Active cellular protrusion: continuum theories and models

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Constitutive equations

The mass and momentum conservation equations are cast in a very general framework that needs to be further constrained to provide closure of the system. These additional prescriptions (the constitutive equations) embody the biological specifications of the cell. For instance, it is likely that the viscosity ν will depend on the cytoskeletal density: a law such as

ν = ν0θn

(10.10)

prescribing a well-defined linear relation between viscosity and network concentration is such a constitutive relation. In principle, this could be verified empirically by investigating the rheology of the cytoplasm at various cytoskeletal concentrations. However, in general such experimental evidence is sparse and often difficult to interpret. One is thus usually reduced to educated guesses for the constitutive laws that govern J (the network formation or cytoskeletal polymerization rate), H (the resistance to solvent flow through the network), (the network stress due to elasticity and static interactions), ν (the network viscosity), and γ (the tension of the cortical membrane). Conversely, the main advantage of this formalism is that it is sufficiently general to accommodate most theories of protrusions: as we shall see below, it all depends on the proper adjustment of the constitutive equations.

Cytoskeletal theories of cellular protrusion

As has been touched on, it appears that polymerization of large amounts of actin in the vicinity of a membrane causes outward force and protrusion. It also appears that this phenomenon is probably not directly dependent on molecular motors such as myosins, especially as their contractile activity tends to ‘pull’ rather than ‘push’ the cytoskeleton. This has led to theories of protrusion such as the Brownian ratchet model, in which the free energy released by the addition of monomers to a filament is transduced to generate a pressure against a membrane that sterically interferes with the reaction. Without going into the specifics, however, it is clear that such cytoskeletal theories of protrusion can be categorized into two classes:

Network–membrane interaction theories in which the cytoskeleton and the membrane repel one another through a force field. The classic Brownian ratchet model belongs to this class, as it relies on the hard-core potential of actin monomers pushing on the membrane (Peskin et al. 1993).

Network–network interaction theories in which the cytoskeleton interacts with itself, resulting in a repulsive force. This could be due to electrostatic interactions (actin is negatively charged) or thermal agitation.

In what follows we shall formalize these classes of theory in a way that enables linkage to the RIF approach.

We wish to emphasize that we are only discussing free protrusions – that is, protrusions that emerge from the cell body without adhesion to an external substrate. When adhesion occurs, additional classes of theory become tenable, but these are not considered here.