Mathematical contributions to dynamics and optimization of gene-environment networks†

Abstract

This article contributes to a further introduction of continuous optimization in the field of computational biology which is one of the most challenging and emerging areas of science, in addition to foundations presented and the state-of-the-art displayed in [C.A. Floudas and P.M. Pardalos, eds., Optimization in Computational Chemistry and Molecular Biology: Local and Global Approaches, Kluwer Academic Publishers, Boston, 2000]. Based on a summary of earlier works by the coauthors and their colleagues, it refines the model on gene-environment patterns by a problem from generalized semi-infinite programming (GSIP), and characterizes the condition of its structural stability. Furthermore, our paper tries to detect and understand structural frontiers of our methods applied to the recently introduced gene-environment networks and tries to overcome them. Computational biology is interdisciplinary, but it also looks for its mathematical foundations. From data got by DNA microarray experiments, non-linear ordinary differential equations are extracted by the optimization of least-squares errors; then we derive corresponding time-discretized dynamical systems. Using a combinatorial algorithm with polyhedra sequences we can detect the regions of parametric stability, contributing to a testing the goodness of data fitting of the model. To represent and interpret the dynamics, certain matrices, genetic networks and, more generally, gene-environment networks serve. Here, we consider n genes in possible dependence with m special environmental factors and a cumulative one. These networks are subject of discrete mathematical questions, but very large structures, such that we need to simplify them. This is undertaken in a careful optimization with constraints, aiming at a balanced connectedness, incorporates any type of a priori knowledge or request and should be done carefully enough to be robust against disturbation by the environment. In this way, we take into account attacks on the network, knockout phenomena and catastrophies, but also changes in lifestyle and effects of education as far as they can approximately be quantified. We characterize the structural stability of the GSIP problem against perturbations like changes between data series or due to outliers. We give explanations on the numerics and the use of splines. This study is an attempt to demonstrate some beauty and applicabilty of continuous optimization which might together one day give a support in health care, food engineering, biomedicine and -technology, including elements of bioenergy and biomaterials. †Dedicated to Prof. Dr H. Th. Jongen on the occasion of his 60th birthday in admiration of his scientific work.