= = . (8.11)
╬║¤Ç 2 n2 6p
Polymer-based models of cytoskeletal networks 159
Similar scaling arguments to those above lead us to expect that the typical transverse
amplitude of a segment of length is approximately given by
in the absence of applied tension. The precise coef´¬ücient for the mean-square ampli-
tude of the midpoint between ends separated by (with vanishing de´¬‚ection at the
ends) is 1/24.
For a ´¬ünite tension ¤„ , however, there is an extension of the chain (toward full
extension) by an amount
╬┤= Ôł’ = , (8.13)
╬║¤Ç 2 n 2 (n 2 + ¤ć)
where ¤ć = ¤„ 2 /(╬║¤Ç 2 ) is a dimensionless force. The characteristic force ╬║¤Ç 2 / 2 that
enters here is the critical force in the classical Euler buckling problem (Landau and
Lifshitz, 1986). Thus, the force-extension curve can be found by inverting this rela-
tionship. In the linear regime, this becomes
╬┤= ¤ć = ¤„, (8.14)
p¤Ç 90 p ╬║
that is, the effective spring constant for longitudinal extension of the chain segment
is 90╬║ p / 4 . The scaling form of this could also have been anticipated, based on very
simple physical arguments similar to those above. In particular, given the expected
dominance of the longest wavelength mode ( ), we expect that the end-to-end contrac-
tion scales as ╬┤ Ôł╝ (Ôł‚u/Ôł‚ x)2 Ôł╝ u 2 / . Thus, ╬┤ 2 Ôł╝ Ôł’2 u 4 Ôł╝ Ôł’2 u 2 2 Ôł╝ 4 / 2 , p
which is consistent with the effective (linear) spring constant derived above. The full
nonlinear force-extension curve can be calculated numerically by inversion of the
expression above. This is shown in Fig. 8-4. Here, one can see both the linear regime
for small forces, with the effective spring constant given above, as well as a diver-
gent force near full extension. In fact, the force diverges in a characteristic way, as
the inverse square of the distance from full extension: ¤„ Ôł╝ |╬┤ Ôł’ |Ôł’2 (Fixman and
We have calculated only the longitudinal response of semi´¬‚exible polymers that
arises from their thermal ´¬‚uctuations. It is also possible that such ´¬ülaments will
actually lengthen (in arc length) when pulled on. This we can think of as a zero-
temperature or purely mechanical response. After all, we are treating semi´¬‚exible
polymers as little bendable rods. To the extent that they behave as rigid rods, we
might expect them to respond to longitudinal stresses by stretching as a rod. Based
on the arguments above, it seems that the persistence length p determines the length
below which ´¬ülaments behave like rods, and above which they behave like ´¬‚exible
polymers with signi´¬ücant thermal ´¬‚uctuations. Perhaps surprisingly, however, even
for semi´¬‚exible ´¬ülament segments as short as Ôł╝ 3 a 2 p , which is much shorter than
the persistence length, their longitudinal response can be dominated by the entropic
force-extension described above, that is, in which the response is due to transverse
thermal ´¬‚uctuations (Head et al., 2003b).
160 F.C. MacKintosh
0.03 0.1 0.3 1
Fig. 8-4. The dimensionless force ¤ć as a function of extension ╬┤ , relative to maximum extension
. For small extension, the response is linear.
Dynamics of single chains
The same Brownian forces that give rise to the bent shapes of ´¬ülaments such as in
Fig. 8.1 also govern the dynamics of these ´¬‚uctuating ´¬ülaments. Both the relaxation
dynamics of bent ´¬ülaments, as well as the dynamic ´¬‚uctuations of individual chains
exhibit rich behavior that can have important consequences even at the level of bulk
solutions and networks. The principal dynamic modes come from the transverse
motion, that is, the degrees of freedom u and v above. Thus, we must consider time
dependence of these quantities. The transverse equation of motion of the chain can
be found from Hbend above, together with the hydrodynamic drag of the ´¬ülaments
through the solvent. This is done via a Langevin equation describing the net force per
unit length on the chain at position x,
0 = Ôł’╬Â u(x, t) Ôł’ ╬║ 4 u(x, t) + ╬żÔŐą (x, t), (8.15)
which is, of course, zero within linearized, inertia-free (low Reynolds number) hy-
drodynamics that we assume here.
Here, the ´¬ürst term represents the hydrodynamic drag per unit length of the ´¬ül-
ament. We have assumed a constant transverse drag coef´¬ücient that is independent
of wavelength. In fact, given that the actual drag per unit length on a rod of length
L is ╬Â = 4¤Ç╬·/ln (AL/a), where L/a is the aspect ratio of the rod, and A is a con-
stant of order unity that depends on the precise geometry of the rod. For a ´¬ülament
´¬‚uctuating freely in solution, a weak logarithmic dependence on wavelength is thus
expected. In practice, the presence of other chains in solution gives rise to an effective
screening of the long-range hydrodynamics beyond a length of order the separation
between chains, which can then be taken in place of L above. The second term in the
Langevin equation above is the restoring force per unit length due to bending. It has
been calculated from Ôł’╬┤ Hbend /╬┤u(x, t) with the help of integration by parts. Finally,
we include a random force ╬żÔŐą that accounts for the motion of the surrounding ´¬‚uid
Polymer-based models of cytoskeletal networks 161
A simple force balance in the Langevin equation above leads us to conclude that
the characteristic relaxation rate of a mode of wavevector q is (Farge and Maggs,
¤ë(q) = ╬║q 4 /╬Â. (8.16)
The fourth-order dependence of this rate on q is to be expected from the appearance of
a single time derivative along with four spatial derivatives in Eq. 8.15. This relaxation
rate determines, among other things, the correlation time for the ´¬‚uctuating bending
modes. Speci´¬ücally, in the absence of an applied tension,
u q (t)u q (0) = .
That the relaxation rate varies as the fourth power of the wavevector q has important
consequences. For example, while the time it takes for an actin ´¬ülament bending mode
of wavelength 1 ┬µm to relax is of order 10 ms, it takes about 100 s for a mode of
wavelength 10 ┬µm. This has implications, for instance, for imaging of the thermal
´¬‚uctuations of ´¬ülaments, as is done in order to measure p and the ´¬ülament stiffness
(Gittes et al., 1993). This is the basis, in fact, of most measurements to date of the
stiffness of DNA, F-actin, and other biopolymers. Using Eq. 8.17, for instance, one
can both con´¬ürm thermal equilibrium and determine p by measuring the mean-
square amplitude of the thermal modes of various wavelengths. However, in order
both to resolve the various modes as well as to establish that they behave according
to the thermal distribution, one must sample over times long compared with 1/¤ë(q)
for the longest wavelengths ╬» Ôł╝ 1/q. At the same time, one must be able to resolve
fast motion on times of order 1/¤ë(q) for the shortest wavelengths. Given the strong
dependence of these relaxation times on the corresponding wavelengths, for instance,
a range of order a factor of 10 in the wavelengths of the modes corresponds to a range
of 104 in observation times.
Another way to look at the result of Eq. 8.16 is that a bending mode of wavelength
╬» relaxes (that is, fully explores its equilibrium conformations) in a time of order
╬Â ╬»4 /╬║. Because it is also true that the longest (unconstrained) wavelength bending
mode has by far the largest amplitude, and thus dominates the typical conformations
of any ´¬ülament (see Eqs. 8.10 and 8.17), we can see that in a time t, the typical or
dominant mode that relaxes is one of wavelength ÔŐą (t) Ôł╝ (╬║t/╬Â )1/4 . As we have seen
above in Eq. 8.12, the mean-square amplitude of transverse ´¬‚uctuations increases
with ´¬ülament length as u 2 Ôł╝ 3 / p . Thus, in a time t, the expected mean-square
transverse motion is given by (Farge and Maggs, 1993; Amblard et al., 1996)
u 2 (t) Ôł╝ ( / Ôł╝ t 3/4 ,
ÔŐą (t)) (8.18)
because the typical and dominant mode contributing to the motion at time t is of
wavelength ÔŐą (t). Equation 8.18 represents what can be called subdiffusive motion
because the mean-square displacement grows less strongly with time than for diffusion
or Brownian motion. Motion consistent with Eq. 8.18 has been observed in living
cells, by tracking small particles attached to microtubules (Caspi et al., 2000). Thus,
in some cases, the dynamics of cytoskeletal ´¬ülaments in living cells appear to follow
the expected motion for transverse equilibrium thermal ´¬‚uctuations in viscous ´¬‚uids.
162 F.C. MacKintosh
The dynamics of longitudinal motion can be calculated similarly. It is found that
the means-square amplitude of longitudinal ´¬‚uctuations of ´¬ülament of length are
also governed by (Granek, 1997; Gittes and MacKintosh, 1998)
╬┤ (t)2 Ôł╝ t 3/4 , (8.19)
where this mean-square amplitude is smaller than for the transverse motion by a
factor of order / p . Thus, both for the short-time ´¬‚uctuations as well as for the static
´¬‚uctuations of a ´¬ülament segment of length , a ´¬ülament end explores a disk-like
region with longitudinal motion smaller than perpendicular motion by this factor.
Although the amplitude of longitudinal motion is smaller than for transverse, the
longitudinal motion of Eq. 8.19 can explain the observed high-frequency viscoelastic
response of solutions and networks of biopolymers, as discussed below.
