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INTRODUCTION |
Many biological movements are based on the
dynamics of the actin network in cells; this is the case, for instance,
for the process of the lamellipodia extension in crawling cells
(Wang, 1985
) or the motion of the neural cone during
dendrite growth (Bray et al., 1991). To produce the
forces required for the motion, actin is often associated with myosin,
as in muscles or in the cell contractile ring (Bray,
1992). However, the presence of motor proteins like myosins is
not necessary to produce a motile force; this issue has been discussed
for ameboid motion (Mitchison and Cramer, 1996) and has
been demonstrated several times in the case of the bacterium
Listeria monocytogenes, for which motor proteins associated
with its actin tail have been sought but not found (Mounier et
al., 1990; Marchand et al., 1995;
Southwick and Purich, 1998; Loisel et al.,
1999). It is therefore widely accepted that the process of
actin polymerization itself is sufficient to induce cell movements.
Physical models of actin-based motility have been suggested in the case
of ameboid motion (Evans, 1993; Alt and Dembo,
1999) and by Mogilner and Oster (1996), who have
proposed a generic Brownian ratchet mechanism that also describes the
growth of microtubules (Mogilner and Oster, 1999).
In this paper, we will focus on the case of Listeria, a
system for which a lot of experimental data are available. The
characteristic numbers for Listeria motion and the notations
that will be used in the following are listed in Table
1. To move within cell cytoplasm and
spread from cell to cell through the cytoplasmic membranes, L. monocytogenes induces the assembly of a tail (Fig.
1), which is an actin gel made of
cross-linked filaments and which forms a tubular structure
(Tilney and Portnoy, 1989). The actin is recruited from
the pool of the infected cell. The tail is used as an anchor in the
cytoplasm, so that as new polymerized actin is added between the
bacterium surface and the older polymerized gel, the organism is
propelled forward (Theriot et al., 1992). This type of
motion has recently been shown to be more general than initially
thought: besides the Gram-positive L. monocytogenes and the
Gram-negative Shigella flexneri (Clerc and
Sansonetti, 1987), the intracellular movements of the vaccinia
virus (Cudmore et al., 1995) and even of some vesicles
(Merrifield et al., 1999) are based on the formation of
an actin tail. In Listeria, the presence of a single
transmembrane protein, ActA, has been shown to be required and
sufficient to trigger actin polymerization and thus to induce the
motion (Kocks et al., 1992; Smith et al.,
1995). Immunolabeling experiments have shown that ActA is
present everywhere around the moving Listeria, except at the
front pole (defined by the direction of motion) and that ActA
colocalizes with the production of actin filaments (Kocks et
al., 1993). All of the other proteins like actin, the cross-linkers such as 
-actinin (Dabiri et al., 1990)
and other proteins required for the actin polymerization (like Arp2/3;
Welch et al., 1997), are provided by the infected
medium. Recently the motility of Listeria has been observed
in a medium reconstituted from pure proteins (Loisel et al.,
1999). Although several biochemical scenarios have been
suggested (Southwick and Purich, 1998; Cossart and Kocks, 1994), the enzymatic reactions responsible for the local shift of the actin monomeric/polymeric equilibrium around the
bacterium have not been completely elucidated. Some progress has been
made in determining the kinetics of nucleation and of the growth of the
actin filaments (Welch et al., 1998), but the time
constants for the release from the surface and the cross-linking of the
filaments are not yet known.
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FIGURE 1
(a) Observation of Listeria
moving in platelet extract, observed by phase-contrast microscopy. The
bacteria move at ~8 µm · min 1. The tail can be
more than 100 µm long when the depolymerization is slow enough.
Bars = 5 µm. (b) Elastic model of the propulsion of
the bacterium: the new filaments are polymerized at the bacterium
surface and expand the older layers, inducing a stress in the actin
gel, which is viewed as a continuous medium. The motion of the
bacterium is due to the relaxation of the strain in the tail.
(c) Heuristic model: the system is simplified in a two-gel
model; the internal gel is produced on the back hemisphere at the
polymerization speed vp1, and the external
gel is produced on the cylindrical surface at
vp2. The gel is a single structure that
moves away from the bacterium at the homogeneous speed v.
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In a recent paper (Gerbal et al., 2000) we presented
results on some of the elastic properties of the actin filament tail. We found that the tail is firmly attached to the bacterium; applying a
force in the picoNewton range between the bacterium and its tail for
several minutes failed to pull the bacterium from its tail. We have
also shown in this paper that the tail behaves like a genuine gel:
optical tweezers were used to apply stresses, and we found that the
tail's response is elastic. This finding is consistent with the
presence of actin-binding cross-linkers (Dabiri et al.,
1990). The strong attachment of the bacterium to the tail puts
limits on the thermal ratchet model of propulsion, where actin
filaments simply push on the bacterium to which they are not attached
(Mogilner and Oster, 1996). The observed attachment suggests that the force between the bacterium and its tail results from
the distortions of a continuous elastic medium (a gel) anchored to and
growing from the bacterium surface. The bending modulus K of
the tail was measured and found to be on the order of 100-1000 Pa
(µm)4 (persistence length ~0.1 m), suggesting a
Young's modulus Y = [K/
(rb4/4)] = 103 to 104 Pa for the actin gel.
