On optimization, dynamics and uncertainty: A tutorial for gene-environment networks

An emerging research area in computational biology and biotechnology is devoted to mathematical modeling and prediction of gene-expression patterns; to fully understand its foundations requires a mathematical study. This paper surveys and mathematically expands recent advances in modeling and prediction by rigorously introducing the environment and aspects of errors and uncertainty into the genetic context within the framework of matrix and interval arithmetic. Given the data from DNA microarray experiments and environmental measurements we extract nonlinear ordinary differential equations which contain parameters that are to be determined. This is done by a generalized Chebychev approximation and generalized semi-infinite optimization. Then, time-discretized dynamical systems are studied. By a combinatorial algorithm which constructs and follows polyhedra sequences, the region of parametric stability is detected. Finally, we analyze the topological landscape of gene-environment networks in terms of structural stability. This pioneering work is practically motivated and theoretically elaborated; it is directed towards contributing to applications concerning better health care, progress in medicine, a better education and more healthy living conditions.