Solutions of semi´¬‚exible polymer
Because of their inherent rigidity, semi´¬‚exible polymers interact with each other
in very different ways than ´¬‚exible polymers would, for example, in solutions of
the same concentration. In addition to the important characteristic lengths of the
molecular dimension (say, the ´¬ülament diameter 2a), the material parameter p , and
the contour length of the chains, there is another important new length scale in a
solution ÔÇ“ the mesh size, or typical spacing between polymers in solution, ╬ż . This
can be estimated as follows in terms of the molecular size a and the polymer volume
fraction ¤ć (Schmidt et al., 1989). In the limit that the persistence length p is large
compared with ╬ż , we can approximate the solution on the scale of the mesh as one
of rigid rods. Hence, within a cubical volume of size ╬ż , there is of order one polymer
segment of length ╬ż and cross-section a 2 , which corresponds to a volume fraction ¤ć
of order (a 2 ╬ż )/╬ż 3 . Thus,
╬ż Ôł╝ a/ ¤ć. (8.20)
This mesh size, or spacing between ´¬ülaments, does not completely characterize
the way in which ´¬ülaments interact, even sterically with each other. For a dilute
solution of rigid rods, it is not hard to imagine that one can embed a long rigid rod
rather far into such a solution before touching another ´¬ülament. A true estimate of the
distance between typical interactions (points of contact) of semi´¬‚exible polymers must
account for their thermal ´¬‚uctuations (Odijk, 1983). As we have seen, the transverse
range of ´¬‚uctuations ╬┤u a distance away from a ´¬üxed point grows according to
╬┤u 2 Ôł╝ 3 / p . Along this length, such a ´¬‚uctuating ´¬ülament explores a narrow cone-
like volume of order ╬┤u 2 . An entanglement that leads to a constraint of the ´¬‚uctuations
of such a ´¬ülament occurs when another ´¬ülament crosses through this volume, in which
case it will occupy a volume of order a 2 ╬┤u, as ╬┤u . Thus, the volume fraction
and the contour length between constraints are related by ¤ć Ôł╝ a 2 /( ╬┤u). Taking
the corresponding length as an entanglement length, and using the result above for
╬┤u = ╬┤u 2 , we ´¬ünd that
¤ć Ôł’2/5 ,
Ôł╝ a4 (8.21)
which is larger than the mesh size ╬ż in the semi´¬‚exible limit ╬ż.
Polymer-based models of cytoskeletal networks 163
These transverse entanglements, separated by a typical length e , govern the elas-
tic response of solutions, in a way ´¬ürst outlined in Isambert and Maggs (1996). A
more complete discussion of the rheology of such solutions can be found in Morse
(1998b) and Hinner et al. (1998). The basic result for the rubber-like plateau shear
modulus for such solutions can be obtained by noting that the number density of
entropic constraints (entanglements) is thus n / c Ôł╝ 1/(╬ż 2 e ), where n = ¤ć/(a 2 ) is
the number density of chains of contour length . In the absence of other energetic
contributions to the modulus, the entropy associated with these constraints results in
a shear modulus of order G Ôł╝ kT /(╬ż 2 e ) Ôł╝ ¤ć 7/5 . This has been well established in
experiments such as those of Hinner et al. (1998).
With increasing frequency, or for short times, the macroscopic shear response of
solutions is expected to show the underlying dynamics of individual ´¬ülaments. One
of the main signatures of the frequency response of polymer solutions in general is
an increase in the shear modulus with increasing frequency. This is simply because
the individual ´¬ülaments are not able to fully relax or explore their conformations on
short times. In practice, for high molecular weight F-actin solutions of approximately
1 mg/ml, this frequency dependence is seen for frequencies above a few Hertz. Initial
experiments measuring this response by imaging the dynamics of small probe particles
have shown that the shear modulus increases as G(¤ë) Ôł╝ ¤ë3/4 (Gittes et al., 1997;
Schnurr et al., 1997), which has since been con´¬ürmed in other experiments and by
other techniques (for example, Gisler and Weitz, 1999).
If, as noted above, this increase in stiffness with frequency is due to the fact that
´¬ülaments are not able to fully ´¬‚uctuate on the correspondingly shorter times, then
we should be able to understand this more quantitatively in terms of the dynamics
described in the previous section. In particular, this behavior can be understood in
terms of the longitudinal dynamics of single ´¬ülaments (Morse, 1998a; Gittes and Mac-
Kintosh, 1998). Much as the static longitudinal ´¬‚uctuations ╬┤ 2 Ôł╝ 4 / 2 correspond
to an effective longitudinal spring constant Ôł╝ kT 2 / 4 , the time-dependent longitudi-
nal ´¬‚uctuations shown above in Eq. 8.19 correspond to a time- or frequency-dependent
compliance or stiffness, in which the effective spring constant increases with increas-
ing frequency. This is because, on shorter time scales, fewer bending modes can relax,
which makes the ´¬ülament less compliant. Accounting for the random orientations of
´¬ülaments in solution results in a frequency-dependent shear modulus
G(¤ë) = ¤ü╬║ (Ôł’2i╬Â /╬║)3/4 ¤ë3/4 Ôł’ i¤ë╬·, (8.22)
where ¤ü is the polymer concentration measured in length per unit volume.
In a living cell, there are many different specialized proteins for binding, bundling,
and otherwise modifying the network of ´¬ülamentous proteins. Many tens of actin-
associated proteins alone have been identi´¬üed and studied. Not only is it important
to understand the mechanical roles of, for example, cross-linking proteins, but as we
shall see, these can have a much more dramatic effect on the network properties than
is the case for ´¬‚exible polymer solutions and networks.
164 F.C. MacKintosh
The introduction of cross-linking agents into a solution of semi´¬‚exible ´¬ülaments
introduces yet another important and distinct length scale, which we shall call the
╬ż , individual
cross-link distance c . As we have just seen, in the limit that p
´¬ülaments may interact with each other only infrequently. That is to say, in contrast
with ´¬‚exible polymers, the distance between interactions of one polymer with its
neighbors ( e in the case of solutions) may be much larger than the typical spacing
between polymers. Thus, if there are biochemical cross-links between ´¬ülaments, these
may result in signi´¬ücant variation of network properties even when c is larger than ╬ż .
Given a network of ´¬ülaments connected to each other by cross-links spaced an
average distance c apart along each ´¬ülament, the response of the network to macro-
scopic strains and stresses may involve two distinct single-´¬ülament responses: (1)
bending of ´¬ülaments; and (2) stretching/compression of ´¬ülaments. Models based on
both of these effects have been proposed and analyzed. Bending-dominated behavior
has been suggested both for ordered (Satcher and Dewey, 1996) and disordered (Kroy
and Frey, 1996) networks. That individual ´¬ülaments bend under network strain is per-
haps not surprising, unless one thinks of the case of uniform shear. In this case, only
rotation and stretching or compression of individual rod-like ´¬ülaments are possible.
This is the basis of so-called af´¬üne network models (MacKintosh et al., 1995), in
which the macroscopic strain falls uniformly across the sample. In contrast, bending
of constituents involves (non-af´¬üne deformations, in which the state of strain varies
from one region to another within the sample.
We shall focus mostly on random networks, such as those studied in vitro. It has
recently been shown (Head et al., 2003a; Wilhelm and Frey, 2003; Head et al., 2003b)
that which of the af´¬üne or non-af´¬üne behaviors is expected depends, for instance, on
´¬ülament length and cross-link concentration. Non-af´¬üne behavior is expected either
at low concentrations or for short ´¬ülaments, while the deformation is increasingly
af´¬üne at high concentration or for long ´¬ülaments. For the ´¬ürst of these responses, the
network shear modulus (Non-Af´¬üne) is expected to be of the form
G NA Ôł╝ ╬║/╬ż 4 Ôł╝ ¤ć 2 (8.23)
when the density of cross-links is high (Kroy and Frey 1996). This quadratic de-
pendence on ´¬ülament concentration c is also predicted for more ordered networks
(Satcher and Dewey 1996).
For af´¬üne deformations, the modulus can be estimated using the effective single-
´¬ülament longitudinal spring constant for a ´¬ülament segment of length c between
cross-links, Ôł╝╬║ p / 4 , as derived above. Given an area density of 1/╬ż 2 such chains
passing through any shear plane (see Fig. 8-5), together with the effective tension of
order (╬║ p / 3 ) , where is the strain, the shear modulus is expected to be
G AT Ôł╝ . (8.24)
This shows that the shear modulus is expected to be strongly dependent on the
density of cross-links. Recent experiments on in vitro model gels consisting of F-actin
with permanent cross-links, for instance, have shown that the shear modulus can vary
from less than 1 Pa to over 100 Pa at the same concentration of F-actin, by varying
the cross-link concentration (Gardel et al., 2004).
Polymer-based models of cytoskeletal networks 165
╬ż Ôëł c AÔł’1/ 2
¤â = G╬Ş
Fig. 8-5. The macroscopic shear stress ¤â depends on the mean tension in each ´¬ülament, and on the
area density of such ´¬ülaments passing any plane. There are on average 1/╬ż 2 such ´¬ülaments per unit
area. This gives rise to the factor ╬ż Ôł’2 in both Eqs. 8.24 and 8.25. The macroscopic response can also
depend strongly on the typical distance c between cross-links, as discussed below.
In the preceding derivation we have assumed a thermal/entropic (Af´¬üne and
Thermal) response of ´¬ülaments to longitudinal forces. As we have seen, however,
for shorter ´¬ülament segments (that is, for small enough c ), one may expect a me-
chanical response characteristic of rigid rods that can stretch and compress (with a
modulus ┬µ). This would lead to a different expression (Af´¬üne, Mechanical) for the
G AM Ôł╝ 2 Ôł╝ ¤ć, (8.25)
which is proportional to concentration. The expectations for the various mechanical
regimes is shown in Fig. 8.6 (Head et al., 2003b).
In contrast with most polymeric materials (such as gels and rubber), most biologi-
cal materials, from the cells to whole tissues, stiffen as they are strained even by a
few percent. This nonlinear behavior is also quite well established by in vitro studies
of a wide range of biopolymers, including networks composed of F-actin, colla-
gen, ´¬übrin, and a variety of intermediate ´¬ülaments (Janmey et al., 1994; Storm et al.,
2005). In particular, these networks have been shown to exhibit approximately ten-fold
Fig. 8-6. A sketch of the expected diagram showing the
various elastic regimes in terms of cross-link density and
polymer concentration. The solid line represents the rigid-
ity percolation transition where rigidity ´¬ürst develops from
a solution at a macroscopic level. The other, dashed lines
indicate crossovers (not thermodynamic transitions). NA
indicates the non-af´¬üne regime, while AT and AM refer to
af´¬üne thermal (or entropic) and mechanical, respectively.
