Poisson channel with binary Markov input and average sojourn time constraint

TL;DR

This paper models gene expression as a Poisson channel with a binary Markov input, deriving its capacity and analyzing how different constraints influence optimal promoter states, revealing OFF-favoring and ON-favoring regimes.

Contribution

It provides an analytical expression for the mutual information of a Poisson channel with Markov input and explores the effects of various constraints on optimal promoter states.

Findings

OFF-favoring optima under all three constraints

Constraint (i) shows a region favoring the ON state

Constraint (iii) exhibits ON-favoring local optima

Abstract

A minimal model for gene expression, consisting of a switchable promoter together with the resulting messenger RNA, is equivalent to a Poisson channel with a binary Markovian input process. Determining its capacity is an optimization problem with respect to two parameters: the average sojourn times of the promoter's active (ON) and inactive (OFF) state. An expression for the mutual information is found by solving the associated filtering problem analytically on the level of distributions. For fixed peak power, three bandwidth-like constraints are imposed by lower-bounding (i) the average sojourn times (ii) the autocorrelation time and (iii) the average time until a transition. OFF-favoring optima are found for all three constraints, as commonly encountered for the Poisson channel. In addition, constraint (i) exhibits a region that favors the ON state, and (iii) shows ON-favoring local optima.