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Active cellular protrusion: continuum theories and models

205

Fig. 10-1. The standard cartoon of cellular protrusion

As a general rule such cartoons are powerful but dangerous instruments of knowledge. On the one hand, this cartoon is a means of communicating a complex mechanism in a compact, readily intelligible way. It makes clear the following important points.

1.The cytoplasm is an inhomogeneous medium; its properties at the leading edge of a protrusion are different than they are elsewhere in the cell.

2.Cytoplasmic dynamics require that there exist a simultaneous forward flow of material in the form of water and cytoskeletal monomers and backward flow of cytoskeletal material in the form of filaments (this is the treadmilling).

3.There is a net flow of cytoplasmic volume into the protrusion (otherwise, the protrusion would not grow!).

On the other hand, the apparent simplicity of the cartoon glosses over important qualitative and quantitative issues that need to be addressed with rigor for the whole scheme to stand scrutiny as follows. First, momentum conservation (or force balance) is not evident; when extending a pseudopod, a cell has to exert an outward-directed force, if only against the cortical tension that tends to minimize the surface area (but there may be other opposing forces). By the principle of action and reaction, this requires some sort of bracing. Without a thrust plate supporting the outward-directed force, the pseudopod would collapse back into the main body of the cell. Second, the cartoon is inevitably silent on the mode of force production. The intuitive picture of a growing scaffolding pushing out a “tent” of membrane can be misleading and in any case gives little quantitative information on the protrusive force.

Even before going into details, it is clear from these issues that a theory of cellular protrusion must embody certain attributes: cytoplasmic inhomogeneity; differential flow of cytoskeleton and cytosol; and volume and momentum conservation. The Reactive Interpenetrative Flow (RIF) formalism described next is a natural choice to address these constraints in a rigorous, quantitative manner.

The RIF formalism

Consider the three principal structural components of animal cells:

The cortical membrane defines the boundary of the cell by controlling (and often preventing) volume fluxes with the external world. It is furthermore highly

206 M. Herant and M. Dembo

flexible, fluid, and virtually inextensible. Together, these three properties make it a good conductor of stress.

The cytoskeleton resists deformation through viscoelastic properties and is able to generate active forces through molecular motors (for example myosin) or other interactions (such as electrostatic).

The cytosol flows passively through the cytoskeleton; it is a medium for the propagation of signals. Furthermore, it can be converted to cytoskeleton via the polymerization of dissolved monomers (for example G-actin F-actin) and vice versa.

Note that in general, many chemical entities will simultaneously contribute to the cytoskeletal and cytosolic phases, the best example being actin in filamentous or globular form. However, given biomolecules can be classified as being either part of the cytoskeleton, where they are able to transmit stresses, or part of the cytosol, where they are able to diffuse, but not both. Note also that this classification ignores membrane-enclosed organelles. In particular, the contribution to the mechanical properties of the cell of the largest of those, the nucleus, can occasionally be important.

If one makes the key assumption that at the mesoscopic scale, that is, at a scale small compared to the whole cell but large compared to individual molecules, the properties of the cell can be represented by continuous fields. Then the general framework of continuum mechanics can be applied to animal cells just as it is done with any other material. More specifically, one can write down a closed set of equations to compute the evolution of θn (x, t) the network phase (cytoskeleton) volume fraction, θs (x, t) the solvent phase (cytosol) volume fraction, vn (x, t) the network velocity field, vs (x, t) the solvent velocity field, where x is the position vector and t is the time. The scalar fields θn and θs and the vector fields vn and vs are thus defined on a simply connected, compact domain in Euclidian space that defines the physical extent of the cell. The boundary of this domain and constraints associated with it through boundary conditions are then to be a representation of the physical cortical membrane. This is the method of Reactive Interpenetrative Flows (see Dembo et al. 1986), in other words ‘reactive’ because it allows conversion of one phase into another, and ‘interpenetrative’ because it allows for different velocity fields for each phase.

The evolution equations for the quantities θn , θs , vn , and vs are determined by the laws of mass and momentum conservation.

Mass conservation

The fact that we have only two phases (cytoskeleton and cytosol) mandates that the sum of their volume fractions is unity:

θn + θs = 1.

(10.1)

Net cytoplasmic volume flow is given by the sum of the flow of cytosolic volume and cytoskeletal volume, that is, v = θn vn + θs vs . Because the cytoplasm is in a condensed phase, it is to an excellent approximation incompressible ( · v = 0), so

Active cellular protrusion: continuum theories and models

207

that the incompressibility condition yields:

 

· (θn vn + θs vs ) = 0.