We are therefore led to a mesoscopic description of the
Listeria motility at a length scale larger than that of
individual proteins. Although there has been a great effort expended in
the discovery of many of the microscopic aspects of the polymerization process and the specific proteins involved, the propulsion mechanism that arises from the growth and cross-linking of the filaments is
important and interesting in its own right. It is also necessary for
the basic understanding of the mechanisms that control the speed of the
bacterium, the maximum force it can overcome, and the effect of the
obstacles and forces it encounters during its motion. In this paper we
describe the gel as a continuous medium that can be treated in the
framework of the linear theory of elasticity. We propose that the
addition of new actin filaments induces elastic deformations in the
gel: the buildup of a new polymerized layer at the bacterium surface
compresses the previously formed layers. Thus, the free energy produced
by actin polymerization is not directly used for the propulsion but is
rather first stored as elastic energy (Fig. 1 b). The
problem is significantly complicated by the actual geometry of the
bacterial surface, which produces the filaments. Consider the
cylindrical surface of the bacterium. If the filaments were not
cross-linked, they would grow radially outward, hindering rather than
aiding the propulsion. Because the filaments are cross-linked, the
outward growth can only be accomplished by an extension of tangential
cross-links as they are forced to a larger radius. This costs a large
elastic energy (
Rgel3), produces a
large stress on the cell surface, and eventually leads to a cessation
of growth, as has been demonstrated experimentally on beads
(Noireaux et al., 2000; Gerbal et al.,
1999). The shape of the gel and, therefore, the motion of the
bacterium depend on the way the gel adopts the lowest energy
conformation. In fact, for most geometries there are no steady-state
solutions that allow continued growth without a steady increase in
elastic energy. The only geometry that allows continued growth for a
cross-linked system corresponds to one-dimensional growth, i.e., to a
long region of constant cross section, such as the Listeria tail.
For the sake of simplicity, we will discuss this problem, using a
two-gel model (Fig. 1 c), in which the gel is artificially divided into two parts: the internal gel, produced from the back part
of the bacterium (gel number 1: light gray in the figure), and the external gel, produced on the cylindrical surface (gel number
2: dark gray). First, we will consider the simplest case of
a bacterium producing only the internal gel. Then we will consider the
case of a bacterium pushed only by the external gel. Finally, we will
present the complete model in which a bacterium is pushed by both gels.
Despite its simplicity, this two-gel model provides important insights
into the distortion of a gel produced in quasi-3D geometry before being
constrained to a 1D geometry. Furthermore, the two-gel model allows us
to understand the behavior of a Listeria mutant
(ActA
21-97) obtained by Lasa et al.
(1997): the speed of this mutant oscillates periodically
between a very slow and a fast phase. In the last part of the paper we
extend the steady-state equations of the elastic model to describe
time-dependent processes and show that the resulting model can also
explain the mechanism of these oscillations.
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STEADY-STATE MOTION |
One-dimensional model
The easiest mechanism of propulsion to imagine would occur if the
actin were simply polymerized at a constant polymerization speed
vp1 (index 1 is used for the description of
the internal gel, and index 2 will be used for the external gel
produced at the cylindrical surface) from a flat region at the end of
the bacterium (Fig. 2). We will describe
Listeria as a cylinder with a circular cross section of area
Sb = rb2 and
assume in the 1D model that the gel is produced only at the back of the
bacterium. The elastic deformations of the gel relevant for propulsion
occur on distances much smaller (typically the size of the bacterium)
compared to the length scale on which the depolymerization of the actin
in the tail becomes relevant (<10 µm). The tail is therefore modeled
as an infinitely long tube of cross-sectional area
St1, made of homogeneous elastic material. It is characterized by a compressional modulus Y and an
axial elastic strain 
1. In the reference frame of the
bacterium, the tail moves away at a speed v. This is the
parameter that we want to determine as a function of
Fext, the external force applied on the
bacterium. We will neglect viscous forces due to the friction against
the outer medium; for cytoplasm viscosity on the order of
102 Pa · s, this force is ~10 fN, which is
negligible in comparison with the forces involved in the propulsion of
Listeria.
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FIGURE 2
Schematic representation of the forces applied on the
bacterium in the one-dimensional model. (a) The tail exerts
the force F1 = Fmot1 on
the bacterium, which moves against the external force
Fext. (b) We solve the problem in the
reference frame of the bacterium; the tail moves away at the speed
v. Fext is also exerted on any section of the
tail of surface St1, inducing an axial
stress  zz.
