 |
INTRODUCTION |
Actin polymerization plays a crucial role in cell
motility. One of the most widely studied examples (Lackie, 1986
;
Stossel, 1993) is the crawling movement of eukaryotic cells by
protrusion of actin-rich lamellipodia in front of the cell that tract
the cell forward when the microfilament network reorganizes. The force induced by the growth of actin filaments is sufficient to stress and
deform cell membranes. A similar system of force generation is also
responsible for the movement of Listeria monocytogenes once
in the cytoplasm of infected cells. By virtue of producing an F-actin
filled-tail, Listeria constitute a simple model for studying
movement induced by actin polymerization. The tail is made of
microfilaments cross-linked together (Cossart, 1995) and oriented with
their plus end (favored for polymerization) toward the bacterium. The
protein responsible for F-actin nucleation has been identified (Kocks
et al., 1992; Domann et al., 1992) as a transmembrane protein of 69 kDa
called ActA. The mechanism of actin filament formation from
Listeria has been studied in cell-free extracts of
Xenopus eggs or human cell extracts, and shows that the
recruitment of eukaryotic proteins is necessary for the motility of
Listeria (Lasa and Cossart, 1996; Welch et al., 1997). Our
goal was to experimentally study the role of topology on actin
polymerization by conceiving a system that resembles that of
Listeria but permits testing various parameters such as geometry or density of actin nucleators. We prepared and purified a
recombinant ActA and grafted it covalently onto beads of different diameters. When added to cell-free extracts prepared from HeLa cells,
the beads acquired an F-actin gel structure. The thickness of this
actin gel around the beads was found to be dependent in a reproducible
manner upon the diameter of the bead. Indeed, the gel is built by
addition of G-actin at the surface of the bead, which necessarily
creates a stress in a spherical geometry. This stress is sufficient to
limit the growth of the actin gel. In a first approximation (i.e., if
we neglect treadmilling), one can state that the polymerization process
stops when the chemical energy gain in the polymerization
(E
) is balanced by the elastic energy cost
for adding a new monomer (Eel). If we designate 
the average distance between nucleating proteins (nucleators) on
the bead (the density of nucleators is then 1/2), 
µ
the chemical energy released in the polymerization process, ri the radius of the bead, then
E = 1/2 µ × 4
ri2. The work of the force for adding a
monomer is 
rra per unit area, where
rr is the radial component of the stress and
a is the size of a G-actin monomer, then the elastic energy
can be expressed as Eel = rra × 4ri2. From the
expression of rr derived in this paper (Eq. 17), we deduce Eel 
C(e/ri)2 a × 4ri2, where C is the elasticity
modulus of the actin gel and e is its thickness. Writing
Eel = E
gives e = ri
, which expresses that
e is an increasing function of the bead size, as
experimentally observed. This simple model applies for equilibrium
situations. In the experimental system we analyze here, the
polymerization process is stationary but not at equilibrium, and the
model presented takes into account the simultaneous
polymerization/depolymerization process (or treadmilling) as well as
the stress created by the actin gel on the spherical bead.
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MATERIALS AND METHODS |
Construction, expression, and purification of the GST-ActA-His
variant
DNA manipulations were performed by routine procedures (Sambrook
et al., 1989). In a first step, the sequence
5'-GGATCCGGTCTAGAGAAGCTTCCCGAATTC-3' (encoding
XbaI and HindIII within BamHI and
EcoRI) was inserted in the pGEX2T expression vector
(Amersham Pharmacia Biotech, Uppsala, Sweden) yielding pGEX2T-adaptor.
In a second step, an oligonucleotide 5'-CTAGACCCGGGCCCATCACCATCACCATCACTG-3' (encoding six histidines and
comprising SmaI and EcoRI sites at the 5' and 3'
ends, respectively) was inserted in the pGEX2T-adaptor yielding
pGEX2T-his. The third step consisted of introducing a DNA fragment
encoding the ActA protein truncated of its transmembrane anchor and the
signal peptide. The pCB6-actA1 (Friederich et al., 1995) expression
vector was linearized by EcoRI and filled in using the
Klenow fragment of DNA polymerase I. After heat-inactivation of the
enzyme, the DNA was digested with XbaI and an ~1.75-kb
XbaI-EcoRI blunt-ended actA gene fragment was
isolated. Finally, the actA fragment was ligated into
XbaI-SmaI-digested pGEXT2T-his. The final
pGEX2T-actA-his construct encodes GST fused to ActA comprising a
six-histidine tag in the C-terminal region (see Fig. 1 A).
GST-ActA-His was produced in Escherichia coli strain BL21
(DE3) and purified successively on a nickel agarose matrix (purchased
from Qiagen GmbH, Hilden, Germany) and on a Sepharose glutathione
matrix (purchased from Amersham Pharmacia Biotech, Uppsala, Sweden).
