemical Networks With Parameter Uncertainties and Disturbances

Biochemical regulatory networks including genes, proteins and other regulatory molecules suffer from internal parametrical fluctuations (thermal, transcriptional, and splicing) as well as external noises (environmental and intercellular). Robustness is an essential property of intracellular biochemical regulatory networks to attenuate the effects of internal fluctuation and external noise. In this study, several system control schemes are proposed for the robust circuit control design of stochastic linear and nonlinear biochemical regulatory networks. First, the robust stability of genetic and proteomic regulatory networks is discussed under internal fluctuations. Then, the filtering ability of external noises is analyzed for stochastic biochemical regulatory networks. For the case where a biochemical regulatory network is not sufficiently robust to tolerate internal fluctuation and does not have enough filtering ability to filter the external noise, how to improve the robustness and noise filtering ability of stochastic biochemical regulatory networks by engineered control mechanisms is also proposed via biochemical circuit design. The proposed robust gene circuit design principles have potential applications for robust biosynthetic network design. Finally, two design examples are given in-silico to illustrate the design procedure and to confirm the performance of the proposed robust circuit design method.

A Dominated Coupling From The Past algorithm for the stochastic simulation of networks of biochemical reactions

BackgroundIn recent years, stochastic descriptions of biochemical reactions based on the Master Equation (ME) have become widespread. These are especially relevant for models involving gene regulation. Gillespie’s Stochastic Simulation Algorithm (SSA) is the most widely used method for the numerical evaluation of these models. The SSA produces exact samples from the distribution of the ME for finite times. However, if the stationary distribution is of interest, the SSA provides no information about convergence or how long the algorithm needs to be run to sample from the stationary distribution with given accuracy. ResultsWe present a proof and numerical characterization of a Perfect Sampling algorithm for the ME of networks of biochemical reactions prevalent in gene regulation and enzymatic catalysis. Our algorithm combines the SSA with Dominated Coupling From The Past (DCFTP) techniques to provide guaranteed sampling from the stationary distribution. The resulting DCFTP-SSA is applicable to networks of reactions with uni-molecular stoichiometries and sub-linear, (anti-) monotone propensity functions. We showcase its applicability studying steady-state properties of stochastic regulatory networks of relevance in synthetic and systems biology.ConclusionThe DCFTP-SSA provides an extension to Gillespie’s SSA with guaranteed sampling from the stationary solution of the ME for a broad class of stochastic biochemical networks.

Optimal signal processing in small stochastic biochemical networks.

We quantify the influence of the topology of a transcriptional regulatory network on its ability to process environmental signals. By posing the problem in terms of information theory, we do this without specifying the function performed by the network. Specifically, we study the maximum mutual information between the input (chemical) signal and the output (genetic) response attainable by the network in the context of an analytic model of particle number fluctuations. We perform this analysis for all biochemical circuits, including various feedback loops, that can be built out of 3 chemical species, each under the control of one regulator. We find that a generic network, constrained to low molecule numbers and reasonable response times, can transduce more information than a simple binary switch and, in fact, manages to achieve close to the optimal information transmission fidelity. These high-information solutions are robust to tenfold changes in most of the networks' biochemical parameters; moreover they are easier to achieve in networks containing cycles with an odd number of negative regulators (overall negative feedback) due to their decreased molecular noise (a result which we derive analytically). Finally, we demonstrate that a single circuit can support multiple high-information solutions. These findings suggest a potential resolution of the “cross-talk” phenomenon as well as the previously unexplained observation that transcription factors that undergo proteolysis are more likely to be auto-repressive.