Duration of switch activation and a biochemical timer

In analyzing the temporal behavior of a biochemical switching molecule, we can study either of the equivalent models of the phosphorylation-dephosphorylation cycle or the GTPase signaling module. In particular, we are interested in the duration of each activation event at the single-molecule level. 

For the GTPase system, every GTPase molecule can be in one of the four states. Using 1,2, 3, and 4 to represent the Ggdp, Ggdp GEF, Ggtp, and Ggtp GAP, we have from Equation (5.27) 

States 3 and 4 correspond to GTPase with GTP bound. Hence we are interested in the dwell time in state 3 and 4 (3 U 4). If we can determine the probability of a molecule remaining in state 3 U 4 at time t, given that the system was in state 3 U 4 at t = 0, then we will have the distribution of dwell times in the state. From the kinetics, the probabilities satisfy the equations 

This is a system of two linear ordinary differential equations. The eigenvalues of the system are 

If the system is in state 3 U 4 at t = 0, then the probability of remaining in the state at time t, pmm(i) in Equation (5.31), is the probability that the dwell time is greater than t. That is, pvjx(t) = Pr T t. The probability density function for the dwell time, T, then is 

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