振荡微管聚合模型
Modeling oscillatory Microtubule--Polymerization
摘要：
微管的聚合在生物细胞中普遍存在，并且在某些条件下，它会随时间振荡。在这里简单的反应模型进行分析，捕捉这种振荡以及微管的长度分布。我们假设反应条件是固定的许多振荡周期，这是一个霍普夫分岔，导致持续振荡微管聚合在这些模型中。导出了分岔阈值和振荡频率与反应速率的解析表达式，并给出了它们随参数变化的典型趋势。依赖于{ \it鸟苷三磷酸}（GTP）配体微管蛋白二聚体密度的突变率和延迟反应（如收缩微管的解聚或低聚物的衰变）都支持振荡。对于低于阈值的微管蛋白二聚体浓度振荡微管聚合发生瞬态的路线上的一个稳定状态，如所示的模型方程的数值解。在阈值附近导出了振幅方程，并证明了微管振荡的分岔是超临界的。
英文摘要：
Polymerization of microtubules is ubiquitous in biological cells and under certain conditions it becomes oscillatory in time. Here simple reaction models are analyzed that capture such oscillations as well as the length distribution of microtubules. We assume reaction conditions that are stationary over many oscillation periods, and it is a Hopf bifurcation that leads to a persistent oscillatory microtubule polymerization in these models. Analytical expressions are derived for the threshold of the bifurcation and the oscillation frequency in terms of reaction rates as well as typical trends of their parameter dependence are presented. Both, a catastrophe rate that depends on the density of {\it guanosine triphosphate} (GTP) liganded tubulin dimers and a delay reaction, such as the depolymerization of shrinking microtubules or the decay of oligomers, support oscillations. For a tubulin dimer concentration below the threshold oscillatory microtubule polymerization occurs transiently on the route to a stationary state, as shown by numerical solutions of the model equations. Close to threshold a so--called amplitude equation is derived and it is shown that the bifurcation to microtubule oscillations is supercritical.
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