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The forces exerted on the bacterium must balance each other, so we have
Fmot1 = 
Fext,
where Fmot1 is the force from the tail on
the bacterium. From Newton's second law (action = reaction), Fext is also exerted on any cross section of the
tail and must be balanced by the elastic stresses ij in
the tail that must fulfill the condition
iij = 0 (Landau and Lifchitz,
1967). Consistent with the neglect of viscous frictional
forces, both the radial and the shear components of the stress must
vanish on the cylindrical surface of the tail. The axial component of
the stress is therefore
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(1)
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Linear elasticity relates this stress to the longitudinal strain
in the tail:
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(2)
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For the sake of simplicity, we assume that the tail is
incompressible, i.e., that the volume of a material element is
conserved under deformation (Poisson ratio = 1/2). This is
consistent with elasticity measurements which showed that the Poisson
ratio of a fibroblast actin cortex is ~0.4 (Sackmann, personal
communication). Using volume conservation relates the
elongation of a cylindrical element of nondeformed material (of length
z and cross-sectional area Sb)
with that of a similar element in the tail (of length z'):
zSb = z'St1,
and we obtain
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(3)
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Another requirement of continuous production of actin fibers is
that they must go somewhere. Polymerization takes place at the
bacterium surface and actin depolymerizes along the tail, because the
infected cytoplasm cell sets a new equilibrium in favor of the
nonpolymerized form of actin. Between these two regions we can account
for the quantity of F-actin filaments by conservation of flow. The rate
at which filaments are produced on the entire surface of the bacterium
is equal to the total flux through any cross section of the tail
(before depolymerization becomes important):
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(4)
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We can now write the force-velocity equation:
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(5)
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The plot of this equation is shown in Fig.
3. The characteristic scale of elastic
force in our model is therefore given by YSb = 1 nN, which is several orders of
magnitude greater than the typical force that is encountered by the
bacterium when it moves through an infected medium. For instance, the
force required to deform a membrane is on the order of 50 pN
(Evans and Yeung, 1989). In vivo, the linear form of the
above equation is therefore more sensible:
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(6)
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FIGURE 3
Force-velocity curve given by the one-dimensional
model. The ratio of bacterium speed to polymerization speed is plotted
versus the external force applied on the bacterium. - - -,
Constant polymerization speed.  , The polymerization speed depends
on the stress normal to the bacterium surface.
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The polymerization speed varies with the stress
We shall now take into account the fact that the polymerization
speed (vp) at which the filaments are produced
also depends on the applied forces. The external force
Fext induces at the molecular level a normal
force fext on the growing filaments stuck between the bacterium surface and the cross-linked gel. If the filaments are compressed by the force fext, then
the additional work required to add a monomer of size a is
W = fexta. Thermodynamics tells us that the effect of the force on both the off rate and the on
rate of monomer addition at the tip of a growing filament can be
described by introducing appropriate Boltzmann factors (Hill,
1987), such that the polymerization rate can be written as
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(7)
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where 0 
x 1 is an adjustable parameter.
Such a dependence has been demonstrated experimentally on microtubules
by Dogterom and Yurke (1997), but their statistics were
not sufficient to determine a value for x. We use Hill's
thermodynamic description because it is generic and does not assume a
precise mechanistic model of polymerization. It is not necessary to
know x to determine the stall force
fs given for vp = 0. Equation 7 gives fs = (kBT/a)ln (V+/V) = (kBT/a)ln
(G/kBT), where G = 14kBT (Mogilner and Oster,
1996) is the free energy of the reaction of polymerization of
one monomer. The stall force per filament should then be 1 pN. If we
assume between 100 and 1000 filaments per Listeria tail section, the force required to stop the bacterium falls in the range of
a nanoNewton, which is also the scale of the elastic forces in the
problem. Thus the speed of the bacterium does not depend much on the
external forces it encounters in vivo. It is determined by the
polymerization rate and by the internal stresses exerted in the actin
network.
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FIGURE 4
Thin sections of Listeria monocytogenes in
infected cells inducing actin polymerization (courtesy Kocks et
al., 1993). The bacterium is surrounded with actin, except at
the front pole. The thickness of the gel over its cylindrical surface
equals ~1/2 to 1 bacterium radius. Bar = 1 µm.
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In our equation, for the sake of simplicity, we will set x = 1, i.e., only the on rate is assumed to be affected by the stress. Thus, the polymerization speed is given by
vp1 = vp0
(1 1)e(
1/0)
with the longitudinal strain 0 = kBT/Yad2, where d is
the mesh size of the gel. Assuming d 
50 nm,
0 1.
