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.

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.

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/.

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Rapid clonal expansion of antigen-specific T cells is a fundamental feature of adaptive immune responses. It enables the outgrowth of an individual T cell into thousands of clonal descendants that diversify into short-lived effectors and long-lived memory cells. Clonal expansion is thought to be programmed upon priming of a single naive T cell and then executed by homogenously fast divisions of all of its descendants. However, the actual speed of cell divisions in such an emerging “T cell family” has never been measured with single-cell resolution. Here, we utilize continuous live-cell imaging in vitro to track the division speed and genealogical connections of all descendants derived from a single naive CD8+ T cell throughout up to ten divisions of activation-induced proliferation. This comprehensive mapping of T cell family trees identifies a short burst phase, in which division speed is homogenously fast and maintained independent of external cytokine availability or continued T cell receptor stimulation. Thereafter, however, division speed diversifies, and model-based computational analysis using a Bayesian inference framework for tree-structured data reveals a segregation into heritably fast- and slow-dividing branches. This diversification of division speed is preceded already during the burst phase by variable expression of the interleukin-2 receptor alpha chain. Later it is accompanied by selective expression of memory marker CD62L in slower dividing branches. Taken together, these data demonstrate that T cell clonal expansion is structured into subsequent burst and diversification phases, the latter of which coincides with specification of memory versus effector fate.

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MotivationStochastic 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.ResultsIn 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 NFκB 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.Availability and implementationThe sMA is implemented in the open-source MATLAB toolbox CERENA and is available from https://github.com/CERENADevelopers/CERENA.Supplementary information Supplementary data are available at Bioinformatics online.

Rapid clonal expansion of antigen specific T cells is a fundamental feature of adaptive immune responses. It enables the outgrowth of an individual T cell into thousands of clonal descendants that diversify into short-lived effectors and long-lived memory cells. Clonal expansion is thought to be programmed upon priming of a single naive T cell and then executed by homogenously fast divisions of all of its descendants. However, the actual speed of cell divisions in such an emerging "T cell family" has never been measured with single-cell resolution. Here, we utilize continuous live-cell imaging in vitro to track the division speed and genealogical connections of all descendants derived from a single naive CD8+ T cell throughout up to ten divisions of activation-induced proliferation. This comprehensive mapping of T cell family trees identifies a short burst phase, in which division speed is homogenously fast and maintained independent of external cytokine availability or continued T cell receptor stimulation. Thereafter, however, division speed diversifies and model-based computational analysis using a novel Bayesian inference framework for tree-structured data reveals a segregation into heritably fast and slow dividing branches. This diversification of division speed is preceded already during the burst phase by variable expression of the interleukin-2 receptor alpha chain. Later it is accompanied by selective re-expression of memory marker CD62L in slower dividing branches. Taken together, these data demonstrate that T cell clonal expansion is structured into subsequent burst and diversification phases the latter of which coincides with specification of memory vs. effector fate.