Active cellular protrusion: continuum theories and models

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near the tip of a protrusion is increased, and that the permeability of the plasma membrane is sufficient to allow significant inflow from the extracellular environment. It is then possible for the volume of the protrusion to grow, and the filling with polymerized cytoskeleton is considered to take place after the fact for structural reasons. There are many problems with such a model – possibly explaining why it has lain fallow for a while, now. To mention just one problem, to the extent we are able to ascertain it, it appears that cellular volume does not change appreciably during the extension of protrusions, even big ones.

Shearing motor protrusion. At a most elementary level, myosin motors are shearing motors in the sense that they actively slide filaments parallel to one another. If one imagines a reasonably stiff assembly of cytoskeletal filaments perpendicular to the plasma membrane, it is conceivable that this structure could be driven out by a shearing motor mechanism as shown in Fig. 10-4 (Condeelis, 1993). Once of some popularity, this model seems more or less abandoned in the context of free protrusions, probably because there is evidence that molecular motors are not required – although we would caution that in our view, the case is far from being experimentally airtight.

Numerical implementation of the RIF formalism

A detailed discussion of the numerical strategies that can be used to solve the evolution equations is beyond the scope of this chapter. We will therefore limit ourselves to a brief outline of the methodology. Because it is well suited to free-boundary problems in the low-Reynolds-number limit, we use a Galerkin finite element scheme implemented in two spatial dimensions (for problems with cylindrical symmetry) on a mesh of quadrilateral cells. Grid and mass advection are implemented following cannonical methods that can be found in standard texts and reviews.

Briefly, the calculation is advanced over a time-step t determined by the Courant condition or other fast time scale of the dynamics. We evolve over t by means of sequential operations (this is operator splitting):

1.We advect the mesh boundary according to the network flow and then reposition mesh nodes for optimal resolution while preserving mesh topology, boundaries, and interfacial surfaces (Knupp and Steiberg, 1994).

2.We advect mass from the old mesh positions to the new mesh using a general Eulerian-Lagrangian scheme with upwind interpolation (Rash and Williamson, 1990).

3.We use constitutive laws to compute necessary quantities such as viscosities and surface tensions.

4.Finally, the momentum equations and the incompressibility condition together with the applicable boundary conditions are discretized using the Galerkin approach and the resulting system is solved for the pressure, network velocity, and solvent velocity on the advected mesh using an Uzawa style iteration (Temam, 1979).