The elution solution was dialyzed in buffer D (0.2 M boric acid, pH
8.5) and stored in aliquots at 
80°C. Protein concentration was
determined as described by Bradford (1976), (reagents purchased from
Bio-Rad, Hercules, CA).
Fluorescent actin preparation
Rabbit skeletal muscle actin was prepared according to the
method described by Spudich and Watt (1971). Actin was labeled with
rhodamine using the procedure of Kreis et al. (1982). Aliquots were
stored at 80°C at a concentration of 2 mg/ml (40 µM). Phalloidin was purchased from Sigma-Aldrich (St. Quentin Fallavier, France) and
directly added to the samples to a final concentration of 0.3 µg/ml.
Latex beads
Latex beads were purchased from Polyscience, Inc. (Warrington,
PA): we used carboxylate functionalized latex beads for covalent grafting (25 µEq/g of carboxylate groups). Diameters of particles were chosen in the 1-to-10 µm range. Proteins were covalently grafted
via EDAC (1-ethyl-3-(-3-dimethylaminopropyl)carbodiimide)) as described
by the manufacturer. ActA-grafted beads were stored at 4°C in a
storage buffer (20 mM phosphate buffer, pH = 7.4, 1% BSA, 150 mM
NaCl, 20 mM NaN2, 0.5% glycerol). We made two series of
ActA-grafted beads. The first series (we will call this series "ActA
saturated beads") with beads of various diameters (1 µm, 2 µm, 10 µm) was prepared by incubating the beads in excess of ActA to
saturate the protein surface concentration. The amount of protein
coupled to the beads was determined by subtracting the quantity of
protein remaining in the supernatant after incubation from the initial
amount of protein in the solution. The total surface of the beads was
deduced from their volume and size. The surface concentration of
grafted proteins was then given by the ratio between the total amount
of coupled proteins versus the total surface of the beads. As an
example, the concentration of ActA on 10-µm-diameter beads was
estimated at (5.6 ± 0.6) × 10
2
protein/nm2 (we reproducibly measured that ~14 ± 3 µg protein could be bound to 100 µl of a suspension of 2.5% of
10-µm-diameter latex beads), assuming a molecular mass of
GST-ActA-His of 92,890 Da. The second series of beads (we will call
this series "ActA concentration beads") was prepared to vary the
surface density of ActA protein. The beads were incubated in a mixture
of ActA and BSA at a ratio of 10, 30, 50, 70, and 90% of ActA. The
total amount of coupled proteins was determined by the same method as
the previous series, and the amount of ActA proteins coupled to the
beads was deduced, according to the manufacturer instructions, by
SDS-PAGE analysis of proteins remaining in solution (see Table 1). We
found that the surface density of ActA did not correspond to the
relative amount of BSA/ActA in the solution, due to the different
affinity of BSA and ActA to carboxylated functions.
Listeria bacteria
We used a modified L. monocytogenes strain described
by Lasa et al. (1995). This strain carries a deletion removing the actA gene and was transformed with a multicopy plasmid encoding ActA.
Cell line
The human HeLa S3 cell line was grown in Dulbecco's minimum
essential medium (DMEM) supplemented with 10% fetal calf serum, at
37°C, under 5% CO2.
HeLa cell-free extracts
Cytosolic extracts were prepared following a modification of the
procedure described by Paschal and Gerace (1995). About 109
cells were centrifuged at 300 × g for 10 min and
washed twice in PBS, resuspended in 5 ml buffer A (5 mM HEPES, pH = 7.4, 5 mM potassium acetate, 2 mM magnesium acetate, 1 mM EGTA, and a cocktail of protease inhibitors including Pefablock, leupeptin, pepstatin, and aprotinin at 1 µM each), and stirred slowly at 4°C
for 20 min on a rotary shaker. The solution was passed five times in a
cell cracker and centrifuged for 30 min at 40,000 × g,
4°C. The supernatant was clarified by centrifugation for 60 min at
100,000 × g. Finally, aliquots of cytosolic extracts
(12 mg/ml) were frozen in liquid nitrogen and stored at 80°C.
Gel electrophoresis and immunoblotting
Proteins were analyzed by SDS-PAGE. Immunoblotting was made by
use of the antibody anti-ActA2 against the N-terminus of ActA (Golsteyn
et al., 1997). Transfer to nitrocellulose and antibody incubation were
performed according to the method described by Burnette (1981).
Methods of observation
Beads were directly taken from the storage buffer.
Listeria were first suspended in half the volume of Xb
buffer (10 mM Hepes, pH = 7.7, 100 mM KCl, 1 mM MgCl2,
0.1 mM CaCl2, 50 mM sucrose) before adding to cell-free
extracts supplemented with 30 mM creatine phosphate, and 1 mM ATP as
described by Marchand et al. (1995). In each case the volume increase
did not exceed 15% of the initial volume of the extracts.