Taking into account the variation of the polymerization speed, Eq. 6
becomes
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(8)
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The slope of this new force-velocity curve (Fig. 3) is about twice
the slope of the one obtained by assuming that the velocity is not
stress dependent. This indicates that the variation of the
polymerization speed affects the speed of the bacterium as much as the
elastic compression of its tail. This concept will be required to
understand the model of a bacterium pushed by a full tail (see Fig. 4).
The three-dimensional model
Bacterium pushed only by the external gel
Before considering the complete two-gel model, this section deals
with a bacterium producing a gel only from the side, i.e., on the
cylindrical surface. Although seemingly artificial, this model is based
on observations of real systems. The bacterium S. flexneri
uses the same trick as Listeria monocytogenes to propel itself forward: it produces an actin tail that appears to be hollow when observed by confocal microscopy (P. Cossart, personal
communication). Moreover, Merrifield et al. (1999) showed that
osmotic shock on a cell culture can induce endocytotic vesicles that
move by forming an actin tail. This tail also appears to be hollow
under a confocal microscope (personal communication).
We assume that actin is polymerized at the speed
vp2 normal to the cylindrically symmetrical
surface of a bacterium. In addition, we assume that the filaments are
immediately cross-linked and that no strain exists at first. As new
material is continuously added at the bacterium surface, polymerization
has to expand the older layer outward. One can view this model as a
stack of rubber bands on a rigid cylinder, in which new bands are added
from underneath, at the cylinder/rubber interface. If no symmetry
breaking occurs, such a mechanism stops as the elastic energy increases
and diverges like r3, where r is the
radius of the external layer. Such a cessation of growth has been
demonstrated experimentally with spherical colloidal particles
(Noireaux et al., 2000; Gerbal et al.,
1999). Moreover, some Listeria do not produce a tail
and are only surrounded by an actin sheath (Lasa et al.,
1997). Alternatively, if the symmetry is broken, the radial
energy built up around the bacterium relaxes in the tail. Between the
bacterium and the tail, the gel changes its conformation to fulfill the
new boundary conditions. It is not clear whether the symmetry breaking
of the distribution of the gel around Listeria is
spontaneous (as for Merrifield's vesicles) or is triggered by an
external factor such as the bacterium division, as shown in the case of
the bacteria S. flexneri (Goldberg et al.,
1994).
In the calculation below we describe the strain in the gel, using
minimization of energy. Notations are shown in Fig.
5. On the cylindrical surface we assume
the tangential speed of the gel v0 to be
constant. In the tail the gel speed becomes v. The gel grows
continuously and reaches a maximum height rm at
the rear of the bacterium. Far away in the tail (assumed to be
infinite), the gel thickness is rout rin. Parameters and 2
are the
dimensionless thicknesses of the gel above the bacterium and in the
tail, respectively (see Table 1).
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FIGURE 5
Notation for the 3D model, a bacterium producing only
an external gel. (a) The gel is polymerized at speed
vp2 all over its cylindrical surface (with
symmetry of revolution). In the reference frame of the bacterium, the
gel moves at the speed v0 and reaches the
maximum external radius rm over the bacterium.
The tail is hollow and has inner and external radii
rin and rout. It moves
away from the bacterium at a speed v. (b) The force
F2 exerted by the external gel on the bacterium
has two components: Ffric, due to the dynamic
connection of the gel to the bacterium surface, and
Fmot2, due to the stress exerted by the gel
on the back hemisphere.
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Calculation of strains and stresses in the gel: the stacked rubber
band model
The expression of the radial stress in a stacked rubber band
system produced at rb and expanding to an outer
radius rext is computed in Appendix I. At radius
r the radial stress is
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(9)
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This equation is valid for the gel above the bacterium and in the
tail. In the tail, no solid surface restrains the inner radius. The
radial component of the stress must vanish on the external and internal
boundaries at r = rin and at r = rout, respectively. The half-thickness of the gel
in the tail is given by
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(10)
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The gel minimizes its elastic energy so that the axial stress in
the tail must satisfy the equation (Landau and Lifchitz, 1967)
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(11)
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where E/V is the elastic energy per unit volume.
Neglecting the nondiagonal terms like zr (they must be
of second order because they both vanish at the internal and external
layers), the elastic energy has an axial and a radial contribution (see Appendix II):
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(12)
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We assume again here that the gel structure is homogeneous and
that the Poisson ratio is 1/2. This is clearly a simplification, inasmuch as several works have shown that the filament orientation is
not isotropic, although there is no clear consensus on this point
(Sechi et al., 1997; Zhukarev et al.,
1995). A small cylinder around the bacterium, despite the
deformations, must have the same volume when it reaches the tail:
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(13)
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Using the dimensionless height of the gel 
(rm rb)/rb, this equation
becomes
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(14)
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so that Eq. 12 can be written in term of 
zz only:
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(15)
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As in the 1D model, another equation is provided by flux (
)
conservation between the surface around the bacterium (of length L) and the section of the gel at the rear of the bacterium
and through any section in the tail before depolymerization becomes important:
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(16)
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This gives a relation between the speeds and the gel thicknesses:
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(17)
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The solution of Eq. 11 is then
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(18)
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Assuming that the speed of the gel does not change much from the
bacterium to the tail (v/v0 1), to the
first order in v/v0 1 the expression
becomes
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(19)
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For a thin gel above the bacterium ( < 1), the expression
is now v/v0 1 Fext/2Y · Sb, which
is exactly equivalent to the expression found for the internal gel.
