**Evaluation of departure from equilibrium at cellular scale**

Carlo Bianca and Annie Lemarchand

J. Chem. Phys. 141, 144102 (2014)

The complex spatiotemporal structures that appear in biological systems require farfrom equilibrium conditions which lead to the circulation of reaction fluxes. Recent developments in nonequilibrium statistical physics propose a theoretical framework for estimating these reaction fluxes. In particular, the time asymmetry of fluctuations could be a priori exploited. Fluorescence imaging gives directly access to the observation of the dynamics of cellular events. The temporal ordering of three proteins in the endocytic pathway has been extracted and reaction flux has been estimated for an assumed mechanism with linear dynamics [D. R. Sisan, D. Yarar, C. M. Waterman, and J. S. Urbach, Biophys. J. 98, 2432 (2010)]. However, endocytosis is known to involve complex regulation mechanisms and our aim is to warn against the blind use of the simple relation between reaction flux and crosscorrelation function of concentration fluctuations found for linear deterministic dynamics [W. J. Heuett and H. Qian, J. Chem. Phys. 124, 044110 (2006)].

In the biologically relevant case of a reactive system which may admit periodic oscillations, our results show that the amplitude of the crosscorrelation functions is not only proportional to the reaction flux but also to a specific parameterdependent function which diverges as the Hopf bifurcation approaches. In order to harvest the determination of correlation functions for reaction flux estimation in a given chemical system, it is therefore essential first to identify the reactionmechanism and secondly to evaluate the associated rate constants. If these demanding requirements may be fulfilled, the stochastic differential equations of Langevin type governing the fluctuating

dynamics of concentrations provides a reliable, analytical formula relating the reaction flux and the correlations of fluctuations. From the theoretical viewpoint, the interplay between fluctuations and nonlinearities of deterministic dynamics is subtle and leads to specific formulas for the time crosscorrelations

of concentration fluctuations, that depend on the details of dynamics.

Difference of time cross-correlation functions of concentration fluctuations, *I*(t)=<(*X*(0)-*X*_{S})(Y(t)-Y_{S})>-<(X(t)-X_{S})(Y(0)-Y_{S})>,

in the case of the Brusselator model, known to possess a stationary state (*X*_{S},*Y*_{S})* *and a Hopf bifurcation. The concentrations of species A, B, and C are fixed and the concentrations *X*(t) and *Y*(t) of species X and Y are variable. The results of the numerical solution of the master equation (blue dashed line) confirm the analytical approach by Langevin equations (red dashed line) and invalidate the result obtained when assuming that dynamics is linear (black dotted line).