Markov chain as a model of Hill's theory on circulation

Establishing a model for various kinds of vivid metabolic systems of living organisms is an subject, uneasy but of great significance. Hill has constructed a theoretical framework for discussions of metabolic systems, such as active transport, muscle contraction, etc. The basic method of his framework is diagram calculation for the cycle flux on the metabolic cycles of those systems [2, 7] He has successively found that the result from diagram calculations agrees with the data of the numbers of completing different cycles given by random test (Monte Carlo test), but has not yet proved that the former is just the circulation rate in the sense of trajectory. In this paper, the model of a Markov chain is used to define the circulation rate in the sense of trajectories and, by deriving the remaining Markov chains, we calculate the expression of circulation rate which coincides with Hill's result obtained from diagrams. And hence we verify that Hill's cycle flux is equivalent to the circulation rate defined by trajectory. Besides, this paper establishes a connection between Hill's free energy of small systems and the potential of a Markov chain which is suggested by Hou Zhenting.