It has recently been shown that structural conditions on the reaction network, rather than a ` fine-tuning' of system parameters, often suffice to impart ` absolute concentration robustness' on a wide class of biologically relevant, deterministically modeled mass-action systems [Shinar and Feinberg, Science, 2010]. Many biochemical networks, however, operate on a scale insufficient to justify the assumptions of the deterministic mass-action model, which raises the question of whether the long-term dynamics of the systems are being accurately captured when the deterministic model predicts stability. We show here that fundamentally different conclusions about the long-term behavior of such systems are reached if the systems are instead modeled with stochastic dynamics and a discrete state space. Specifically  we characterize a large class of models which exhibit convergence to a positive robust equilibrium in the deterministic setting, whereas trajectories of the corresponding stochastic models are necessarily absorbed by a set of states that reside on the boundary of the state space . If the time to absorption is large relative to the relevant time-scales of the system, the process will very likely seem to settle down to an equilibrium long before the resulting instability will appear . This quasi-stationary distribution is considered for two systems taken from the literature, and results consistent with absolute concentration robustness are recovered by characterizing the discrepancy between the quasi-stationary distribution for the robust species and a Poisson distribution.
It has recently been shown that structural conditions on the reaction network, rather than a ' fine-tuning' of system parameters, often suffice to impart ' absolute concentration robustness' on a wide class of biologically relevant, deterministically modeled mass-action systems [Shinar and Feinberg, Science, 2010]. We show here that fundamentally different conclusions about the long-term behavior of such systems are reached if the systems are instead modeled with stochastic dynamics and a discrete state space. Specifically , we characterize a large class of models that exhibit convergence to a positive robust equilibrium in the deterministic setting, whereas trajectories of the corresponding stochastic models are necessarily absorbed by a set of states that reside on the boundary of the state space , i.e. the system undergoes an extinction event. If the time to extinction is large relative to the relevant time-scales of the system, the process will appear to settle down to a stationary distribution long before the inevitable extinction will occur . This quasi-stationary distribution is considered for two systems taken from the literature, and results consistent with absolute concentration robustness are recovered by showing that the quasi-stationary distribution of the robust species approaches a Poisson distribution.
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Many biochemical networks, however, operate on a scale insufficient to justify the assumptions of the deterministic mass-action model, which raises the question of whether the long-term dynamics of the systems are being accurately captured when the deterministic model predicts stability.
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