166 F.C. MacKintosh
Fig. 8-7. The differential modulus K = d¤â/d╬│ describes the increase in the stress ¤â with strain
╬│ in the nonlinear regime. This was measured for cross-linked actin networks by small-amplitude
oscillations at low frequency, corresponding to a nearly purely elastic response, after applying a
constant prestress ¤â0 . This was measured for four different concentrations represented by the various
symbols. For small prestress ¤â0 , the differential modulus K is nearly constant, corresponding to a
linear response for the network. With increasing ¤â0 , the network stiffens, in a way consistent with
theoretical predictions (MacKintosh et al., 1995; Gardel et al., 2004), as illustrated by the various
theoretical curves. Speci´¬ücally, it is expected that in the strongly nonlinear regime, the stiffening
increases according to the straight line, corresponding to d¤â/d╬│ Ôł╝ ¤â 3/2 . Data taken from Gardel
et al., 2004.
stiffening under strain. Thus these materials are compliant, while being able to with-
stand a wide range of shear stresses.
This strain-stiffening behavior can be understood in simple terms by looking at
the characteristic force-extension behavior of individual semi´¬‚exible ´¬ülaments, as
described above. As can be seen in Fig. 8-4, for small extensions or strains, there
is a linear increase in the force. As the strain increases, however, the force is seen
to grow more rapidly. In fact, in the absence of any compliance in the arc length of
the ´¬ülament, the force strictly diverges at a ´¬ünite extension. This suggests that for a
network, the macroscopic stress should diverge, while in the presence of high stress,
the macroscopic shear strain is bounded and ceases to increase. In other words, after
being compliant at low stress, such a material will be seen to stop responding, even
under high applied stress.
This can be made more quantitative by calculating the macroscopic shear stress
of a strained network, including random orientations of the constituent ´¬ülaments
(MacKintosh et al., 1995; Kroy and Frey, 1996; Gardel et al., 2004; Storm et al.,
2005). Speci´¬ücally, for a given shear strain ╬│ , the tension in a ´¬ülament segment of
length c is calculated, based on the force-extension relation above. This is done within
the (af´¬üne) approximation of uniform strain, in which the microscopic strain on any
such ´¬ülament segment is determined precisely by the macroscopic strain and the
´¬ülamentÔÇ™s orientation with respect to the shear. The contribution of such a ´¬ülamentÔÇ™s
tension to the macroscopic stress, in other words, in a horizontal plane in Fig. 8.5,
also depends on its orientation in space. Finally, the concentration or number density
of such ´¬ülaments crossing this horizontal plane is a function of the overall polymer
concentration, and the ´¬ülament orientation.
Polymer-based models of cytoskeletal networks 167
The full nonlinear shear stress is calculated as a function of ╬│ , the polymer con-
centration, and c , by adding all such contributions from all (assumed random) orien-
tations of ´¬ülaments. This can then be compared with macroscopic rheological studies
of cross-linked networks, such as done recently by Gardel et al. (2004). These ex-
periments measured the differential modulus, d¤â/d╬│ versus applied stress ¤â , and
found good agreement with the predicted increase in this modulus with increasing
stress (Fig. 8-7). In particular, given the quadratic divergence of the single-´¬ülament
tension shown above (Fixman and Kovac, 1973), this modulus is expected to increase
as d¤â/d╬│ Ôł╝ ¤â 3/2 , which is consistent with the experiments by Gardel et al. (2004).
This provides a strong test of the underlying mechanism of network elasticity.
In addition to good agreement between theory and experiment for densely
cross-linked networks, these experiments have also shown evidence of a lack of
strain-stiffening behavior of these networks at lower concentrations (of polymer or
cross-links), which may provide evidence for a non-af´¬üne regime of network response
Cytoskeletal ´¬ülaments play key mechanical roles in the cell, either individually (for
example, as paths for motor proteins) or in collective structures such as networks.
The latter may involve many associated proteins for cross-linking, bundling, or cou-
pling the cytoskeleton to other cellular structures like membranes. Our knowledge
of the cytoskeleton has improved in recent years through the development of new
experimental techniques, such as in visualization and micromechanical probes in
living cells. At the same time, combined experimental and theoretical progress on in
vitro model systems has provided fundamental insights into the possible mechanical
mechanisms of cellular response.
In addition to their role in cells, cytoskeletal ´¬ülaments have also proven remarkable
model systems for the study of semi´¬‚exible polymers. Their size alone makes it
possible to visualize individual ´¬ülaments directly. They are also unique in the extreme
separation of two important lengths, the persistence length p and the size of a single
monomer. In the case of F-actin, p is more than a thousand times the size of a
single monomer. This makes for not only quantitative but also qualitative differences
from most synthetic polymers. We have seen, for instance, that the way in which
semi´¬‚exible polymers entangle is very different. This makes for a surprising variation
of the stiffness of these networks with only changes in the density of cross-links, even
at the same concentration.
In spite of the molecular complexity of ´¬ülamentous proteins as compared with
conventional polymers, a quantitative understanding of the properties of single ´¬üla-
ments provides a quantitative basis for modeling solutions and networks of ´¬ülaments.
In fact, the macroscopic response of cytoskeletal networks quite directly re´¬‚ects, for
example, the underlying dynamics of an individual semi´¬‚exible chain ´¬‚uctuating in
its Brownian environment. This can be seen, for instance in the measured dynamics
of microtubules in cells (Caspi et al., 2000).
In developing our current understanding of cytoskeletal networks, a crucial role has
been played by in vitro model systems, such as the one in Fig. 8-1. Major challenges,
168 F.C. MacKintosh
however, remain for understanding the cytoskeleton of living cells. In the cell, the
cytoskeleton is hardly a passive network. Among other differences from the model
systems studied to date is the presence of active contractile or force-generating ele-
ments such as motors that work in concert with ´¬ülamentous proteins. Nevertheless,
in vitro models may soon permit a systematic and quantitative study of various actin-
associated proteins for cross-linking and bundling (Gardel et al., 2004), and even
contractile elements such as motors.
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9 Cell dynamics and the actin cytoskeleton
James L. McGrath and C. Forbes Dewey, Jr.
This chapter focuses on the mechanical structure of the cell and how it is affected
by the dynamic events that shape the cytoskeleton. We pay particular attention to actin because
the actin structure turns over rapidly (on the order of tens of seconds to tens of minutes) and is
strongly correlated with dynamic events such as cell crawling. The chapter discusses the way
in which actin and its associated binding proteins provide the dominant structure within the
cell, and how the actin is organized. Models of the internal structure that attempt to provide a
quantitative picture of the stiffness of the cell are given, followed by an in-depth discussion of
the actin polymerization and depolymerization mechanics. The chapter provides a tour of the
experiments and models used to determine the speci´¬üc effects of associated proteins on the
actin cycle and contains an in-depth exposition of how actin dynamics play a pivotal role in
cell crawling. Some conclusions and thoughts for the future close the chapter.
Introduction: The role of actin in the cell
Eukaryotic cells are wonderful living engines. They sustain themselves by bringing
in nutrients across their membrane shells, manufacturing thousands of individual
protein species that are needed to sustain the cellÔÇ™s function, and communicating with
the surrounding environment using a complex set of receptor molecules that span the
membrane and turn external chemical and mechanical signals into changes in cell
function and composition.
The cell membrane is ´¬‚exible and allows the cell to move ÔÇ“ and to be moved ÔÇ“ by
changes in the internal cytoskeletal structure. The cytoskeleton is a spatially sparse
tangled matrix of rods and rod-like elements held together by smaller proteins. One of
the characteristics of this structure is that it is continuously changing. These dynam-
ics are driven by thermal energy and by phosphorylation of the major proteins that
are constituents of the cell. Even though the bond strengths in most cases are many
folds larger than the mean thermal energy kT, on rare occasions the bonds will be
broken and a new mechanical arrangement will replace the old. There are additional
mechanisms that grow the long ´¬ülaments from monomers in the cell cytoplasm. Com-
plimentary reactions remove molecules from the ends of the ´¬ülaments and also cleave
the ´¬ülaments, creating additional free ends. If these growth and turnover mechanisms
have a bias direction, the cell will crawl.
Cell dynamics and the actin cytoskeleton 171
The many different cellular states that occur can be examined, with varying degrees
of dif´¬üculty. The most dif´¬ücult state is when the cells of interest are embedded in a
tissue matrix in a living animal, such as chondrocytes within the matrix of joint
cartilage. Some cells, such as the endothelial cells that line the cardiovascular tree
of mammals, are more accessible because they lie closely packed along the surface
of blood vessels. This con´¬üguration is called the endothelial monolayer because the
equilibrium condition for these cells is to form a single-cell-thick layer in which all
cells are in contact with their neighbors. Much study has been given to endothelial cells
because of their putative role in controlling the events that can cause atherosclerosis
and subsequent heart disease. An additional attractiveness of studying endothelial
cells is the ease with which they can be grown in culture (Gimbrone, 1976). One can
study the in´¬‚uence of many physical and chemical properties in a setting having strong
similarities to the in vivo conditions, while varying the environment one parameter
at a time. Many of the data in this chapter were derived from experiments using
endothelial cells in culture.
The interior of the cell between the cytoskeletal members is ´¬ülled with cytoplasm, a
water-based slurry composed primarily of electrolytes and small monomeric proteins.
