Bounding Mean First Passage Times in Population Continuous-Time Markov Chains

We consider the problem of bounding mean first passage times and reachability probabilities for the class of population continuous-time Markov chains, which capture stochastic interactions between groups of identical agents. The quantitative analysis of such models is notoriously difficult since typically neither state-based numerical approaches nor methods based on stochastic sampling give efficient and accurate results. Here, we propose a novel approach that leverages techniques from martingale theory and stochastic processes to generate constraints on the statistical moments of first passage time distributions. These constraints induce a semi-definite program that can be used to compute exact bounds on reachability probabilities and mean first passage times without numerically solving the transient probability distribution of the process or sampling from it. We showcase the method on some test examples and tailor it to models exhibiting multimodality, a class of particularly challenging scenarios from biology

Inference for stochastic chemical kinetics using moment equations and system size expansion

Quantitative mechanistic models are valuable tools for disentangling biochemical pathways and for achieving a comprehensive understanding of biological systems. However, to be quantitative the parameters of these models have to be estimated from experimental data. In the presence of significant stochastic fluctuations this is a challenging task as stochastic simulations are usually too time-consuming and a macroscopic description using reaction rate equations (RREs) is no longer accurate. In this manuscript, we therefore consider moment-closure approximation (MA) and the system size expansion (SSE), which approximate the statistical moments of stochastic processes and tend to be more precise than macroscopic descriptions. We introduce gradient-based parameter optimization methods and uncertainty analysis methods for MA and SSE. Efficiency and reliability of the methods are assessed using simulation examples as well as by an application to data for Epo-induced JAK/STAT signaling. The application revealed that even if merely population-average data are available, MA and SSE improve parameter identifiability in comparison to RRE. Furthermore, the simulation examples revealed that the resulting estimates are more reliable for an intermediate volume regime. In this regime the estimation error is reduced and we propose methods to determine the regime boundaries. These results illustrate that inference using MA and SSE is feasible and possesses a high sensitivity

A scalable moment-closure approximation for large-scale biochemical reaction networks.

Motivation: Stochastic molecular processes are a leading cause of cell-to-cell variability. Their dynamics are often described by continuous-time discrete-state Markov chains and simulated using stochastic simulation algorithms. As these stochastic simulations are computationally demanding, ordinary differential equation models for the dynamics of the statistical moments have been developed. The number of state variables of these approximating models, however, grows at least quadratically with the number of biochemical species. This limits their application to small-and medium-sized processes. Results: In this article, we present a scalable moment-closure approximation (sMA) for the simulation of statistical moments of large-scale stochastic processes. The sMA exploits the structure of the biochemical reaction network to reduce the covariance matrix. We prove that sMA yields approximating models whose number of state variables depends predominantly on local properties, i.e. the average node degree of the reaction network, instead of the overall network size. The resulting complexity reduction is assessed by studying a range of medium-and large-scale biochemical reaction networks. To evaluate the approximation accuracy and the improvement in computational efficiency, we study models for JAK2/STAT5 signalling and NFjB signalling. Our method is applicable to generic biochemical reaction networks and we provide an implementation, including an SBML interface, which renders the sMA easily accessible

Modeling of stochastic biological processes with non-polynomial propensities using non-central conditional moment equation.

Biological processes exhibiting stochastic fluctuations are mainly modeled using the Chemical Master Equation (CME). As a direct simulation of the CME is often computationally intractable, we recently introduced the Method of Conditional Moments (MCM). The MCM is a hybrid approach to approximate the statistics of the CME solution. In this work, we provide a more comprehensive formulation of the MCM by using non-central conditional moments instead of central conditional moments. The modified formulation allows for additional insight into the model structure and for extensions to higher-order reactions and non-polynomial propensity functions. The properties of the non-central MCM are analyzed using a model for the regulation of pili formation on the surface of bacteria, which possesses rational propensity functions

CERENA: ChEmical REaction Network Analyzer - a toolbox for the simulation and analysis of stochastic chemical kinetics.

Gene expression, signal transduction and many other cellular processes are subject to stochastic fluctuations. The analysis of these stochastic chemical kinetics is important for understanding cell-to-cell variability and its functional implications, but it is also challenging. A multitude of exact and approximate descriptions of stochastic chemical kinetics have been developed, however, tools to automatically generate the descriptions and compare their accuracy and computational efficiency are missing. In this manuscript we introduced CERENA, a toolbox for the analysis of stochastic chemical kinetics using Approximations of the Chemical Master Equation solution statistics. CERENA implements stochastic simulation algorithms and the finite state projection for microscopic descriptions of processes, the system size expansion and moment equations for meso- and macroscopic descriptions, as well as the novel conditional moment equations for a hybrid description. This unique collection of descriptions in a single toolbox facilitates the selection of appropriate modeling approaches. Unlike other software packages, the implementation of CERENA is completely general and allows, e.g., for time-dependent propensities and non-mass action kinetics. By providing SBML import, symbolic model generation and simulation using MEX-files, CERENA is user-friendly and computationally efficient. The availability of forward and adjoint sensitivity analyses allows for further studies such as parameter estimation and uncertainty analysis. The MATLAB code implementing CERENA is freely available from http://cerenadevelopers.github.io/CERENA/

Method of conditional moments (MCM) for the chemical master equation: A unified framework for the method of moments and hybrid stochastic-deterministic models.

The time-evolution of continuous-time discrete-state biochemical processes is governed by the Chemical Master Equation (CME), which describes the probability of the molecular counts of each chemical species. As the corresponding number of discrete states is, for most processes, large, a direct numerical simulation of the CME is in general infeasible. In this paper we introduce the method of conditional moments (MCM), a novel approximation method for the solution of the CME. The MCM employs a discrete stochastic description for low-copy number species and a moment-based description for medium/high-copy number species. The moments of the medium/high-copy number species are conditioned on the state of the low abundance species, which allows us to capture complex correlation structures arising, e.g., for multi-attractor and oscillatory systems. We prove that the MCM provides a generalization of previous approximations of the CME based on hybrid modeling and moment-based methods. Furthermore, it improves upon these existing methods, as we illustrate using a model for the dynamics of stochastic single-gene expression. This application example shows that due to the more general structure, the MCM allows for the approximation of multi-modal distributions

LNA++: Linear noise approximation with first and second order sensitivities.

The linear noise approximation (LNA) provides an approximate description of the statistical moments of stochastic chemical reaction networks (CRNs). LNA is a commonly used modeling paradigm describing the probability distribution of systems of biochemical species in the intracellular environment. Unlike exact formulations, the LNA remains computationally feasible even for CRNs with many reactions. The tractability of the LNA makes it a common choice for inference of unknown chemical reaction parameters. However, this task is impeded by a lack of suitable inference tools for arbitrary CRN models. In particular, no available tool provides temporal cross-correlations, parameter sensitivities and efficient numerical integration. In this manuscript we present LNA++, which allows for fast derivation and simulation of the LNA including the computation of means, covariances, and temporal cross-covariances. For efficient parameter estimation and uncertainty analysis, LNA++ implements first and second order sensitivity equations. Interfaces are provided for easy integration with Matlab and Python. Implementation and availability: LNA++ is implemented as a combination of C/C++, Matlab and Python scripts. Code base and the release used for this publication are available on GitHub (https://github.com/ICB-DCM/LNAplusplus ) and Zenodo (https://doi.org/10.5281/zenodo.1287771 )