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0
filament velocity

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0
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Fig. 9-9. Actin dynamics in crawling cells. (A) For steady crawling, actin polymerization and
depolymerization must complete a balanced cycle. However, the demand for assembling monomers
at the leading edge causes a spatial segregation of these processes and ¬‚ows. Actin that assembles
at the leading edge but does not incorporate into a protrusion ¬‚ows in retrograde fashion as it
is disassembled. The emerging G-actin population is presumably returned to the leading edge by
diffusion. (B) The graphs show possible pro¬les for polymerization, monomer concentration, and
retrograde ¬‚ow across the front of a crawling cell. The calculated numbers combine measurements of
actin dynamics in ¬broblasts (Vallotton et al., 2004; Watanabe and Mitchison, 2002) and the leading
edge monomer demand for these cells (Abraham et al., 1999), and assumes monomer is returned by
diffusion with a diffusivity of 6 — 10’8 cm2 /s. From McGrath et al., 1998a.


Pollard and Borisy propose the lamellipodia are ¬lled with a neatly segregated, highly
dynamic network (Pollard and Borisy, 2003). In their model, ¬laments are ¬rst gener-
ated by activated Arp2/3 complex to form brush-like networks within one micron of
the advancing plasma membrane. The newly born ¬laments remain short because their
growth is rapidly terminated by abundant capping protein. The continuous assembly
of ATP-actin at the leading edge explains why the ADP-actin speci¬c ADF/co¬lins are
excluded from this region (Svitkina and Borisy, 1999). ADF/co¬lin-family proteins
Cell dynamics and the actin cytoskeleton 191

take up residence at distances one micron and more from the leading edge where
they bind to aged ADP-actin, disassembling the network into monomers for rapid
recycling and further polymerization (Svitkina and Borisy, 1999). More recent and
sophisticated analysis of actin ¬lament dynamics are consistent with a segregation
of the lamellipod into a ∼1 micron membrane-proximal region dominated by poly-
merization and an immediately adjacent region where signi¬cant depolymerization
occurs (Vallotton et al., 2004).
A leading theory for ¬lopodia generation proposes that these structures emerge
from a rearrangement of the dendritic networks of the lamellipod (Svitkina et al.,
2003). In the convergence/elongation theory, convergent ¬laments with barbed ends
that abut the plasma and are protected against capping are zippered together by fascin
as they grow to several microns in length. Consistent with this model, recent knock-
down studies reveal that ¬lopodia-rich phenotypes occur in capping-protein-depleted
cells and point to an essential role for the anticapping activies of the Ena/VASP family
in ¬lopodia formation (Mejillano et al., 2004).
Nearer the cell body, ¬laments surviving the destructive actions of ADF/co¬lins
mature into a contractile network. In fast-moving cells like ¬sh keratocytes, the surviv-
ing network remains ¬xed with respect to the substrate as the cell crawls past (Theriot
and Mitchison, 1991). In slower-moving cells like ¬broblasts, polymerization exceeds
the rate of cell advancement, and much of the network ¬‚ows toward the cell center
(Theriot and Mitchison, 1992). Filament survival is facilitated by association with the
long, side-binding protein tropomyosin, which blocks the association of ADF/co¬lins
(Bernstein and Bamburg, 1982; Cooper, 2002; DesMarais et al., 2002), and growth to
several microns is likely facilitated by the anticapping activities of formins (Higashida
et al., 2004). In ¬broblasts, the polarity of ¬laments is graded such that all ¬laments
at the leading edge are oriented with their barbed ends facing the periphery, but the
interior bundles have a well-mixed polarity (Cramer et al., 1997). The gradation ap-
pears to facilitate both pushing at the edge of cells and myosin-based contractions by
muscle-like ¬laments sliding within the cell interior. Just as some of the ¬laments of
the lamellipod mature into contractile stress ¬bers, the focal contacts that transmit
these stresses to surfaces also begin life in lamellipodia as nascent focal complexes
and mature into focal contacts as they become part of the more central structures of
the cell (DeMali et al., 2002).


Monomer recycling: the other ˜actin dynamics™
For steady migration a cell must have a constant supply of monomer delivered to
its leading edge. This monomer certainly derives from ¬lament disassembly at more
interior regions, and so the rates of assembly and migration are tied to the rate of
monomer supply. If monomer is provided by diffusion, the supply rate is equal to the
product of the diffusion coef¬cient and the gradient of the monomer concentration.
The diffusion coef¬cient of the fastest of two kinetically distinguishable populations
of actin tracers is ∼ 6 — 10’8 cm2 /s (Giuliano and Taylor, 1994; Luby-Phelps et al.,
1985; McGrath et al., 1998a). Assigning this diffusion coef¬cient to actin monomer
has caused dif¬culties for modelers of the lamellipod (Abraham et al., 1999; Mogilner
and Edelstein-Keshet, 2002). If the assembly of monomers at the leading edge of a
192 J.L. McGrath and C.F. Dewey, Jr.

