Optimization and dynamics of gene-environment networks with intervals

There are a few areas of science and technology which are only as challenging, emerging and promising as computational biology. This area is looking for its mathematical foundations, for methods of prediction while guaranteeing robustness, and it is of a rigorous interdisciplinary nature. In this paper, we deepen and extend the approach of learning gene-expression patterns in the framework of gene-environment networks by optimization, especially, generalized semi-infinite optimization (GSIP). With respect to research done previously, we additionally imply the fact that there are measurement errors in the microarray technology and in the environmental data likewise; moreover, the effects which exists among the genes and environmental items can seldom be precisely quantified. Furthermore, we present the well-established matrix algebra for our extended model space, and we indicate further new approaches. Based on data from DNA microarray experiments, nonlinear ordinary differential equations are extracted by least-squares and, then, time-discretized dynamical systems are derived. Using a combinatorial algorithm which constructs and observes polyhedra sequences, the region of parametric stability is detected. This supports the testing of the quality of data fitting. For the parameter estimation we apply a GSIP problem; we characterize its structural stability. Hopefully, this pioneering study will serve and lead to a more realistic understanding and forecast in biomedicine, food engineering, and biotechnology. The inclusion of error and imprecision intervals may lead to a more careful evaluation of the experimental data in the forthcoming years, especially, when the microarray technology becomes more and more refined.