The small proteins diffuse through the viscous cytoplasm with a diffusion coef´¬ücient
that can be measured with modern ´¬‚uorescence techniques. For actin monomer in
the cytoplasm of a vascular endothelial cell, the diffusion coef´¬ücient D is about
3 Ôł’ 6 ├— 10Ôł’8 cm2 /s (Giuliano and Taylor, 1994; McGrath et al., 1998b). By con-
trast, water molecules in the liquid have a self-diffusion coef´¬ücient three orders of
Of the three types of cytoskeletal polymers ÔÇ“ actin ´¬ülaments, intermediate ´¬ülaments,
and microtubules ÔÇ“ that determine endothelial cell shape, actin ´¬ülaments are the most
abundant and are located in closest proximity to the cell membrane. Con´¬‚uent en-
dothelial cells assemble Ôł╝70 percent of their 100 ┬µM total actin into a rich meshwork
of just over 50,000 actin ´¬ülaments that are on average Ôł╝3 ┬µm long (McGrath et al.,
2000b). Cross-linking proteins organize actin ´¬ülaments into viscoelastic gels that
connect to transmembrane proteins and signaling complexes located at intercellular
attachment sites and extracellular matrix adhesion sites. Of particular importance are
the direct connections of actin ´¬ülaments to ╬▓ integrin tails by talin (Calderwood and
Ginsberg, 2003) and ´¬ülamins (Stossel et al., 2001), and to cadherins by vinculin and
catenins (Wheelock and Knudsen, 1991). During cell locomotion and shape change
events, the actin cytoskeleton is extensively remodeled (Satcher et al., 1997; Theriot
and Mitchison, 1991), primarily by adding and subtracting subunits at free ´¬ülament
Monomeric actin has a molecular weight of 42.8 KDaltons and is found in all
eukaryotic cells. It is the major contributor to the mechanical structure of the cell,
and the degree to which it is polymerized into long ´¬ülaments changes dramatically
depending on the conditions in which the cell ´¬ünds itself. The properties of en-
dothelial cells, those that line blood vessels, are illustrated by the typical values for
the actin polymerization and other mechanical properties summarized in Table 9-1.
Similar numbers have been found for many eukaryotic cells (see Stossel et al.
172 J.L. McGrath and C.F. Dewey, Jr.
Table 9-1. Mechanical properties of vascular endothelium
Property Typical value Reference
Ôł╝ 50 ┬µm dia.
Cell size ÔÇ“ subcon´¬‚uent Gimbrone, 1976
Ôł╝ 40 ┬µm dia.
20 ├— 60 ┬µm
Cell size ÔÇ“ aligned, con´¬‚uent Dewey et al., 1981
Crawling speed ÔÇ“ subcon´¬‚uent Tardy et al., 1997
ÔÇ“ con´¬‚uent Osborn et al., 2004
ÔÇ“ con´¬‚uent + ´¬‚ow 0.05 to 0.15 ┬µm/min.
Total polymerized actin (F-actin) 10ÔÇ“20 mg/ml Hartwig et al., 1992
Fraction of actin polymerized
ÔÇ“ subcon´¬‚uent 35ÔÇ“40% McGrath et al., 2000b
ÔÇ“ con´¬‚uent 65ÔÇ“80%
Total actin content (calculated) 20ÔÇ“40 mg/ml
YoungÔÇ™s modulus for actin ´¬ülaments 2.3 GPa Gittes et al., 1993
Interaction of the cell cytoskeleton with the outside environment
Eukaryotic cells are composed of a semistructured interior and an enclosing mem-
brane that separates the interior from the environment. In Fig. 9-1A, it can be seen
how the external membrane covers the cell. Removing the membrane and the interior
cytoplasm with a suitable solvent reveals the actin cytoskeleton as seen in Fig. 9-1B.
These views are from the apical side of cultured endothelial cells. One can see rem-
nants of the membrane at junctions where the bounding bilayer membrane shell was
attached to the underlying structure with large protein complexes. A diagram of the
current view of these complicated attachments is given in Fig. 9-2. These complexes
´¬ügure prominently in transducing mechanical signals from the outside, responding to
external forces on the membrane such as the shear stress produced by ´¬‚ow.
Endothelial cells are found to be strongly attached to a substrate. In vivo, the cells
attach to the artery wall, which is covered with a basement membrane of collagen and
other proteins. The attachment consists of large protein complexes that connect the
substrate to the internal actin cytoskeleton through the intermediary of transmembrane
proteins. The transmembrane proteins attach to ligands in the substrate. In equilibrium,
the cell is pulled into a ´¬‚at con´¬üguration varying in thickness from about 2ÔÇ“3 ┬µm in
the cytoplasmic periphery to around 4ÔÇ“6 ┬µm over the nucleus.
In vitro cell attachment in culture medium is similar, except that there is no basement
membrane; the substrate is normally covered with ´¬übrinogen or collagen and the
transmembrane proteins attach directly to this layer. Within twenty-four hours in
culture, the cells begin to excrete their own substrate proteins, and this forms a surface
to which the cells stick tenaciously. Fluid shear stresses up to 40 dynes/cm2 have no
ability to detach the cells. On the other hand, the individual attachment complexes
turn over continuously, as shown by exquisite confocal microscopy experiments, with
a time scale on the order of ´¬üfteen minutes (Davies et al., 1993; Davies et al., 1994).
Although the mechanosensitive molecular mechanisms that determine shear-stress-
mediated endothelial shape change are poorly de´¬üned, a growing body of evi-
dence supports a decentralized, integrated signaling network in which force-bearing
cytoskeletal polymers attach to transmembrane proteins where conformational
Cell dynamics and the actin cytoskeleton 173
0.5 ┬µm 0.2 ┬µm
Fig. 9-1. Endothelial cells in culture, showing intact membrane (A) and the underlying actin cy-
toskeleton (B) after solubilization to remove most of the membrane. Note that the two pictures are
at different magni´¬ücations. From Satcher et al., 1997.
changes in connected proteins initiate signaling events (Helmke et al., 2003; Kamm
and Kaazempur-Mofrad, 2004). Recently, heterogeneous ┬µm-scale displacements of
cytoskeletal structures have been described in endothelial cells that, when converted
to strain maps, reveal forces applied at the lumen being transmitted through the cell
to the basal attachments (Helmke et al., 2003). Changes in the number, type, and
structure of cytoskeletal connections alter the location and magnitude of transmitted
forces and may modify the speci´¬üc endothelial phenotype, depending on the spatial
and temporal microstimuli that each endothelial cell senses (Davies et al., 2003).
Fig. 9-2. A schematic representation of the focal adhesion complex joining the extracellular matrix to
the cytoskeleton across the membrane. This diagram would represent the basal focal adhesion sites.
Apical sites would have integrin receptor pairs without attachment to collagen or other materials.
The cell could be activated by ligands binding to the integrin pair and causing the cytoplasmic tails
to produce new biological reactions in the cytoplasm. From Brakebush and F┬Ę ssler, 2003.
174 J.L. McGrath and C.F. Dewey, Jr.
Clearly, the nature of the cytoskeleton and its ability to transmit these forces plays a
key role in cell function.
Properties of actin ´¬ülaments
Eukaryotic cells exhibit three distinct types of internal polymerized actin structures.
The ´¬ürst and arguably most important is the distributed actin lattice that supports most
cells and appears to be very well distributed throughout the cell. That con´¬üguration
is shown in Fig. 9-1B. The individual ´¬ülaments are too small to be resolved with
optical microscopy, and staining for ´¬‚uorescence of actin can at most show a dull
glow throughout the cell when staining this component.
The second type of polymerized actin is found in ´¬ülament bundles called stress
´¬übers. These stress ´¬übers are often seen to have ends coincident with the location
of attachment complexes at the cell-substrate boundary, and also at cell-cell junction
complexes. These stress ´¬übers are collections of ´¬üve to twenty individual actin ´¬ül-
aments tightly bound together by other proteins. Their large size and high density
makes them prominently visible in actin ´¬‚uorescence experiments. Because the ´¬ül-
aments are visible and also change with the stress conditions of the cell, it is often
believed that these stress ´¬übers re´¬‚ect the main role of actin in a cell. Careful es-
timates of the fraction of total cellular actin associated with stress ´¬übers, however,
suggest that they only play a small role in maintaining cell shape and providing the
cytoskeleton of the cell (Satcher and Dewey, 1996). The distributed actin lattice is
much more important to cell shape and mechanical properties.
A third type of actin structure is a lattice similar in appearance to Fig. 9-1B that
is con´¬üned to a small region of the cell just under the surface membrane. This is
termed cortical actin, and is found for example in red blood cells. Red blood cells are
devoid of any distributed lattice within the rest of the cell interior, and their reliance on
cortical structure is not typical of most other cells, including endothelium (Hartwig
and DeSisto, 1991).
Single actin ´¬ülaments have been studied intensively for over twenty-´¬üve years.
Reviews of their properties can be found in the excellent treatises by Boal (2002)
and Preston et al. (1990). An actin ´¬ülament in a thermally active bath of surround-
ing molecules will randomly deform from a straight rod into a curved shape. The
characteristic length, ╬żp , over which the curvature of an isolated actin ´¬ülament can
become signi´¬ücant is very long, about 10ÔÇ“20 ┬µm at 37 C (Boal, 2002; Janmey et al.,
1994; MacKintosh et al., 1995). This is comparable to the dimensions of the cell,
whereas the distance between points where the ´¬ülaments are in contact is much shorter.
Typical actin ´¬ülament lengths are less than 1 ┬µm, so that the cytoskeleton appears to
be a tangle of fairly stiff rods. A most important feature is the fact that the small actin
´¬ülaments are bound to one another by special actin-binding proteins. The most promi-
nent of this class is Filamin A, and its characteristics are described in the next two
paragraphs. For the purposes of classifying the mechanical properties of this mixture,
one can visualize a relatively dense packing of ´¬ülaments bound together at relatively
large angles by attached protein bridges. An examination of Fig. 9-1B suggests that
the ´¬ülament spacing is typically 100 nm, the intersection angles of the ´¬ülaments vary
Cell dynamics and the actin cytoskeleton 175
Fig. 9-3. Functional properties of ´¬ülamin. The bar is 200 nm. Text and drawing courtesy of J.H.
Hartwig; micrographs courtesy of C.A. Hartemink, unpublished, 2004.
signi´¬ücantly but are most often closer to 90 degrees than to 45 degrees, and that the
spacing will vary with the density of ´¬ülaments.
The role of ´¬ülamin A (FLNa)
Cellular actin structure is controlled at different levels. Of particular importance are
proteins that regulate actin ´¬ülament assembly/disassembly reactions and those that
regulate the architecture of F-actin and/or attach it to the plasma membrane. One
such protein is ´¬ülamin A (FLNa), a product of the X-chromosome (Gorlin et al.,
1993; Gorlin et al., 1990; Stossel et al., 2001). Filamin A was initially isolated and
characterized in 1975 (Stossel and Hartwig, 1976) and subsequent research has con-
tinued to ´¬ünd important new functions for the protein (Nakamura et al., 2002).
