Kinetics of autocatalysis in small systems

Abstract

Autocatalysis is a ubiquitous chemical process that drives a plethora of biological phenomena, including the self-propagation of prions etiological to the Creutzfeldt-Jakob disease and bovine spongiform encephalopathy. To explain the dynamics of these systems, we have solved the chemical master equation for the irreversible autocatalytic reaction A+B-->2A. This solution comprises the first closed form expression describing the probabilistic time evolution of the populations of autocatalytic and noncatalytic molecules from an arbitrary initial state. Grand probability distributions are likewise presented for autocatalysis in the equilibrium limit (A+B <==>2A), allowing for the first mechanistic comparison of this process with chemical isomerization (B<==>A) in small systems. Although the average population of autocatalytic (i.e., prion) molecules largely conforms to the predictions of the classical "rate law" approach in time and the law of mass action at equilibrium, thermodynamic differences between the entropies of isomerization and autocatalysis are revealed, suggesting a "mechanism dependence" of state variables for chemical reaction processes. These results demonstrate the importance of chemical mechanism and molecularity in the development of stochastic processes for chemical systems and the relationship between the stochastic approach to chemical kinetics and nonequilibrium thermodynamics.