Motility, the remarkable ability to move spontaneously via consumption of energy delivered by the internal metabolism, is a fascinating feature of living systems. Cell migration and motility is an active research topic in biology, and it has fascinated physicists, materials scientists, and applied mathematicians. The fundamental question here is to understand how local interactions among individual components lead to the observed collective behavior and the formation of highly organized entities.

 

The phase-field method had been originally introduced to model solidification processes, but later on was applied to a diversity of systems including domain growth, microstructure evolution, and structural transformations in materials science, elastic surface instabilities and fracture mechanics, the fluidization transition in granular media, and finally biological systems such as vesicles and growing actin gels. This list clearly demonstrates the method’s versatility, which meanwhile has developed into a standard tool in the vast field of multiphase/composite materials. The main idea is the following: instead of tracing the boundary (i.e., the membrane) explicitly, which poses the problems discussed above for the sharp interface models, one uses an auxiliary field to describe the interface implicitly. Associated with this field, the phase field ρ(r, t), is a certain ‘free energy functional’ with two local energy minima.

 

Over a number of years, we have developed a phase-field model to describe the movement of many self-organized, interacting cells. The model takes into account the main mechanisms of cell motility – acto-myosin dynamics, as well as substrate-mediated and cell-cell adhesion. It predicts that collective cell migration emerges spontaneously as a result of inelastic collisions between neighboring cells: collisions lead to a mutual alignment of the cell velocities and to the formation of coherently-moving multi-cellular clusters. Small cell-to-cell adhesion, in turn, reduces the propensity for large-scale collective migration, while higher adhesion leads to the formation of moving bands.

 

Current research is focused on understanding of confinement and substrate topography effects on 3D cell migration. On planar substrates, most cells move by forming lamellipodia, thin sheets containing actin filaments that polymerize and branch towards the cell membrane, thereby exerting protrusions forces. A long-standing question had been whether these motile structures dominating the 2D motion also exist and are relevant in 3D. There is growing evidence that this is indeed the case for fibroblasts and neutrophils, at least under certain external conditions. We are developing a simple, physics-based modeling framework that transfers the knowledge of 2D lamellipodia-based cell motion to 3D settings. The use of the phase field allows to model motion in, in principle, arbitrarily shaped and even topologically nontrivial environments.

 

Funding:

The US Department of Energy, Basic Energy Sciences, Division of Materials Science and Engineering

 

Publications:

Igor S Aranson (ed.), “Physical Models of Cell Motility”, Springer-Verlag, 2016

Falko Ziebert and Igor S. Aranson, “Computational approaches to substrate-based cell motility.” npj Computational Materials 2 (2016): 16019

J Löber, F Ziebert, IS Aranson, “Collisions of deformable cells lead to collective migration” Scientific reports 5 (2015) 9172

 

Figure 1. | Illustration of inelastic collisions between the cells. (a) Strongly inelastic collision of two canoe-shaped cells, leading to an effective alignment of the directions of motion. (b) An almost elastic collision of two bell-shaped cells (c) Effect of the incidence angle on the cells’ center of mass trajectories. (d) Effect of adhesion strength κ on the cells’ center of mass trajectories: increasing adhesion reduces the effective alignment of cells.

Figure 2 | Multiplicity of cell migration modes on various 3D substrates.