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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., Comment: 21 pages, 3 figures

Biochemical covalent modification networks exhibit a remarkable suite of steady state and dynamical properties such as multistationarity, oscillations, ultrasensitivity and absolute concentration robustness. This paper focuses on conditions required for a network to have a species with absolute concentration robustness. We find that the robustness in a substrate is endowed by its interaction with a bifunctional enzyme, which is an enzyme that has different roles when isolated versus when bound as a substrate-enzyme complex. When isolated, the bifunctional enzyme promotes production of more molecules of the robust species while when bound, the same enzyme facilitates degradation of the robust species. These dual actions produce robustness in the large class of covalent modification networks. For each network of this type, we find the network conditions for the presence of robustness, the species that has robustness, and its robustness value. The unified approach of simultaneously analyzing a large class of networks for a single property, i.e. absolute concentration robustness, reveals the underlying mechanism of the action of bifunctional enzyme while simultaneously providing a precise mathematical description of bifunctionality., Comment: 24 pages

Reaction networks are widely used models to describe biochemical processes. Stochastic fluctuations in the counts of biological macromolecules have amplified consequences due to their small population sizes. This makes it necessary to favor stochastic, discrete population, continuous time models. The stationary distributions provide snapshots of the model behavior at the stationary regime, and as such finding their expression in terms of the model parameters is of great interest. The aim of the present paper is to describe when the stationary distributions of the original model, whose state space is potentially infinite, coincide exactly with the stationary distributions of the process truncated to finite subsets of states, up to a normalizing constant. The finite subsets of states we identify are called copies and are inspired by the modular topology of reaction network models. With such a choice we prove a novel graphical characterization of the concept of complex balancing for stochastic models of reaction networks. The results of the paper hold for the commonly used mass-action kinetics but are not restricted to it, and are in fact stated for more general setting.

In a reaction network, the concentration of a species with the property of dynamic absolute concentration robustness (dynamic ACR) converges to the same value independent of the overall initial values. This property endows a biochemical network with output robustness and therefore is essential for its functioning in a highly variable environment. It is important to identify structure of the dynamical system as well as constraints required for dynamic ACR. We propose a power-engine-load form of dynamic ACR and obtain results regarding convergence to the ACR value based on this form.

Autocatalytic systems are very often incorporated in the "origin of life" models, a connection that has been analyzed in the context of the classical hypercycles introduced by Manfred Eigen. We investigate the dynamics of certain networks called bimolecular autocatalytic systems. In particular, we consider the dynamics corresponding to the relative populations in these networks, and show that they can be analyzed by studying well-chosen autonomous polynomial dynamical systems. Moreover, we find that one can use results from reaction network theory to prove persistence and permanence of several types of bimolecular autocatalytic systems called autocatalytic recombination networks., Comment: 24 pages, 6 figures

We introduce the notion of order of magnitude reversibility (OM-reversibility) in Markov chains that are parametrized by a positive parameter $\ep$. OM-reversibility is a weaker condition than reversibility, and requires only the knowledge of order of magnitude of the transition probabilities. For an irreducible, OM-reversible Markov chain on a finite state space, we prove that the stationary distribution satisfies order of magnitude detailed balance (analog of detailed balance in reversible Markov chains). The result characterizes the states with positive probability in the limit of the stationary distribution as $\ep \to 0$, which finds an important application in the case of singularly perturbed Markov chains that are reducible for $\ep=0$. We show that OM-reversibility occurs naturally in macroscopic systems, involving many interacting particles. Clustering is a common phenomenon in biological systems, in which particles or molecules aggregate at one location. We give a simple condition on the transition probabilities in an interacting particle Markov chain that characterizes clustering. We show that such clustering processes are OM-reversible, and we find explicitly the order of magnitude of the stationary distribution. Further, we show that the single pole states, in which all particles are at a single vertex, are the only states with positive probability in the limit of the stationary distribution as the rate of diffusion goes to zero., 22 pages, 3 figures