crawling cell is driven by mass action, then the concentration of monomer at the
leading edge must exceed 15 µM for a single ¬lament to keep pace with the plasma
membrane in keratocytes. Fick™s law, in combination with a monomer diffusion co-
ef¬cient of 6 — 10’8 cm2 /s, requires a gradient of 20 µM/µm moving toward the
interior of the cell. If this gradient persists over 10 microns, as modelers have assumed,
then a maximum concentration of ∼ 200 µM G-actin near the cell body is prohibitively
high because cells typically carry less than 100 µM of actin total.
Assuming that the more diffusive population of actin is strictly monomer may
be wrong, as short, diffusing ¬laments from recent severing and growth events are
likely. Indeed, investigators have proposed that the diffusion of small oligomers may
explain why the diffusion coef¬cient for the mobile actin in cytoplasm is ∼6 times
slower than the value for monomer-sized ¬coll (Luby-Phelps et al., 1987). Through
comparisons with the diffusion of sugar particles of various sizes, one concludes that
actin diffuses as a molecule ¬ve times bigger than its hydrodynamic radius (Luby-
Phelps et al., 1987), and so the discrepancy cannot be explained by the fact that
G-actin is complexed with smaller molecules of thymosin or pro¬lin. The possibility
of ¬lament diffusion appears to justify the use of the high G-actin diffusion coef¬cient
of 3 — 10’7 cm2 /s in models (Abraham et al., 1999; Mogilner and Edelstein-Keshet,
2002). The value is that for the diffusion of monomer-sized ¬coll and leads to the
derivation of G-actin pro¬les with more gradual gradients of ∼ 4 µM/µm. However it
must be noted that all proteins studied diffuse slower in cytoplasm than size-matched
˜inert™ sugars, even those that do not oligomerize (Luby-Phelps et al., 1985), and so
the most reasonable explanation for the discrepancy between the diffusion of actin
and sugar particles is that the sugars are not good models for protein diffusivity.
In support of the interpretation of the fast-diffusing species as monomer, roughly
half of the tracer actin is in this population consistent with biochemical fractiona-
tion (McGrath et al., 2000c). Further, when cells are treated with jasplankinolide,
a membrane-permeating toxin that blocks ¬lament depolymerization (Bubb et al.,
2000), all actin becomes immobile, indicating that the fast-diffusing species is assem-
bly competent (McGrath et al., 1998a; Zicha et al., 2003). As noted, recent speckle
microscopy experiments indicate that the major zone of depolymerization in lamel-
lipodia is found 2 microns behind the leading edge (Ponti et al., 2004; Watanabe and
Mitchison, 2002), and so the gradient may in fact be steep but con¬ned to a dynamic
region considerably smaller than the 10 microns assumed in the models. Over this
small distance the experimentally determined values for monomer diffusion should
be suf¬cient for the steady resupply of monomer to the leading edge.
Recent data also suggest that G-actin is returned to the leading edge of cells by
active transport mechanisms. This prospect is raised by the recent results of Grahm
Dunn and colleagues (Zicha et al., 2003). Using a modi¬cation of the photobleaching
technique, these experimentalists marked a population of actin several microns behind
the leading edge of a protruding cell and found that some of the marked actin incor-
porates into new protrusions at rates that defy simple diffusion. They provide some
evidence that the phenomenon is halted by myosin inhibitors and does not depend on
microtubules. This suggests that the privileged monomer does not ride as cargo on
a motor complex, but that it happens to be near a convective channel. The dif¬culty
with the prospect of motors moving G-actin as cargo is that the association between
Cell dynamics and the actin cytoskeleton 193

actin and motor proteins would likely be speci¬c. The prospect of a return mechanism
that does not discriminate among molecules is more attractive because the problem
of recycling cytoskeletal pieces is not limited to G-actin; whatever mechanism is at
work must be able to carry all the building blocks to the leading edge.


The biophysics of actin-based pushing
While compelling support for actin polymerization forces have existed for decades
(Tilney et al., 1973), the mechanism by which polymerization leads to pushing remains
unclear. Today there are two leading theories: (1) a series of related ˜ratchet™ models
that explain pushing as a natural consequence of the polymerization of semi¬‚exible
¬laments against a membrane (Dickinson and Purich, 2002; Mogilner and Oster,
1996; Mogilner and Oster, 2003; Peskin et al., 1993) and (2) a mesoscopic model that
explains how pushing forces derive from the formation of actin networks on curved
surfaces.
Understanding of the biophysics of actin-based pushing in the Listeria system has
progressed through a steadily tightening cycle of theory and experiment that continues
to this day. In large part, these efforts have centered around the actin-based motility
of the bacterium Listeria monocytogenes. This intracellular pathogen invades host
cytoplasm and hi-jacks the same force-producing mechanisms that drive leading-
edge motility. Riding a wave of actin polymerization, the bacterium becomes motile
so that it can eventually exit the dying infected cell for an uninfected neighbor.
The ¬rst ratchet model proposed that Listeria was a 1-D Brownian particle blocked
from rearward diffusion by the presence of the growing actin tail (Mogilner and
Oster, 1996). Observations that Shigella move at speeds similar to Listeria despite
being more than twice as large violated a prediction of the Brownian Ratchet and
motivated a new theory. The “Elastic Ratchet” proposed that ¬laments, rather than
Listeria, ¬‚uctuate due to thermal excitation (Mogilner and Oster, 1996). Filaments
¬‚uctuate away from the Listeria surface to allow space for polymerization. Lengthened
¬laments apply propulsive pressure as they relax to unstrained con¬gurations.
More recent biophysical measurements established that Listeria are tightly bound
to their tails (Gerbal et al., 2000a; Kuo and McGrath, 2000), rigorously eliminating
the Brownian Ratchet theory and demanding a revision of the Elastic Ratchet theory.
One study also established that Listeria motion is discontinuous, with frequent pauses
and nanometer-sized steps (Kuo and McGrath, 2000). The developments led to the
¬rst proposal for how elastic ¬laments could push Listeria while attached. In the
˜Actoclampin™ model (Dickinson and Purich, 2002), ¬laments diffuse axially due to
bending ¬‚uctuations within a surface-bound complex. The complex binds ATP-bound
subunits and releases the ¬lament upon hydrolysis. In this scheme, ¬‚exed ¬laments
push the Listeria and lagging ¬laments act as tethers (Fig. 9-10A).
More than molecular stepping, the force-velocity curve of the polymerization en-
gine constrains theoretical models, but current data appear to be in disagreement.
Recently, we published a curve for Listeria monocytogenes (McGrath et al., 2003),
using methylcellulose to manipulate the viscoelasticity of extracts, particle track-
ing to determine viscoelastic parameters near motile Listeria, and a modi¬ed Stokes
equation to infer forces. In 2003, Mogilner and Oster published an evolution of the
194 J.L. McGrath and C.F. Dewey, Jr.




