Gene expression in individual cells is an inherently stochastic process with large fluctuations. Here we present a comprehensive analysis for stochastic gene expression kinetics in a minimal coupled gene circuit with positive-plus-negative feedback. Our theory unifies and generalizes the discrete and continuous gene expression models proposed previously by viewing the latter as various macroscopic limits of the former. Two types of macroscopic limits are obtained : the Kurtz limit applies to proteins with large burst frequencies and the L\'{e}vy limit applies to proteins with large burst sizes. We also derive the analytic steady-state distributions of the protein abundance  for both the discrete chemical master equation model and its two macroscopic limits . Furthermore, we ] obtain the analytic time-dependent distribution of the protein concentration for the classical Friedman-Cai-Xie random bursting model   . Our analytic results reveal a strong synergistic interaction between positive and negative feedback loops and a critical phase-transition-like phenomenon in the regime of slow promoter switching. Our theory is also applied to study the intrinsic noise structure of stochastic gene expression in coupled gene circuits and a complete decomposition of noise in terms of five different biophysical origins is provided.
Here we investigate single-cell stochastic gene expression kinetics in a minimal coupled gene circuit with positive-plus-negative feedback. A triphasic stochastic bifurcation upon the increasing ratio of the positive and negative feedback strengths is observed, which reveals a strong synergistic interaction between positive and negative feedback loops. We discover that coupled positive-plus-negative feedback amplifies gene expression mean but reduces gene expression noise over a wide range of feedback strengths when promoter switching is relatively slow, stabilizing gene expression around a relatively high level. In addition, we study two types of macroscopic limits of the discrete chemical master equation model : the Kurtz limit applies to proteins with large burst frequencies and the L\'{e}vy limit applies to proteins with large burst sizes. We derive the analytic steady-state distributions of the protein abundance in a coupled gene circuit for both the discrete model and its two macroscopic limits , generalizing the results obtained in Chaos 26:043108, 2016]. We also obtain the analytic time-dependent protein distribution for the classical Friedman-Cai-Xie random bursting model proposed in Phys. Rev. Lett. 97:168302, 2006 . Our analytic results are further applied to study the structure of gene expression noise in a coupled gene circuit and a complete decomposition of noise in terms of five different biophysical origins is provided.
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Gene expression in individual cells is an inherently stochastic process with large fluctuations. Here we present a comprehensive analysis for
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Our theory unifies and generalizes the discrete and continuous gene expression models proposed previously by viewing the latter as various macroscopic limits of the former. Two
A triphasic stochastic bifurcation upon the increasing ratio of the positive and negative feedback strengths is observed, which reveals a strong synergistic interaction between positive and negative feedback loops. We discover that coupled positive-plus-negative feedback amplifies gene expression mean but reduces gene expression noise over a wide range of feedback strengths when promoter switching is relatively slow, stabilizing gene expression around a relatively high level. In addition, we study two
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reveal a strong synergistic interaction between positive and negative feedback loops and a critical phase-transition-like phenomenon in the regime of slow promoter switching. Our theory is also
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