(10.2)

Finally, conservation of cytoskeleton implies that the rate of change of network concentration at a given point in space (Eulerian derivative) is the sum of an advective transport term describing the net inflow of network, and a source term J , which represents the net rate of in situ cytoskeletal production by polymerization:

∂ θn

= − · (θn vn ) + J .

(10.3)

t

Obviously J depends on a prescription for local chemical activity that needs to be provided separately. Eq. 10.3 naturally has a counterpart for the solvent

∂ θs

= − · (θs vs ) J ,

(10.4)

t

which, when taken together with Eq. 10.1, unsurprisingly reduces to Eq. 10.2. As a result, only Eqs. 10.1, 10.2, and 10.3 are needed and Eq. 10.4 is redundant.

Momentum conservation

The momentum equations for the solvent and network phases are simplified by two observations. First, due to the small dimensions and velocities involved, the inertial terms are negligible in comparision with typical cellular forces. Second, the essentially aqueous nature of the cytosol implies that its characteristic viscosity is not very different from that of water (0.02 poise). Because this is much less than typical cytoplasmic viscosities (of order 1000 poise) we shall assume that the entire viscous stress is carried by the cytoskeletal (network) phase, while the cytosolic (solvent) phase remains approximately inviscid.

Within such an approximation, the only two forces that act on the solvent are pressure gradients and solvent-network drag – that is the drag force that occurs when the solvent moves through the network because of mismatched velocities. In the spirit of Darcy’s law, the solvent momentum equation can then be written

θs P + s θn (vn vs ) = 0.

(10.5)

P is the cytoplasmic pressure, and it is assumed that, as for the partial pressures of a mixture of gases, it is shared by the cytosolic and cytoskeletal phases according to concentrations (volume fractions). H is the solvent-network drag coefficient more familiar as the product θn H, which represents the hydraulic conductivity that appears in the usual form of Darcy’s equation. Theoretical considerations (for example see Scheidegger 1960) as well as experiments on polymer networks (Tokita and Tanaka 1991) give estimates of H that lead to small drag forces compared to other forces acting within the cytoplasm, chief among them the cytoskeletal vicosity. This is not surprising, because H should be approximately proportional to the solvent viscosity, which is small compared to network viscosity.

The smallness of H in turn implies that pressure gradients will be small, or that the pressure is close to uniform inside the cell. Thus from the point of view of overall cell

208M. Herant and M. Dembo

shape and motion that is determined by cytoskeletal dynamics (Eq. 10.6), the precise value H does not matter as long as it is sufficiently small. However, from the point of view of internal cytosolic flow, which can play an important transport role, the value of H does matter and pressure gradients, even though small, are not negligible.

It is in the network (cytoskeleton) momentum equation that the rich complexity of cell mechanics becomes evident. Aside from pressure gradients and solvent-network drag, the network is also subject to viscous, elastic, and interaction forces and the network momentum equation can therefore be written:

θn P s θn (vn vs ) + · ν vn + ( vn )T − · = 0, (10.6)

Here, ν is the network viscosity and is the part of the network stress tensor remaining under static conditions. The latter can include interfilament interactions (such as contractility due to actin myosin assembly), filament-membrane interactions (such as Brownian ratchets), elastic forces due to deformations, and so forth.

Boundary conditions

These partial differential equations must of course be complemented by boundary conditions and this is where the characteristics of the plasma membrane come into play. From a mass conservation point of view, the key issue is that of permeability. In most circumstances, it seems reasonable that the membrane remains impermeable to the cytoskeleton (which may even be anchored to the membrane) so that therefore:

vM · n = vn · n

(10.7)

where vM is the velocity of the boundary and n is the outward normal unit vector. If we also assume that the membrane is impermeable to the cytosol (which appears to be true in some cases and not in others) we also have

vM · n = vs · n,

(10.8)

but this condition can certainly be relaxed to allow a net volume flux through the boundary.

From a momentum conservation point of view, there are two main possibilities: either the boundary is constrained by interaction with a solid surface, as in the case of a cell/dish or cell/pipette interface; or it is free membrane bathed by an inviscid external medium. In the former case, the boundary condition boils down to constraints on the normal (and in the case of no slip, tangential) components of the velocities. In the latter case, the boundary condition amounts to a stress continuity requirement:

ν vn + ( vn )T · n − · n Pn = −2γ κ n Pextn,

(10.9)

where γ is the surface tension, κ is the mean curvature of the membrane, and n is the outward normal to the membrane. The surface tension (and sometimes the permeability to the cytosol) is thus the main contribution of the cortical membrane to the governing evolution equations.