As shown in Fig. 9-3, this large protein binds actin ´¬ülaments, thus de´¬üning the
cytoskeletal architecture, and attaches them to membrane by also binding to a number
176 J.L. McGrath and C.F. Dewey, Jr.
of membrane adhesive receptors including ╬▓1 and ╬▓7 integrin and GP1b╬± (Andrews
and Fox, 1991; Andrews and Fox, 1992; Fox, 1985; Sprandio et al., 1988; Takafuta
et al., 1998).
With many binding partners now described, FLNa participates in signaling cascades
by spatially collecting and concentrating signaling proteins at the plasma membraneÔÇ“
cytoskeletal junction and may possibly function as an organizing center for actin
network rearrangements (see Fig. 9-3). Important partner interactions that may be
dependent on ´¬ülamin include GTPase targeting and charging and linkage of the actin
cytoskeleton to membrane glycoproteins such as GP1b╬± and ╬▓-integrins. FLNa is
part of a larger family of proteins that include FLNb and FLNc, whose genes are on
chromosomes 3 and 7, respectively (Br┬Ę cker and al, 1999; Krakow et al., 2004; Sheen
et al., 2002; Thompson et al., 2000).
FLNa is an elongated homodimer (Hartwig and Stossel, 1981). Each subunit has an
N-terminal actin-binding site joined to twenty-four repeat motifs, each Ôł╝100 residues
in length. Repeats are ╬▓╬▓-barrel structures that are believed to interconnect like beads
on a string. Subunits self-associate into dimers using only the most C-T repeat motif.
The location of known binding partner proteins along each FLNa subunit is indicated
in Fig. 9-3. Molecules are 160 nm in length in the electron microscope (Fig. 9-3,
bottom right) but can organize actin ´¬ülaments into branching networks (Fig. 9-3,
The FLNa concentrations in endothelial and other cells is normally such that there
are many times more FLNa molecules than junctions in the cell cytoskeleton. This
can be ascertained by measuring the amount of FLNa in the soluble portion of the cell,
computing the molecular concentration per unit cell volume, and then comparing that
to the concentration of ´¬ülament junctions per unit volume of cytoskeleton observable
in electron microscopy (see Fig. 9-1).
The role of cytoskeletal structure
The internal structure of the cell has several functions. One is to provide a suf´¬ücient
amount of rigidity so that the cell can withstand external forces. Figure 9-4 illustrates
the functions that the cytoskeleton performs when the cell is subjected to ´¬‚uid shear
stress. A balance of forces requires that the cytoskeleton transmit the entire applied
force to the substrate.
The second function to be served is that the cytoskeleton must be malleable enough
to allow the cell to accommodate new environmental parameters such as imposed me-
chanical forces from ´¬‚uid shear stress and mechanical deformation of the artery. There
are two separate time scales to be considered; the ´¬ürst is short time behavior, where
´¬‚uctuations such as systolic and diastolic changes in ´¬‚ow must be accommodated,
and longer times, where the actin matrix can completely disassociate and reform.
This latter scale is typically on the order of tens of minutes. More discussion of these
mechanisms is given later in this chapter.
The following section will draw upon the ultrastructure presented above and de-
scribe how simple mechanical models of the lattice can be used to predict the me-
chanical properties of the cell.
Cell dynamics and the actin cytoskeleton 177
Fig. 9-4. Schematic representation of an endothelial cell subjected to ´¬‚uid shear stress. The force
exerted on the surface must be transmitted through the cell by the cytoskeleton to the substrate to
which the cell is attached. Some structural ´¬ülaments attach to the cell nucleus, so that the force is
transmitted through the nucleus as well as the extranuclear part of the cell. The lower portion of the
´¬ügure suggests some of the biochemical cascades triggered by the force. From Davies, 1995.
The complicated combination of semi´¬‚exible ´¬ülaments and binding proteins that form
the structural matrix within a cell presents a formidable challenge to the scienti´¬üc
investigator. Many of the key interactions, such as the details of the FLNa binding to
actin polymers, are still subjects of active research. The fraction of actin molecules
polymerized into structural ´¬übers varies from cell to cell, and the mean can range
from 35 percent to 80 percent depending on the state of stress, the degree of cell-
cell attachment, and time. As will be described in the section on actin dynamics, the
actin ´¬ülaments polymerize and depolymerize continuously, so that the whole internal
structure is replaced within a time scale that is tens of minutes to hours. To make the
situation even more interesting, thermal ´¬‚uctuations of the ´¬ülaments could contribute
substantially to the apparent cell stiffness. Yet just the simplicity of the randomness
tempts one to ´¬ünd models that can at least scale the behavior of the mixture and arrive
at working conclusions regarding the structural rigidity of the cell when exposed to
178 J.L. McGrath and C.F. Dewey, Jr.
It is attractive to look at two bodies of literature for examples from other ´¬üelds. The
´¬ürst is polymer physics, where a body of literature is presented on various thermally
driven models. These works are well summarized in the book by Boal (2002). Included
in these models are ´¬‚oppy chains, semi´¬‚exible chains, and welded chains. A second
´¬üeld is the study of porous solids, which range from structured anisotropic systems
such as honeycombs to foams with random isotropic bubble inclusions. This ´¬üeld is
reviewed in the classic treatise by Gibson and Ashby in 1988, with a second edition
in 1997 (Gibson and Ashby, 1997; Gibson and Ashby, 1988).
It is possible to derive a simple model for the elastic modulus of the actin cytoskele-
ton by picturing it as a collection of relatively stiff elements attached to one another
with stiff joints. In quantitative terms, following nomenclature from the polymer lit-
erature (Gittes and MacKintosh, 1998),1 this means that the so-called persistence
length over which the ´¬ülaments will bend because of thermal agitation, lp , is long
compared to the distance between attachment points (or mesh size), l, and l is in turn
large compared to the characteristic thickness of the structural elements, t. Then the
intracellular structure can be compressed, stretched, and sheared by applying forces
to the ´¬ülament ensemble through its attachments to the membrane, to the substrate,
and to the adjacent cells.
This approach was taken by Satcher and Dewey (1996) who used the analogy of
a porous solid to obtain numerical values for the stiffness of cellular actin networks.
A simple representative model originally proposed by Gibson and Ashby (1988)
considers a three-dimensional rectangular meshwork of short rods connected to the
lateral sides of adjacent elements as shown in Fig. 9-5. The key feature of the model
is that the structural elements are placed into bending by applied forces. Although the
bonds between the ´¬ülaments are shown in the model as being rigid and the prototypical
geometry is taken to be cubic cells, similar scaling of the rigidity of the matrix would
be expected if the joints were simply pinned and the prototypical geometry were
triangular. In that case, individual elements would buckle with applied stress. In the
matrix described by Fig. 9-5, the ratio of the apparent density of the matrix ¤ü Ôł— to the
density of the solid ´¬ülament material, ¤üs , varies as (¤ü Ôł— /¤üs ) Ôł╝ (t/l )2 .
Comparing the idealization of Fig. 9-5 with the actual cytoskeletal con´¬üguration
of Fig. 9-1B, one can see many simpli´¬ücations. First, the angles with which the ´¬üla-
ments come together to make the matrix vary considerably. One should recognize that
Fig. 9-1B is a view looking down into a three-dimensional structure, and the actual
lattice is much more open than is apparent from the micrograph. The vast majority of
junctions have an angle between the intersecting ´¬ülaments that is greater than 45 de-
grees, and many approach right angles. What is important to the representation is that
it puts individual structural elements into bending, thereby causing a de´¬‚ection ╬┤ that
can be simply computed from beam theory. The scaling of the geometry then allows
the overall elastic modulus to be computed as a function of the density of ´¬ülaments and
their bending stiffness. Because the characteristic length l of the cellular structures is
on the order of 100 nm and the individual structural elements have a typical dimension
t of about 7 nm, one would expect the beams to be reasonably stiff in bending.
Gittes and MacKintosh use the symbols ╬ż for the characteristic dimension of the lattice and a for the
characteristic thickness of the structural element.
Cell dynamics and the actin cytoskeleton 179
Fig. 9-5. The porous solid model of Gibson and Ashby ap-
plied to represent the structure of actin ´¬ülaments within the
cell. (a) is the undistorted lattice and (b) shows the action of
an applied force. The vertices of the ´¬ülaments are assumed
to be at right angles and tightly bound for purposes of the
calculation. In reality, the angles between ´¬ülaments vary and
the lattice spacing is not uniform; further, the bonds between
individual ´¬ülaments are not necessarily rigid, but may simply
pin the joint. From Satcher and Dewey, 1996.
The results of the calculation show that strain ╬µ in the lattice, ╬µ = ╬┤/l, is propor-
tional to the stress per unit area, ¤â , of the matrix, and the proportionality constant
is the effective YoungÔÇ™s modulus, E Ôł— , for the material. Using bending theory for the
strain ╬µ, it is found (Satcher and Dewey, 1996) that
(E Ôł— /E s ) Ôëł C1 (¤ü Ôł— /¤üs )2
where C1 is a constant and E Ôł— is the YoungÔÇ™s modulus of solid actin, which is taken to be
2.3 GPa (Gittes et al., 1993). In order to complete the quantitative calculation, C1 must
be determined. That was done using empirical tabulations for polymer foams ranging
over many orders of magnitude in density and structural properties as represented
in Gibson and Ashby (1988). The result is that C1 Ôëł 1. An identical scaling law is
obtained if the lattice element is put into shear rather than compression, with the
front-running constant of 3/8 instead of one.
180 J.L. McGrath and C.F. Dewey, Jr.
This scaling can be compared to the results derived from semi´¬‚exible polymers
with crosslinking. The latest results in this ´¬üeld are discussed in detail in Chap-
ter 8. Published theories by MacKintosh and colleagues (Gittes and MacKintosh,
1998; MacKintosh et al., 1995; Chapter 8) show that the shear modulus G for densely
cross-linked gels scales as G Ôł╝ (¤ü Ôł— /¤üs )5/2 . Although the power law is similar, the
value of G depends on the ÔÇťentanglement length,ÔÇŁ which is measured by thermal
´¬‚uctuations of the ´¬ülaments. It seems plausible that for open structures for which
the typical dimensions are lp l t the thermal agitation would not play a strong
role and the so-called enthalpic contribution would be negligible. From the point of
view of existing experimental data taken in dilute solutions (Janmey et al., 1991),
it is very dif´¬ücult to decide between the effective modulus scaling as (¤ü Ôł— /¤üs )2 or
(¤ü Ôł— /¤üs )5/2 .
