Liu Yuewu, Zou Xiufen
School of Mathematics and Statistics, Computational Science Hubei Key Laboratory, Wuhan University, Wuhan, 430072, China; College of Science, Hunan Agricultural University, Hunan, 410128, China.
School of Mathematics and Statistics, Computational Science Hubei Key Laboratory, Wuhan University, Wuhan, 430072, China.
Math Biosci. 2017 Jun;288:46-51. doi: 10.1016/j.mbs.2017.02.010. Epub 2017 Feb 22.
In vitro, the recombinant HIV-1 Gag protein can generate spherical particles with a diameter of 25-30 nm in a fully defined system. It has approximately 80 building blocks, and its intermediates for assembly are abundant in geometry. Accordingly, there are a large number of nonlinear equations in the classical model. Therefore, it is difficult to compute values of geometry parameters for intermediates and make the mathematical analysis using the model. In this work, we develop a new model of HIV-like particle assembly in vitro by using six-fold symmetry of HIV-like particle assembly to decrease the number of geometry parameters. This method will greatly reduce computational costs and facilitate the application of the model. Then, we prove the existence and uniqueness of the positive equilibrium solution for this model with 79 nonlinear equations. Based on this model, we derive the interesting result that concentrations of all intermediates at equilibrium are independent of three important parameters, including two microscopic on-rate constants and the size of nucleating structure. Before equilibrium, these three parameters influence the concentration variation rates of all intermediates. We also analyze the relationship between the initial concentration of building blocks and concentrations of all intermediates. Furthermore, the bounds of concentrations of free building blocks and HIV-like particles are estimated. These results will be helpful to guide HIV-like particle assembly experiments and improve our understanding of the assembly dynamics of HIV-like particles in vitro.
在体外，重组HIV-1 Gag蛋白能够在一个完全明确的系统中生成直径为25 - 30纳米的球形颗粒。它大约有80个构建模块，其组装中间体在几何形状上种类丰富。因此，经典模型中有大量非线性方程。所以，计算中间体几何参数的值并使用该模型进行数学分析很困难。在这项工作中，我们利用类HIV颗粒组装的六重对称性开发了一种新的体外类HIV颗粒组装模型，以减少几何参数的数量。这种方法将大大降低计算成本并便于模型的应用。然后，我们证明了这个具有79个非线性方程的模型正平衡解的存在性和唯一性。基于这个模型，我们得出了一个有趣的结果，即平衡时所有中间体的浓度与三个重要参数无关，这三个参数包括两个微观结合速率常数和成核结构的大小。在达到平衡之前，这三个参数影响所有中间体的浓度变化率。我们还分析了构建模块初始浓度与所有中间体浓度之间的关系。此外，还估计了游离构建模块和类HIV颗粒浓度的范围。这些结果将有助于指导类HIV颗粒组装实验，并增进我们对体外类HIV颗粒组装动力学的理解。