Fluorescence microscopy
Observations were made of beads or bacteria in extracts
containing a final concentration of 0.5 µM rhodamine actin. A 1-µl suspension of 2.5% beads in storage buffer (or from the resuspended Listeria in Xb buffer) was resuspended in 10 µl of
cell-free extracts supplemented with rhodamine actin, creatine
phosphate, and ATP. Five µl of the mixture was squashed between a
microscope slide and a 22-mm-square coverslip sealed with varnish.
Samples were observed by fluorescence microscopy with an inverted
microscope (IX70, Olympus Optical Co. Gmbh, Hamburg, Germany).
Electron microscopy
Samples for observation were prepared as described by Tilney and
Portnoy, 1989. A number n µl of 2.5% bead solution in
storage buffer (or 8 µl resuspended Listeria in 10 times
less volume of Xb) was added to 100 µl of cell-free extracts
supplemented with ATP and creatine phosphate. n was
calculated to have the same total bead surface for the different
samples: n = 4 or 8, respectively, for 1- and
2-µm-diameter beads; for 10-µm-diameter beads, we took 8 µl
from 5× reconcentrated beads in storage buffer. The mixture was
incubated 4 h at room temperature. The latex beads and/or the
bacteria were pelleted in a horizontal centrifuge (3000 RPM for 3 min),
and incubated for 40 min at 4°C with 1% glutaraldehyde, 0.5% of
tannic acid in phosphate buffer, 50 mM, pH = 6.3, rinsed twice in
phosphate buffer, incubated 20 min at 4°C in phosphate buffer
containing 0.5% osmium, rinsed three times with water, and stained
with an aqueous solution of 2% uranyl acetate for 1 h at 4°C.
The samples were then dehydrated in alcohol and embedded in epon. Gold
labeling was performed by incubating 4 µl beads in a solution (2.5%
BSA in PBS) containing 1/150 of the polyclonal rabbit antibody
anti-ActA2 against the N-terminus of ActA (Golsteyn et al., 1997).
Then, after washing, the beads were incubated with protein A coupled to
10 nm gold particles (PAG10) purchased from Dr. J. W. Slot,
Department of Cell Biology, Utrecht University, The Netherlands.
Ultrathin sections (thickness 70 ± 10 nm) stained with ethanolic
uranyl acetate and lead citrate were observed in a Philips CM 120 electron microscope at 80 kV.
| |
RESULTS |
Purification of GST-ActA-His
The protein ActA was first purified on a nickel agarose matrix and
further on a Sepharose glutathione matrix. The purity of the protein
was confirmed by SDS-PAGE under reducing conditions. Coomassie staining
of the gel (Fig. 1 B) revealed
one protein band migrating at a position corresponding to an apparent
molecular mass of 120 kDa that was higher than expected from the amino
acid sequence (see Fig. 1 A): 92 kDa. It is likely that this
apparent migration behavior is due to the high proline content of ActA (Kocks et al., 1992). Antibodies against ActA reacted with this band,
confirming that indeed this band corresponded to ActA.
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FIGURE 1
(A) Diagram describing GST-ActA-His, the
variant of ActA used for our experiments. Amino acid numbers are shown.
(B) SDS-PAGE analysis of the purified protein GST-ActA-His
(right column) and molecular weight marker proteins marker
(left column).
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Experimental assay: Listeria in cell-free extracts prepared from
HeLa cells
We set up an assay for studying Listeria in vitro in
cell-free extracts prepared from HeLa cells. Cell-free extracts were supplemented with ATP, creatine phosphate, and rhodamine actin as
described in Materials and Methods. During the incubation ~40% of
Listeria developed an F-actin tail, and the length of the
comet-like tail ranged from 15 to 30 µm. The velocity of
Listeria varied between 0.75 ± 0.5 µm/min within a
population of 30 moving Listeria. Some of the comets
displayed a periodic density (see Fig.
2). A similar phenomenon has been
described for an ActA truncated variant of Listeria (Lasa et
al., 1997). Immobile Listeria were surrounded with an
isotropic F-actin cloud and did not form any comet-like tail.
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FIGURE 2
Mixture of beads 2 µm in diameter (round
objects) coated with GST-ActA-His and Listeria
monocytogenes (elongated objects) a few hours after
incubation in HeLa cell-free extracts. Fluorescence microscopy.
