Single-cell gene expression is inherently stochastic , its emergent behavior can be defined in terms of the chemical master equation describing the evolution of the mRNA and protein copy numbers as the latter N\rightarrow\infty . We establish two types of "macroscopic limits": the Kurtz limit is consistent with the classical chemical kinetics, while the L\'evy limit provides a theoretical foundation for an empirical equation proposed in [Phys. Rev. Lett. 97:168302, 2006]. Furthermore, we clarify the biochemical implications and ranges of applicability for various macroscopic limits and calculate a comprehensive analytic expression for the protein concentration distribution in autoregulatory gene networks. The relationship between our work and modern population genetics is discussed.
Single-cell gene expression is inherently stochastic ; its emergent behavior can be defined in terms of the chemical master equation describing the evolution of the mRNA and protein copy numbers as the latter tends to infinity . We establish two types of "macroscopic limits": the Kurtz limit is consistent with the classical chemical kinetics, while the L\'{e limit provides a theoretical foundation for an empirical equation proposed in [Phys. Rev. Lett. 97:168302, 2006]. Furthermore, we clarify the biochemical implications and ranges of applicability for various macroscopic limits and calculate a comprehensive analytic expression for the protein concentration distribution in autoregulatory gene networks. The relationship between our work and modern population genetics is discussed.
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N\rightarrow\infty
tends to infinity
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L\'evy
L\'{e