As the total number of molecules in a system decreases, transitions in the qualitative behavior of autocatalytic chemical reaction dynamics may appear. By adapting a method based on  the discrete-time Markov process , we provide here a general analytic tool for the reaction dynamics of systems with a small number of molecules. Bistability induced by the small-number effect in a two-component model is analyzed and excellent agreement with the simulated mean switching time is demonstrated. A novel transition involving the reversal of the chemical reaction flow is also found in a three-component model and quantified analytically using the proposed method .
Transitions in the qualitative behavior of chemical reaction dynamics with a decrease in molecule number have attracted much attention. Here, a method based on a Markov process with a tridiagonal transition matrix is applied to the analysis of this transition in reaction dynamics. The transition to bistability due to the small-number effect and the mean switching time between the bistable states are analytically calculated in agreement with numerical simulations. In addition, a novel transition involving the reversal of the chemical reaction flow is found in the model under an external flow, and also in a three-component model . The generality of this transition and its correspondence to biological phenomena are also discussed .
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As the total number of molecules in a system decreases, transitions in
Transitions in
D
autocatalytic
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may appear. By adapting
with a decrease in molecule number have attracted much attention. Here,
A
a Markov process with a tridiagonal transition matrix is applied to the analysis of this transition in reaction dynamics. The transition to bistability due to
D
discrete-time Markov process , we provide here a general analytic tool for the reaction dynamics of systems with a small number of molecules. Bistability induced by the
R
in a two-component model is analyzed and excellent agreement with the simulated
and the
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is demonstrated. A
between the bistable states are analytically calculated in agreement with numerical simulations. In addition, a
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also found in
found in the model under an external flow, and also in
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and quantified analytically using the proposed method
. The generality of this transition and its correspondence to biological phenomena are also discussed