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

Overview of software packages for stochastic modeling and their capabilities.

<p>Overview of software packages for stochastic modeling and their capabilities.</p

Approximation error of MM and MCM of various orders with various moment closures for the three-stage model of gene expression.

<p>Relative errors of mean and variance of the protein concentration at the steady state are depicted for different truncation orders and moment closures. The truncation order <i>m</i> means that moments up to order <i>m</i> are simulated. For moment orders and closures for which the numerical simulation could not be completed, i.e. derivative matching, no approximation error is reported.</p

Workflow of CERENA.

<p>(a) CERENA can be used to study (multi-compartment) chemical reaction networks. (b) The reaction network can be defined in MATLAB, or alternatively, imported from SBML. (c) The system of equations for different modeling approaches implemented in CERENA is generated, and optionally stored as MATLAB functions for numerical simulation using MATLAB ODE solvers. Furthermore, the representation of the system can be exported to the estimation toolbox Data2Dynamics. (d) The symbolic representation of the system of equations together with the initial conditions is stored in a MATLAB script. (e) Based on the symbolic representation, 1^st and 2^nd order sensitivity equations are derived. MEX-files, which use CVODES and IDAS packages of SUNDIALS for the numerical simulation of the models, are compiled. (f) The generated MEX-files are used for numerical simulation, and can be integrated with other software for parameter estimation. (g) Various aspects of the simulation results can be visualized using CERENA.</p

Simulation results for the JAK-STAT signaling pathway.

<p>(a) Schematic of the simplified JAK-STAT signaling pathway. The intermediate states npSTAT1 to npSTAT5 are used to model the delayed export of STAT from the nucleus. (b) The mean (left) and variance (right) of dimerized phosphorylated STAT concentration, obtained using several methods. SSA simulation results serve as the reference solution.</p

Computation time for different sensitivity analysis methods.

<p>The objective function gradient for MM2 simulation is evaluated for an increasing number of parameters. The computation times of finite differences, forward sensitivity analysis, and adjoint sensitivity analysis are shown.</p

Simulation of the three-stage model of gene expression.

<p>(a) Schematic of the three-stage model of gene expression. (b) Mean (left) and variance (right) of the number of protein molecules obtained using different orders of SSE, MM and MCM. (c) Marginal probabilities of promotor states (left), the mean of protein molecule numbers conditioned on the promotor state (middle), and the variance of protein molecule numbers conditioned on the promotor state (right) predicted by MCM of order 3. (b,c) FSP results serve as the reference solution. Low dispersion closure was used for MM and MCM. MM2, MM3, MCM2 and MCM3 denote the second- and third-order MM and the second- and third-order MCM.</p

Inference methods for stochastic processes.

<p>(a) Single-cell snapshot data collected using a high-throughput technique, such as flow cytometry. (b) Empirical density functions for SSA runs (black —) and experimental data (blue —), the difference is used as distance measure in Approximate Bayesian Computing. (c) Instantaneous probability distribution computed using FSP (black —) to evaluate the likelihood of the observing the individual cells (blue ×). (d) Mean computed using MA/SSE (black —) as well as measured mean and its uncertainty (blue —). (e) Summary of the properties of the displayed methods.</p

Analysis of model selection and rejection criteria.

<p>(a) and (b) Median AIC weight for EMRE and IOS at respective estimated parameters. A green color indicates that the EMRE and IOS description is more probable and a blue color indicates the RRE and LNA description is more probable. (c) and (d) area in which the models can on average be rejected based on a chi-square test to confidence level 0.01. The coloring indicates the method to which the area corresponds.</p

Heritable changes in division speed accompany the diversification of single T cell fate

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.ISSN:0027-8424ISSN:1091-649