Here the gel is produced at the back of the bacterium from a surface
2Sb (instead of Sb) at
the speed v0 (instead of
vp1). The last term on the right-hand side
of Eq. 19 shows that the strain is mainly radial when the gel is around
the bacterium and becomes axial when it moves to the tail.
Force balance on the bacterium
We now must calculate the force exerted by the external gel on the
bacterium. This is given by the integration of the stresses tangential
and normal to the surface (Fig. 5 b):
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(20)
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Let us first consider the normal term:
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(21)
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By symmetry of revolution, this expression vanishes when it is
integrated over the cylindrical part. However, there is a nonvanishing
contribution to the integral from the part of the back hemisphere where
the gel is still in contact with the bacterium. The integration is
carried out in Appendix III; the result is
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(22)
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For small deformations (v 
v0) and
small, we have, from Eq. 16, = 2, so we find a scaling
law for the force:
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(23)
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This equation implies that the speed of the bacterium is no longer
determined by the polymerization rate as in the 1D model. We call this
force the "soap effect," because it recalls the rapid motion of a
wet bar of soap slipping away as it is slowly squeezed by hand. If such
a force seems unlikely at first sight, it is supported by the
impressive motion of the mutant presented in the last part of this paper.
Let us now consider the tangential term of the stress. We show in
Appendix V that for slow motion of the gel relative to the bacterium
surface, the tangential force can be approximated by a friction
law:
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(24)
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The experimental observation that the bacterium is linked to the
tail (Gerbal et al., 2000) shows that the friction
coefficient is larger than a minimum value discussed in the following.
We assume that each actin filament is coupled to the bacterium surface before it is released and moves to the tail with the gel. On a sufficiently long time scale during which many filaments attach to and
detach from the bacterium surface, these transient links result in a
friction force. Our experiments showed that the forces developed by an
optical tweezer or by an electric field on the bacterium-tail
connection were not sufficient to detach the bacterium from its tail.
Thus, we were not able to measure 
experimentally. However, it is
possible to put a lower bound on its value: assuming v v0 0.2 µm · s1, the
force exerted by an electric field on the bacterium in our experiment
was ~1 pN applied over a typical time of 100 s. Observed by
video microscopy, no separation between the tail and the bacterium (to
accuracy 0.5 µm) was detected; therefore, > 104 Pa · m · s. We can also propose an
upper limit by requiring the consistency of our model: the bacterium
can move forward if the motile force exceeds the friction:
F2 = Fmot2 + Ffric >0, so that < SbY/v < 1 Pa · m · s.
The calculations presented below were performed for various values of
in this range.
Also, as for the one-dimensional model, we suppose that the
polymerization speed depends on the stress normal to the surface, so we
have
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(25)
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Using the expression for the stress given by Eq. 9 and taking
roughly the average value rm/2 for the height of
the gel over the bacterium surface, we have
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(26)
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with 0 kBT/Yad2 1.
The consistency of our model requires that we take into account the
dependence of the polymerization speed on the stress: in the
computation shown below, the axial strain remains lower than 0.1 in the
tail (it is at maximum for = 1 Pa · m · s), so
that the deformations are sufficiently small to be correctly treated by
linear elasticity theory. This is not the case if the actin
polymerization velocity is set constant in the computations leading to
strains on the order of 1.
Force-velocity curves
The force-velocity curve is obtained when we solve for the speed
in the force balance equation:
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(27)
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Details on the calculations are given in Appendix IV. Fig.
6 shows the numerical solutions for
various values of .
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FIGURE 6
Velocity (bottom) and gel thickness (top) of a bacterium pushed only by an external gel, as
functions of the external force. The curves are plotted for various
values of the friction parameter . Dashed line: force-velocity curve
computed from the 1D model.
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These calculations demonstrate that increasing the internal friction
() or increasing the external force opposed to the motion have
qualitatively the same effects: the bacterium slows down, the gel has
time to grow thicker, and a larger stress builds up, which increases
the driving force. The force-velocity curve of a bacterium driven by an
external gel only is therefore also very stable, in agreement with the
observed steady motion of Shigella. In the range of values
of shown here, the computed gel thickness is in good agreement with
the sizes observed in the electron micrographs provided by the
literature. The remarkable difference with the 1D model is that the
bacterium speed is no longer equal to the polymerization rate at zero
external load: here v can be either greater than or less
than vp0, depending on the value of .
