Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom.
J Chem Phys. 2011 Aug 14;135(6):065105. doi: 10.1063/1.3608916.
Self-assembly processes resulting in linear structures are often observed in molecular biology, and include the formation of functional filaments such as actin and tubulin, as well as generally dysfunctional ones such as amyloid aggregates. Although the basic kinetic equations describing these phenomena are well-established, it has proved to be challenging, due to their non-linear nature, to derive solutions to these equations except for special cases. The availability of general analytical solutions provides a route for determining the rates of molecular level processes from the analysis of macroscopic experimental measurements of the growth kinetics, in addition to the phenomenological parameters, such as lag times and maximal growth rates that are already obtainable from standard fitting procedures. We describe here an analytical approach based on fixed-point analysis, which provides self-consistent solutions for the growth of filamentous structures that can, in addition to elongation, undergo internal fracturing and monomer-dependent nucleation as mechanisms for generating new free ends acting as growth sites. Our results generalise the analytical expression for sigmoidal growth kinetics from the Oosawa theory for nucleated polymerisation to the case of fragmenting filaments. We determine the corresponding growth laws in closed form and derive from first principles a number of relationships which have been empirically established for the kinetics of the self-assembly of amyloid fibrils.
在分子生物学中，经常观察到导致线性结构的自组装过程，其中包括功能性纤维的形成，如肌动蛋白和微管蛋白，以及通常无功能的纤维，如淀粉样纤维聚集物。尽管描述这些现象的基本动力学方程已经建立，但由于它们的非线性性质，除了特殊情况之外，很难推导出这些方程的解。一般解析解的可用性为从宏观实验测量的生长动力学分析除了滞后时间和最大生长速率等已经可以从标准拟合程序获得的现象参数之外，提供了从分子水平过程的速率确定的途径。我们在这里描述了一种基于不动点分析的分析方法，该方法为丝状结构的生长提供了自洽的解决方案，除了伸长之外，丝状结构还可以经历内部断裂和单体依赖性成核，作为充当生长位点的新游离端的生成机制。我们的结果将核化聚合理论中用于描述指数生长动力学的解析表达式推广到了片段丝状结构的情况。我们以封闭形式确定相应的生长规律，并从第一性原理推导出一些已经在淀粉样纤维自组装动力学中得到经验验证的关系。
