Scaling relations of energy dissipation rate in nonequilibrium reaction systems*

Many biochemical systems operate in nonequilibrium steady states that carry out certain biological functions. These systems constantly dissipate energy, and the dissipation rate could be determined by the underlying reaction network. For complex systems with numerous microscopic states, however, the system could only be measured at a coarse-grained level, and such a coarse-grained description leads to underestimation of the dissipation rate. To quantify how energy dissipation is associated across scales, we develop a coarse-graining process in the state space and a corresponding renormalization procedure for reaction rates, an approach conceptually inspired by the real space renormalization group. We find that the energy dissipation rate has an inverse power-law dependence on the number of microscopic states in a coarse-grained state, with an exponent that depends on both the network structure and the probability flux correlation. We demonstrate the existence of the scaling relation in realistic biochemical models such as biochemical oscillators and microtubule-kinesin active flow systems and discuss its relation with the Kolmogorov cascade in Turbulence. Finally, we report on the exact solution of the scaling exponent in square lattice and regular lattices of higher dimensions.