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

Context in Synthetic Biology: Memory Effects of Environments with Mono-molecular Reactions

Synthetic biology aims at designing modular genetic circuits that can be assembled according to the desired function. When embedded in a cell, a circuit module becomes a small subnetwork within a larger environmental network, and its dynamics is therefore affected by potentially unknown interactions with the environment. It is well-known that the presence of the environment not only causes extrinsic noise but also memory effects, which means that the dynamics of the subnetwork is affected by its past states via a memory function that is characteristic of the environment. We study several generic scenarios for the coupling between a small module and a larger environment, with the environment consisting of a chain of mono-molecular reactions. By mapping the dynamics of this coupled system onto random walks, we are able to give exact analytical expressions for the arising memory functions. Hence, our results give insights into the possible types of memory functions and thereby help to better predict subnetwork dynamics.Comment: 14 pages, 6 figures Accepted Versio

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

Guidelines for the use and interpretation of assays for monitoring autophagy (4th edition)1.

In 2008, we published the first set of guidelines for standardizing research in autophagy. Since then, this topic has received increasing attention, and many scientists have entered the field. Our knowledge base and relevant new technologies have also been expanding. Thus, it is important to formulate on a regular basis updated guidelines for monitoring autophagy in different organisms. Despite numerous reviews, there continues to be confusion regarding acceptable methods to evaluate autophagy, especially in multicellular eukaryotes. Here, we present a set of guidelines for investigators to select and interpret methods to examine autophagy and related processes, and for reviewers to provide realistic and reasonable critiques of reports that are focused on these processes. These guidelines are not meant to be a dogmatic set of rules, because the appropriateness of any assay largely depends on the question being asked and the system being used. Moreover, no individual assay is perfect for every situation, calling for the use of multiple techniques to properly monitor autophagy in each experimental setting. Finally, several core components of the autophagy machinery have been implicated in distinct autophagic processes (canonical and noncanonical autophagy), implying that genetic approaches to block autophagy should rely on targeting two or more autophagy-related genes that ideally participate in distinct steps of the pathway. Along similar lines, because multiple proteins involved in autophagy also regulate other cellular pathways including apoptosis, not all of them can be used as a specific marker for bona fide autophagic responses. Here, we critically discuss current methods of assessing autophagy and the information they can, or cannot, provide. Our ultimate goal is to encourage intellectual and technical innovation in the field

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