Two common features between the models are important. First, the elastic modulus
is independent of frequency, at least for times over which the cross-links do not turn
over and for which the frequencies are suf´¬üciently low so as to not change the basic
mechanisms of deformation of the ´¬ülaments. Second, the existence of cross-linking
is crucial; without cross-linking, the effective modulus would be substantially lower
than the observed values.
A practical consequence of the predictions of the theory is that a drop in the fraction
of actin polymerized can have a very profound effect on the rigidity of the cell. In
separate experiments, we have observed a drop by factors of two to three in the fraction
of actin polymerized following changes from a packed monolayer to freely crawling
cells (McGrath et al., 2000b; McGrath et al., 1998b). We therefore ´¬ünd that the freely
crawling cells have a much lower effective modulus and a higher motility. This has
potential implications for wound healing, endothelialization of graft materials, and
the integrity of monolayers subjected to ´¬‚uid-´¬‚ow forces.
We have examined the theory and observations relative to the internal cytoskeleton
of a static collection of cells. Cells are in a dynamic state, with the internal cytoskeleton
changing continuously. The following section explores the interesting dynamics that
occur within the cell and presents quantitative models for the processes that are at
Abundant, essential, and discovered more than a half-century ago, actin is one of the
most studied of all the proteins. Many investigations have focused on the dynamic
character of actin, leading to a rich quantitative understanding of actin assembly and
disassembly. In this section we brie´¬‚y review this history and summarize the modern
understanding of actin dynamics and its regulation by key binding proteins.
The emergence of actin dynamics
The appreciation that actin has both dynamic and mechanical properties can be
traced to the work of its discoverer, F.B. Straub (Mommaerts, 1992). Trying to under-
stand the difference between a highly viscous mixture of ÔÇ˜myosin BÔÇ™ and a less viscous
Cell dynamics and the actin cytoskeleton 181
mixture of ÔÇ˜myosin A,ÔÇ™ Straub discovered these were not different myosins at all, but
that the myosin B preparations were ÔÇ˜contaminatedÔÇ™ by another protein that ACTivated
myosIN to make the viscous solution (Feuer et al., 1948). StraubÔÇ™s laboratory later
revealed that the contaminating actin could itself convert between low- and high-
viscosity solutions with the introduction of physiological salts and/or ATP (Straub and
Feuer, 1950). Further data from Straub demonstrated that the phase change occurred
because a globular protein (ÔÇ˜G-actinÔÇ™) polymerized into long ´¬ülaments (ÔÇ˜F-actinÔÇ™) and
that the ´¬ülamentous form could hydrolyze ATP. With the discovery of sarcomere struc-
ture (Huxley and Hanson, 1954) and the sliding ´¬ülament model of muscle contraction
in the 1950ÔÇ™s (Huxley, 1957), actinÔÇ™s role as the structural thin ´¬ülament of muscle was
in place. The signi´¬ücance of actinÔÇ™s dynamic properties, however, remained unclear.
Over the next several decades actin was identi´¬üed in every eukaryotic cell inves-
tigated. Extracts formed from macrophages (Stossel and Hartwig, 1976b) or anan-
thamoeba (Pollard and Ito, 1970) could be made to ÔÇ˜gelÔÇ™ in a manner that involved
actin polymerization. Actin ´¬ülaments were found to be concentrated near ruf´¬‚ing
membranes and in ´¬übroblast ÔÇ˜stress ´¬übersÔÇ™ (Goldman et al., 1975; Lazarides and
Weber, 1974), and cell movements could be halted with the actin-speci´¬üc cy-
tochalasins (Carter, 1967). Cytochalasins were found to block actin polymerization
(Brenner and Korn, 1979), to inhibit the gelation of extracts (Hartwig and Stossel,
1976; Stossel and Hartwig, 1976; Weihing, 1976), and to reduce the strengths of re-
constituted actin networks (Hartwig and Stossel, 1979). With these ´¬ündings, actinÔÇ™s
role as a structural protein appeared universal. However, unlike in muscle cells, actinÔÇ™s
dynamic properties allow nonmuscle cells to tailor diverse and dynamic mechanical
structures. Elucidating the intrinsic and regulated dynamics of actin was rightly viewed
as a fundamental question in cell physiology, and became a lifelong pursuit for many
talented biologists and physicists.
The intrinsic dynamics of actin
The pioneering work on protein self-assembly by Fumio Oosawa (see Fig. 9-6) led
to some of the earliest insights into the process of actin polymerization. Using light
scattering to follow the progress of polymerization, Oosawa noticed that actin poly-
merization typically began several minutes after the addition of polymer-inducing
salts. He concluded that during the early ÔÇ˜lagÔÇ™ phase of polymerization, monomers
form nuclei in an unfavorable reaction, but that once formed, nuclei are rapidly sta-
bilized by monomer addition (Oosawa and Kasai, 1962). Because the addition of
monomers to ´¬ülaments is a highly favorable reaction, the lag phase could be over-
come by adding preformed ´¬ülaments as nuclei. OosawaÔÇ™s studies indicated that the
nucleus was formed of three monomers, a number that has stood further scrutiny
(Wegner and Engel, 1975).
Oosawa described the mechanism of actin assembly by constructing some of the
´¬ürst kinetic models of the process (Oosawa and Asakura, 1975). Because experiments
revealed actin ´¬ülaments to have biochemically distinct ÔÇ˜barbedÔÇ™ and ÔÇ˜pointedÔÇ™ ends
(Woodrum et al., 1975), and because ATP- and ADP-bound monomers are capable
of assembling at these ends, Oozawa considered four assembly and four disassembly
182 J.L. McGrath and C.F. Dewey, Jr.
f il a
Fig. 9-6. Intrinsic actin dynamics. Oosawa and colleagues found that actin polymerizes in two phases:
(a) an unfavorable nucleation phase involving an actin trimer and (b) a rapid assembly phase in which
actin monomers add to both barbed and pointed ends. In the presence of excess ATP, actin does not
polymerize to equilibrium but to a ÔÇ˜treadmillingÔÇ™ steady-state in which monomers continuously add
to the barbed end and fall off of the pointed end.
reactions. Oozawa also assumed that spontaneous ´¬ülament fragmentation and anneal-
ing ultimately determined the number of ´¬ülaments in solution; this was later validated
with detailed modeling (Murphy et al., 1988; Sept et al., 1999).
Obtaining reliable constants for the assembly and disassembly rates in the Oosawa
model was a challenge that would occupy many years. Indirect assays gave con´¬‚icting
results because of their inability to control or accurately measure the number of
´¬ülaments in solutions. In 1986, PollardÔÇ™s direct electron microscopic measurements
of growth on individual nuclei produced numbers that are now the most widely cited
A decade before PollardÔÇ™s experiments, Albert Wegner proposed that actin assembly
does not proceed to equilibrium, but to a steady-state in which assembled subunits
continuously traverse ´¬ülaments from the barbed to the pointed end (Wegner and Engel,
1975). His now-classic experiments revealed that even after bulk polymerization had
halted, ´¬ülaments continued to hydrolyze ATP. Wegner decided that the results could
only be explained if ATP-carrying monomers had a higher af´¬ünity for barbed ends
than for pointed ends, and that the energy of ATP hydrolysis must be used to sustain
the imbalance. Wegner suggested that, on average, ATP-G-actin assembles at barbed
ends, hydrolyzes ATP in the ´¬ülament interior, and disassembles bound to ADP at
pointed ends. Thus Wegner was the ´¬ürst to introduce the concept of actin ´¬ülament
ÔÇ˜treadmilling,ÔÇ™ although the term itself must be credited to Kirschner some years later
(Kirschner, 1980). While PollardÔÇ™s rate constants were not measured at steady-state,
they clearly suggested barbed ends had a higher af´¬ünity for monomer in the presence
of ATP and thus supported the existence of treadmilling.
A dif´¬üculty with both the Pollard and the Wegner studies was the lack of an accurate
measure of the ATP hydrolysis rate on F-actin. Within the same year of PollardÔÇ™s paper,
Carlier and colleagues reported that ATP hydrolysis occurred less than a second after
subunit addition (Carlier et al., 1987). Because this time scale for ATP hydrolysis
Cell dynamics and the actin cytoskeleton 183
is comparable to the time between monomer assembly events in most experiments,
PollardÔÇ™s assumption that ATP-actin was the disassembling species in his experiments
must be reevaluated. Also signi´¬ücant were data revealing that inorganic phosphate
was released from the cleaved nucleotide several minutes after hydrolysis (Carlier and
Pantaloni, 1986). The latter discovery meant that three species needed to be considered
for actin dynamics: ATP, ADP, and a long-lived intermediate ADP┬·Pi. Furthermore,
an ADP┬·Pi monomer species should be generated by disassembly to either reassemble
or release inorganic phosphate (Pi) and become a source of ADP-G-actin. Unfortu-
nately, there has been no focused effort to determine all twelve assembly/disassembly
rate constants and the two rates of Pi release (G-actin and F-actin) needed to properly
update OosawaÔÇ™s model. One reason for the missing effort is that both biochemical
and structural data indicate that ADP┬·Pi and ATP-F-actin are similar (Otterbein et al.,
2001; Rickard and Sheterline, 1986; Wanger and Wegner, 1987), so that distinguishing
between the two species may be unnecessary in many contexts. Assuming equiva-
lence between ADP┬·Pi and ATP-actin species, we have recently published a broad
mathematical model of the steady-state actin cycle that predicts a broad range of
experimentally observed behaviors (Bindschadler et al., 2004). While this agreement
is encouraging, it does not replace the need for newly designed experiments that
de´¬ünitively establish rates.
