Biochemical systems and processes in living cells generally operate far from equilibrium. This review presents an overview of a statistical thermodynamic treatment for such systems, with examples from several key components in cellular signal transduction. Open-system nonequilibrium steady-state (NESS) models are introduced. The models account quantitatively for the energetics and thermodynamics in phosphorylation-dephosphorylation switches, GTPase timers, and specificity amplification through kinetic proofreading. The chemical energy derived from ATP and GTP hydrolysis establishes the NESS of a cell and makes the cell—a mesoscopic–biochemical reaction system that consists of a collection of thermally driven fluctuating macromolecules—a genetically programmed chemical machine.

A unifying mathematical theory of nonequilibrium thermodynamics of stochastic systems in terms of master equations is presented. As generalizations of isothermal entropy and free energy, two functions of state play central roles: the Gibbs entropy S and the relative entropy F, which are related via the stationary distribution of the stochastic dynamics. S satisfies the fundamental entropy balance equation dS/dt=ep$-${}hd/T with entropy production rate ep$\ge${}0 and heat dissipation rate hd, while dF/dt=$-${}fd$\le${}0. For closed systems that satisfy detailed balance: Tep(t)=fd(t). For open systems, one has Tep(t)=fd(t)+Qhk(t), where the housekeeping heat, Qhk$\ge${}0, was first introduced in the phenomenological nonequilibrium steady-state thermodynamics put forward by Oono and Paniconi. Qhk represents the irreversible work done by the surrounding to the system that is kept away from reaching equilibrium. Hence, entropy production ep consists of free energy dissipation associated with spontaneous relaxation (i.e., self-organization), fd, and active energy pumping that sustains the open system Qhk. The amount of excess heat involved in the relaxation Qex=hd$-${}Qhk=fd$-${}T(dS/dt). Two kinds of irreversibility, and the meaning of the arrow of time, emerge. Quasistationary processes, adiabaticity, and maximum principle for entropy are also generalized to nonequilibrium settings.

Schlögl's model is the canonical example of a chemical reaction system that exhibits bistability. Because the biological examples of bistability and switching behaviour are increasingly numerous, this paper presents an integrated deterministic, stochastic and thermodynamic analysis of the model. After a brief review of the deterministic and stochastic modelling frameworks, the concepts of chemical and mathematical detailed balances are discussed and non-equilibrium conditions are shown to be necessary for bistability. Thermodynamic quantities such as the flux, chemical potential and entropy production rate are defined and compared across the two models. In the bistable region, the stochastic model exhibits an exchange of the global stability between the two stable states under changes in the pump parameters and volume size. The stochastic entropy production rate shows a sharp transition that mirrors this exchange. A new hybrid model that includes continuous diffusion and discrete jumps is suggested to deal with the multiscale dynamics of the bistable system. Accurate approximations of the exponentially small eigenvalue associated with the time scale of this switching and the full time-dependent solution are calculated using M atlab . A breakdown of previously known asymptotic approximations on small volume scales is observed through comparison with these and Monte Carlo results. Finally, in the appendix section is an illustration of how the diffusion approximation of the chemical master equation can fail to represent correctly the mesoscopically interesting steady-state behaviour of the system.