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ActA saturated beads in cell-free extracts prepared from HeLa cells
GST-ActA-His was covalently grafted to polystyrene beads
functionalized with COOH (of 1 µm, 2 µm, and 10 µm in diameter). These beads were added to cell-free extracts supplemented with rhodamine actin as described in the experimental assay for
Listeria. We confirmed that the quantity of ActA detached
from the beads was negligible (<5%) by analyzing the extracts after
incubation with the 2-µm-diameter beads by immunoblotting with ActA
specific antibodies. Within 30 min the beads were surrounded by a
fluorescent staining (see beads of 2-µm-diameter on Fig. 2), whose
intensity increased with time, indicating the accumulation of actin
around the beads. The beads were easily detected by phalloidin staining (0.3 µg/ml final concentration), which revealed that this actin structure was composed of F-actin. Growth of the actin gel was stabilized after 4 h, as confirmed by video time-lapse microscopy observations. At this time the beads reduced their Brownian motion and
became stuck to the bottom slide.
To test whether the quantity of F-actin accumulated around beads and
around bacteria was the same, we mixed beads and Listeria in
one preparation (Fig. 2): the fluorescence intensity was indeed equivalent for beads and bacteria (within a relative error of 10%).
After 4-5 h the actin gel around the beads stopped growing, whereas
Listeria continued to develop an F-actin tail.
The three following tests confirm the specificity of ActA for F-actin
gel growth:
- The same type of carboxylate latex beads (2 µm in diameter)
grafted with BSA did not recruit rhodamine actin (the quantity of
grafted BSA was estimated at 2 × 101
molec./nm2) under the same conditions;
- uncoupled carboxylated beads did not nucleate an actin gel as well;
- we tested that actin nucleation was not a simple effect due to the
presence of lysine in ActA (although the global charge of ActA is
expected to be negative in physiological conditions, since the
calculated isoelectric point of GST-ActA-His is 5), as polylysine,
under certain experimental conditions, is reported to nucleate actin
polymerization (filaments are oriented with their pointed end toward
the nucleating surface) (Brown and Spudich, 1979): we repeated the
experimental assay in cell-free extracts with beads coated with
polylysine instead of ActA. We used three types of polylysine (ref.
P8920, P0899, P1149 purchased from Sigma). In none of the three
samples, under similar experimental conditions, did we observe any
fluorescence due to actin assembly (long filaments, ~50 µm,
appearing after 24 h are of a very different nature). We checked
that our polylysine was indeed functional: beads grafted with
polylysine placed in a buffer containing 0.5 M KCl, 0.5 mM MgCl2, 0.2 mg/ml G-actin did nucleate F-actin, as described
by Brown and Spudich (1979). Under these conditions, beads grafted with
ActA did not generate actin filaments when placed in buffer containing
0.5 M KCl, 0.5 mM MgCl2, 0.5 mM rhodamine actin,
supplemented with G-actin to a final concentration of 0.2 mg/ml actin.
This is consistent with the report that incubation of
Listeria with actin alone did not result in actin
association with bacteria (Welch et al., 1997).
We cut the GST part of GST-ActA-His with thrombin: we obtained
four segments including the GST segment, but three others as well
despite the fact that no other site of ActA was supposed to react with
thrombin. We did the same experiments on the ActA-grafted latex beads
with GST-ActA-His, and got the GST segment only. Fluorescence microscopy observations of these latter beads in the extracts (actin
marked with rhodamine) showed the same behavior as nontreated beads:
this shows that the presence of GST does not affect, in a significant
way, the polymerization process.
We examined the protein-grafted beads and the Listeria,
incubated in cell-free extracts for 4 h, by electron microscopy to observe their surrounding actin gel in detail. Two important
observations were made (see Fig. 3).
First, neither the 2-µm-diameter beads grafted with BSA nor the
uncoupled carboxylated beads produced an actin gel: this confirms the
observations made by fluorescence microscopy. Second, the thickness of
the actin gel produced by GST-ActA-His-grafted beads was found to vary
with the radius of the beads. Third, we examined actin structures and
confirmed that both the beads coated with ActA and Listeria
produced similar F-actin gels.
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FIGURE 3
Electron micrographs of latex beads (or
Listeria) incubated in HeLa cell-free extracts.
(a) Latex beads ( = 2 µm) coated with BSA;
(b) latex beads ( = 1 µm) coated with
GST-ActA-His; (c) latex beads ( = 2 µm) coated
with GST-ActA-His; (d) latex beads ( = 10 µm)
coated with GST-ActA-His; (e) bacterium with its actin comet
tail. One can observe that the beads are slightly deformed, probably
because of processing for EM analysis.
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The effect of ActA concentration on beads in cell-free extracts
The role of ActA density was studied by preparing beads that
contained different ratios of BSA and ActA. These beads were incubated
in supplemented cell-free extracts and observed both by fluorescence
microscopy and electron microscopy. The estimated densities of ActA are
given in Table 1. We confirmed that the density of ActA around one bead
was homogeneous by gold labeling (see Fig.