FL fw


fa




Fig. 9-10. Theories of force production in actin-based motility. (A“B) In Molecular Ratchet
models, ¬lament ¬‚uctuations and growth near the surface combine to create motility. (A) In the
Actoclampin model (from Dickinson and Purich, 2002) all ¬laments are similarly attached to the
motile surface, but some are compressed and others are stretched taut. (B) The Tethered Ratchet
(from Mogilner and Oster, 2003), considers distinct attached and pushing (working) ¬laments. (C“G)
In the Elastic Propulsion theory, elastic stresses lead to symmetry breaking and motion. (C) The
¬rst layer of actin polymerized at the surface is pushed outward (D) by the next layer, creating hoop
stresses in the gel and normal pressure on the sphere (E). Fluctuations in stress levels and strengths
cause a local fracturing event (F) and unraveling of the gel that leads to a motile state (G) in which
stresses build and relax periodically as the particle moves forward. (H) Large particles create tails
with periodic actin density (˜hopping™), suggesting stress building and relaxation. Bar is 10 microns.
From Bernheim-Groswasser, 2003.



Elastic Ratchet model (Mogilner and Oster, 2003) that quantitatively predicts the
force-velocity curve of McGrath et al. (2003). The “Tethered Ratchet” features “work-
ing” ¬laments that push as Elastic Ratchets and “tethered” ¬laments anchored at
complexes that also nucleate dendritic branches (Fig. 9-10B).
While Molecular Ratchets appear to account for the force-velocity data in McGrath
et al. (2003), shallower force-velocity curves obtained by Wiesner et al. (2003) are
interpreted in terms of a very different theory termed Elastic Propulsion (Gerbal
et al., 2000b) (Fig. 9-10C“H). The theory describes stress build-up in continuum,
elastic actin networks that grow on curved surfaces. During nucleation, older layers
are displaced radially by newer polymerization at the nucleating surface. Because the
displaced layers must stretch, they generate ˜hoop™ or ˜squeezing™ stresses around the
Cell dynamics and the actin cytoskeleton 195

curved particle or bacterium. Propulsive tails emerge from symmetric actin ˜clouds™
because the stresses eventually exceed gel strengths and cause the network to partially
unravel. In the motile con¬guration, hoop stresses continue to build due to surface
curvature, and frictional tractions hold the tail on the particle (Fig. 9-10G). Once
the propulsive stresses exceed frictional resistances, the object slips forward and tail
stresses relax. The process repeats as polymerization continues and stresses rebuild.
Further evidence supporting the Elastic Propulsion theory is a report by Bernheim-
Groswasser et al. (2002) that large (>4 micron) VCA-coated particles advance in
a micron-scale ˜hopping™ pattern (Bernheim-Groswasser et al., 2002). In this phe-
nomenon, the density of actin in tails varies in a periodic fashion to give tails a
banded pattern (Fig. 9-10H), periods of high actin intensity are also periods of low
velocities and vice versa (Bernheim-Groswasser et al., 2002). The pattern is inter-
preted as the build up and release of squeezing stresses in Elastic Propulsion. The
pattern is prominent on large particles because it takes longer to build the critical
stresses for slipping on surfaces with lower curvature.
Despite evidence supporting Elastic Propulsion, we found that it cannot be the only
mechanism for generating pushing forces in reconstitution experiments. Recognizing
that the nucleating surface must be curved for the Elastic Propulsion mechanism but
not for Molecular Ratchets, we tested whether actin-based motility could occur on
¬‚at surfaces. In Schwartz et al. (2004) we created ¬‚at particles by compressing heated
polystyrene spheres. Not only did we ¬nd that ¬‚at surfaces could be substrates for
pushing forces, we found that disks pushed on ¬‚at faces moved faster than did the
coated versions of the spheres from which they were manufactured.


Conclusion
This chapter on cell dynamics and the role of the actin cytoskeleton should be con-
sidered as a snapshot of a rapidly evolving ¬eld of research. In the last three years,
PubMed lists 4,000 articles with the two key words actin and cytoskeleton. Indeed,
many of the fascinating topics we work on today “ such as actin ¬lament branching and
severing, connection of the actin cytoskeleton to membrane-associated protein com-
plexes, and the behavior of actin bundles commonly termed stress ¬bers “ receive
negligible mention here. The ¬eld is very rich, and the inquiries are diverse. It is
a fruitful ¬eld of research with much fundamental biology and biophysics to be
discovered.
Can we point to speci¬c therapies that could be in¬‚uenced by research in the areas
described in this chapter? One possibility is understanding how the endothelium acts
to realign and create small-vessel proliferation in cancerous tissue. Finding means
to defeat the invagination of tumors would create the possibility of starving grow-
ing tumors and killing cancerous tissue selectively. The motility of the endothelium
depends intimately on the organization and turnover of the actin cytoskeleton.
A second grand challenge to which this research points is understanding the mech-
anisms of atherosclerosis proliferation. A key step is the trans-endothelial migration
of leukocytes and an in¬‚ammatory response cycle that leads to intimal smooth muscle
cell proliferation. In order for the leukocytes to cross an intact layer of endothelial
cells, the cells must ¬rst stick to the endothelium and then induce the underlying
196 J.L. McGrath and C.F. Dewey, Jr.

