In this paper, we formulate the space-dependent variable-order fractional master equation to model clustering of particles, organelles, inside living cells. We find its solution in the long-time limit describing non-uniform distribution due to a space-dependent fractional exponent. In the continuous space limit, the solution of this fractional master equation is found to be exactly the same as the space-dependent variable-order fractional diffusion equation. In addition, we show that the clustering of lysosomes, an essential organelle for healthy functioning of mammalian cells, exhibit space-dependent fractional exponents. Furthermore, we demonstrate that the non-uniform distribution of lysosomes in living cells is accurately described by the asymptotic solution of the space-dependent variable-order fractional master equation. Finally, Monte Carlo simulations of the fractional master equation validate our analytical solution.

This article is part of the theme issue ‘Transport phenomena in complex systems (part 1)’.

在本文中，我们制定了空间相关的可变阶分数主方程来模拟活细胞内粒子、细胞器的聚类。我们在描述由于空间相关的分数指数而导致的非均匀分布的长时间限制中找到了它的解决方案。在连续空间极限下，发现该分数阶主方程的解与空间相关变阶分数阶扩散方程完全相同。此外，我们还表明溶酶体是哺乳动物细胞健康功能的重要细胞器，其聚集表现出空间依赖性分数指数。此外，我们证明了溶酶体在活细胞中的非均匀分布可以通过空间相关可变阶分数主方程的渐近解来准确描述。最后，

本文是主题问题“复杂系统中的传输现象（第 1 部分）”的一部分。