4) for the beads with the highest and the
lowest density. The amount of PAG10 around one bead was found to be
63.3 ± 30 PAG/bead on the 3.8 ± 0.6 × 1016 prot./m2 beads by counting 13 beads on
ultrathin sections that crossed the bead through its center.
Fluorescence microscopy revealed that the fluorescence intensity
decreased when the ActA density decreased.
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FIGURE 4
Immuno-gold (10 nm) labeling of beads grafted with
3.8 ± 0.6 × 1016 ActA/m2. Beads are
deformed due to EM processing.
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The thickness e of the actin gel layer around both ActA
saturated beads and ActA concentration beads is given in Table 1; e was measured on images taken at the same magnification
(×28,000). We examined ~100 beads and performed a statistical
analysis on a random population of 15 beads, making 20 measurements per
bead. If the ultrathin section did not cross the bead right through its
center, a gray ring of width d (projection of the bead edge) was visible around the latex beads on the images. The radial thickness e of the actin gel was deduced from:
where 
is the F-actin thickness measured on the image, and
= 70 ± 10 nm the thickness of the section (see Materials
and Methods). The correcting factor 1/[
] does not qualitatively change the
experimental results, but reduces the dispersion. The variation of the
bead size for the 1- and 2-µm beads was within 4% (see Table 1).
Although the size range of the 10-µm calibrated beads was within
10%, it happened that we observed beads as large as 20 µm, and we
took advantage of this variation to make measurements of actin gel size
around one of these larger beads. We estimated its radius R
from the electron microscopy image by using geometrical arguments,
which give:
where r is the radius of the bead section on the image,
and d is defined above.
The experimental values in Table 1 show
that 1) the consequence of a decrease in ActA surface density is that
no actin gel is formed, below a density of the order of 3 × 1016 prot./m2; and 2) the thickness of the
actin gel around ActA saturated beads is an increasing function of the
radius of the bead.
| |
DISCUSSION |
Let us now try to understand quantitatively why, in a spherical
geometry, the polymerization stops when a given thickness of actin gel
is reached. In this context, the word "gel" means that actin
filaments are cross-linked in a network that resists both static and
shear compression. Any addition of G-actin material at the particle/gel
interface requires the buildup of a stress able to push away the
already formed gel. Within the time scales over which the
polymerization process takes place the gel is clearly in mechanical
equilibrium. The following model simultaneously takes into account that
actin is polymerizing at the surface of the sphere (i),
depolymerizing at the outer end (e), and that the gel is
constrained. Our notations are summarized in Fig.
5.
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FIGURE 5
Notations used in the text. ri
is the radius of the bead, r and  the spherical
coordinates, e is the thickness of the gel layer.
re = ri + e.
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Let dni/dt and
dne/dt be, respectively, the number
of monomers added to the gel per unit time at the surface of the bead
(i) and at the outer surface of the gel (e);
according to van't Hoff (1884) one can write:
|
(1a)
|
|
(1b)
|
where ci and ce are,
respectively, the concentration of free G-actin monomers at the surface
of the bead and at the outer surface. 
1b and
1p are the polymerization rates for barbed
(b) ends and pointed (p) ends, and
2b and 2p are the probability for
a monomer to leave the filament at the barbed (b) and
pointed (p) end. This notation (b and
p) implies that the actin filaments are oriented with their
barbed end toward the beads surface, and their pointed end toward the
outer part of the gel. This assumption is in agreement with electron
microscopy observations on Listeria comets (Tilney et al.,
1992) but we have no direct experimental evidence of actin filament
polarity on beads. However, as an indirect proof, the model based on
this assumption accounts for experimental results.
Considering that polymerization of actin filaments occurs on the
surface of the bead, the inner part of the gel is stressed, which means
that the coefficients 1b and 2b
of dni/dt depend on the stress
= rr(ri). First we
calculate the stress created by the actin gel on a bead. Second, we
theoretically describe the general situation that includes diffusion of
G-actin monomers and treadmilling of actin filaments. We discuss the
different possible regimes and show that in our experimental
conditions, treadmilling is essentially negligible. We show that, in
this limit, the thickness of the grown gel depends on the radius of the
bead, as measured experimentally. We end up in discussing the possible
conditions under which beads can nucleate a comet tail made of actin filaments.
Expression of the stress (r, t) as a
function of ri and the Young's modulus of the
gel
The radial component of the stress
rr(r, t) and the tangential component
(r, t) of the stress
(r, t), must obey the equilibrium equation (in spherical
coordinates).
|
(2)
|
A spherical layer of area 4
ri2 and
volume 4ri2dri,
initially polymerized (synthesized) and cross-linked at the particle
surface at time t', is converted after a time (t t') to a spherical layer of area
4r(t)2 and volume 4r(t)2
dr(t). The tangential component of the stress can be simply
evaluated as (Landau and Lifchitz, 1967):
|
(3)
|
where C is the elasticity modulus of the gel.
