Mapping dynamical systems into chemical reactions

Tomislav Plesa
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Abstract

Dynamical systems with polynomials on the right-hand side can model a wide range of physical processes. A subset of such dynamical systems that can model chemical reactions under mass-action kinetics are called chemical systems. A central problem in synthetic biology is to map general polynomial dynamical systems into dynamically similar chemical ones. In this paper, we present a novel map, called the quasi-chemical map, that can systematically solve this problem. The quasi-chemical map introduces suitable state-dependent perturbations into any given polynomial dynamical system which then becomes chemical under suitably large translation of variables. We prove that this map preserves robust dynamical features, such as generic equilibria and limit cycles, as well as temporal properties, such as periods of oscillations. Furthermore, the resulting chemical systems are of only at most one degree higher than the original dynamical systems. We demonstrate the quasi-chemical map by designing relatively simple chemical systems with exotic dynamics and pre-defined bifurcation structures.
将动力系统映射到化学反应中
右侧有多项式的动力学系统可以模拟更广泛的物理过程。这类动力学系统的一个子集可以模拟质量作用动力学下的化学反应,被称为化学系统。合成生物学的一个核心问题是将一般多项式动力学系统映射为动力学上相似的化学系统。在本文中,我们提出了一种新的映射,称为准化学映射,它可以系统地解决这个问题。准化学映射在任何给定的多项式动力系统中引入了适当的状态相关扰动,然后在变量的适当大转换下,多项式动力系统就变成了化学系统。我们证明,准化学映射保留了稳健的动力学特征,如一般均衡和极限循环,以及时间特性,如振荡周期。我们通过设计具有奇异动力学和预定分岔结构的相对简单的化学系统来证明准化学图谱。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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