Abstract

This paper discusses mathematical modeling of artificial gene networks. We consider a phenomenological model of the simplest gene network, called a repressilator. This network contains three elements unidirectionally coupled into a ring. More specifically, the first element inhibits the synthesis of the second, the second inhibits the synthesis of the third, and the third, which closes the cycle, inhibits the synthesis of the first one. The interaction of protein concentrations and mRNA (messenger RNA) concentration is surprisingly similar to the interaction of six ecological populations, three predators and three preys. This makes it possible to propose a new phenomenological model, which is represented by a system of unidirectionally coupled ordinary differential equations. We investigate the problem of existence and stability of a relaxation periodic solution that is invariant with respect to cyclic permutations of coordinates. To find the asymptotics of this solution, a special relay system is constructed. We prove that the periodic solution of the relay system gives the asymptotic approximation of the orbitally asymptotically stable relaxation cycle of the problem under consideration.

中文翻译：

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基因网络建模的新方法

摘要

本文讨论了人工基因网络的数学建模。我们考虑最简单的基因网络的现象学模型，称为再压制。该网络包含单向耦合成环的三个元素。更具体地说，第一种元素抑制第二种元素的合成，第二种元素抑制第三种元素的合成，而第三种元素则关闭循环，抑制了第一种元素的合成。蛋白质浓度和mRNA（信使RNA）浓度的相互作用令人惊讶地类似于六个生态种群，三个捕食者和三个猎物的相互作用。这使得有可能提出一种新的现象学模型，该模型以单向耦合常微分方程组为代表。我们研究了松弛周期解的存在性和稳定性问题，该周期解关于坐标的循环置换是不变的。为了找到该解决方案的渐近性，构建了一个特殊的中继系统。我们证明中继系统的周期解给出了所考虑问题的轨道渐近稳定松弛周期的渐近逼近。