Inference of temporally evolving network dynamics with applications in biological systems

Modeling of biological signal pathways forms the basis of systems biology. Also, the problem of identifying dynamics of biological networks is of critical importance in order to understand biological systems. In this paper, we propose a data-driven inference scheme to identify dynamics with a local point of view, the Jacobian matrix. A graph model is a natural way to represent a biological signal pathway and doesn´t require any constraints on dynamics such as mass action kinetics or Hill function representations, used in Ordinary Differential Equation (ODE) models. A graph is a set of vertices which represents state, and a set of edges which depicts the relationship or connection between two or more states. Once a system is abstracted by a graph, in order to identify the activity level of the corresponding interactions based on a given data set, we reformulate the problem as a Linear Quadratic (LQ) Optimal Control problem by transforming the unknown entries of the activity of edges into the control inputs of the LQ setting. In the formulation of the LQ problem, we use an adjacency map as a priori information and define a performance index which both drives the connectivity of the graph to match the biological data as well as generates a sparse network. Through simulation studies on simple examples, it is shown that this scheme can help to capture the topological change of a biological signal pathway and show the influence or activity of each edge over time. Also, we sketch briefly the potential application of this approach to correcting the graph model.