The Memory Lag Regulates the Threshold Effect Within the Repressor Gene Module

The abundance level of the lambda phage repressor is randomly fluctuated by the influence of multiple ligand signals, inducing its phenotypic diversity. However, the mechanism of how the memory lag of fluctuations and the nonlinear structure of the expression system regulate the phenotypic diversity is still obscure. Here, we try to investigate a prototypical regulatory network coupled with random fluctuations (noise) and nonlinear structures to focus on the impact of protein abundance fluctuations within bacteriophages on cellular phenotypic transitions and energy dissipation throughout the process. Our findings reveal that there exists a threshold of the CI protein abundance to regulate the switching from the lysogenic to lytic states of the lambda phage, influencing its reproductive strategy. Specifically, an increase in the memory lag of multiplicative noise leads to a delayed transition from the lysogenic to lytic states. Additive noise exerts an effect that is nearly the opposite of that of multiplicative noise. Furthermore, we reconstruct the effective equivalent topological network to calculate the energy consumption cost of these switching. It is indicated that it only dissipates the lower energy to achieve the bimodality in a low-noise environment. In contrast, it needs to dissipate more energy to maintain the stability of the expression system in larger fluctuations. Comprehensive analysis suggests that lambda phages can optimize their survival strategies by modulating the cellular microenvironment, specifically through adjusting noise intensity and memory lag.