Moment Analysis of Linear Time-Varying Dynamical Systems with Renewal Transitions

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. In summary, theory developed here provides novel tools for characterizing moments in an important class of stochastic dynamical systems that arises naturally in diverse application areas.

Time-triggered stochastic hybrid systems with two timer-dependent resets

“…A special case of the Theorem is when h 2 (τ ) = 0 and the system reduces to a single family of resets as considered in [42]. For completeness we provide this result as a Corollary.…”

“…and the corresponding pdf of τ is given by [42]. Moreover, the uncentered statistical moments of τ are related to moments of τ s as per…”

“…and ⊗ stands the Kronecker Product [42]. When the first family (i = 1) or the second family (i = 2) of resets occurs, µ changes as per the map…”

“…Depending on the protein synthesis rate and partitioning noise, noise in protein concentration may decrease, increase or remain invariant of the noise in the cell-cycle timing. Steady-state Fano factor of protein concentration as a function of the noise in the cell-cycle timing for a Gamma-distributed τs and synthesis rate(42). If the coefficient k2 is zero, Fano factor of protein concentration remains constant with respect to changes in CV 2τs .…”

Time triggered stochastic hybrid system with nonlinear continuous dynamics

“…Let random variable τ s denote the time between two successive resets, then having a probabilisitic rate as per (3) gauarantes that τ s is an iid random variable following an arbitrary continuous positively-valued distribution g [10]. With such a formulation, one can capture events like the duration of cell cycle that typically follow a lognormal or gamma distribution [34]- [38].…”