Unveiling the hidden structure of complex stochastic biochemical networks.

Complex Markov models are widely used and powerful predictive tools to analyze stochastic biochemical processes. However, when the network of states is unknown, it is necessary to extract information from the data to partially build the network and estimate the values of the rates. The short-time behavior of the first-passage time distributions between two states in linear chains has been shown recently to behave as a power of time with an exponent equal to the number of intermediate states. For a general Markov model we derive the complete Taylor expansion of the first-passage time distribution between two arbitrary states. By combining algebraic methods and graph theory approaches it is shown that the first term of the Taylor expansion is determined by the shortest path from the initial state to the final state. When this path is unique, we prove that the coefficient of the first term can be written in terms of the product of the transition rates along the path. It is argued that the application of our results to first-return times may be used to estimate the dependence of rates on external parameters in experimentally measured time distributions.