Moment analysis of linear time-varying dynamical systems with renewal transitions

TL;DR

This paper develops a mathematical framework for analyzing the stability and steady-state moments of linear time-varying stochastic hybrid systems with renewal events, with applications to gene expression and cell division.

Contribution

It provides necessary and sufficient conditions for moment stability and exact formulas for steady-state moments in TTSHS, a class of systems modeling stochastic hybrid dynamics.

Findings

Derived stability conditions for moments of TTSHS.

Obtained explicit formulas for steady-state mean and variance.

Applied results to gene expression, revealing counterintuitive noise reduction insights.

Abstract

Stochastic dynamics of several systems can be modeled via piecewise deterministic time evolution of the state, interspersed by random discrete events. Within this general class of systems, we consider time-triggered stochastic hybrid systems (TTSHS), where the state evolves continuously according to a linear time-varying dynamical system. Discrete events occur based on an underlying renewal process (timer), and the intervals between successive events follow an arbitrary continuous probability density function. Moreover, whenever the event occurs, the state is reset based on a linear affine transformation that allows for the inclusion of state-dependent and independent noise terms. Our key contribution is derivation of necessary and sufficient conditions for the stability of statistical moments, along with exact analytical expressions for the steady-state moments. These results are illustrated on an example from cell biology, where deterministic synthesis and decay of a gene product (RNA or protein) is coupled to random timing of cell-division events. As experimentally observed, cell-division events occur based on an internal timer that measures the time elapsed since the start of cell cycle (i.e., last event). Upon division, the gene product level is halved, together with a state-dependent noise term that arises due to randomness in the partitioning of molecules between two daughter cells. We show that the TTSHS framework is conveniently suited to capture the time evolution of gene product levels, and derive unique formulas connecting its mean and variance to underlying model parameters and noise mechanisms. Systematic analysis of the formulas reveal counterintuitive insights, such as, if the partitioning noise is large then making the timing of cell division more random reduces noise in gene product levels.