In Bindschadler et al. (2004) we carefully tabulated consensus rate constants for
intrinsic actin dynamics. While controversies persist concerning the mechanism of
ATP hydrolysis and the rate of nucleotide exchange on G-actin, a growing consensus
on these topics can be inferred from agreements by independent laboratories. As men-
tioned, the rates for the ADP┬·Pi-actin species remain unmeasured. Also unaddressed
is the fact that the rate constants must depend on the nucleotide content of ´¬ülaments
and so the single values reported by Pollard cannot constitute the complete story.
Regulation of dynamics by actin-binding proteins
Many efforts over the past twenty years have focused on elucidating the mechanisms
by which actin-associated proteins modulate actin dynamics. These efforts are essen-
tial because intracellular signaling pathways do not modify actin itself, and so the
regulation of binding proteins provides an indirect route for changing cytoskeletal
structure and cell shape. Unlike solutions of pure actin, cells contain short, dynamic
´¬ülaments in highly structured networks and often a large fraction of unpolymerized
actin. Here we review properties of actin-binding proteins thought to account for the
major differences between the dynamics of cellular and puri´¬üed actin (see Fig. 9-7).
ADF/co´¬ülin: targeting the rate-limiting step in the actin cycle
Named for their activity as ÔÇťactin depolymerizing factorsÔÇŁ and their ability to form
co´¬ülamentous structures with F-actin, the ADFs and co´¬ülins form two subgroups of a
family of proteins (ADF/co´¬ülins or ACs) expressed in most mammalian cells. Efforts
to understand AC function have been challenged by a multiplicity of AC functions
that differ slightly between the ADFs and co´¬ülins and also among species (Bamburg,
1999). In general, ACs bind to both monomeric and ´¬ülamentous actin with rather
184 J.L. McGrath and C.F. Dewey, Jr.
barbed end capping disassembly by cofilin
by capping protein
polymerization by profilin
filament severing by
cofilin and gelsolin
new filament generation by
Arp2/3 complex, gelsolin
Fig. 9-7. Regulation of actin dynamics. Actin-binding proteins modulate every phase of the actin
cycle including assembly and disassembly kinetics, nucleotide exchange, and ´¬ülament number and
exclusive af´¬ünity for the ADP-bound conformations (Carlier et al., 1997; Maciver
and Weeds, 1994). On ´¬ülaments, structural data indicate that ACs bind the sides
of ´¬ülaments and destablize the most interior interactions between subunits (Bobkov
et al., 2004; Galkin et al., 2003; McGough et al., 1997). AC-decorated ´¬ülaments break
along their length (Maciver et al., 1991) and rapidly disassemble at their ends (Carlier
et al., 1997). Despite debate over whether ACs should be thought of primarily as
´¬ülament-severing proteins or catalysts of ADP subunit disassociation (Blanchoin and
Pollard, 1999; Carlier et al., 1997), both effects may occur as manifestations of the
same structural instability on ´¬ülaments. ACs have been shown to increase the rate of
Pi release on ´¬ülaments, and to slow the rate at which ADP monomers recharge with
ATP on monomer (Blanchoin and Pollard, 1999). Thus ACs both hasten the production
of ADP-actin and stabilize the ADP form. Some AC proteins bind ADP-G-actin with
a much higher af´¬ünity than ADP-F-actin to create a thermodynamic drive toward the
ADP-G-actin state (Blanchoin and Pollard, 1999). Like their multiple functions, ACs
have multiple avenues for regulation including pH sensitivity, inactivation by PIP2
binding, and serine phosphorylation (Bamburg, 1999).
Given this seemingly perfect arsenal of disassembling functions in vitro, observa-
tions that ACs trigger polymerization and generate new barbed ends in cells (Ghosh
et al., 2004) certainly appear contradictory. There are at least two likely explanations
for the paradox. First, with a large pool of sequestered ATP-actin available to assemble
at free barbed ends (see the discussion of thymosins), the conditions inside a cell are
primed for assembly (Condeelis, 2001). Thus ´¬ülaments generated by AC severing may
not have suf´¬ücient time to disassemble before they become nuclei for new ´¬ülament
Cell dynamics and the actin cytoskeleton 185
growth. Supporting this, enhanced assembly occurs at early time points in vitro when
ACs are added to solutions containing an excess of ATP-G-actin. (Blanchoin and
Pollard, 1999; Du and Frieden, 1998; Ghosh et al., 2004).
The cellular data on co´¬ülin-mediated growth should not be interpreted to mean that
ACs are not involved in ´¬ülament dissolution in vivo. Indeed, theoretical calculations
indicate that severing alone cannot explain how ´¬ülament turnover in cells occurs
orders of magnitude faster than unregulated actin (Carlier et al., 1997). Thus, the
second explanation for the paradox is that AC-mediated disassembly is directly linked
to ´¬ülament assembly. If pointed-end disassembly is rate-limiting for the actin cycle
in vivo as it is in vitro, the enhanced production of ADP-G-actin should lead to a
larger supply of ATP-monomer and enhanced polymerization elsewhere. Thus both
of the ÔÇ˜destructiveÔÇ™ activities of ACs ÔÇ“ severing and enhanced disassembly ÔÇ“ can lead
to ´¬ülament renewal and rapid turnover in the cellular environment.
Pro´¬ülin: a multifunctional protein to close the loop
Pro´¬ülin was the ´¬ürst monomer-binding protein discovered and originally thought to
sequester G-actin in a nonpolymerizable form (Tobacman and Korn, 1982; Tseng and
Pollard, 1982). However, later data made clear that pro´¬ülins do not prevent actin as-
sembly, but actually drive the assembly phase of the actin cycle (Pollard and Cooper,
1984). Today pro´¬ülins are known to catalyze the rate of nucleotide exchange on
G-actin by as much as an order of magnitude (Goldschmidt-Clermont et al., 1991;
Selden et al., 1999). In cells this means that newly released ADP-G-actin is recharged
to the ATP state shortly after binding to pro´¬ülin. Signi´¬ücantly, the pro´¬ülin-G-actin
complex is capable of associating with ´¬ülament barbed ends, but not with pointed
ends (Pollard and Cooper, 1984). Pro´¬ülin binds to a structural hinge on the barbed
end of an actin ´¬ülament and slightly opens the hinge to expose the nucleotide-binding
pocket and promote nucleotide exchange (dos Remedios et al., 2003; Schutt et al.,
1993). Pro´¬ülin binding in this region also sterically blocks association of G-actin
with pointed ends. Because there is no evidence that pro´¬ülin blocks actin assem-
bly at barbed ends, pro´¬ülin presumably instantly disassociates from monomer af-
ter assembly. Further, the pro´¬ülin-actin complex assembles at barbed ends at the
same rates as ATP-G-actin alone (Kang et al., 1999; Pantaloni and Carlier, 1993).
With these properties, pro´¬ülin-bound actin becomes a subpopulation of barbed-end-
speci´¬üc monomer (Kang et al., 1999). In our published analysis of the actin cycle
(Bindschadler et al., 2004), we found that pro´¬ülinÔÇ™s functions provide the perfect com-
plement to co´¬ülinÔÇ™s disassembly functions. Together the two proteins appear to over-
come every major barrier to increasing the rate of ´¬ülament treadmilling (Bindschadler
et al., 2004).
Arp2/3 complex and formins: making ´¬ülaments anew
A perplexing and important question for cell biologists in the 1980s and 1990s was
ÔÇťhow are new ´¬ülaments created in cells?ÔÇŁ One answer was that new ´¬ülaments are
generated when existing ´¬ülaments are ´¬ürst severed and then elongate; however there
was no reasonable mechanism for the de novo generation of ´¬ülaments in cells. In
the late 1990s it became clear that the Arp2/3 complex was dedicated to this task
(Mullins et al., 1998; Pollard and Beltzner, 2002). The two largest members of this
186 J.L. McGrath and C.F. Dewey, Jr.
seven-protein complex are the actin-related proteins Arp2 and Arp3 (Machesky et al.,
1994). Like the other members of the Arp family, Arp2 and Arp3 share a strong
structural similarity to actin. In the Arp2/3 complex, these similarities are used to
create a pocket that recruits an actin monomer to form a pseudotrimer that nucleates
a new ´¬ülament (Robinson et al., 2001). The complex holds the growing ´¬ülament at
its pointed end, leaving the barbed end free for rapid assembly (Mullins et al., 1998).
For robust nucleation of ´¬ülaments, the Arp2/3 complex requires activation, ´¬ürst by
WASp/Scar family proteins (Machesky et al., 1999), and secondarily by binding to
preexisting F-actin (Machesky et al., 1999). The arrangement gives autocatalytic
growth of branched ´¬ülament networks in vitro: new ´¬ülaments grow from the sides of
old ones to create a Ôł╝70Ô—¦ included angle (Mullins et al., 1998). This same network
geometry is found at the leading edge of cells and the Arp2/3 complex localizes to
branch points in the cellular networks (Svitkina and Borisy, 1999).
While it is now clear that the Arp2/3 complex is an essential ingredient of the actin
cytoskeleton, its discovery is new enough that many details of its mechanism are
clouded in controversy. The most visible controversy has been over the nature of the
Arp2/3 complex/F-actin interaction. Carlier and colleagues argue that the complex
incorporates at the barbed ends of actin ´¬ülaments to create a bifurcation in ´¬ülament
growth (Pantaloni et al., 2000); however, using direct visualization of ´¬‚uorescently
labeled ´¬ülaments, several laboratories have demonstrated that new ´¬ülaments can grow
from the sides of preexisting ´¬ülaments (Amann and Pollard, 2001a; Amann and
Pollard, 2001b; Fujiwara et al., 2002; Ichetovkin et al., 2002). One paper appears
to resolve the confusion with data indicating that branching occurs primarily from
the sides of ATP-bound regions of the mother ´¬ülament very near the barbed end
(Ichetovkin et al., 2002). However, others believe that branching can occur on any
subunit but ´¬ülaments release or debranch rapidly from ADP segments of the mother
´¬ülaments. This idea is supported by a correlation between the kinetics of debranching
and Pi release on the mother ´¬ülament (Dayel and Mullins, 2004), but contradicted
by a report indicating that ATP hydrolysis on Arp2 is the trigger for debranching
(Le Clainche et al., 2003). A disheartened reader looking for clearer understandings
should consult the most recent reviews on the Arp2/3 complex.
