标题： 一种对磷酸化循环无限多稳定性证明 显示英文标题

标题： A proof of unlimited multistability for phosphorylation cycles

摘要： 多级无能循环是一个磷酸化系统，在该系统中，分子底物可能通过酶促机制依次被磷酸化n次。 该系统已经通过反应网络理论和常微分方程进行了数学研究。 已知对于某些参数选择，该系统可能具有至少2[n/2]+1个稳态（其中[x]表示x的整数部分）。 此外，对于简单的无能循环和双无能循环（n=1,2），稳态的稳定性已经被确定，即简单无能循环的唯一稳态是全局稳定的，而对于某些参数值，双无能循环可以存在两个渐近稳定和一个不稳定的稳态。 对于一般的n，已经提供了证据表明可能的渐近稳定稳态的数量随着$n$的增加而增加，这导致了一个猜想，即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，其余的稳态是不稳定的。 我们在这里通过首先将系统简化为一个更小的系统来证明这个猜想，在该系统中我们找到了一个参数值的选择，该选择产生一个具有2[n/2]+1重数的唯一稳态。 利用几何奇摄动理论的论点以及对该稳态中心流形的详细分析，我们得出了所需的结果。 显示英文摘要

摘要： 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.