endothelial cells to retract, paving a path through which the leukocyte can enter the
arterial wall. Understanding the cytoskeleton dynamics associated with this process
could lead to therapies preventing intimal proliferation and subsequent plaque buildup
in the arteries.
The chapter makes clear the tremendous detail with which science now under-
stands both the dynamics and biophysics of the actin machinery that controls cell
shape. Applying this knowledge to disease will require not only continued discovery
of mechanisms and rates, but also the organization of the vast information into a
predictive computational model. Thanks to decades of investigation by scientists em-
phasizing quantitative experiments, modeling of the actin cytoskeleton is advanced
compared to the hundreds of other subcellular systems required to quantitatively de-
scribe cellular life. Models of other systems will inevitably join models of the actin
cytoskeleton over the next few years to begin the broad integration of knowledge.
The possibility exists to use actin models as examples, and to begin today to design
information architectures that can handle such massive amounts of information. Only
with a quantitative means of describing the highly nonlinear interaction between the
many important cellular systems can we hope to represent the complex and highly
nonlinear behavior within cells, and eventually tissues and organs.


Acknowledgements
The authors wish to thank John H. Hartwig for many years of fruitful collaboration
on problems related to cell mechanics and the cytoskeleton. His insights with regard
to the role of actin-binding proteins has been especially valuable. JLM acknowledges
support from Whitaker Research Grant #RG-01-033 and thanks Mr. Ian Schwartz for
Fig. 9-8. CFD is grateful for many years of support by the NHLBI, National Institutes
of Health.


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10 Active cellular protrusion: continuum
theories and models

Marc Herant and Micah Dembo




This chapter attempts to develop a general perspective on the phenomenon of
ABSTRACT:
active protrusion by ameboid cells. Except in rare special cases, principles of mass and mo-
mentum conservation require that one consider at least two phases to explain the process of
cellular protrusion. These phases are best identi¬ed with the cytosolic and cytoskeletal com-
ponents of the cytoplasm. A continuum mechanical formalism of Reactive Interpenetrative
Flows (RIF) is contructed. It is general enough to encompass within its framework a large
range of theories of active cell deformation and movement. Most physically plausible theories
of protrusion fall into one of two classes: protrusion driven by cytoskeletal self-interactions;
or protrusion driven by cytoskeletal“membrane interactions. These are described within the
RIF formalism. The RIF formalism is cast in a form that is amenable to computer simulations
through standard numerical algorithms. An example is next given of the numerical study of the
most elementary protrusive event possible: the formation of a single pseudopod by an isolated
round cell. Some ¬nal thoughts are offered on the role of modeling in understanding cellular
mechanical activity.




Cellular protrusion: the standard cartoon
Over the past decades, experiments have consistently demonstrated that active protru-
sion in animal cells is accompanied by a local increase in cytoskeletal density through
active polymerization. A review of the evidence for this is beyond the scope of this
chapter, but key points include the high concentration of ¬lamentous actin observed
by ¬‚uorescence or EM at the leading edge of growing protrusions, as well as the
abrogation of protrusion by nearly any disruption of actin polymerization, such as
that caused by cytochalasin. In addition, in most cases of ameboid free protrusion,
it does not appear that molecular motors such as myosins are mandatory; instead,
simple polymerizing activity seems suf¬cient.
This has led to a “standard cartoon” model of cellular protrusion that ¬gures promi-
nently in many reviews or textbooks of cell biology (see Fig. 10-1). In this picture,
cytoskeletal monomers are added by polymerization at the leading edge and removed
by depolymerization at the base of the protrusion. The free monomers then diffuse
back to the front to be reincorporated in the cytoskeleton in a process that has been
called “treadmilling.”
204
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 knowl-
edge. On the one hand, this cartoon is a means of communicating a complex mech-
anism 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 ¬‚ow of
material in the form of water and cytoskeletal monomers and backward ¬‚ow of
cytoskeletal material in the form of ¬laments (this is the treadmilling).
3. There is a net ¬‚ow 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
¬‚ow 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:
r The cortical membrane de¬nes the boundary of the cell by controlling (and
often preventing) volume ¬‚uxes with the external world. It is furthermore highly
206 M. Herant and M. Dembo

¬‚exible, ¬‚uid, and virtually inextensible. Together, these three properties make
it a good conductor of stress.
r 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).
r The cytosol ¬‚ows 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 ¬lamentous or
globular form. However, given biomolecules can be classi¬ed 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 classi¬cation ig-
nores membrane-enclosed organelles. In particular, the contribution to the mechan-
ical 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 ¬elds. Then the general framework of
continuum mechanics can be applied to animal cells just as it is done with any other
material. More speci¬cally, 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 ¬eld, vs (x, t) the
solvent velocity ¬eld, where x is the position vector and t is the time. The scalar ¬elds θn
and θs and the vector ¬elds vn and vs are thus de¬ned on a simply connected, compact
domain in Euclidian space that de¬nes 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 ¬elds 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 ¬‚ow is given by the sum of the ¬‚ow 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 con-
centration at a given point in space (Eulerian derivative) is the sum of an advective
transport term describing the net in¬‚ow 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 simpli¬ed 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 + Hθs 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 coef¬cient 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
208 M. 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 suf¬ciently small. However, from the point of
view of internal cytosolic ¬‚ow, 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 ’ Hθs 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 inter¬lament interactions (such as
contractility due to actin myosin assembly), ¬lament-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:

v M · n = vn · n (10.7)

where v M 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

v M · n = vs · n, (10.8)

but this condition can certainly be relaxed to allow a net volume ¬‚ux 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 κn ’ Pext n, (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.
Active cellular protrusion: continuum theories and models 209

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 speci¬cations of the
cell. For instance, it is likely that the viscosity ν will depend on the cytoskeletal
density: a law such as

