Fragility in a Togashi–Kaneko stochastic model with mutations

Abstract

The Togashi–Kaneko (TK) stochastic model is a prototypical example of an autocatalytic reaction network exhibiting dramatic switching behavior. The desire to understand this unusual behavior has attracted considerable attention in recent years. In this paper, we study the TK model with additional mutations. We establish a rigorous stochastic averaging principle that describes slow dynamics in terms of certain ergodic means of fast variables. Beginning with two species, we demonstrate a sensitivity of the model to even slight departures from symmetry in the autocatalytic reactions. We accomplish this through a detailed analysis of the stationary distribution of the fast process when the state of the slow process is fixed. We call this high sensitivity property “fragility”. We give some examples of behavior that can occur when there are more than two species. These preliminary explorations for multiple species point to a wealth of open questions for future research. Relevance to Life Sciences. Autocatalysis or self-amplification plays a key role in many biochemical and biological processes, ranging from pattern formation and self-organization, through gene regulation and signaling cascades, to ecological interactions and evolutionary dynamics. Understanding the sensitivity of such stochastic systems to small parameter changes is important for the formulation of models from data and for drawing conclusions for real life systems. In this paper we explore the sensitivity of the prototypical Togashi–Kaneko model with additional mutations. We find a high sensitivity to even slight departures from symmetry in the autocatalytic reactions, which we call fragility. We believe that fragility is an important underappreciated and understudied phenomenon, that will affect the formulation and interpretation of autocatalytic models across a wide variety of applications in the life sciences. Mathematical Content. We develop tools for analyzing and understanding the dynamic stochastic behavior of autocatalytic reaction systems, especially in asymmetric situations, by considering an extension of the standard TK model with additional mutation reactions. We prove a rigorous stochastic averaging principle that links the slow population dynamics with fast autocatalytic reactions. Through analysis of the ergodic mean of the fast variables (when the slow variables are frozen at a given value) for the two-species model, we find a high sensitivity of the model (which we call fragility) to even a slight departure from symmetry in autocatalytic rates. Furthermore, our preliminary explorations for more than two species suggest that such a phenomenon can occur for four species but not for three. This rather surprising observation suggests a wealth of open problems for future research.