Developing Computational Tools for Moving Boundary Problems : with Applications to Cell Migration

We have extended our 'moving boundaries' parabolic solver to models in which cell kinematics are coupled with intracellular dynamics. The original method, developed by Igor L. Novak and Boris M. Slepchenko [1], that combines an Eulerian approach with tracking an explicit boundary was linked to FronTier, a robust front-tracking technique [2]. The extended algorithm was validated using a set of benchmark problems related to cell migration.

Below, you can see a couple of motile cell examples from the minimal models of actin-based cell motility, originally proposed by Alex Mogilner [3]. Our algorithm, currently in 2D only, is capable of dealing with large deformations while preserving exact (numerical) mass conservation and sufficient accuracy.

small deformation example: a moving cell, starting from an initial circular shape, the cell becomes polarized and moves continuously to the right. pseudo-color represents Myosin distribution; vectors represent actin-velocity field; the dashed red line shows cell centroid at different times. Three snapshots of the computational mesh are also shown at intermediate times (Image by: Masoud Nickaeen).

moderate deformation example: (left) a rotating cell, the cell is unstable with respect to unidirectional motion and falls into a stable rotating mode; (right) blow-up of the snapshots during deformation at intermediate times (Image by: Masoud Nickaeen).

large deformation example: a crazy-for-food cell, starting from a circular initial shape (solid line) cell deforms extensively and eventually falls into a stable rotating mode (not shown here for brevity). Automatically generated Voronoi mesh (shown at some intermediate steps) seamlessly conforms to the boundary as the cell deforms and changes shape (Image by: Masoud Nickaeen).

Where possible, COMSOL Multiphysics® was used to obtain alternative numerical simulations to compare our solutions. The methods used in COMSOL are based on Arbitrary-Lagrangian-Eulerian finite element method. I'll be presenting some of the comparison results in the upcoming COMSOL conference in Boston, October 5-7, 2016 (link to conference).

The 'moving boundaries' algorithm was initially prototyped and tested in Matlab using both explicit and implicit time integration methods. The explicit version of the code is now implemented by our software developers as a production quality C++ code in Virtual Cell (VCell), a computational framework for simulating cellular phenomena in realistic geometries [4]. Once a front end is developed for the solver, the users will be able to simulate processes in cells whose membranes protrude/retract with velocities determined by cell biochemistry. Ultimately, the algorithm will be integrated with codes simulating cell mechanics.

References

[1] http://dx.doi.org/10.1016/j.jcp.2014.03.014

[2] http://dx.doi.org/10.1016/j.jcp.2005.08.034

[3] http://www.pnas.org/content/112/16/5045.long

[4] doi.wiley.com/10.1002/wsbm.165