ν = ν 0 θn (10.10)

prescribing a well-de¬ned linear relation between viscosity and network concentra-
tion is such a constitutive relation. In principle, this could be veri¬ed empirically by
investigating the rheology of the cytoplasm at various cytoskeletal concentrations.
However, in general such experimental evidence is sparse and often dif¬cult to inter-
pret. One is thus usually reduced to educated guesses for the constitutive laws that
govern J (the network formation or cytoskeletal polymerization rate), H (the resis-
tance to solvent ¬‚ow 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 suf¬ciently
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 ¬lament is
transduced to generate a pressure against a membrane that sterically interferes with
the reaction. Without going into the speci¬cs, however, it is clear that such cytoskeletal
theories of protrusion can be categorized into two classes:
r Network“membrane interaction theories in which the cytoskeleton and the mem-
brane repel one another through a force ¬eld. 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).
r 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, pro-
trusions 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.
210 M. Herant and M. Dembo

Network“membrane interactions
The basic idea behind a network“membrane disjoining stress is that there exists
a repulsive force between actin monomers (polymerized or unpolymerized) and the
cortical membrane. For free (G-actin) subunits this has no dynamical consequences, as
redistribution occurs freely in the cytosol. However, once subunits are sequestered into
the cytoskeleton by polymerization, the repulsive force has dynamical consequences
because it endows the cytoskeleton with a macroscopic stress. In other words, while
free monomers cannot push back against the membrane, ¬laments can because they
are braced by the entire inner cytoskeletal scaffolding of the cell (these concepts ¬rst
came to the fore with the work of Hill and Kirschner, 1982).
To put these notions on a more formal footing, let us postulate a mean-¬eld repulsive
force between monomers (free or in a ¬lament) and the cortical membrane that derives
from the interaction potential ψ M (r ) where r is the distance from the membrane
and where we set the constant by requiring limr ’∞ ψ M (r ) = 0. From equilibrium
thermodynamics, we can relate the ratio of free monomer concentration far away
([M free (∞)]) and at distance r with the interaction potential as a Boltzmann factor:

ψ M (r )
[M free (r )]
= exp ’ . (10.11)
[M free (∞)] kB T

Here we can think of ψ M (r ) as the average work required to bring a monomer from
in¬nity to distance r from the membrane.
At the same time we have the chemical reaction between the polymerized and
unpolymerized state of the monomer.

M bound .
M free (10.12)

Taking actin as an example where addition of free monomers into polymers occurs
free
at the barbed ends of ¬laments, there exists a critical free-monomer density [Mcrit ]
above which reaction 10.12 is driven to the right and below which it is driven to the
left.
We need to examine 10.12 in the light of 10.11. Let us assume that the membrane“
monomer interaction potential ψ M (r ) decreases monotonously with distance from
the membrane, and further that it is in¬nite (or very large) at zero distance from
the membrane (see Fig. 10-2; in other words, the monomer is excluded from the
membrane by a hard-core potential). Then it should be clear that in a region near
free
the membrane, [M free (r )] is smaller than [Mcrit ], and hence that there cannot be any
monomers added into polymers in this region. This region is labeled the ˜gap™ in
Fig. 10-2. Advected polymers may appear in the gap, but unless they are stabilized
by capping, they will tend to disassemble.
Further away from the membrane, ψ M (r ) decreases to the point that [M free (r )]
free
is smaller than [Mcrit ] and this allows the elongation of polymers by driving free
monomers from the free to the bound state. Assuming that the polymerization of free
free
monomers is rapid, we then have a region where [M free (r )] [Mcrit ]. This means
that for the region of interest, that is, where polymerization is allowed to take place
(outside the gap) but within the region where membrane“monomer interaction is still
Active cellular protrusion: continuum theories and models 211



networ“kmembrane
ψ
potential



δ


r

rearward outward
cytoskeletal membrane
velocity velocity




polymerization gap
zone

Fig. 10-2. Cartoon of network-membrane interactions.


signi¬cant, we have

[M free (∞)]
ψ (r ) = k B T ln .
M
(10.13)
free
Mcrit
This region is labeled ˜polymerization zone™ in Fig. 10-2; within a network“membrane
interaction model, it is that region that determines the dynamics of protrusion. Of note
is that polymerization could take place further back, but that regions further to the
rear do not have a dynamical impact because there, ψ M (r ) ∼ 0.
The force exerted by the membrane on a monomer (free or bound) is given by
’∇ · ψ M , and by action and reaction, this is the opposite of the force f M exterted by
the monomer on the membrane. If δ is the range of the potential (we assume the gap
region to be small), then we approximately have


[M free (∞)]
kB T
fM = ln (10.14)
n.
δ free
Mcrit
If we further assume that most of the monomers in the vicinity of the membrane are
sequestered in the cytoskeletal phase, the total force-per-unit area of the membrane
is given by the number of monomers within range times the force per unit monomer:

θn θn
[M free (∞)] [M free (∞)]
kB T
= δ— n=
M
ln k B T ln (10.15)
n,
δ free free
VM VM
Mcrit Mcrit
where VM is the volume of a monomer. For actin networks, VM = (4π/3)(2.7 —
10’7 cm)3 , so that for normal temperature conditions

[M free (∞)]
dyn cm’2 .
= 5 — 10 θn ln
M 5
(10.16)
free
Mcrit
212 M. Herant and M. Dembo

Natural logarithms of even very large numbers are seldom more than 10, and maximal
volume fractions of cytoskeleton are ∼2% (see Hartwig and Shevlin, 1986). The
maximum disjoining stress at the membrane/cytoskeleton interface is thus of order
105 dyn cm’2 (104 Pa, 100 cm H2 O).

