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

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.

9 pages, 7 figures

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• Dissipation bounds all steady-state current fluctuations
• Proof of the Finite-Time Thermodynamic Uncertainty Relation for Steady-State Currents
• Spectral theory of fluctuations in time-average statistical mechanics of reversible and driven systems
• Learning nonequilibrium control forces to characterize dynamical phase transitions
• Process interpretation of current entropic bounds
• Nonrenewal statistics in transport through quantum dots
• Waiting time distribution revealing the internal spin dynamics in a double quantum dot
• Universal energy-accuracy tradeoffs in nonequilibrium cellular sensing
• Optimal inference strategies and their implications for the linear noise approximation
• Dispersion of the time spent in a state: General expression for unicyclic model and dissipation-less precision
• Thermodynamic uncertainty relations in a linear system
• Nonequilibrium fluctuation-response relations for state observables