Quantitative mechanistic models based on reaction networks with stochastic chemical kinetics can help elucidate fundamental biological process where random fluctuations are relevant , such as in single cells. The dynamics of such models is described by the master equation , which provides the time course evolution of the probability distribution across the discrete state space consisting of vectors of population levels of the interacting biochemical species. Since solving the master equation exactly is very difficult in general due to the combinatorial explosion of the state space size , several analytical approximations have been proposed. The deterministic rate equation (DRE) offers a macroscopic view of the system by means of a system of differential equations that estimate the average populations for each species, but it may be inaccurate in the case of nonlinear interactions such as in mass-action kinetics . Here we propose finite state expansion (FSE), an analytical method that mediates between the microscopic and the macroscopic interpretations of a chemical reaction network by coupling the master equation dynamics of a chosen subset of the discrete state space with the  population dynamics of the DRE. This is done via an algorithmic translation of a chemical reaction network into a target expanded one where each discrete state is represented as a further distinct chemical species. The translation produces a network with stochastically equivalent dynamics, but the DRE of the expanded network can be interpreted as a correction to the original ones. Through a publicly available software implementation of FSE , we demonstrate its effectiveness in models from systems biology which challenge state-of-the-art techniques due to the presence of intrinsic noise, multi-scale population dynamics , and multi-stability.
Stochastic reaction networks are a fundamental model to describe interactions between species where random fluctuations are relevant . The master equation provides the evolution of the probability distribution across the discrete state space consisting of vectors of population counts for each species. However, since its exact solution is often elusive , several analytical approximations have been proposed. The deterministic rate equation (DRE) gives a macroscopic approximation as a compact system of differential equations that estimate the average populations for each species, but it may be inaccurate in the case of nonlinear interaction dynamics . Here we propose finite state expansion (FSE), an analytical method mediating between the microscopic and the macroscopic interpretations of a stochastic reaction network by coupling the master equation dynamics of a chosen subset of the discrete state space with the mean population dynamics of the DRE. An algorithm translates a network into an expanded one where each discrete state is represented as a further distinct species. This translation exactly preserves the stochastic dynamics, but the DRE of the expanded network can be interpreted as a correction to the original one. The effectiveness of FSE is demonstrated in models that challenge state-of-the-art techniques due to intrinsic noise, multi-scale populations , and multi-stability.
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Quantitative mechanistic models based on reaction networks with stochastic chemical kinetics can help elucidate fundamental biological process
Stochastic reaction networks are a fundamental model to describe interactions between species
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, such as in single cells. The dynamics of such models is described by the master equation , which provides the time course
. The master equation provides the
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levels of the interacting biochemical species. Since solving the master equation exactly is very difficult in general due to the combinatorial explosion of the state space size
counts for each species. However, since its exact solution is often elusive
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offers a macroscopic view of the system by means of a
gives a macroscopic approximation as a compact
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interactions such as in mass-action kinetics
interaction dynamics
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that mediates
mediating
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chemical
stochastic
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mean
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This is done via an algorithmic translation of a chemical reaction network into a target
An algorithm translates a network into an
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chemical species. The translation produces a network with stochastically equivalent
species. This translation exactly preserves the stochastic
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ones. Through a publicly available software implementation of FSE , we demonstrate its effectiveness in models from systems biology which
one. The effectiveness of FSE is demonstrated in models that
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the presence of
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population dynamics
populations