Network dynamics near the membrane
One should be cautious not to assimilate the force-per-unit area of membrane or
stress given in Eq. 10.16 to a protrusive force; in general the net outward force at the
membrane as could theoretically be measured with a constraining spring would be
considerably less. There are two reasons for this.

1. Imperfect cytoskeletal bracing against back¬‚ow: unless there exists some sort
of mechanism to brace the cytoskeleton near the membrane, it will simply
slide back, negating any outward force. In general, such bracing is expected
to be provided by the viscoelastic properties of the cytoskeleton interior to the
boundary layer, which transports stress to whatever is bracing the cell (in other
words, the substratum).
2. Imperfect cytoskeletal decoupling from the membrane: if the cytoskeleton is
somehow anchored to the membrane and cannot ¬‚ow back, the stress FM is
simply carried by the anchors and no outward force results.

We will assume here that the second condition is appropriately ful¬lled, although
experimental evidence is sometimes contradictory (for example, Listeria actin tails
are attached to the Listeria, Gerbal et al., 2000). Let us further assume that the
counteracting force to rear ¬‚ow is provided by interior cytoskeletal viscosity. In that
case, simple dimensional analysis (which can be made more rigorous, see Herant
et al., 2003) gives

δ
M
v (10.17)
ν
where ν is the viscosity and v is the velocity change near the membrane (see
Fig. 10-2). In the case of perfect bracing and no external opposing force, the protrusive
velocity is then v, but in the general case it will be less. In the limit of a stalled
protrusion, the protrusive velocity is zero while the retrograde ¬‚ow of cytoskeleton
is ’ v. Note that within this picture, the important parameter is not the magnitude
of M but rather its product with the range of the membrane“network interaction
δ.
M

Finally, an interesting ¬nding is that, if one assumes that the constitutive laws for
the viscosity and for the network“membrane force have the same dependence on
the cytoskeletal concentration, for example ν = ν0 θn , M ∝ 0 θn , then one notices
M

that Eq. 10.17 implies that v is independent of the network concentration near the
membrane. Using numerical values for ν0 (6 — 106 poise, see Herant et al., 2003)
and 0 (upper limit 5 — 106 dyn cm’2 , see Eq. 10.16), one gets v < δ s’1 . Typical
M

velocities v at the leading edge are at least 10 nm s’1 and may range to as high as
0.5 µm s’1 , (see Theriot and Mitchison 1992), which means that δ, the characteristic
range of interaction, must be greater than 10 nm and even reach 0.5 µm.
Active cellular protrusion: continuum theories and models 213



M M




Position Position

Fig. 10-3. Monomer concentration vs. position (dots “ free monomers; dashes “ bound monomers;
solid “ total monomers).



Special cases of network“membrane interaction: polymerization
force, brownian and motor ratchets
In recent years, the concept of “polymerization force” has gained traction as a putative
explanation for cellular protrusion. While the concept is often used in a vague manner,
it has been formalized on more solid physical grounds in two interesting models: the
Brownian ratchet model (see Mogilner and Oster, 1996) and the clamped ¬lament
elongation model (Dickinson and Purich, 2002). In essence, these models are special
cases of network“membrane interactions in that they rely on the hard-core repulsion
between monomers and membrane as an interaction potential. For instance, Eq. 10.14
is identical to that commonly given for the force produced by a Brownian ratchet (see
for example Howard, 2001), where δ is taken to be the incremental lengthening of the
polymer by addition of a monomer.


Network“network interactions
The basic idea behind a network swelling stress is similar to that of a membrane“
network disjoining stress and we shall follow the approach of the previous section.
One begins with the assumption that there exists a repulsive force between actin
monomers, free or bound. Again, for free (G-actin) subunits, this has no dynamical
consequences, as redistribution occurs freely in the cytosol. However, once subunits
are sequestered into the cytoskeleton by polymerization, the repulsive force has dy-
namical consequences because it endows the cytoskeleton with a macroscopic stress.
Under these conditions, one can intuitively perceive how the energy of the chemical
process of polymerization can be transformed into expansion work.
Fig. 10-3 illustrates this principle. On the left, polymerizing activity is moderate;
wherever there is a mild excess of monomers bound in ¬laments, free monomers are
driven out by the repulsive interaction. The end result is that the total concentration
of monomers (bound and unbound) varies little.
The amount of variation is determined by the relative magnitudes of the time scale
of thermal-driven diffusion of monomer into regions of excess polymer „diff = l 2 /D
and the time scale of force-driven diffusion out of regions of excess polymer „force =
l 2 /[D(ψ/k B T )] (where l is the length scale of the region, D the monomeric diffusion
coef¬cient, and ψ the repulsive potential, and where we have used Einstein™s relation
between viscosity and diffusion coef¬cients).
214 M. Herant and M. Dembo

If, however, polymerization becomes intense “ for instance due to uncapping of
¬laments “ it can drive the number of monomers sequestered in ¬laments above that
of the background, and we have the case depicted on the right of Fig. 10-3. The local
free monomer concentration goes near zero but cannot be negative, so that the total
monomer concentration has a signi¬cant bump. A monomer moving around therefore
sees a repulsive potential force in the bump that is higher than the baseline away from
the bump.
Our formal development here will approximately parallel that of the network“
membrane interaction problem in the previous section. We assume a pairwise repulsive
potential force (potential φ) between actin monomers either free or part of a ¬lament.
The total force exerted on a monomer M is therefore the result of a sum on all other
monomers Mi :

‚φ M Mi
F M Mi = ’ ’∇ψ n . (10.18)
‚r M Mi
i i

Here ψ n is the part of the potential that comes from ¬xed monomers sequestered
in ¬laments. Generally, this will be the dominant contribution wherever network is
highly concentrated, as free monomers will naturally diffuse away and lower the free
monomer concentration (see Fig. 10-3). Just like the case of membrane“network
interactions, we have a Boltzmann factor,

