Optimization of gene-environment networks in the presence of errors and uncertainty with Chebychev approximation

This mathematical contribution is addressed towards the wide interface of life and human sciences that exists between biological and environmental information. Like very few other disciplines only, the modeling and prediction of genetical data is requesting mathematics nowadays to deeply understand its foundations. This need is even forced by the rapid changes in a world of globalization. Such a study has to include aspects of stability and tractability; the still existing limitations of modern technology in terms of measurement errors and uncertainty have to be taken into account. In this paper, the important role played by the environment is rigorously introduced into the biological context and connected with employing the theories of optimization and dynamical systems. Especially, a matrix-vector and interval concept and algebra are used; some special attention is paid to splines. From data got by DNA microarray experiments and environmental measurements we extract nonlinear ordinary differential equations. This is done by Chebychev approximation and 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. This is used for testing and maybe improving the goodness of the achieved model. We analyze the topological landscape of gene-environment networks in terms of structural stability which we characterize. This pioneering practically motivated and theoretically elaborated work is devoted to a contribution to better health care, progress in medicine, better education, and to recommending more healthy living conditions. The present paper mainly bases on the authors’ and their coauthors’ contributions of the last few years, it critically discusses structural frontiers and future challenges, while respecting related research contributions, giving access and referring to alternative concepts that exist in the literature.      

On generalized semi-infinite optimization of genetic networks

Since some years, the emerging area of computational biology is looking for its mathematical foundations. Based on modern contributions given to this area, our paper approaches modeling and prediction of gene-expression patterns by optimization theory, with a special emphasis on generalized semi-infinite optimization. Based on experimental data, nonlinear ordinary differential equations are obtained by the optimization of least-squares errors. The genetic process can be investigated by a time-discretization and a utilization of a combinatorial algorithm to detect the stability regions. We represent the dynamical systems by means of matrices which allow biological-medical interpretations, and by genetic or new gene-environment networks. For evaluating these networks we optimize them under constraints imposed. For controlling the connectedness structure of the network, we introduce GSIP into this modern application field which can lead to important services in medicine and biotechnology, including energy production and material science.