The validity of Eq. 3 requires that the gel deformation is small
enough, that it can be considered in the linear elasticity regime.
Considering that the observed thicknesses are of the order of a few
hundred nanometers for several micron diameter spheres, this is a
reasonable approximation.
Let us define the outer radius of the gel by
re(t). At time t = 0, re(t = 0) = ri.
Furthermore, at any given time, the absence of external stress on the
gel surface is expressed by:
|
(4)
|
Making use of Eqs. 2-4, one can calculate the radial component of
the stress as a function of radius vector r:
|
(5)
|
As a result, the gel exerts a stress
rr(r = ri, t) = (t) on the bead surface, given by:
|
(6)
|
This stress in turn controls the polymerization rate at the bead surface.
Steady-state treadmilling regime
In the stationary regime, the gel thickness e = re ri is independent of
time; polymerization at the inner surface exactly balances
depolymerization at the outer surface and a monomer diffusive flux
transports the monomers from the outer surface to the inner one. This
implies:
|
(9)
|
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(10)
|
and
|
(11)
|
in which JC(r) is the flux
(algebraic value) of monomeric actin (G-actin), and the average
distance between ActA molecules.
As usual, the diffusion flux can be expressed in terms of the gradient
of the monomeric concentration C(r), and the monomer diffusion coefficient D:
|
(12)
|
If we can take D as a constant, then Eqs. 10 and 12
give:
|
(13)
|
where ce and ci
stand for the concentration of monomers outside the gel and at the bead
surface, respectively.
Combining Eqs. 9, 11, and 13, we obtain:
|
(14)
|
That is, with Eqs. 1 and after elimination of
ci:
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(15)
|
Equation 15 determines the gel thickness e as a
function of the polymerization rates, their stress dependence, the
diffusion coefficient D, the ActA density
2, the external monomeric concentration
ce, and the particle radius
ri. This is a treadmilling regime in which the
polymerization rate is governed by the stress buildup and monomer
diffusion, rather than by an adjustment of the monomer concentration,
as would be the case in solution (Carlier et al., 1997).
Gel thickness at steady state
The rates 1b() and
2b() can be related to the stress-free rates
1b(0) and 2b(0) by a simple use
of Kramers or Eyring rate theories (Eyring, 1935; Kramers, 1940) in
which the potential barriers to be overcome for either adding or
subtracting a monomer are shifted by the mechanical work against
addition or for subtraction of the monomer at the barrier maximum. As
usual, k is the Boltzmann constant and T the
temperature (S.I. unit).
Hence:
|
(16a)
|
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(16b)
|
The force acting on a single filament is 2, and
a1, a2 are the distances over which
the force produces work to reach the maximum of the potential barrier.
In a simple picture, a1 + a2 a, where a is the size of a G-actin monomer.
Equation 6 expressing the stress can be simplified when
e 
ri:
|
(17)
|
If we further remark that under most practical circumstances
1pce
2p, we can rewrite Eq. 15 in the form:
|
(18)
|
where two important lengths clearly emerge: the bead radius
ri and the diffusion length e* = D2ce/2p. They
correspond to two different possibilities of reaching steady state:
either the stress buildup is so large that the polymerization rate
essentially drops to zero, or the diffusion becomes so slow that the
monomer concentration at the bead surface becomes small enough that it
is balanced by the depolymerization at the outer surface.
Diffusion-limited regime
Let us first consider ri 
e* (i.e., essentially flat surfaces, no stress can build
up); knowing from Eq. 18 that e < e*, Eq. 18
simplifies to:
|
(19)
|
or
|
(20)
|
If we further remark that under usual circumstances the
stress-free initial polymerization rate ce
1b(0) is much larger than both the depolymerization
rate at the pointed end 2p and the stress-free
depolymerization rate at the barbed end 2b(0), then
|
(21)
|
Note that at steady state in this regime, the polymerization rate
is ci1b(0) 2b = 2p
(ci ce). The
concentration at the bead surface reaches the steady-state treadmilling
concentration obtained in solution (Carlier et al., 1997).