This is a consequence of the geometric change that the gel undergoes when moving from the vicinity of the bacterium to the tail, and of the
nonlinearity introduced by the "soap effect." This raises the
question of what happens in the complete system of a bacterium pushed
by two gels that are cross-linked and leave the bacterium at the same speed.
The force-velocity curve for a bacterium pushed by both the external
and the internal gel (Fig.
7 a) is given by the solution of the equation
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(28)
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In the presence of the two gels, the elastic force balances must
remain the same: F1 = 1St1 and
F2 = 2St2, but with the new condition
F1 + F2 = Fext. This implies the presence of shear strain in the
tail: rz 
0. Furthermore, a more realistic
description requires the continuity of the radial stress
rr at the interface between the internal and external
gels. Taking these conditions into account would have complicated the
model without providing much insight into the mechanism, and so they
are neglected for the sake of simplicity. Of course, we require that
the two different parts of the gel have the same speed v.
The complete set of equations used for the numerical calculations is
presented in Appendix IV.
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FIGURE 7
(a) Force-velocity curve for the two-gel
model for various values of the friction parameter (solid
lines). The dashed line is the curve obtained for the 1D model.
(b) Force exerted by the various parts of the gel on the
bacterium versus the external force Fext. For a
small external force, they exert antagonistic stresses on the
bacterium (c) and cancel each other out.
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The results of the calculations are shown in Fig. 7, for various values
of . Our model provides semiquantitative predictions and the order
of magnitude of the forces involved in the motion. A more precise
quantitative calculation would require us to take into account
all of the terms of the stress tensors and the possibility of local
plastic deformations.
The dependence of the polymerization rate on the normal stress solves
the problem of the different velocity of the internal and external gels
when considered independently: the linkage of the gel in the tail
induces normal stresses on the bacterium surface that locally modulate
the rate of actin polymerization. Removing the condition of the
adaptive polymerization rate with the stress leads to huge nonrealistic
strains in our calculations. This shows that it is not possible to
change smoothly the configuration of a gel from a spherical to a linear
geometry. The problem is solved if the gel growth speed is not constant
along the surface. As a consequence, in the full model, the internal
gel forces the bacterium velocity to be close toalthough not equal
tothe polymerization rate, when the external force is null.
The curve given by the full model is qualitatively similar to the
one-dimensional curve, but it is even flatter: the velocity is even
more constant when the motion of the bacterium is regulated by the two
gels. As already predicted by the 1D model the amount of force required
to slow down significantly the bacterium
(YSb 1 nN) is much larger than the
typical force it encounters in vivo (10-50 pN). This result is
consistent with the observation of the very constant velocity of the
wild-type Listeria and the fact that we were not able to
induce any change in the velocity of the bacterium when exerting forces
with optical tweezers or applying an electric field in the medium (in
the range of 1-10 pN). The important new feature provided by the 3D
model is that very strong antagonistic forces are applied on the
bacterium by the different parts of the gel, as shown in Fig. 7,
b and c. These forces almost compensate for each
other, to provide the driving force opposing a weak external one. If
the bacterium needs more force to propel itself forward, there is
sufficient power in reserve: some parts of the gel become more
compressed and increase the driving force. The amount of power
available is not limiting; it is provided by the host cell through the
hydrolysis of ATP caused by the actin polymerization reaction. The
presence of these internal forces also shows that the gel holds the
bacterium very tightly and that together they form a very robust system.
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DYNAMIC DESCRIPTION OF THE LISTERIA MOTION |
The hopping Listeria
The antagonistic forces predicted by the 3D model are a key
concept in understanding the oscillatory motion of the
Listeria mutant ActA21-97 (Lasa et
al., 1997) described below.
To determine which subdomains of the ActA protein Listeria
give the ability to induce the polymerization of actin, Lasa et al.
performed genetic deletions of various parts of the acta
gene (Lasa et al., 1995,
1997). By doing so, they isolated various types of
Listeria mutants. Some cannot polymerize actin at all; others are still able to do so, but are stuck in an actin sheath and do
not produce a tail. Most amazing is the mutant
ActA21-97, which we named the "hopping
Listeria," because it seems to move by jumps in a very
discontinuous way, as shown in Fig. 8.
The tail shows periodic spots of dense actin of ~2 µm in length
(the size of a bacterium) spaced by distances varying from less than 1 to 4 µm. We have studied the motion of three of these hopping Listeria. The bacteria are stopped most of the time, and
they achieve the greatest part of their displacement by bursts of speed that can be up to four times faster than the wild type (up to 1 µm · s1). However, the average speed of the
mutant is only 0.1 µm · s1, about half the
wild-type speed under the same conditions. At the beginning of a cycle,
the bacterium is almost stopped. It is progressively surrounded by a
fluorescent halo, showing clearly an accumulation of actin, and it
seems to be stuck into a sheath. Meanwhile, the bacterium moves slowly
until it starts to emerge from the sheath, at which point it
accelerates very abruptly. The top speed is reached approximately when
the bacterium fully emerges from the sheath (because the raw data are
noisy, the top speed cannot be determined with an accuracy better than
a few microns). Then the speed decreases down to zero and the mutant starts another cycle.
