Stochastic dynamics in biochemical systems: from single-molecule kinetics to Chemical master equation
Stochasticity is unavoidable in the open biochemical systems which have sustained source(s) and sink(s). First, we introduce the single-molecule enzyme kinetics in terms of the theory of nonequilibrium steady state, including cycle fluxes, waiting cycle times, generalized Haldane equality and fluctuation theorem. Then we continue to show several well-known phenomena in classical enzyme kinetics, e.g., kinetic proofreading and non-Michaelis-Menten-cooperativity.
Next we present an analytical theory for the one-dimensional chemical master equation (CME) of mesoscopic cellular biochemical processes. We take the phosphorylation-dephosphorylation signaling modules with positive feedback as an illustrative example. An emergent landscape plays the key role for understanding the stochastic dynamics. For systems with nonlinear bistability, there are three different time scales: (a) individual biochemical reactions, (b) nonlinear network dynamics approaching to attractors, and (c) cellular evolution (transitions among discrete stochastic attractors).
For high dimensional case, although studying the chemical master equation meets technical difficulties, Freidlin-Wentzell's large deviation theorem for singularly perturbed diffusion processes has been known for decades. Their stochastic dynamics are qualitatively similar. In the case of the corresponding ordinary differential equation having a stable limit cycle, the stationary solution of the singularly perturbed diffusion processes would exhibit a clear separation of the exponential terms and the prefactor of algebraic small contributions. The large deviation rate function must achieve its minimum zero on the entire stable limit cycle, while the leading term of the prefactor is inversely proportional to the velocity of the non-uniform periodic oscillation on the cycle. Further, limiting our discussion of dynamics on a circle with multiple attractors, both a local and a global landscapes arise and in the limit of zero noise, a Markov jumping process emerges. The nonequilibrium steady state corresponds to a discontinuity in the local landscape.
Hao Ge is Associate Professor at the School of Mathematical Sciences and Centre for Computational Systems Biology, Fudan University. For more information, please visit: http://homepage.fudan.edu.cn/~gehao/English%20version/main.htm