Nonequilibrium Lyapunov function and a fluctuation relation for stochastic systems: Poisson-representation approach

Discontinuous phase transition in chemotactic aggregation with density-dependent pressure

Many small organisms such as bacteria can attract each other by depositing chemical attractants. At the same time, they exert repulsive force on each other when crowded, which can be modeled by effective pressure as an increasing function of the organisms' density. As the chemical attraction becomes strong compared to the effective pressure, the system will undergo a phase transition from homogeneous distribution to aggregation. In this work, we describe the interplay of organisms and chemicals on a two-dimensional disk with a set of partial differential equations of the Patlak-Keller-Segel type. By analyzing its Lyapunov functional, we show that the aggregation transition occurs discontinuously, forming an aggregate near the boundary of the disk. The result can be interpreted within a thermodynamic framework by identifying the Lyapunov functional with free energy.

Master equations and the theory of stochastic path integrals

This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a 'generating functional', which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a 'forward' and a 'backward' path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from them. Upon expanding the forward and the backward path integrals around stationary paths, we then discuss and extend a recent method for the computation of rare event probabilities. Besides, we also derive path integral representations for processes with continuous state spaces whose forward and backward master equations admit Kramers-Moyal expansions. A truncation of the backward expansion at the level of a diffusion approximation recovers a classic path integral representation of the (backward) Fokker-Planck equation. One can rewrite this path integral in terms of an Onsager-Machlup function and, for purely diffusive Brownian motion, it simplifies to the path integral of Wiener. To make this review accessible to a broad community, we have used the language of probability theory rather than quantum (field) theory and do not assume any knowledge of the latter. The probabilistic structures underpinning various technical concepts, such as coherent states, the Doi-shift, and normal-ordered observables, are thereby made explicit.

Noise-induced multistability in the regulation of cancer by genes and pseudogenes

We extend a previously introduced model of stochastic gene regulation of cancer to a nonlinear case having both gene and pseudogene messenger RNAs (mRNAs) self-regulated. The model consists of stochastic Boolean genetic elements and possesses noise-induced multistability (multimodality). We obtain analytical expressions for probabilities for the case of constant but finite number of microRNA molecules which act as a noise source for the competing gene and pseudogene mRNAs. The probability distribution functions display both the global bistability regime as well as even-odd number oscillations for a certain range of model parameters. Statistical characteristics of the mRNA's level fluctuations are evaluated. The obtained results of the extended model advance our understanding of the process of stochastic gene and pseudogene expressions that is crucial in regulation of cancer.

Oscillations in probability distributions for stochastic gene expression

The phenomenon of oscillations in probability distribution functions of number of components is found for a model of stochastic gene expression. It takes place in cases of low levels of molecules or strong intracellular noise. The oscillations distinguish between more probable even and less probable odd number of particles. The even-odd symmetry restores as the number of molecules increases with the probability distribution function tending to Poisson distribution. We discuss the possibility of observation of the phenomenon in gene, protein, and mRNA expression experiments.