Prevalence of multistationarity and absolute concentration robustness in reaction networks

TL;DR

This paper studies how often multistationarity and absolute concentration robustness occur in reaction networks, showing they are generally rare and rarely occur together in large, random reversible networks.

Contribution

It provides probabilistic thresholds for the emergence of multistationarity and ACR in random networks, highlighting their rarity and the conditions under which they co-occur.

Findings

Multistationarity appears at certain edge probabilities in random networks.

ACR becomes rare as networks grow larger.

Both properties rarely occur together in large, random reversible networks.

Abstract

For reaction networks arising in systems biology, the capacity for two or more steady states, that is, multistationarity, is an important property that underlies biochemical switches. Another property receiving much attention recently is absolute concentration robustness (ACR), which means that some species concentration is the same at all positive steady states. In this work, we investigate the prevalence of each property while paying close attention to when the properties occur together. Specifically, we consider a stochastic block framework for generating random networks, and prove edge-probability thresholds at which - with high probability - multistationarity appears and ACR becomes rare. We also show that the small window in which both properties occur only appears in networks with many species. Taken together, our results confirm that, in random reversible networks, ACR and multistationarity together, or even ACR on its own, is highly atypical. Our proofs rely on two prior results, one pertaining to the prevalence of networks with deficiency zero, and the other "lifting" multistationarity from small networks to larger ones.