Modern tools for the time-discrete dynamics and optimization of gene-environment networks

In this study, we discuss the models of genetic regulatory systems, so-called gene-environment networks. The dynamics of such kind of systems are described by a class of time-continuous ordinary differential equations having a general form (E) over dot = M(E)E, where E is a vector of gene-expression levels and environmental factors and M(E) is the matrix having functional entries containing unknown parameters to be optimized. Accordingly, time-discrete versions of that model class are studied and improved by introducing 3rd-order Heun's method and 4th-order classical Runge-Kutta method. The corresponding iteration formulas are derived and their matrix algebras are obtained. After that, we use nonlinear mixed-integer programming for the parameter estimation in the considered model and present the solution of a constrained and regularized given mixed-integer problem as an example. By using this solution and applying both the new and existing discretization schemes, we generate corresponding time-series of gene-expressions for each method. The comparison of the experimental data and the calculated approximate results is additionally done with the help of the figures to exercise the performance of the numerical schemes on this example. (C) 2011 Elsevier B.V. All rights reserved.