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Abstract Continuous time Markov chains are commonly used as models for the stochastic behavior of chemical reaction networks. More precisely, these Stochastic Chemical Reaction Networks (SCRNs) are frequently used to gain a mechanistic understanding of how chemical reaction rate parameters impact the stochastic behavior of these systems. One property of interest is mean first passage times (MFPTs) between states. However, deriving explicit formulas for MFPTs can be highly complex. In order to address this problem, we first introduce the concept of$$coclique\, level\, structure$$and develop theorems to determine whether certain SCRNs have this feature by studying associated graphs. Additionally, we develop an algorithm to identify, under specific assumptions, all possible coclique level structures associated with a given SCRN. Finally, we demonstrate how the presence of such a structure in a SCRN allows us to derive closed form formulas for both upper and lower bounds for the MFPTs. Our methods can be applied to SCRNs taking values in a generic finite state space and can also be applied to models with non-mass-action kinetics. We illustrate our results with examples from the biological areas of epigenetics, neurobiology and ecology.
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Comparison Theorems for Stochastic Chemical Reaction Networks
Abstract Continuous-time Markov chains are frequently used as stochastic models for chemical reaction networks, especially in the growing field of systems biology. A fundamental problem for these Stochastic Chemical Reaction Networks (SCRNs) is to understand the dependence of the stochastic behavior of these systems on the chemical reaction rate parameters. Towards solving this problem, in this paper we develop theoretical tools called comparison theorems that provide stochastic ordering results for SCRNs. These theorems give sufficient conditions for monotonic dependence on parameters in these network models, which allow us to obtain, under suitable conditions, information about transient and steady-state behavior. These theorems exploit structural properties of SCRNs, beyond those of general continuous-time Markov chains. Furthermore, we derive two theorems to compare stationary distributions and mean first passage times for SCRNs with different parameter values, or with the same parameters and different initial conditions. These tools are developed for SCRNs taking values in a generic (finite or countably infinite) state space and can also be applied for non-mass-action kinetics models. When propensity functions are bounded, our method of proof gives an explicit method for coupling two comparable SCRNs, which can be used to simultaneously simulate their sample paths in a comparable manner. We illustrate our results with applications to models of enzymatic kinetics and epigenetic regulation by chromatin modifications.
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- PAR ID:
- 10404969
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Bulletin of Mathematical Biology
- Volume:
- 85
- Issue:
- 5
- ISSN:
- 0092-8240
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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