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A proof of unlimited multistability for phosphorylation cycles
Nonlinearity ( IF 1.6 ) Pub Date : 2020-09-30 , DOI: 10.1088/1361-6544/ab9a1e
Elisenda Feliu 1 , Alan D Rendall 2 , Carsten Wiuf 1
Affiliation  

The multiple futile cycle is a phosphorylation system in which a molecular substrate might be phosphorylated sequentially n times by means of an enzymatic mechanism. The system has been studied mathematically using reaction network theory and ordinary differential equations. It is known that the system might have at least as many as 2[n/2]+1 steady states (where [x] is the integer part of x) for particular choices of parameters. Furthermore, for the simple and dual futile cycles (n=1,2) the stability of the steady states has been determined in the sense that the only steady state of the simple futile cycle is globally stable, while there exist parameter values for which the dual futile cycle admits two asymptotically stable and one unstable steady state. For general n, evidence that the possible number of asymptotically stable steady states increases with $n$ has been given, which has led to the conjecture that parameter values can be chosen such that [n/2]+1 out of 2[n/2]+1 steady states are asymptotically stable and the remaining steady states are unstable. We prove this conjecture here by first reducing the system to a smaller one, for which we find a choice of parameter values that give rise to a unique steady state with multiplicity 2[n/2]+1. Using arguments from geometric singular perturbation theory, and a detailed analysis of the centre manifold of this steady state, we achieve the desired result.

中文翻译:

磷酸化循环无限多重稳定性的证明

多重无效循环是一种磷酸化系统,其中分子底物可以通过酶促机制依次磷酸化 n 次。已使用反应网络理论和常微分方程对该系统进行了数学研究。众所周知,对于特定的参数选择,系统可能至少有 2[n/2]+1 个稳态(其中 [x] 是 x 的整数部分)。此外,对于简单和双重无效循环(n = 1,2),稳态的稳定性已经确定,因为简单无效循环的唯一稳态是全局稳定的,而存在参数值双无用循环允许两种渐近稳定状态和一种不稳定稳定状态。对于一般 n,已经给出了渐近稳定稳态的可能数量随 $n$ 增加的证据,这导致了以下猜想:可以选择参数值使得 [n/2]+1 出 2[n/2]+1稳态是渐近稳定的,其余的稳态是不稳定的。我们通过首先将系统简化为更小的系统来证明这一猜想,为此我们找到了一个参数值的选择,这些参数值会产生具有多重性 2[n/2]+1 的独特稳态。使用几何奇异摄动理论的论点,以及对该稳态中心流形的详细分析,我们达到了预期的结果。这导致了这样的猜想:可以选择参数值,使得 2[n/2]+1 个稳态中的 [n/2]+1 个是渐近稳定的,而其余的稳态是不稳定的。我们通过首先将系统缩小到一个更小的系统来证明这个猜想,为此我们找到了一个参数值的选择,这些参数值会产生一个具有多重性 2[n/2]+1 的独特稳态。使用几何奇异摄动理论的论点,以及对该稳态中心流形的详细分析,我们达到了预期的结果。这导致了这样的猜想:可以选择参数值,使得 2[n/2]+1 个稳态中的 [n/2]+1 个是渐近稳定的,而其余的稳态是不稳定的。我们通过首先将系统缩小到一个更小的系统来证明这个猜想,为此我们找到了一个参数值的选择,这些参数值会产生一个具有多重性 2[n/2]+1 的独特稳态。使用几何奇异摄动理论的论点,以及对该稳态中心流形的详细分析,我们达到了预期的结果。
更新日期:2020-09-30
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