关于多重无效循环中稳态个数的一个注记
A remark on the number of steady states in a multiple futile cycle
摘要：
多位点磷酸化-去磷酸化循环是在细胞信号转导中重复使用的基序。这个基序本身可以产生各种动态行为，如双稳态和超灵敏度，而无需直接的正反馈。本文研究了一般多位点磷酸化-去磷酸化循环的正稳态数目，以及正稳态数目如何随生物学参数的变化而变化。我们用解析的方法证明：（1）对于某些参数范围，至少有n+1个（如果n是偶数）或n（如果n是奇数）稳态;（2）永远不会有超过2n-1稳态（特别地，这意味着对于n=2，包括单水平的MAPK级联，存在至多三个稳态）;（3）对于接近标准Michaelis-Menten准稳态条件的参数，存在至多n+1个稳态;以及（4）对于远离标准Michaelis-Menten准稳态条件的参数，存在至多一个稳态。
英文摘要：
The multisite phosphorylation-dephosphorylation cycle is a motif repeatedly used in cell signaling. This motif itself can generate a variety of dynamic behaviors like bistability and ultrasensitivity without direct positive feedbacks. In this paper, we study the number of positive steady states of a general multisite phosphorylation-dephosphorylation cycle, and how the number of positive steady states varies by changing the biological parameters. We show analytically that (1) for some parameter ranges, there are at least n+1 (if n is even) or n (if n is odd) steady states; (2) there never are more than 2n-1 steady states (in particular, this implies that for n=2, including single levels of MAPK cascades, there are at most three steady states); (3) for parameters near the standard Michaelis-Menten quasi-steady state conditions, there are at most n+1 steady states; and (4) for parameters far from the standard Michaelis-Menten quasi-steady state conditions, there is at most one steady state.
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