Markovian dynamics on complex reaction networks

Abstract

Complex networks, comprised of individual elements that interact with each other through reaction channels, are ubiquitous across many scientific and engineering disciplines. Examples include biochemical, pharmacokinetic, epidemiological, ecological, social, neural, and multi-agent networks. A common approach to modeling such networks is by a master equation that governs the dynamic evolution of the joint probability mass function of the underling population process and naturally leads to Markovian dynamics for such process. Due however to the nonlinear nature of most reactions, the computation and analysis of the resulting stochastic population dynamics is a difficult task. This review article provides a coherent and comprehensive coverage of recently developed approaches and methods to tackle this problem. After reviewing a general framework for modeling Markovian reaction networks and giving specific examples, the authors present numerical and computational techniques capable of evaluating or approximating the solution of the master equation, discuss a recently developed approach for studying the stationary behavior of Markovian reaction networks using a potential energy landscape perspective, and provide an introduction to the emerging theory of thermodynamic analysis of such networks. Three representative problems of opinion formation, transcription regulation, and neural network dynamics are used as illustrative examples.

Minimal moment equations for stochastic models of biochemical reaction networks with partially finite state space

“…For details of these results, the reader is referred to the provided references. [7][8][9] In this paper, I will consider a subclass of general reaction networks. Specifically, I will focus on systems in which the structure of the reaction network ensures that some of the chemical species can only be present in finite amounts of molecules.…”

“…11, which means that a direct comparison of the number of moment equations obtained here and in the reference is possible. Noting that for this example we have that n = 2 and m = 1, we can directly compare the number of moment equations using (4), (8), and (11). The results are listed in Table I.…”

“…[4][5][6] An approach that has recently gained popularity is based on deriving systems of differential equations from the CME that only describe the time evolution of momentsup to some desired order L-of the underlying probability distribution. [7][8][9] The solution of such moment equations can often be computed or at least approximated in cases where the whole CME is intractable. 10 However, if more than just moments of very low order are desired (i.e., L ≫ 1), the size of the system of moment equations increases quickly in the number of chemical species that play a role for the system.…”

Intrinsic noise and two-dimensional maps: Quasicycles, quasiperiodicity, and chaos

“…When setting up such a model, one is often faced with the choice of whether the model should depict the individual components present in the system (be they molecules, animals, or other entities), or simply describe the population by a macroscopic concentration. A popular choice is to take a mesoscopic approach, where instead of tracking each individual one is content to describe the fraction of individuals of each 'species' present in the system [1,2]. The stochasticity, present in the interactions between individual elements in the system is, however, retained.…”

Pathway structure determination in complex stochastic networks with non-exponential dwell times

“…[1][2][3][4][5] Since the functioning of these complex systems is strongly influenced by their structures, the most important step in theoretical analysis is to determine the topology of underlying networks. Despite recent strong advances in understanding the dynamical and structural properties of complex chemical and biological networks, [6][7][8][9][10][11][12][13][14] revealing the hidden structures of networks and their relations to dynamics remains a challenging task.…”