TRANSITION AND TRANSIT-TIME DISTRIBUTIONS FOR TIME-DEPENDENT REACTIONS WITH APPLICATION TO BIOCHEMICAL NETWORKS

Temporal aspects of the dynamic behavior of biochemical pathways in st ationary states have been described by a transition time tau, which is the ratio of the sum of the pool concentrations of chemical intermedi ates to the flux for a given stationary state. In this paper, a relate d random variable is introduced, the transit rime theta, which is defi ned as the age of (metabolic) intermediates at the time of leaving the system. The theory, based on a semi-stochastic approach, leads to cal culations of the probability distributions of the ages of the intermed iates, as functions of time. By assuming that the kinetics of the path way is described by mass-action laws, a system of partial differential equations is derived for the distribution function of the transit tim e. By using the method of characteristics the solving of the evolution equations for the distribution function is reduced to the solving of the kinetic equations of the process. The method is applied to a simpl e enzyme-substrate reaction operated in two different regimes: (1) wit h a constant input of reagent and (2) with a periodically varying inpu t. In the first case the transit time probability distributions in the steady state are calculated both analytically and numerically. The me an transit time, calculated as the first moment of the distribution, c oincides with the transition time calculated in the literature. In add ition, the presented approach provides information concerning the fluc tuations of the transit time. For a periodic input, the distribution f unction of transit times can be evaluated semianalytically by using th e technique of Green functions. We show that in this case the distribu tion oscillates in time, and both the distribution of the transit time and its different moments and cumulants oscillate.