Linear piecewise-deterministic Markov processes with families of random discrete events

TL;DR

This paper analyzes a class of linear piecewise-deterministic Markov processes with random discrete events, providing explicit conditions for finite moments and exact formulas, with applications in systems biology and noise analysis.

Contribution

It introduces explicit conditions and formulas for moments of a new class of stochastic hybrid systems with linear dynamics and random resets, expanding analytical tools for such models.

Findings

Derived exact stationary moments for the processes.

Provided explicit formulas for mean and noise in biological systems.

Analyzed how noise contributions vary with cell division randomness.

Abstract

We consider a class of piecewise-deterministic Markov processes where the state evolves according to a linear dynamical system. This continuous time evolution is interspersed by discrete events that occur at random times and change (reset) the state based on a linear affine map. In particular, we consider two families of discrete events, with the first family of resets occurring at exponentially-distributed times. The second family of resets is generally-distributed, in the sense that, the time intervals between events are independent and identically distributed random variables that follow an arbitrary continuous positively-valued probability density function. For this class of stochastic systems, we provide explicit conditions that lead to finite stationary moments, and the corresponding exact closed-form moment formulas. These results are illustrated on an example drawn from systems biology, where a protein is expressed in bursts at exponentially-distributed time intervals, decays within the cell-cycle, and is randomly divided among daughter cells when generally-distributed cell-division events occur. Our analysis leads to novel results for the mean and noise levels in protein copy numbers, and we decompose the noise levels into components arising from stochastic expression, random cell-cycle times, and partitioning. Interestingly, these individual noise contributions behave differently as cell division times become more random. In summary, the paper expands the class of stochastic hybrid systems for which statistical moments can be derived exactly without any approximations, and these results have applications for studying random phenomena in diverse areas.