Very recently, it has become clear that the Arp2/3 complex is not the only molecule
capable of de novo ´¬ülament generation in cells. In yeast, members of the formin
family of proteins generate actin ´¬ülament bundles needed for polarized growth
(Evangelista et al., 2002), and in mammalian cells formin family members help
generate stress ´¬übers (Watanabe et al., 1999) and actin bundles involved in cy-
tokinesis (Wasserman, 1998). Dimerized FH2 domains of formins directly nucle-
ate ´¬ülaments in a most remarkable manner (Pruyne et al., 2002; Zigmond et al.,
2003). The FH2 dimer remains attached to the barbed ends of actin ´¬ülaments even
as it allows insertion of new subunits at that same end (Kovar and Pollard, 2004;
Pruyne et al., 2002). By tracking the barbed ends of growing ´¬ülaments, formins
block associations with capping protein (Kovar et al., 2003; Zigmond et al., 2003).
In cells where capping protein and cross-linking proteins are abundant, formin-
based nucleation should naturally lead to bundles of long ´¬ülaments, while Arp2/3-
complex-generated ´¬ülaments should naturally arrange into branched networks of short
Cell dynamics and the actin cytoskeleton 187
Capping protein: ÔÇ˜decommissioningÔÇ™ the old
Capping protein is an abundant heterodimeric protein that binds with high af´¬ünity to
the barbed ends of actin ´¬ülaments to block both assembly and disassembly at these
ends (Cooper and Pollard, 1985; Isenberg et al., 1980). Vertebrates express multiple
isoforms of both the ╬± and ╬▓ subunits (Hart et al., 1997; Schafer et al., 1994). With
the conditions of cells favoring polymerization at free barbed ends, capping protein
is essential to control the degree of polymerization. The association rates of capping
protein with barbed ends in combination with high cellular concentrations of capping
protein (Ôł╝ 2 ┬µM) should only allow a newly crated, unprotected barbed end to grow for
Ôł╝ 1 s (Schafer et al., 1996). On the other hand, because the residency time of capping
proteins on barbed ends is Ôł╝ 30 minutes (Schafer et al., 1996), short capped ´¬ülaments
will depolymerize from their pointed ends in cells. Capping proteins are thought to
be integral to the recycling of monomers in dendritically arranged ´¬ülaments at the
leading edge of cells (Pollard et al., 2000). Consistent with this idea are ´¬ündings that
perturbations of capping activity dramatically alter the geometry of Arp2/3-complex-
induced networks in reconstitution studies (Pantaloni et al., 2000; Vignjevic et al.,
In addition to blocking barbed-end dynamics, capping protein diminishes the lag
phase of actin polymerization (Pollard and Cooper, 1984). In this ÔÇ˜nucleatingÔÇ™ ac-
tivity, capping protein is probably stabilizing small oligomers rather than generating
´¬ülaments de novo (Schafer and Cooper, 1995). Because the growing ´¬ülaments are
capped at their barbed end, this function is probably not active in cells with abundant
sequestering proteins that can prevent assembly at pointed ends. The only known
regulation of capping protein activity is by phospholipids. Phospholipids can both
inactivate free capping protein (Heiss and Cooper, 1991) and remove bound capping
protein from barbed ends (Schafer et al., 1996).
Gelsolin: rapid remodeling in one or two steps
If the job of actin-binding proteins is to remodel the actin cytoskeleton, then gelsolin
has exceptional quali´¬ücations. Activated by micromolar Ca2+ (Yin and Stossel, 1979),
gelsolin binds to the sides of actin ´¬ülaments and severs them (Yin et al., 1980).
However, unlike co´¬ülin, gelsolin remains attached to the new barbed end created by
severing to block further polymerization (Yin et al., 1981; Yin et al., 1980). Because
gelsolin has nM af´¬ünity for barbed ends, it functions as a permanent cap that can
only be removed through subsequent binding by phospholipids (Janmey and Stossel,
1987). In platelets and neutrophils, activated gelsolin remodels actin in two steps
(Barkalow et al., 1996; Glogauer et al., 2000). Because the majority of ´¬ülaments in
resting cells are capped, cellular activation ´¬ürst leads to gelsolin severing to create
a large number of dynamically stable ´¬ülaments. Shortly thereafter, these ´¬ülaments
become nuclei for new growth as phospholipid levels increase to result in massive
While gelsolin seems built for acute remodeling, expression studies clearly indi-
cate a role for gelsolin at steady-state. Gelsolin null ´¬übroblasts have impaired motil-
ity, reduced membrane ruf´¬‚ing, slow ´¬ülament turnover, and abundant stress ´¬übers
(Azuma et al., 1998; McGrath et al., 2000a; Witke et al., 1995), and gelsolin over-
expression produces the opposite trends (Cunningham et al., 1991). With its high
188 J.L. McGrath and C.F. Dewey, Jr.
Ca2+ requirements, it is unclear if these steady-state effects are mediated by inter-
mittent and localized gelsolin activity, or constant but low levels of activity. Gelsolin
also has a fascinating role in apoptosis where caspases cleave the protein to create
an unregulated severing peptide (Kothakota et al., 1997). Continuous severing by this
peptide helps create a mechanically compromised cell that eventually detaches from
its substrate (Kothakota et al., 1997).
╬▓4-thymosin: accounting (sometimes) for the other half
The 15 kD protein ╬▓4-thymosin is present in mammalian cells at levels that equal
or exceed actin itself (Safer and Nachmias, 1994). Its discovery appeared to resolve
the critical question of how mammalian cells maintained nearly half of their actin
in an unpolymerized form despite intracellular conditions that favored polymeriza-
tion. ╬▓4-thymosins are unstructured in solution and partially wrap around the G-actin
monomer to control its associations (Safer et al., 1997). ╬▓4-thymosins bind to ATP-
G-actin (but not ADP-G-actin) with an af´¬ünity comparable to the pointed end ATP
critical concentration, but less than the barbed end ATP critical concentration (Carlier
et al., 1993). In this way when barbed ends are mostly capped, ╬▓4-thymosin func-
tions as a ÔÇ˜sequesteringÔÇ™ protein that maintains a pool of non´¬ülamentous actin, but
when free barbed ends are abundant the pool diminishes. Pro´¬ülin can compete for
the ATP-G-actin pool maintained by ╬▓4-thymosin (Carlier et al., 1993), possibly by
emerging with the charged monomer after forming a complex that includes all three
proteins (Yarmola et al., 2001). Thus with or without pro´¬ülin, ╬▓4-thymosin helps
to reserve ATP-G-actin for future assembly at barbed ends. The high concentration
of ╬▓4-thymosin can support extensive and sudden conversions from G-actin to ´¬ül-
aments. This conversion is likely occurring in the dramatic polymerization-induced
shape change of both platelets (Safer et al., 1990) and neutrophils (Cassimeris et al.,
1992), both of which contain abundant ╬▓4-thymosin/G-actin complex at rest. How-
ever ╬▓4-thymosin apparently is not required for the continuous shape change during
crawling, because motile amoebae are thought to be void of thymosin-family proteins
(Pollard et al., 2000).
Dynamic actin in crawling cells
In this section we explore a most conspicuous and well-studied function of the actin
cytoskeleton: its ability to serve as the engine for cell crawling. By driving the expan-
sion of the plasma membrane in the direction of cell advancement, actin polymeriza-
tion initiates the crawling cycle (Fig. 9-8). The networks formed by polymerization
evolve to structures that provide mechanical support for cell extensions; that link the
cell to its substrate; and that support the myosin-based contractions needed for cell
translation. The network must also disassemble to recycle its constituents for further
rounds of assembly. Thus the actin network at the leading edge of motile cells pro-
vides both the structure and the forces needed for crawling (see Fig. 9-9). Here we
review the current understanding of the geometry and dynamics of these networks,
and address the important question of how polymerization might lead to pushing
Cell dynamics and the actin cytoskeleton 189
Fig. 9-8. The four steps in cell migration. The classic schematic of crawling breaks the process
down into a four-step cycle. The cycle begins with the protrusion of the leading edge driven by actin
polymerization. The extended cell forms new attachments in advance of its body and then contracts
against this attachment to break tail adhesions and translate forward. From Mitchison and Cramer,
Actin in the leading edge
The extension of the plasma membrane that interrogates new regions of substrate can
come in several forms. Mammalian cells crawling in culture environments extend both
´¬ünger-like projections, called ´¬üllopodia, and broad, thin, veil-like projections called
lamellipodia. Which structure occurs more frequently is a strong function of cell type
and substrate conditions (Pelham and Wang, 1997). Cells that crawl in amoeboid
fashion ÔÇ“ a class that includes the leukocytes of the mammalian immune system ÔÇ“ use
bulkier protrusions known as pseudopodia. By all accounts, the initiator of ´¬ülament
assembly in each of these cellular protrusions is the Arp2/3 complex activated by
membrane-bound WASp/Scar family proteins. Pollard and Borisy have offered the
most detailed proposal for how actin networks evolve in lamellipodia (Pollard and
Borisy, 2003), and ´¬ülopodia appear to be triggered from rearrangements of a lamel-
lipodial network (Svitkina et al., 2003). Because the three-dimensional character of
pseudopodia makes them less amenable to ultrastructural and ´¬‚uorescence studies,
far less is known about the geometry and dynamics of pseudopodial networks.
Synthesizing data from electron micrographs of cytoskeletal structure in the lamel-
lipodia of fast-moving keratocytes (Svitkina and Borisy, 1999), immunochemical
analysis of Arp2/3 complex, ADF/co´¬ülin, and capping protein location in these same
samples (Svitkina and Borisy, 1999), and live cell ´¬‚uorescence data revealing regions
of actin assembly and disassembly in ´¬übroblasts (Watanabe and Mitchison, 2002),
190 J.L. McGrath and C.F. Dewey, Jr.
(monomers / s)
relative toleading edge concentration