Mathematical models and algorithms for genetic regulatory networks

Abstract

Genetic regulatory network is an important research topic in bioinformatics, which considers the on-off switches and rheostats of a cell operating at the gene level. Mathematical modeling and computation are indispensable in such studies, especially for the complex patterns of behavior which needs high industrial payoffs and is difficult to get the information through experimental methods. Boolean networks (BNs) and probabilistic Boolean networks (PBNs) are proposed to model the interactions among the genes and have received much attention in the biophysics community. The study in this thesis is based on the BN and PBN models. With the BN model, several algorithms using gene ordering and feedback vertex sets are developed to identify small attractors which correspond to cell types and cell states. The average case time complexities of some proposed algorithms are analyzed. Extensive computational experiments are performed which are in good agreement with the theoretical results. A simple and complete proof for showing that finding an attractor with the shortest period is NP-hard is given. Finding global states incoming to a specified global state is useful for the preprocessing of finding control actions in BNs and for identifying the basin of attraction for a given attractor. This problem is NP-hard in general. New algorithms based on the algorithms for finding small attractors are developed, which are much faster than the naive exhaustive search-based algorithm. Based on the PBN model, an efficient method for the construction of the sparse transition probability matrix is proposed. Power method is then applied to compute the steady-state probability distribution. With this method, the sensitivity of the steady-state distribution to the influences of input genes, gene connections and Boolean functions is studied with simulations. An approximation method is proposed to further reduce the time complexity for the construction of the transition probability matrix by neglecting some BNs with very small probabilities. An error analysis of this approximation method is given and theoretical result on the distribution of BNs in a PBN with at most two Boolean functions for one gene is also presented. Numerical experiments are given to demonstrate the efficiency of the proposed method. The ultimate goal of studying the long-term behavior of the genetic regulatory network is to study the control strategies such that the system can go into the desirable states with larger probabilities. A control model is also proposed for gene intervention here. The problem is formulated as a minimization problem with integer variables to minimize the amount of control cost for a genetic network over a given period of time. Experimental results show that the proposed formulation is efficient and effective for solving the control problem of gene intervention.