Hammele Martin, Zimmermann Walter
Theoretical Physics, University of Saarland, D-66041 Saarbrücken, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2003 Feb;67(2 Pt 1):021903. doi: 10.1103/PhysRevE.67.021903. Epub 2003 Feb 14.
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, and typical trends of their parameter dependence are presented. Both, a catastrophe rate that depends on the density of guanosine triphosphate 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.
微管的聚合在生物细胞中普遍存在，在某些条件下，它会随时间振荡。在此，我们分析了简单的反应模型，这些模型能够捕捉到此类振荡以及微管的长度分布。我们假设反应条件在多个振荡周期内保持稳定，在这些模型中，是霍普夫分岔导致了持续的振荡微管聚合。我们根据反应速率推导出了分岔阈值和振荡频率的解析表达式，并展示了它们对参数依赖的典型趋势。依赖于鸟苷三磷酸结合微管蛋白二聚体密度的灾变率以及延迟反应，例如收缩微管的解聚或寡聚物的衰变，都能支持振荡。对于低于阈值的微管蛋白二聚体浓度，振荡微管聚合在达到稳态的过程中会短暂出现，如模型方程的数值解所示。接近阈值时，我们推导了一个所谓的振幅方程，并表明微管振荡的分岔是超临界的。
