# American Institute of Mathematical Sciences

October  2020, 7(4): 253-268. doi: 10.3934/jdg.2020018

## Foundations of semialgebraic gene-environment networks

 1 Christerweg, 12, 83624 Otterfing, Germany 2 Faculty of Engineering Management, Poznan University of Technology, Poznan, Poland 3 Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey 4 Department of Industrial Engineering, Mazandaran University of Science and Technology, Babol, Iran 5 Department of Industrial and Systems Engineering, Istinye University, Istanbul, Turkey

* Corresponding author: Erik Kropat

Received  December 2019 Published  October 2020 Early access  July 2020

Gene-environment network studies rely on data originating from different disciplines such as chemistry, biology, psychology or social sciences. Sophisticated regulatory models are required for a deeper investigation of the unknown and hidden functional relationships between genetic and environmental factors. At the same time, various kinds of uncertainty can arise and interfere with the system's evolution. The aim of this study is to go beyond traditional stochastic approaches and to propose a novel framework of semialgebraic gene-environment networks. Foundation is laid for future research, methodology and application. This approach is a natural extension of interconnected systems based on stochastic, polyhedral, ellipsoidal or fuzzy (linguistic) uncertainty. It allows for a reconstruction of the underlying network from uncertain (semialgebraic) data sets and for a prediction of the uncertain futures states of the system. In addition, aspects of network pruning for large regulatory systems in genome-wide studies are discussed leading to mixed-integer programming (MIP) and continuous programming.

Citation: Erik Kropat, Gerhard-Wilhelm Weber, Erfan Babaee Tirkolaee. Foundations of semialgebraic gene-environment networks. Journal of Dynamics & Games, 2020, 7 (4) : 253-268. doi: 10.3934/jdg.2020018
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Gene-environment networks: Interactions between genes (red), environmental factors and genes (green) and between environmental factors (blue).
Gene-environment networks with cluster partition: Interactions between genetic clusters (red), environmental clusters and genetic clusters (green) and between environmental clusters (blue).
Gene-environment network under semialgebraic uncertainty: Interactions between genetic clusters and/or environmental clusters as well as the corresponding semialgebraic uncertainty sets.
Comparison of measurements and predictions. The computation of the parameter vector $\theta \in \Theta$ depends on $K$ direct comparisons of the intersection between semialgebraic measurements and predictions

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