The effect of coupled stochastic processes in a two-state biochemical switch.

Cell signaling pathways consist of multiple connections of different types of gene, mRNA and protein networks. It is not a trivial task to follow the signals flowing through these networks. The difficulty comes from considering the entire biological structure as a single network without breaking it into connected modules. The study of these networks simplifies if the complex system is reduced to a hierarchy of interconnected modules. Out of many potential modules, a specific one, namely the Goldbeter-Koshland switch, was encountered by the authors during their study of the Mammalian Heat Shock Response Network (MHSRN) where the switch acts as a stress sensor. Usually, only the steady state behavior of the switch is studied, in which the phosphorylated protein is given as a function of the enzyme concentration. Experimental results show that the heat shock response is still present 20 h after the temperature stress had ended. Thus, it is useful to analyze the transient behavior of the switch that couples the environment to the MHSRN. A stochastic model for the switch is proposed using the Master Equation which is subsequently transformed into an equation for the factorial cumulant generating function. This generating function can be easily read from a graphical representation of the stochastic switch. The second order approximation of the equation for the factorial cumulant generating function is solved and the time dependence of the transient regime of the mean and standard deviation is readily obtained. Using the mean and standard deviation of the switch's output as a function of the stochastic input signals that represent the environment, we classify the switches according to different criteria. The switches differ by the numerical values of the parameters that characterize the switch's chemical reactions. The classifying criteria will distinguish the switches by the levels of the response for a given transition time and by the sensitivity of the response to the enzyme levels. It is also found that the environment can drastically change the response of the switch, which has important biological consequences.