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Topological techniques are powerful tools for characterizing the complexity of many dynamical systems, including the commonly studied area-preserving maps of the plane. However, the extension of many topological techniques to higher dimensions is filled with roadblocks preventing their application. This article shows how to extend the homotopic lobe dynamics (HLD) technique, previously developed for 2D maps, to volume-preserving maps of a three-dimensional phase space. Such maps are physically relevant to particle transport by incompressible fluid flows or by magnetic field lines. Specifically, this manuscript shows how to utilize two-dimensional stable and unstable invariant manifolds, intersecting in a heteroclinic tangle, to construct a symbolic representation of the topological dynamics of the map. This symbolic representation can be used to classify system trajectories and to compute topological entropy. We illustrate the salient ideas through a series of examples with increasing complexity. These examples highlight new features of the HLD technique in 3D. Ultimately, in the final example, our technique detects a difference between the 2D stretching rate of surfaces and the 1D stretching rate of curves, illustrating the truly 3D nature of our approach.

The spatial localization or sequestering of motile cargo and their dispersal within cells is an important process in a number of physiological contexts. The morphology of the cytoskeletal network, along which active, motor-driven intracellular transport takes place, plays a critical role in regulating such transport phases. Here, we use a computational model to address the existence and sensitivity of dynamic sequestering and how it depends on the parameters governing the cytoskeletal network geometry, with a focus on filament lengths and polarization away or toward the periphery. Our model of intracellular transport solves for the time evolution of a probability distribution of cargo that is transported by passive diffusion in the bulk cytoplasm and driven by motors on explicitly rendered, polar cytoskeletal filaments with random orientations. We show that depending on the lengths and polarizations of filaments in the network, dynamic sequestering regions can form in different regions of the cell. Furthermore, we find that, for certain parameters, the residence time of cargo is non-monotonic with increasing filament length, indicating an optimal regime for dynamic sequestration that is potentially tunable via filament length. Our results are consistent with in vivo observations and suggest that the ability to tunably control cargo sequestration via cytoskeletal network regulation could provide a general mechanism to regulate intracellular transport phases.

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