# Numerical analysis of gene networks models

## Abstract

Gene networks are sets of genes functioning in a coordinated manner, their major components are the biopolymers DNA, RNA, and proteins. In this work, we consider a small network with several genes (they can be considered standalone) and observe the concentration of the corresponding proteins. Studying the dynamic behavior of gene networks is a complicated problem. Therefore, we research the principles of their functioning on special hypothetical model. These models can be describe by nonlinear system of differential equations. We are very interested in the questions on existence and stability of attractors in such models. Particular attention is paid to some multidimensional symmetric systems with negative and positive feedbacks. These systems are studied with a specially developed algorithm, which reduces them to a single equation with delayed argument. Cycles in this equation searched by Andronov-Hopf theorem and stability analyzes using the first Lyapunov coefficient. Computer modelling and numerical experiments with our models has been implemented in our special computer program called PhasePortraitAnalyzer. The mathematical core is developed in R, and the interface part is created using C# + WPF. Its software can build the graphical representation of the gene networks phase portraits, search equilibrium points, simulate the trajectory and perform other calculations for analyzing the target system. We can prove the correctness of the numerical solutions using some theorems on topology of the models.

## Reference

A A Akinshin

“Numerical analysis of gene networks models”(2013) DOI: 10.1111/febs.12396

```
@Inproceedings{akinshin2013febs,
abstract = {Gene networks are sets of genes functioning in a coordinated manner, their major components are the biopolymers DNA, RNA, and proteins. In this work, we consider a small network with several genes (they can be considered standalone) and observe the concentration of the corresponding proteins. Studying the dynamic behavior of gene networks is a complicated problem. Therefore, we research the principles of their functioning on special hypothetical model. These models can be describe by nonlinear system of differential equations. We are very interested in the questions on existence and stability of attractors in such models. Particular attention is paid to some multidimensional symmetric systems with negative and positive feedbacks. These systems are studied with a specially developed algorithm, which reduces them to a single equation with delayed argument. Cycles in this equation searched by Andronov-Hopf theorem and stability analyzes using the first Lyapunov coefficient. Computer modelling and numerical experiments with our models has been implemented in our special computer program called PhasePortraitAnalyzer. The mathematical core is developed in R, and the interface part is created using C\# + WPF. Its software can build the graphical representation of the gene networks phase portraits, search equilibrium points, simulate the trajectory and perform other calculations for analyzing the target system. We can prove the correctness of the numerical solutions using some theorems on topology of the models.},
annote = {SW06.W29-32},
author = {Akinshin, A A},
booktitle = {8th FEBS Congress, Saint Petersburg, Russia, July 6-11, 2013},
doi = {10.1111/febs.12396},
language = {english},
pages = {547},
title = {Numerical analysis of gene networks models},
url = {https://apps.webofknowledge.com/full\_record.do?product=UA\&search\_mode=GeneralSearch\&qid=1\&SID=U1kYDgZoTuCyURIaqPx\&page=1\&doc=4 https://publons.com/publon/34688524/},
volume = {280},
year = {2013}
}
```