Moment dynamics for gene regulation with rational rate laws

This aim of this paper is mainly to investigate the performance of two typical moment closure schemes in gene regulatory master equations of rational rate laws. When the reaction rate is polynomial, the error bounds between the authentic and approximate moments obtained by schemes of Gaussian moment closure and log-normal moment closure are explicitly given. When the reaction rate is not polynomial, it is shown that the two schemes both behave well in the absence of active-inactive state switch, but in the presence of active-inactive state switch the log-normal closure scheme is far superior to the Gaussian closure scheme in capturing the asymptotic ensemble statistics. Moreover, the accuracy of the log-normal closure method is further confirmed by steady-state analytic results and the conditional Gaussian closure method. It is also disclosed that optimal negative feedback exists in suppressing protein noise in the presence of the on-off switch control.

Figure 1

A gene autoregulation model. Protein $X$ is produced at a rate $k_{p}f\left(p\right)$ following binding of RNA polymerase to a promotor site and degraded at a rate $d_p$. For simplicity, the stages of transcription and translation are lumped together.

Figure 2

Moment dynamics for the autoregulation gene without active-inactive state switch. The time evolution of the mean (a), (c) and standard deviation (b), (d) of protein copy numbers. In (a) and (b), the negative feedback strength is weak, that is, $a=0.02,$ and in (c) and (d), the negative feedback strength is strong, that is, $a=0.5$. The other parameters are taken as $k_p=5.5{\mathrm{h}}^{-1}, d_p=0.1{\mathrm{h}}^{-1}$, and $V=0.5$. The initial conditions are chosen as $p\left(0\right)=10$.

Figure 3

A two-state gene regulation model. Due to the binding and unbinding of an activator protein $Y$ to a promotor site, the model illustrates that a gene can be in ON or OFF states with the transition rates $k_0$ and $k_1$. Protein $X$ is produced at a rate $k_{p}f\left(p\right)$ and degraded at a rate $d_p$ in the ON state, while there is no protein production in the OFF state. For simplicity, the stages of transcription and translation are lumped together.

Figure 4

Moment dynamics for an autoregulation gene with state switch. The time evolution of the mean (a), (c) and standard deviation (b), (d) of protein copy numbers. In (a) and (b), the negative feedback strength is weak, that is, $a=0.02,$ and in (c) and (d), the negative feedback strength is strong, that is, $a=0.5$. The other parameters are taken as $k_0=0.5{\mathrm{h}}^{-1}, k_1=0.1{\mathrm{h}}^{-1}, k_p=5.5{\mathrm{h}}^{-1}, d_p=0.1{\mathrm{h}}^{-1}$, and $V=0.5$. The initial conditions are chosen as $p\left(0\right)=10$ and $g\left(0\right)=1$.

Figure 5

Dependence of the steady-state statistics obtained from log-normal moment closure for the autoregulation gene without state switch on the feedback strength under different regulation coefficients: $V=1.5$ (blue solid), $V=1$ (green dashed), and $V=0.5$ (red dot). The analytical statistics are denoted by black plus symbols. The parameters are the same as in Fig. 2.

Figure 6

Dependence of the steady-state statistics obtained from log-normal moment closure for the autoregulation gene with state switch on the feedback strength under different regulation coefficients: $V=1.5$ (blue solid), $V=1$ (green dashed), and $V=0.5$ (red dot). The other parameters are the same as in Fig. 4.

Figure 7

The time evolution of the low-order moments of the autoregulation gene with a cooperative effect (with Hill coefficient $n=2$). The middle green curve stands for the mean value of the protein copy number, the upper blue curve denotes the mean plus standard deviation, and the lower red curve represents the mean minus the standard deviation. In the pictures, the solid curves are the results from the log-normal moment method, and the circles are results from SSA, with the first row without active-inactive state switch (a), (b) and the second row with active-inactive state switch (c), (d). In (a) and (b), the negative feedback strength is weak, that is, $a=0.02,$ and in (c) and (d), the negative feedback strength is strong, that is, $a=0.5$.