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FIGURE 8
(a) Snapshots from a videotape of the mutant
ActA21-97 (courtesy of P. Cossart et al.),
seen at the same time by phase-contrast and fluorescent microscopy. The
numbers indicate the time in seconds. (b) Kinematics record
of the same mutant: Speed (µm · s1) and
curvilinear position (µm) as functions of time (s). The data shown
have been filtered to suppress the high-frequency noise due to the
uncertainty on the bacterium center in the video. (c) Speed
and measurement of the video gray level intensity along the tail from
the snapshots at time 108 s as a function of the position.
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Several groups observed that wild-type Listeria may
also have an oscillatory velocity when infecting some medium (Hela cell cytoplasm extract). Its tail also seems to be dashed. However, the
speed variations are much less dramatic than for the mutant.
Most of the biological systems showing a highly nonlinear time
dependence have found an explanation at the chemical level through
enzymatic reaction (Goldbeter, 1996). Here we propose that the mechanism of the oscillations is physical rather than chemical, that they are due to the breaking of some connections between
the bacterium and the gel as a consequence of the internal forces
exerted at the bacterium surface predicted by the 3D model. A
statistical model of the connection between the bacterium and the
filaments is presented in Appendix V. There we demonstrate that the
average connection time between the filaments and the surface depends
on the "natural" chemical kinetics and on the force exerted by the
tail. This force lowers the potential barrier that links the actin to
the surface. Above a threshold, these links may break, thus inducing a
dynamic instability.
The deletion of some peptides in the ActA protein may affect
the enzymatic reaction in many different ways. It could change the
energy potential and/or the kinetics of the links. One simple explanation is that if the turnover rate of the connections is slowed
down, the mutant remains stuck within the new layer of polymerized
actin. This would be in contrast with the wild type, which moves
without breaking the transient links because their connection-release
kinetics is faster. This means that the mutant has to store a larger
amount of elastic energy to break the sheath. Slowing down the kinetics
of the filament connections results, at a mesoscopic scale, in an
increase in the effective friction parameter (see Appendix V). The
force required to break a link should be on the order of
G/a = 10kBT/5 nm 10 pN, where a is the size of an actin monomer and
G is a typical free energy associated with a coupling
reaction (14kBT for actin
polymerization). In Fig. 9, the
amplitudes of the internal forces predicted by the 3D model are plotted
as a function of . One can see that for very low values of , the
internal tail is pulled by a force on the order of 200 pN
(YSb 1 nN). For large values of ,
forces that can reach 1 nN are exerted on the bacterium-external gel links. These forces are sufficient to break the linkage between the
bacterium to the internal or the external gel if the numbers of active
connections are 20 and 100 filaments, respectively.
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FIGURE 9
Diagram of the forces exerted by the internal gel
() and the absolute value of the friction exerted by the external
gel (- - -) on the bacterium as a function of the friction
parameter (log scale).
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The time-dependent equations
In this section the equations of the elastic model are modified
slightly to account for the dynamics of the system. Calling S(z, t) the cross-sectional area of the external gel at the
position z, the flux conservation equation is
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(29)
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This can be simplified by assuming that the gel has a constant
thickness 
(dimensionless variable) above the cylindrical part (the
gel cross section is S
= rb2(2 1)) before it
reaches the thickness above the back hemisphere (the surface area
of which is S
= rb2(2 1)) (Fig.
10). The flux conservation equations
are therefore
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(30)
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(31)
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where the polymerization speed vp2
still depends on the normal stress:
vp2() = vp0e2.
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FIGURE 10
(a) Diagram of notation for the
time-dependent flux conservation equation. Plot of the gel thickness
(over the cylindrical part) and (over the back hemisphere).
(b) Result of the numerical calculation, showing the
transitory regime of the bacterium motion. and (thin
lines) and the bacterium velocity (thick line) are
plotted versus time for the case = 0.1.
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Another simplification can be used if we assume that the gel speed does
not vary much when leaving the bacterium side:
v0 v. Notice that the steady-state
solution is then simply
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(32)
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We have not modified the flux equation for the internal gel; if it
is compressed or extended, the gel returns to equilibrium on a length
scale of a few mesh sizes. Thus, any hysteresis generated by a
nonuniform strain in the internal gel would occur on a time scale too
short to be relevant for the oscillations we want to describe.
