Fig. 1
From: The Chemical Fluctuation Theorem governing gene expression
Chemical Fluctuation Theorem (CFT) applied to various transcription models. Equation (1) relates the variance of mRNA number to the TCF of the transcription rate, whose mathematical form depends on the transcription network model. a In Model I, the transcription rate is given by constant κ. In Model II, the gene state switches stochastically between the active state (ξ = 1) and the inactive state (ξ = 0), for which the transcription rate is given by \(\kappa \xi\), with κ being a constant and ξ being a stochastic variable. In Model III, the transcription rate is given by \(\kappa (\it\Gamma )\xi\), where \(\kappa (\it\Gamma )\) is a stochastic variable dependent on the cell state \(\it\Gamma\). For each transcription model, the CFT yields the variance in the mRNA number. Non-Poisson mRNA noise for single-gene transcription is defined by \(\left( {\sigma _{n,1}{\mathrm{/}}\left\langle n \right\rangle _1} \right)^2 - \left\langle n \right\rangle _1^{ - 1}\), where \(\sigma _{n,1}^2\) and \(\langle n\rangle _1\) denote the variance and mean of the number of mRNA produced by a single-gene copy, denoted by \(Q_{n,1}{\mathrm{/}}\left\langle n \right\rangle _1\) in the main text (\(Q_{n,1}\) denotes \(\sigma _{n,1}^2{\mathrm{/}}\left\langle n \right\rangle _1 - 1\)). The duration time of each repressed and unrepressed gene state is reported to be an exponentially distributed random variable36,56; the mean is, respectively, denoted by \(k_{\mathrm{on}}^{ - 1}\) and \(k_{\mathrm{off}}^{ - 1}\). For mRNA lifetimes, we assume an arbitrary distribution, \(\psi _{{ d}}({t})\). The analytic expression of the non-Poisson mRNA noise obtained from Eq. (1) is tabulated for each model (Supplementary Methods). \(\eta _q^2\) denotes the relative variance, \(\left\langle {\delta q^2} \right\rangle {\mathrm{/}}\left\langle {q^2} \right\rangle\), of variable \(q\), \((q \in \{ \kappa ,\xi \} )\). The respective susceptibilities \(\chi _{n\xi }\), \(\chi _{n\kappa }\), and \(\chi _{n(\kappa ,\xi )}\) of the mRNA noise to \(\eta _\xi ^2\), \(\eta _\kappa ^2\), and \(\eta _\xi ^2\eta _\kappa ^2\) are determined by TCFs of the fluctuations in transcription rate factors, κ and ξ, and the survival probability of mRNA (see the text below Eq. (2)). b Non-Poisson mRNA noise among cells with multiple gene copies. The gene copy number, g, is a stochastic variable. For this system, the non-Poisson mRNA noise is defined by \(\left( {\sigma _n{\mathrm{/}}\left\langle n \right\rangle } \right)^2 - \left\langle n \right\rangle ^{ - 1}\left( { \equiv Q_n{\mathrm{/}}\left\langle n \right\rangle } \right)\). \(Q_n\) denotes \(\sigma _n^2{\mathrm{/}}\left\langle n \right\rangle - 1\), with \(\sigma _n^2\) and \(\langle n\rangle\) being the variance and mean, respectively, in the number of mRNA copies across the cells with gene copy number variation. For all the three models, Model I–III, additional non-Poisson mRNA noise emerges from the gene copy number variation. \(\eta _g^2\) denotes the relative variance \(\sigma _g^2{\mathrm{/}}\left\langle g \right\rangle ^2\) in the gene copy number. For Model III, the non-Poisson mRNA noise among cells with gene copy number variation also emerges from the environment-induced correlation between the transcription levels of different gene copies. C n denotes the mean-scaled correlation between the number \(n_1\) of mRNAs produced by the first gene copy and the number \(n_j\) of mRNA produced by another gene copy, e.g., the j-th, i.e., \(C_n=\left\langle{\delta n_1\delta n_j}\right\rangle{\mathrm{/}}\left\langle{n_1}\right\rangle\left\langle{n_j}\right\rangle\,\,\left({j\ne1}\right)\)
