Large Differences Between Stochastic and Deterministic Kinetics in a Simple Autocatalytic Reaction Network

In small systems, quantitative discrepancies between stochastic and deterministic descriptions of chemical kinetics can be significant, with their magnitude depending on the specific reaction network. Here, we study the Finke-Watzky model-an irreversible autocatalysis, A + B -- > 2B, supplemented by an irreversible first-order process, A -- > B. This model has been used to describe the formation of transition metal nanoparticles and protein misfolding and aggregation, but it may also serve as a minimal model for the spread of a non-fatal but incurable disease. We show that, for certain parameter values, exceptionally large deviations can arise between stochastic and deterministic kinetics of the Finke-Watzky model. Moreover, its stochastic time evolution may be highly sensitive to initial conditions. These properties are retained in the generalization of the model to reversible reactions. To quantify the differences between the predictions of deterministic and stochastic kinetics, we derive the explicit analytical solution of the Chemical Master Equation for the Finke-Watzky model. This solution also allows us to derive analogous solutions for two related reaction networks: A + A -- > A + B, A -- > B, and A + A -- > A + B, A + B -- > 2B. Our findings may have implications for modeling epidemics and intracellular chemical processes, and more broadly for models of population dynamics.

Tomasz Bednarek, Jakub Jędrak · 2024