In this work, we propose a generalized Langevin equation-based model to describe the lateral diffusion of a protein in a lipid bilayer. The memory kernel is represented in terms of a viscous (instantaneous) and an elastic (noninstantaneous) component modeled through a Dirac δ function and a three-parameter Mittag-Leffler type function, respectively. By imposing a specific relationship between the parameters of the three-parameter Mittag-Leffler function, the different dynamical regimes—namely ballistic, subdiffusive, and Brownian, as well as the crossover from one regime to another—are retrieved. Within this approach, the transition time from the ballistic to the subdiffusive regime and the spectrum of relaxation times underlying the transition from the subdiffusive to the Brownian regime are given. The reliability of the model is tested by comparing the mean-square displacement derived in the framework of this model and the mean-square displacement of a protein diffusing in a membrane calculated through molecular dynamics simulations.

在这项工作中，我们提出了一个基于广义 Langevin 方程的模型来描述蛋白质在脂质双层中的横向扩散。记忆内核用通过狄拉克δ建模的粘性（瞬时）和弹性（非瞬时）分量表示函数和三参数 Mittag-Leffler 类型函数。通过在三参数 Mittag-Leffler 函数的参数之间施加特定关系，可以检索不同的动力学状态，即弹道、次扩散和布朗，以及从一种状态到另一种状态的交叉。在这种方法中，给出了从弹道到次扩散状态的转变时间以及从次扩散到布朗状态转变的弛豫时间谱。通过比较在该模型的框架中得出的均方位移和通过分子动力学模拟计算的扩散在膜中的蛋白质的均方位移来测试模型的可靠性。