Branchings and time evolution of reaction networks

In this thesis I analyze flows in reaction networks in terms of branchings in a digraph. If the coupled differential equations governing the rate of change of probabilities X of a state or species are finite-differenced in time, a matrix equation (I+ADelta t)X(t + Deltat) = X(t) results, where X( t) is a vector giving the probabilities at time t and X(t + Deltat) is a vector giving the probabilities at time t + Delta t. I demonstrate that the matrix (I + A Deltat) may be written as the product of an incidence matrix and a weight matrix for a directed graph (digraph) representing the network. From this I demonstrate that individual diagonal element of the inverse matrix (I + ADeltat)-1 may be written as a sum of the exponential weight of all branchings rooted at the vertex corresponding to the root vertex in the digraph divided by the sum of the exponential weight of all branchings. I also demonstrate that the individual element of the inverse matrix at row i, column j is the sum of exponential weights of all branchings rooted at vertex i but with a path from vertex i to vertex j in the digraph divided by the sum of exponential weights of all branchings. From this I demonstrate how to compute X( t + Deltat) from X(t ) in terms of sums of branchings and how to compute effective transition rates. I then consider long-term solutions and demonstrate how to condense linear networks that obey detailed balance. This provides a useful connection to equilibrium analysis of the network. I then consider some implications of the branching analysis for the statistical mechanics of reaction networks, and I extend the analysis to nonlinear networks. Finally I provide some example applications. I conclude that branchings in network digraphs hold promise for analyzing complicated reaction flows, and I list some future directions of possible research.