Exact results for queuing models of stochastic transcription with memory and crosstalk

Gene transcription is a complex multistep biochemical process, which can create memory between individual reaction events. On the other hand, many inducible genes, when activated by external cues, are often coregulated by several competitive pathways with crosstalk. This raises an unexplored question: how do molecular memory and crosstalk together affect gene expressions? To address this question, we introduce a queuing model of stochastic transcription, where two crossing signaling pathways are used to direct gene activation in response to external signals and memory functions to model multistep reaction processes involved in transcription. We first establish, based on the total probability principle, the chemical master equation for this queuing model, and then we derive, based on the binomial moment approach, exact expressions for statistical quantities (including distributions) of mRNA, which provide insights into the roles of crosstalk and memory in controlling the mRNA level and noise. We find that molecular memory of gene activation decreases the mRNA level but increases the mRNA noise, and double activation pathways always reduce the mRNA noise in contrast to a single pathway. In addition, we find that molecular memory can make the mRNA bimodality disappear.

Figure 1

A schematic diagram for a gene model with crosstalk and memory, where the gene is activated by one of two signaling pathways with crosstalk, and functions $f_1\left(t\right)$, $f_2\left(t\right),$ and $h\left(t\right)$ model possible multistep processes involved in gene expression.

Figure 2

Influence of molecular memory on the mRNA noise, where solid lines represent theoretical results and circles represent numerical results obtained by a numerical method [40] (see Appendix pp2 for a detailed description). (a) mRNA noise intensity via memory index; (b) skewness via memory index; (c) kurtosis via memory index. Parameter values are set as $I=2\alpha$, $\beta =0.1,$ and $\mu =10$, and the mean waiting time of γ distribution is set as a constant. Numerical results are obtained by averaging 500 simulations (the total time of each simulation is set as $t=1000$).

Figure 3

Influence of molecular memory on the mRNA distribution, where solid lines represent theoretical results and circles represent numerical results obtained by a numerical method [40] (see Appendix pp2 for a description). Parameter values are set as $\beta =0.1$ and $\mu =10$ with (a) $I=1,$ (b) $I=4,$ (c) $I=20$, and $I=2\alpha$ (since the mean waiting time of γ distribution is fixed).

Figure 4

The joint effect of crosstalk and memory on the mRNA expression. These figures show how the difference of the mRNA noise intensity for double channels minus that for a single channel depends on pathway-selecting probability ($q$) and memory index ($I_1$). The top three panels correspond to big differences of mean off dwell time in the case of double pathways: ${\alpha }_1=I_1$, ${\alpha }_2=3I_2/\phantom{3I_{2}5}5$, where the values of memory index $I_2$ are set as (a) $I_2=1,$ (b) $I_2=5,$ and (c) $I_2=10.$ The bottom three panels correspond to small differences of mean off dwell time in the case of double pathways: ${\alpha }_1=I_1$, ${\alpha }_2=I_2/\phantom{I_{2}10}10$, where the values of memory index $I_2$ are set as (d) $I_2=1,$ (e) $I_2=5,$ and (f) $I_2=10$. Other parameter values are set as $\beta =0.1$, $\mu =10$, $\alpha =1$, and $I=\alpha \left[q\left(\frac{I_1}{{\alpha }_1}-\frac{I_2}{{\alpha }_2}\right)+\frac{I_2}{{\alpha }_2}\right]$. Here parameter values are chosen so that the parameter conditions in Ref. [43] are satisfied.