Mechanisms and topology determination of complex chemical and biological network systems from first-passage theoretical approach

Abstract

The majority of chemical and biological processes can be viewed as complex networks of states connected by dynamic transitions. It is fundamentally important to determine the structure of these networks in order to fully understand the mechanisms of underlying processes. A new theoretical method of obtaining topologies and dynamic properties of complex networks, which utilizes a first-passage analysis, is developed. Our approach is based on a hypothesis that full temporal distributions of events between two arbitrary states contain full information on number of intermediate states, pathways, and transitions that lie between initial and final states. Several types of network systems are analyzed analytically and numerically. It is found that the approach is successful in determining structural and dynamic properties, providing a direct way of getting topology and mechanisms of general chemical network systems. The application of the method is illustrated on two examples of experimental studies of motor protein systems. © 2013 AIP Publishing LLC.

Pathway structure determination in complex stochastic networks with non-exponential dwell times

“…We have found a universal identity that connects dynamic information, which can be obtained from experimental measurements, with the number of states along the selected path as a first step to unveil the complete structure of the complex network. It is shown that this general result reduces to previously found relations 25,26 for exponential dwell times. Our theoretical predictions are supported by Monte Carlo computer simulations for several networks with different topology.…”

“…It suggests that the first-passage time probability function f ij (t) at early times has a power-law dependence (∼t β ), which is similar to results derived in previous studies with exponential waiting times. 25,26 The corresponding exponent β satisfies the equation…”

“…In the following, we assume that t 0 = 1 s. For γ k = 1 and α = 0 we recover the exponential waiting times from previous studies. 25,26 Equation (10) is, in essence, a mathematical generalization of the Gamma distribution, which can be obtained by setting γ k = 1. We have allowed for the possibility to take γ k = 1 in order to have the most general dwell time distribution that we are able to treat mathematically with the method described in this work.…”

“…25,26 At early time, the probability density of the firstpassage time between two states on a linear network has been found to follow a power law behavior. 25 It was proposed that similar relationships hold for general complex networks, and it was supported by Monte Carlo computer simulations. Later, the conjecture was proved to be correct for the network of any topology by deriving the Taylor series of first-passage time probability density using graph theory methods.…”

“…By applying this theoretical method for several motor protein systems, it was argued that it might be especially useful for analyzing single-molecule experiments in various chemical and biological systems. 25 However, this approach explicitly assumed that all dynamic transitions between states are Poissonian, i.e., the corresponding dwell times on each single state are exponential. However, there is a large number of natural and industrial processes that involve non-exponential waiting times.…”