Large Deviations of Piecewise-Deterministic-Markov-Processes with Application to Stochastic Calcium Waves

TL;DR

This paper establishes a Large Deviation Principle for Piecewise-Deterministic Markov Processes and applies it to model and analyze the probability of stochastic calcium waves in cellular systems.

Contribution

It introduces a Large Deviations framework for PDMPs and applies it to a biological calcium signaling model, providing explicit equations for optimal trajectories.

Findings

Derived explicit Euler-Lagrange equations for PDMPs.

Estimated probabilities of calcium wave generation.

Validated the model with biological relevance.

Abstract

We prove a Large Deviation Principle for Piecewise Deterministic Markov Processes (PDMPs). This is an asymptotic estimate for the probability of a trajectory in the large size limit. Explicit Euler-Lagrange equations are determined for computing optimal first-hitting-time trajectories. The results are applied to a model of stochastic calcium dynamics. It is widely conjectured that the mechanism of calcium puff generation is a multiscale process: with microscopic stochastic fluctuations in the opening and closing of individual channels generating cell-wide waves via the diffusion of calcium and other signaling molecules. We model this system as a PDMP, with $N \gg 1$ stochastic calcium channels that are coupled via the ambient calcium concentration. We employ the Large Deviations theory to estimate the probability of cell-wide calcium waves being produced through microscopic stochasticity.