Stress-limiting regime
Let us now consider the opposite limit e*
ri; anticipating that ri 
e, Eq. 18 reads:
|
(22)
|
Clearly, in this regime the gel thickness is governed by the bead
radius thickness. Whenever ATP hydrolysis is not directly involved in
the polymerization process [note that ATP hydrolysis occurs later,
once the polymerization has taken place and detailed balance should
hold], one can write:
|
(23a)
|
|
(23b)
|
where v is the reaction volume, µ1 the
chemical potential difference per monomer, between the unpolymerized
state and the polymerized state excluding the translational entropy
kT × ln(cev), and
µ the chemical potential difference per monomer including the
translational entropy. µ represents the chemical energy released in the polymerization process. Using Eq. 16 and Eq. 23b we get:
|
(24)
|
Dividing Eq. 22 by 2b, one can extract:
|
(25)
|
In principle, Eq. 25 is only an implicit equation for , and
hence for e/ri. However, the ratio
2p/2b comes only in a logarithm,
and it only appears as a corrective term. If we ignore the logarithm,
Eq. 25 expresses the fact that the polymerization stops in this regime,
when the mechanical work required to add a new monomer equals the
chemical energy gained in the process. The depolymerization at the
pointed end appears as a correction to this basic feature (unless the
depolymerization rate under stress is unexpectedly small).
Transforming Eq. 25 into an equation for the gel thickness by using Eq. 17, we get:
|
(26)
|
in which we have written 
= µ kT × ln(1 + 2p/2b).
The gel thickness is proportional to the bead radius, which simply
expresses that there is one stress value for which steady state is reached.
General case and orders of magnitude
Equation 18 may be easily solved, for instance graphically as
shown in Fig. 6.
The general solution gives values intermediate between e*
and e**. More important are the estimates of e*
and e**. In our experiments, in which the distance between
ActA molecules on the surface is 42-77 Å (see Materials and Methods;
Table 1), we take a mesh size which is the smallest possible length
imposed by the actin filament diameter: = 10 nm; the
concentration in free G-actin in the HeLa cell extracts is expected to
be of the order of 0.5 µM, as suggested by the critical
concentrations of actin filaments dynamics (Carlier, 1991), since we
are in a situation where the pointed ends depolymerize. This gives a
ce value of 3 × 1014
molec./cm3; 2p can be estimated (Theriot
et al., 1992) from our observations on Listeria as the ratio
between the velocity of the bacteria (0.75 µm/min) and the length of
the comet (20 µm): we get 2p 
6 × 104 s1. The value of the monomeric
diffusion coefficient of G-actin has been measured in a buffer (buffer
A: 2 mM Tris (pH = 8), 0.2 mM CaCl2, 1.0 mM ATP, 0.5 mM dithiothreitol [DTT]): Db = 5 × 107 cm2/s (Lanni et al., 1981). Knowing that
the viscosity of cell-free extracts is about three times that of water
(or buffer) (Fushimi and Verkman, 1991), we infer a value D 1.6 × 107 cm2/s in our
experiments. Hence we expect e* to be of the order of 1 mm.
This estimate represents an upper limit, since steric hindrance and
temporary interactions of monomers with the gel proteins could slow
down the diffusion process. In any case this length is large compared
to the experimentally found thicknesses, and one expects the experiment
to correspond to the stress-governed regime.
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|
FIGURE 6
Graphic solutions for Eq. 18 are obtained by rewriting
Eq. 18 using Eqs. 16a and b, taking for the sake of argument
a1 = a2 = a/2, which leads
to:
with
the left and right part of the above equation are, respectively,
represented bold and dashed; the numerical values needed are the ones
given in the text:
as estimated from the slope S of the experimental
curve giving the thickness e versus time: S = 200 (nm)/20 × 60 (s) (V. Noireaux, manuscript in preparation)
by writing ce1b(0) = S/a, then 2b(0) = 2.7 × 108 s1 as deduced from Eq. 24.
(a) Taking e*/ri = 102, e = e* is the solution, as described
in the text; (b) taking e*/ri = 1 gives the solution e/e* = 101; (c) taking
e*/ri = 102, the solution is
e/e* = 0.001, and consequently e e** and
e/ri = 101 as measured
experimentally.
|
|
The radius dependence of the gel thickness observed experimentally
(Table 1) confirms these expectations. Equation 26 gives us a
prescription for estimating the proportionality ratio expected between
e and ri. If we take the gel elastic
modulus C (K/c4) = (kTlp/c4) in which
K = kTlp is the bending elastic modulus of
actin filaments, lp their persistence length,
and c the average distance between cross-links, we get:
|
(27)
|
With µ ~ 14 kT (Gordon et al., 1976),
c 106 cm,
lp 15 µm (Yanagida et al., 1984; Ott et
al., 1993; Gittes et al., 1993; Drögemeier and Eimer, 1994;
Isambert et al., 1995), ap 5 × 107 cm, we obtain (e/ri) 101, which is typically what we observe experimentally.
We can thus conclude that the polymerization process in our experiment
is indeed stopped by the mechanical stress buildup. Note that the above-discussed numbers imply an elastic modulus C (a few
106 Pa, given that 106 cm) large
compared to values measured with actin gels. If we take as an upper
limit of C the largest value measured in the Listeria comet (F. Gerbal et al., submitted for
publication), i.e., C 104 Pa, we are
led with a length 3 × 106 cm, which
implies that not all ActA are functional at the surface. We then find
(e/ri) 0.3, which is still compatible
with our experiments.
