The biological processes necessary for the development and continued survival of any organism are often strongly influenced by the transport properties of various biologically active species. The transport phenomena involved vary over multiple temporal and spatial scales, from organism-level behaviors such as the search for food, to systemic processes such as the transport of oxygen from the lungs to distant organs, down to microscopic phenomena such as the stochastic movement of proteins in a cell. Each of these processes is influenced by many interrelated factors. Identifying which factors are the most important, and how they interact to determine the overall result is a problem of great importance and interest. Experimental observations are often fit to relatively simple models, but in reality the observations are the output of complicated functions of the physicochemical, topological, and geometrical properties of a given system. Herein we use multistate continuous-time random walks and generalized master equations to model transport processes involving spatial jumps, immobilization at defined sites, and stochastic internal state changes. The underlying spatial models, which are framed as graphs, may have different classes of nodes, and walkers may have internal states that are governed by a Markov process. A general form of the solutions, using Fourier–Laplace transforms and asymptotic analysis, is developed for several spatially infinite regular lattices in one and two spatial dimensions, and the theory is developed for the analysis of transport and internal state changes on general graphs. The goal in each case is to shed light on how experimentally observable macroscale transport coefficients can be explained in terms of microscale properties of the underlying processes. This work is motivated by problems arising in transport in biological tissues, but the results are applicable to a broad class of problems that arise in other applications.

任何生物体的发育和持续生存所必需的生物过程通常受到各种生物活性物种的运输特性的强烈影响。所涉及的运输现象在多个时间和空间尺度上有所不同，从有机体水平的行为（例如寻找食物）到系统过程（例如氧气从肺部运输到远处器官），再到微观现象（例如细胞中的蛋白质。这些过程中的每一个都受到许多相互关联的因素的影响。确定哪些因素最重要，以及它们如何相互作用以确定整体结果是一个非常重要和有趣的问题。实验观察通常适用于相对简单的模型，但实际上，观察结果是给定系统的物理化学、拓扑和几何特性的复杂函数的输出。在这里，我们使用多状态连续时间随机游走和广义主方程来模拟涉及空间跳跃、固定在定义位置和随机内部状态变化的运输过程。被构建为图的底层空间模型可能具有不同类别的节点，而步行者可能具有由马尔可夫过程控制的内部状态。使用傅立叶-拉普拉斯变换和渐近分析，为一维和二维空间中的几个空间无限规则格子开发了解决方案的一般形式，并且开发了用于分析一般图上的传输和内部状态变化的理论。每种情况下的目标都是阐明如何根据潜在过程的微观特性来解释实验上可观察到的宏观传输系数。这项工作的动机是生物组织运输中出现的问题，但结果适用于其他应用中出现的一系列问题。