Folding kinetics of proteins with multiple domains: inference of transition rates from extinction times

Abstract

In this paper we are concerned with the folding kinetics of large proteins that may have multiple binding/unbinding domains. The theory underlying such systems have received much less attention compared to small, ‘two-state’ (folded/unfolded) proteins. To analyze this problem, we model the reaction coordinate using a birth-death chain, a well-known stochastic process. Specifically, we consider a protein with N domains where every domain that unfolds (folds) increases (decreases) the reaction coordinate by an integer so that the state-space of the reaction coordinate is the set of integers {0, 1, …, N}. As input data, our method uses (i) the extinction time of trajectories, starting from when the protein leaves the 0 state for the first time and finishing when the protein re-enters state 0 for the first time and (ii) the maximum number of unfolded domains in the said trajectory. Since the maximum value n reached by each trajectory is known, we use the proportion of trajectories that do not exceed n and corresponding mean extinction times (ET) to recover the birth-death rates sequentially from 1 to N. In general, the initial error will propagate with the site number exponentially. However, with sufficient amount of input data, we can recover the rates with relatively small error. For instance, given 50 million ETs of an 11-site BDP, we can recover the rates with a relative error of about 3%.