Spherical symmetry versus "comet"
The above-developed arguments show that if the polymerization
process takes place on a spherically symmetric substrate, the growth
stops automatically at a given thickness. This will always be the case
unless a symmetry-breaking transition takes place. A very rough
estimate of this symmetry-breaking possibility goes as follows: at the
outer surface, although the normal stress vanishes, the tangential
stress (given by Eq. 3) is at its maximum: the gel is under tangential
tension. In general, beyond a given threshold, solid materials under
tension break. The threshold value depends on material properties, but
most of the time it can be expressed as a deformation threshold (i.e.,
a strain threshold), which turns out to be of order one. In other
words, when (re ri)/ri = (e/ri) 1, the gel is very likely to develop a
fracture. This fracture releases a significant amount of the tensile
stress at the interior of the gel layer, and consequently also a
sizable amount of the normal stress at the inner surface. The
polymerization process can then go on, and one can understand that a
comet can result from this initial fracture. For this to occur, one
wants (from Eq. 27):
|
(28)
|
Since µ is essentially of order 10 kT, and
lp, a are not subject to large
changes, one needs a ratio (c2/) as large as
possible to have chances of observing symmetry breaking according to
this mechanism. It is striking to remark that native
Listeria develop comets containing ~103
filaments per cross section of the comet, which corresponds to an
average distance between ActA of ~100 nm: if we assume
c , condition 28 is then essentially
fulfilled. Note, however, that in our experiments c and
are clearly different, since varying moderately can result in
the absence of the gel.
In this argument, the size of the particle does not play a role. This
is correct as long as we ignore fluctuations. The relevant fluctuations
are ActA surface density fluctuations that are frozen during the
grafting process. As a rough rule of thumb they are of the order of
, where N is the total number of ActA
on the bead. Typically one-half of the bead will have an ActA excess of
over the other half: this considerably decreases
the instability threshold. The exact conditions under which a comet
could develop go beyond the scope of this work. We can easily
understand that the smallest bead radius compatible with a gel
formation will be the best, since this corresponds to the largest
relative unbalance. This is confirmed by recent observations made by
the group of J. A. Theriot (Cameron et al., 1999), who propose an
alternative mechanism involving the stochasticity of the polymerization
process of actin filaments (van Oudenaarden and Theriot, 1999).
| |
CONCLUSIONS |
This work confirms the crucial role played by ActA in actin
polymerization and demonstrates the interest of studying this process
in a spherical topology. If there were no cross-linking of the actin
filaments one would observe the growth of a polymerized layer bound
only by the diffusion length e* (see Discussion). Our
results show unambiguously that the factor limiting the thickness of
the actin gel is the mechanical stress exerted by the gel on the bead
surface. The fact that an external force could modify the
polymerization process of microtubules or actin filaments has been
previously analyzed theoretically (Hill and Kirschner, 1982). In our
work, the force per filament necessary to block polymerization is found
to be of the order of 10 pN, which is quite reasonable (i.e., 10 kT per monomer size). It does not provide either strong
support or strong opposition to any molecular theory (Mogilner and
Oster, 1996). It is interesting to realize that the pressure exerted by
the gel on the bead is of the order of one atmosphere. Scaling this
pressure with ActA density allows us to estimate the maximum force a
native Listeria is able to develop: we find a force of the
order of a few nanonewtons, much larger than adverse forces a cell
could oppose [note, however, that buckling of the Listeria
comet would drastically decrease this force (Gerbal et al., 1999)].
Our observations are very close to the one made by M. Dogterom on
microtubules polymerization (Dogterom and Yurke, 1997): the orders of
magnitude are fairly similar. In this last experiment, measurements are
made on single microtubules and the force is due to the existence of an
external obstacle. In our case the force results from the
self-developed stress bound to the spherical topology. Note that a
native Listeria has globally the same topology, so that our
work demonstrates the importance of mechanical stresses in
Listeria as well. Finally, it is interesting to remark that
the incidence of spherical topology on polymeric growth properties has
been pointed out in other contexts; for instance, the problem of
"starburst polymers" (de Gennes and Hervet, 1983), where the
coordination number of the reacting entity should change at a certain
radius because of steric hindrance. In this case, the polymerization
that takes place at the outer edge of the star is not stopped, but the
number of bonds allowed in the reaction decreases.
Address reprint requests to Dr. Cecile Sykes, Laboratoire
Physico-Chimie "Curie," UMR 168, Institute Curie, 11 rue Pierre et
Marie Curie, 75231 Paris Cedex 5, France. Tel.: 33-1-423-46790; Fax:
33-1-405-10636; E-mail: cecile.sykes{at}curie.fr.
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ABSTRACT
INTRODUCTION
