Identifying Kinetic Constants by the Intrinsic Properties of Markov Chain

Abstract

The process underlying the opening and closing of ion channels in biological can be modelled kinetically as a time-homogeneous Markov chain. How to identify the kinetic constants (transition rates) that measure the 'speed' to jump from one state to another plays a very important role in ion channels. Maximum likelihood method is widely employed for estimating the kinetic constants. However it leads to the non-identifiability since the joint probability distributions could be the same to models with different generator matrices, and the estimation could be very rough since it involves the estimating of some latent variables. Here we develop a totally different approach to supply a gap. Our algorithms employ the intrinsic properties of the Markov process and all calculations are simply reduced to the estimation of their PDFs (probability density functions) of lifetime and death-time of observable states. Once we have them, all subsequent calculations are then automatic and exact. In the current paper, four classical mechanisms: star-graph, line,star-graph branch and (reversible) cyclic chain, are considered to single-ion channels. It is found that all kinetic constants are uniquely determined by the PDFs of their lifetime and death-time for partially (a few) observable states. Numerical examples are included to demonstrate the application of our approach to data.