The expressions of the forces pressing on the bacterium remain the same
as in the steady-state description (Eqs. 8, 22, and 24), and the force
balance is again
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(33)
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The time-dependent solutions were computed by simultaneously
integrating the equations, using a fourth-order Runge-Kutta algorithm
starting with the following initial condition: no gel is produced above
the bacterium, and the initial speed is equal to the polymerization
speed of the back gel, i.e., S(t = 0) = S(t = 0) = 0 and v(t = 0) = vp0. Fig. 10 b shows the result of
the computation when no ruptures have been introduced in the
simulation. After an oscillatory period corresponding to the time
required for the external gel to grow over the bacterium and to be
located above the back hemisphere, the solution reaches the steady
state already found in the time-independent equations. The type of
solution remains the same whatever the value of ; only the time
scale changes. Indeed, it is possible to rewrite the set of equations
as
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(34)
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(35)
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and it is easy to show that they are stable. As expected, they are
not sufficient to describe the "hopping Listeria"
because they do not take into account a possible breakage of the links. A complete simulation of the rupture must contain the equations of
Appendix V, particularly the Fokker-Planck equation 62, which controls
the rate of connections between the gel and the surface. We have only
used the quasistatic result giving the friction force as a function of
the speed shown in Fig. 16. Qualitatively, this curve remains valid as
long as no fast breakage occurs, i.e., before the friction law reaches
its maximum. Once the threshold is reached, the links break
catastrophically and a "stick-slip" transition occurs. This is
modeled in our program by introducing a threshold force
Fs2 above which the external gel is no
longer connected and slips along the bacterium. Thus, the program
contains the following conditions: if
|Ffric| > Fs2
then the friction is arbitrarily lowered: slip = stick/100. Proportionally to the number of filaments (or
the surface), a weaker force is sufficient to detach the internal gel
from the bacterium: if Fmot1 > Fs1 = Fs1/8, then the gel 1 ruptures and no
longer exerts a force: Fmot1 = 0.
The reconnection between the gels and the bacterium ( returns to its
initial value and Fmot1 equals its initial
expression) occurs when one of these forces falls below its respective threshold.
The same Runge-Kutta algorithm was used to solve the equations
completed by the breaking conditions. Fig.
11 shows the result in the case where
Fs1 = 0.1YSb
and = 0.5. These values are not fine-tuned because the above
parameters may be chosen in a full domain of values and still produce
qualitatively similar results. Comparison of Fig. 11 with the analysis
of the mutant kinematics in Fig. 7 shows that our model is indeed able
to reproduce the experimental data. Initially (Fig. 11), the same
transitory period occurs as in the wild type described by the stable
equations (Fig. 10 b). In the case of the mutant, it models
the slow phase of a cycle. We found that the first condition required
for the system to oscillate is that the initial friction () is high
enough with regard to the breakage thresholds. If this is the case, the
bacterium moves slowly enough for the gel to accumulate ( increases
steadily) and to strongly squeeze the bacterium on the back hemisphere. If the stress has time to reach the critical value of the yield force
before the end of the transitory period, the linkage to the external
gel breaks. A stick-slip transition occurs, the surface friction drops
suddenly (we move from the right side to the left side of the diagram
in Fig. 9), and the soap effect due to the accumulated gel pushes the
bacterium forward. A second requirement for the oscillations is that
the acceleration is high enoughthe bacterium has to go faster than
the polymerization rateto also break the internal gel connections.
Otherwise, another steady regime is reached in which the bacterium is
simply pushed by the internal gel at about the polymerization rate,
with the external gel loosely connected. But if the internal gel also
detaches, the regime is unstable and we get oscillations: the bacterium is no longer retained by the internal gel and is propelled by the soap
effect much faster than the polymerization rate. Consequently, not
enough actin is polymerized to supply the diminishing gel on the side,
so that its thickness ( and ) decreases. However, our
calculations show that before the gel completely decays the bacterium
has time to run in the fast regime over a distance comparable to
several times its size. Once the gel thickness has vanished, the
driving forces fall under their respective breaking thresholds and the
filaments of the new layer are therefore connected again to the
surface. The system has returned to its initial conditions and the
mutant starts a new cycle. It can be surprising that when all of the
links to both parts of the gel are broken, the bacterium does not free
itself from its tail; although a stick-slip transition occurs, the
external gel still surrounds the bacterium. The friction is reduced but
is still much larger (typically 103 Pa
· m · s) than the friction due to the outer medium
(<106 Pa · m · s). Eventually, separation
of the bacterium from its tail might occur. This is consistent with the
observation of many simply diffusing bacteria coexisting in the medium
with mutants pushed by a dashed tail.
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FIGURE 11
Result of the numerical calculation with a possible
rupture of the gels. Here = 0.5 and
Fs1 = 0.1YSb.
The velocity and the bacterium displacement are shown as a function of
the time (a) and displacement (b). The gel
thicknesses and around the bacterium are also plotted as
function of the bacterium posit |
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and
TOP
ABSTRACT
INTRODUCTION
