Dispersion for Two Classes of Random Variables: General Theory and Application to Inference of an External Ligand Concentration by a Cell

TL;DR

This paper develops a general theoretical framework for calculating dispersion in two classes of random variables in Markov processes and applies it to analyze cellular sensing mechanisms, revealing conditions for finite uncertainty and limits of measurement.

Contribution

It introduces a unified formalism for dispersion of both jump-related and time-in-state random variables, with an application to cellular ligand concentration inference.

Findings

Monitoring time in a state yields finite uncertainty only with dissipation.

Activity-based measurements can reach the Berg-Purcell limit even at equilibrium.

The formalism extends known results to new classes of random variables.

Abstract

We derive expressions for the dispersion for two classes of random variables in Markov processes. Random variables like current and activity pertain to the first class, which is composed by random variables that change whenever a jump in the stochastic trajectory occurs. The second class corresponds to the time the trajectory spends in a state (or cluster of states). While the expression for the first class follows straightforwardly from known results in the literature, we show that a similar formalism can be used to derive an expression for the second class. As an application, we use this formalism to analyze a cellular two-component network estimating an external ligand concentration. The uncertainty related to this external concentration is calculated by monitoring different random variables related to an internal protein. We show that, inter alia, monitoring the time spent in the phosphorylated state of the protein leads to a finite uncertainty only if there is dissipation, whereas the uncertainty obtained from the activity of the transitions of the internal protein can reach the Berg and Purcell limit even in equilibrium.