ψ n (bump)
[M free (bump)]
= exp ’ (10.19)
[M free (baseline)] kB T

where we have assumed that the network concentration outside of the “bump” is so
low as to make ψ n (bump) ψ n (baseline) 0. In this picture, ψ n (bump) is the work
of bringing one free monomer from baseline concentration into bump (Fig. 10-3).
Again, we have the chemical reaction

M bound ,
M free (10.20)

which goes to the right if [M free (x)] > [Mcrit ] and to the left if [M free (x)] < [Mcrit ]
free free
free
where, as before, [Mcrit ] is the critical free monomeric concentration above which
free ends of polymers are lengthened by monomer addition.
In regions of very high network density, ψ n is large, and by Eq. 10.19 this leads to
[M free ] < [Mcrit ]. This drives Eq. 10.20 to the left (depolymerization) so that one can
free

say that such a region cannot be created by polymerization (although external or con-
tractile forces could compress network above such a threshold). It is therefore clear
that the highest network concentration achievable by chemical network polymeriza-
free
tion is that for which [M free (x)] [Mcrit ], and that therefore, the highest achievable
network repulsive stress per monomer is:

[M free (baseline)]
ψ = k B T ln .
n
(10.21)
free
Mcrit

Eq. 10.21 makes it clear that in general, the stress contribution by polymerizing a
single monomer into the bump is at most of order 10 k B T . Let VM be the volume of
Active cellular protrusion: continuum theories and models 215

a monomer, we have
θn (bump)
,
n
(bump) 10 k B T (10.22)
VM
where n is the spatially averaged stress density that will contribute in the network
momentum equation (Eq. 10.6). Note here that a proper treatment would integrate n
from a network concentration of 0 to θn , including a threshold effect (see Fig. 10-3)
and would also yield a θn term instead of θn . Considering the other uncertainties of
2

the problem, we have preferred to sacri¬ce accuracy to simplicity.
If we use typical biological numbers, for example actin, VM = (4π/3)(2.7 —
10’7 cm)3 , and maximal volume fractions of cytoskeleton θn ∼2%, we obtain a maxi-
mum swelling stress of the cytoskeleton of order 105 dyn cm’2 (104 Pa, 100 cm H2 O),
the same as the maximum stress for the network“membrane interaction.

Network dynamics with swelling
It is obvious that network“network repulsion will tend to smooth out nonuniform
cytoskletal distributions by expansion of overdense regions into underdense regions.
It is thus of interest to look at the dynamics of a clump of network in a low-density
environment. If the principal retarding force is taken to be viscosity of the network
itself, simple dimensional analysis shows that the time scale of expansion is:
ν
„ = n. (10.23)

Of note is that there is no intrinsic scale to the problem; instead it is set by the
length scale of the clump of overdense network d, and so the characteristic veloc-
ity is v = d/„ . In addition, if both ν and n have the same functional dependence
(for example, linear) on the network volume fraction θn , then „ becomes indepen-
dent of θn . We have used and provided experimental support for ν = ν0 θn where
ν0 = 6 — 106 poise (Herant et al. 2003) and from the calculation above, the swelling
stress n = 0 θn is such that at most 0 5 — 106 dyn cm’2 (there is no reason
n n

it cannot be much less), which gives a minimal expansion time scale of order one
second.


Other theories of protrusion
For the sake of completeness, we would like to brie¬‚y touch upon alternative theories
of protrusion that are not currently in vogue because they do not ¬t in the standard
cartoon (Fig. 10-1). Of note is that all these theories are also amenable to modeling
within the RIF formalism and that it is out of concern for keeping this survey within
a reasonable length that we do not pursue a quantitative analysis for each of these
models (see Fig. 10-4).

Hydrostatic pressure protrusion. Hydrostatic pressure-driven protrusion is a ven-
erable model (Mast, 1926) that probably has applicability in a limited number of
circumstances. The basic idea is that an increase in internal pressure (presumably due
to contractile activity somewhere in the cell) leads to bulging and protrusion of the
membrane. Hidden behind the apparent simplicity of the concept are a number of
216 M. Herant and M. Dembo




H2O H2O




Cortical Shearing
Contraction/Weakening Hypertonicity Myosins
Permeability

Fig. 10-4. Three alternative theories of cellular protrusion.


factors that merit consideration. In order for there to be a local protrusion of mem-
brane, either the increase of hydrostatic pressure has to be local to the region, or the
compliance of the cortical membrane must increase locally. Except in very large cells
(for example, Amoeba proteus), or in compartmentalized organisms, the former con-
dition is dif¬cult to realize. This is because the hydraulic resistance to cytosolic ¬‚ow
through the cytoskeleton is typically extremely small. As a result, any local pressure
excess tends to be quickly erased by solvent ¬‚ow. The only way around this constraint
is to have a large distance leading to a large hydraulic resistance, or an isolated com-
partment in which pressure can be locally increased without driving cytosolic ¬‚ow to
other parts of the cell.
The alternative is that of a local weakening of the cortical tension driving a local
Marangoni-type of ¬‚ow. Again there are some dif¬culties with such a mechanism.
Recall that cell cortical tension is the result of contributions from the tension of
the plasma membrane and from the cytoskeletal cortex underlying the membrane.
It is unlikely that membrane tension can be lowered locally, because the massless,
¬‚uid-mosaic nature of the plasma membrane should make it a good conductor of
stress that equilibrates surface tension rapidly around the cell. (Experimental and
modeling evidence hint at the membrane tension being a global property of the cell,
Raucher and Sheetz, 1999; Herant et al., 2003.) It is, however, possible for the cy-
toskeletal cortex to be locally weakened (see Lee et al., 1997). In a regime in which
it carries substantial (tensile) stresses, such a weakening may result in pseudopod
formation.

