Marginal process framework: A model reduction tool for Markov jump processes

Markov jump process models have many applications across science. Often, these models are defined on a state-space of product form and only one of the components of the process is of direct interest. In this paper, we extend the marginal process framework, which provides a marginal description of the component of interest, to the case of fully coupled processes. We use entropic matching to obtain a finite-dimensional approximation of the filtering equation, which governs the transition rates of the marginal process. The resulting equations can be seen as a combination of two projection operations applied to the full master equation, so that we obtain a principled model reduction framework. We demonstrate the resulting reduced description on the totally asymmetric exclusion process. An important class of Markov jump processes are stochastic reaction networks, which have applications in chemical and biomolecular kinetics, ecological models and models of social networks. We obtain a particularly simple instantiation of the marginal process framework for mass-action systems by using product-Poisson distributions for the approximate solution of the filtering equation. We investigate the resulting approximate marginal process analytically and numerically.Comment: 16 pages, 5 figures; accepted for publication in Physical Review

A variational approach to moment-closure approximations for the kinetics of biomolecular reaction networks

Approximate solutions of the chemical master equation and the chemical Fokker-Planck equation are an important tool in the analysis of biomolecular reaction networks. Previous studies have highlighted a number of problems with the moment-closure approach used to obtain such approximations, calling it an ad-hoc method. In this article, we give a new variational derivation of moment-closure equations which provides us with an intuitive understanding of their properties and failure modes and allows us to correct some of these problems. We use mixtures of product-Poisson distributions to obtain a flexible parametric family which solves the commonly observed problem of divergences at low system sizes. We also extend the recently introduced entropic matching approach to arbitrary ansatz distributions and Markov processes, demonstrating that it is a special case of variational moment closure. This provides us with a particularly principled approximation method. Finally, we extend the above approaches to cover the approximation of multi-time joint distributions, resulting in a viable alternative to process-level approximations which are often intractable.Comment: Minor changes and clarifications; corrected some typo

Approximation and Model Reduction for the Stochastic Kinetics of Reaction Networks

The mathematical modeling of the dynamics of cellular processes is a central part of systems biology. It has been realized that noise plays an important role in the behavior of these processes. This includes not only intrinsic noise, due to "random" molecular events within the cell, but also extrinsic noise, due to the varying environment of a cellular (sub-)system. These environmental effects and their influence on the system of interest have to be taken into account in a mathematical model. The thesis at hand deals with the (exact or approximate) reduced or marginal description of cellular subsystems when the environment of the subsystem is of no interest, and also with the approximate solution of the forward problem for biomolecular reaction networks in general. These topics are investigated across the hierarchy of possible models for reaction networks, from continuous-time Markov chains to stochastic differential equations to ordinary differential equation models. The first contribution is the derivation of moment closure approximations via a variational approach. The resulting viewpoint sheds light on the problems usually associated with moment closure, and allows one to correct some of them. The full probability distributions obtained from the variational approach are used to find approximate descriptions of heterogeneous rate equations with log-normally distributed extrinsic noise. The variational method is also extended to the approximation of multi-time joint distributions. Finally, the general form of moment equations and cumulant equations for mass-action kinetics is derived in the form of a diagrammatic technique. The second contribution is the investigation of the use of the Nakajima-Zwanzig-Mori projection operator formalism for the treatment of heterogeneous kinetics. Cumulant expansions in terms of partial cumulants are used to obtain approximate convolutional forward equations for the process of interest, with the heterogeneous reaction rates or the environment marginalized out. The performance of the approximation is investigated numerically for simple linear networks. Finally, extending previous work, a marginal description of the subsystem of interest on the process level, for fully bi-directionally coupled reaction networks, is obtained by means of stochastic filtering equations in combination with entropic matching. The resulting approximation is interpreted as an orthogonal projection of the full joint master equation, making it conceptually similar to the projection operator formalism. For mass-action kinetics, a product-Poisson ansatz for the filtering distribution leads to the simplest possible marginal process description, which is investigated analytically and numerically

ROC’n’Ribo: Characterizing a Riboswitching Expression System by Modeling Single-Cell Data

RNA-engineered systems offer simple and versatile control over gene expression in many organisms. In particular, the design and implementation of riboswitches presents a unique opportunity to manipulate any reporter device <i>in cis</i>, executing tight temporal and spatial control at low metabolic costs. Assembled to higher order genetic circuits, such riboswitch-regulated devices may efficiently process logical operations. Here, we propose a hierarchical stochastic modeling approach to characterize an <i>in silico</i> repressor gate based on neomycin- and tetracycline-sensitive riboswitches. The model was calibrated on rich, transient <i>in vivo</i> single-cell data to account for cell-to-cell variability. To capture the effect of this variability on gate performance we employed the well-known ROC-analysis and derived a novel performance indicator for logic gates. Introduction of such a performance measure is necessary, since we aimed to assess the correct functionality of the gate at the single-cell levela prerequisite for its further adaption to a genetic circuitry. Our results may be applied to other genetic devices to analyze their efficiency and ensure their correct performance in the light of cell-to-cell variability