Mathematical modelling in cell migration: tackling biochemistry in changing geometries

Abstract

Directed cell migration poses a rich set of theoretical challenges. Broadly, these are concerned with (1) how cells sense external signal gradients and adapt; (2) how actin polymerisation is localised to drive the leading cell edge and Myosin-II molecular motors retract the cell rear; and (3) how the combined action of cellular forces and cell adhesion results in cell shape changes and net migration. Reaction–diffusion models for biological pattern formation going back to Turing have long been used to explain generic principles of gradient sensing and cell polarisation in simple, static geometries like a circle. In this minireview, we focus on recent research which aims at coupling the biochemistry with cellular mechanics and modelling cell shape changes. In particular, we want to contrast two principal modelling approaches: (1) interface tracking where the cell membrane, interfacing cell interior and exterior, is explicitly represented by a set of moving points in 2D or 3D space and (2) interface capturing. In interface capturing, the membrane is implicitly modelled analogously to a level line in a hilly landscape whose topology changes according to forces acting on the membrane. With the increased availability of high-quality 3D microscopy data of complex cell shapes, such methods will become increasingly important in data-driven, image-based modelling to better understand the mechanochemistry underpinning cell motion.

Modeling and simulation of cell migration on the basis of force equilibrium

“…In most studies on cell dynamics, experiments have been performed in wet labs. A few studies have presented mathematical models through applying partial differential equation to introduce cell migration, 1–3 presuppositions, and simulation methods for understanding cell dynamics; however, these studies have obtained divergent results 4–7 . Numerical studies help elucidate the fundamental mechanism of cell dynamics, which is essential for simulating cell dynamics in various conditions before wet lab testing 8,9 .…”

The Roles of Signaling in Cytoskeletal Changes, Random Movement, Direction-Sensing and Polarization of Eukaryotic Cells

“…Since cells often move in a spatially variable environment, the feedback from the ME can affect the mode of movement dynamically, and far more work is needed to understand how the cell-ME interaction controls the mode of movement. Significant progress has been made on simpler systems such as keratocytes moving on a flat surface [169], and recent techniques that can capture more dynamic shape changes in 3D via interface tracking shows promise [170], but much remains to be done. In the context of swimmers such as shown in Figure 3, a model in which protrusions propagate along the body length can replicate swimming speeds under various conditions [28], but how extension of protrusions is controlled by local fluid properties and other factors is not yet known.…”