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Register a new userProduct-form stationary distributions for deficiency zero chemical reaction networks
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We consider stochastically modeled chemical reaction systems with mass-action kinetics and prove that a product-form stationary distribution exists for each closed, irreducible subset of the state space if an analogous deterministically modeled system with mass-action kinetics admits a complex balanced equilibrium. Feinbergs deficiency zero theorem then implies that such a distribution exists so long as the corresponding chemical network is weakly reversible and has a deficiency of zero. The main parameter of the stationary distribution for the stochastically modeled system is a complex balanced equilibrium value for the corresponding deterministically modeled system. We also generalize our main result to some non-mass-action kinetics.
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We consider the relationship between stationary distributions for stochastic models of reaction systems and Lyapunov functions for their deterministic counterparts. Specifically, we derive the well known Lyapunov function of reaction network theory as a scaling limit of the non-equilibrium potential of the stationary distribution of stochastically modeled complex balanced systems. We extend this result to general birth-death models and demonstrate via example that similar scaling limits can yield Lyapunov functions even for models that are not complex or detailed balanced, and may even have multiple equilibria.
It is well known that stochastically modeled reaction networks that are complex balanced admit a stationary distribution that is a product of Poisson distributions. In this paper, we consider the following related question: supposing that the initial distribution of a stochastically modeled reaction network is a product of Poissons, under what conditions will the distribution remain a product of Poissons for all time? By drawing inspiration from Crispin Gardiners Poisson representation for the solution to the chemical master equation, we provide a necessary and sufficient condition for such a product-form distribution to hold for all time. Interestingly, the condition is a dynamical complex-balancing for only those complexes that have multiplicity greater than or equal to two (i.e. the higher order complexes that yield non-linear terms to the dynamics). We term this new condition the dynamical and restricted complex balance condition (DR for short).
Deficiency zero is an important network structure and has been the focus of many celebrated results within reaction network theory. In our previous paper textit{Prevalence of deficiency zero reaction networks in an ErdH os-Renyi framework}, we provided a framework to quantify the prevalence of deficiency zero among randomly generated reaction networks. Specifically, given a randomly generated binary reaction network with $n$ species, with an edge between two arbitrary vertices occurring independently with probability $p_n$, we established the threshold function $r(n)=frac{1}{n^3}$ such that the probability of the random network being deficiency zero converges to 1 if $frac{p_n}{r(n)}to 0$ and converges to 0 if $frac{p_n}{r(n)}toinfty$, as $n to infty$. With the base ErdH os-Renyi framework as a starting point, the current paper provides a significantly more flexible framework by weighting the edge probabilities via control parameters $alpha_{i,j}$, with $i,jin {0,1,2}$ enumerating the types of possible vertices (zeroth, first, or second order). The control parameters can be chosen to generate random reaction networks with a specific underlying structure, such as closed networks with very few inflow and outflow reactions, or open networks with abundant inflow and outflow. Under this new framework, for each choice of control parameters ${alpha_{i,j}}$, we establish a threshold function $r(n,{alpha_{i,j}})$ such that the probability of the random network being deficiency zero converges to 1 if $frac{p_n}{r(n,{alpha_{i,j}})}to 0$ and converges to 0 if $frac{p_n}{r(n,{alpha_{i,j}})}to infty$.
Reaction networks are commonly used within the mathematical biology and mathematical chemistry communities to model the dynamics of interacting species. These models differ from the typical graphs found in random graph theory since their vertices are constructed from elementary building blocks, i.e., the species. In this paper, we consider these networks in an ErdH os-Renyi framework and, under suitable assumptions, derive a threshold function for the network to have a deficiency of zero, which is a property of great interest in the reaction network community. Specifically, if the number of species is denoted by $n$ and if the edge probability is denote by $p_n$, then we prove that the probability of a random binary network being deficiency zero converges to 1 if $frac{p_n}{r(n)}to 0$, as $n to infty$, and converges to 0 if $frac{p_n}{r(n)}to infty$, as $n to infty$, where $r(n)=frac{1}{n^3}$.
Mass-action kinetics is frequently used in systems biology to model the behaviour of interacting chemical species. Many important dynamical properties are known to hold for such systems if they are weakly reversible and have a low deficiency. In particular, the Deficiency Zero and Deficiency One Theorems guarantee strong regularity with regards to the number and stability of positive equilibrium states. It is also known that chemical reaction networks with disparate reaction structure can exhibit the same qualitative dynamics. The theory of linear conjugacy encapsulates the cases where this relationship is captured by a linear transformation. In this paper, we propose a mixed-integer linear programming algorithm capable of determining weakly reversible reaction networks with a minimal deficiency which are linearly conjugate to a given reaction network.
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