Hypertonic protrusion. Although as far as we are aware they have not been the subject
of much recent study, models based on osmotic swelling accompanied by modi¬ed
membrane permeability were once actively pursued, especially in the context of the
extension of the acrosomal process of the Thyone sperm, a setting that may not have
general applicability to ameboid protrusion (Oster and Perelson, 1987). Here the basic
idea is that through the action of locally activated severing enzymes, osmotic tension
Active cellular protrusion: continuum theories and models 217

near the tip of a protrusion is increased, and that the permeability of the plasma
membrane is suf¬cient to allow signi¬cant in¬‚ow from the extracellular environment.
It is then possible for the volume of the protrusion to grow, and the ¬lling 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 shear-
ing motors in the sense that they actively slide ¬laments parallel to one another. If
one imagines a reasonably stiff assembly of cytoskeletal ¬laments 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 protru-
sions, 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 ¬nite 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.
Brie¬‚y, 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 ¬‚ow 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 ap-
proach 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).
218 M. Herant and M. Dembo

Because of the multiphase nature of the ¬‚ow (one has to solve the triplet vn , vs , and
P rather than just for v and P), this last step requires some modi¬cations from the
usual treatment.
By adding the solvent and network momentum equation (Eq. 10.5 and 10.6) to-
gether, vs can be eliminated to obtain a “bulk” cytoplasmic momentum equation:
’∇ P + ∇ · ν(∇vn + (∇vn )T ) ’ ∇ · = 0. (10.24)
This is to be complemented with the appropriate boundary condition for stress across
the membrane, which, in usual situations, looks like
ν(∇vn + (∇vn )T ) · n ’ Pn ’ · n = ’Pext n ’ 2γ κn, (10.25)
where Pext is the external pressure, γ and κ the surface tension and mean curvature.
The solvent momentum equation (Eq. 10.5) gives an expression for vs :
∇P
vs = vn ’ (10.26)
Hθn
which can then be substituted in the incompressibility condition to yield:
1 θs
∇ · vn ’ ∇P = 0. (10.27)
H θn
In situations where there is zero membrane permeability (in other words, no trans-
membrane solvent ¬‚ow) the boundary condition simpli¬es to:
∇ P = 0. (10.28)
Following the standard Uzawa method, an initial guess for the pressure ¬eld allows
the computation of the network velocity ¬eld by Eq. 10.24. This velocity ¬eld can
then be used to update the pressure ¬eld by Eq. 10.27, and so on through iterations
between the two equations. Once the network velocity ¬eld vn and pressure ¬eld
P have converged to a self-consistent solution, the solvent velocity ¬eld vs can be
trivially extracted through the use of Eq. 10.26 with automatic enforcement of the
incompressibility condition.


An example of cellular protrusion
Probably the simplest possible case of cellular protrusion is the emergence of a single
pseudopod from an isolated, initially round nonadherent cell. This con¬guration has
the advantages of a simple geometry with two-dimensional cylindrical symmetry
and of avoiding potential confounding factors that appear when the mechanics of
adhesion is involved. The formation of such pseudopods has been studied in individual
neutrophils by Zhelev et al. (2004) and modeled in some detail by Herant et al. (2003).
Here we present a simpli¬ed version of this process as a pedagogical introduction to
the basic principles that are involved.
We will describe behaviors under both a cytoskeleton-membrane-repulsion and
cytoskeleton-swelling model. In both cases, we make the following assumptions:
r Initial condition is that of a round cell of diameter 8.5 µm.
r Cortical tension is γ = 0.025 dyn cm’1 .
Active cellular protrusion: continuum theories and models 219

r Equilibrium network (cytoskeleton) volume fraction is θ0 = 0.1% (Watts and
Howard 1993) everywhere inside the cell except:
r Network fraction over a spherical patch of cell surface (diameter 1 µm) is
¬xed to one percent.
r Away from the frontal patch, network fraction evolves to its equilibrium
according to ¬rst order kinetics with time-scale „n = 20 seconds, that is
dθn /dt = (θ0 ’ θn )/„n .
r Network viscosity is given by ν0 θn where ν0 = 6 — 106 poise so that the
baseline interior viscosity of the cell is ν = ν0 θ0 = 6000 poise.

These values are reasonable approximations of the characteristics of human neu-
trophils (see Herant et al., 2003).
The idea is that in both the repulsion and the swelling models, the excess cytoskele-
ton at the activated patch of cortex will drive the formation of a pseudopod. Bracing
by the viscous interior of the cell allows outward protrusion against the restoring force
of the cortical tension that tends to sphericize the cell.


Protrusion driven by membrane“cytoskeleton repulsion
The mechanics of membrane“cytoskeleton repulsion as a driver of protrusion is
straightforward as follows: (i) a region with increased cytoskeletal density appears
due to enhanced polymerization at the leading edge of the pseudopod. (ii) Due to
repulsion from the cortical membrane, cytoskeleton ¬‚ows into the cell while cytosol
is sucked forward. (iii) Cytoskeletal ¬‚ow into the cell is opposed by viscous resistance
of the underlying cytoplasm. By action and reaction, this leads to bulging out of the
membrane, eventually creating and lengthening a pseudopod.
As has been pointed out, the quantity that matters for the dynamics of membrane“
cytoskeleton interaction is the product of the membrane“cytoskeleton interaction
stress density M with the range δ of the interaction. In numerical simulations that
encompass the whole cell, it is not practical to try to accurately model the details of
the cytoskeletal dynamics near the membrane as depicted in Fig. 10-2. Instead, the

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