Stochastic severing of actin filaments by actin depolymerizing factor/cofilin controls the emergence of a steady dynamical regime.
ABSTRACT Actin dynamics (i.e., polymerization/depolymerization) powers a large number of cellular processes. However, a great deal remains to be learned to explain the rapid actin filament turnover observed in vivo. Here, we developed a minimal kinetic model that describes key details of actin filament dynamics in the presence of actin depolymerizing factor (ADF)/cofilin. We limited the molecular mechanism to 1), the spontaneous growth of filaments by polymerization of actin monomers, 2), the ageing of actin subunits in filaments, 3), the cooperative binding of ADF/cofilin to actin filament subunits, and 4), filament severing by ADF/cofilin. First, from numerical simulations and mathematical analysis, we found that the average filament length, L, is controlled by the concentration of actin monomers (power law: 5/6) and ADF/cofilin (power law: -2/3). We also showed that the average subunit residence time inside the filament, T, depends on the actin monomer (power law: -1/6) and ADF/cofilin (power law: -2/3) concentrations. In addition, filament length fluctuations are approximately 20% of the average filament length. Moreover, ADF/cofilin fragmentation while modulating filament length keeps filaments in a high molar ratio of ATP- or ADP-P(i) versus ADP-bound subunits. This latter property has a protective effect against a too high severing activity of ADF/cofilin. We propose that the activity of ADF/cofilin in vivo is under the control of an affinity gradient that builds up dynamically along growing actin filaments. Our analysis shows that ADF/cofilin regulation maintains actin filaments in a highly dynamical state compatible with the cytoskeleton dynamics observed in vivo.
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ABSTRACT: Tight coupling between biochemical and mechanical properties of the actin cytoskeleton drives a large range of cellular processes including polarity establishment, morphogenesis, and motility. This is possible because actin filaments are semi-flexible polymers that, in conjunction with the molecular motor myosin, can act as biological active springs or "dashpots" (in laymen's terms, shock absorbers or fluidizers) able to exert or resist against force in a cellular environment. To modulate their mechanical properties, actin filaments can organize into a variety of architectures generating a diversity of cellular organizations including branched or crosslinked networks in the lamellipodium, parallel bundles in filopodia, and antiparallel structures in contractile fibers. In this review we describe the feedback loop between biochemical and mechanical properties of actin organization at the molecular level in vitro, then we integrate this knowledge into our current understanding of cellular actin organization and its physiological roles.Physiological Reviews 01/2014; 94(1):235-63. · 30.17 Impact Factor
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ABSTRACT: Several cellular processes rely on the fine tuning of actin cytoskeleton. A central component in the regulation of this cellular machinery is the ADF-H domain proteins. Despite sharing the same domain, ADF-H domain proteins produce a diverse functional landscape in the regulation of the actin cytoskeleton. Recent findings emphasize that the functional and structural features of these proteins can differ not only between ADF-H families but even within the same family. The structural and evolutional background of this functional diversity is poorly understood. This review focuses on the specific functional characteristics of ADF-H domain proteins and how these features can be linked to structural differences in the ADF-H domain and also to different conformational transitions in actin. In the light of recent discoveries we pay special attention to the ADF/cofilin proteins to find tendencies along which the functional and structural diversification is governed through the evolution.European Journal of Cell Biology. 01/2014;
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ABSTRACT: Detailed modeling and simulation of biochemical systems is complicated by the problem of combinatorial complexity, an explosion in the number of species and reactions due to myriad protein-protein interactions and post-translational modifications. Rule-based modeling overcomes this problem by representing molecules as structured objects and encoding their interactions as pattern-based rules. This greatly simplifies the process of model specification, avoiding the tedious and error prone task of manually enumerating all species and reactions that can potentially exist in a system. From a simulation perspective, rule-based models can be expanded algorithmically into fully-enumerated reaction networks and simulated using a variety of network-based simulation methods, such as ordinary differential equations or Gillespie's algorithm, provided that the network is not exceedingly large. Alternatively, rule-based models can be simulated directly using particle-based kinetic Monte Carlo methods. This "network-free" approach produces exact stochastic trajectories with a computational cost that is independent of network size. However, memory and run time costs increase with the number of particles, limiting the size of system that can be feasibly simulated. Here, we present a hybrid particle/population simulation method that combines the best attributes of both the network-based and network-free approaches. The method takes as input a rule-based model and a user-specified subset of species to treat as population variables rather than as particles. The model is then transformed by a process of "partial network expansion" into a dynamically equivalent form that can be simulated using a population-adapted network-free simulator. The transformation method has been implemented within the open-source rule-based modeling platform BioNetGen, and resulting hybrid models can be simulated using the particle-based simulator NFsim. Performance tests show that significant memory savings can be achieved using the new approach and a monetary cost analysis provides a practical measure of its utility.PLoS Computational Biology 04/2014; 10(4):e1003544. · 4.87 Impact Factor
Stochastic Severing of Actin Filaments by Actin Depolymerizing
Factor/Cofilin Controls the Emergence of a Steady Dynamical Regime
Jeremy Roland,* Julien Berro,* Alphe ´e Michelot,yLaurent Blanchoin,yand Jean-Louis Martiel*
*Universite ´ Joseph Fourier, TIMC-IMAG Laboratory, Grenoble, France; CNRS UMR 5525, Grenoble, France; INSERM, IRF 130,
Grenoble, France; andyInstitut de Recherches en Technologie et Sciences pour le Vivant, Laboratoire de Physiologie Cellulaire
Ve ´ge ´tale, Commissariat a ` l’Energie Atomique, Centre National de la Recherche Scientifique, Institut National de la Recherche
Agronomique and Universite ´ Joseph Fourier, F38054 Grenoble, France
great deal remains to be learned to explain the rapid actin filament turnover observed in vivo. Here, we developed a minimal
kinetic model that describes key details of actin filament dynamics in the presence of actin depolymerizing factor (ADF)/cofilin.
We limited the molecular mechanism to 1), the spontaneous growth of filaments by polymerization of actin monomers, 2), the
ageing of actin subunits in filaments, 3), the cooperative binding of ADF/cofilin to actin filament subunits, and 4), filament
severing by ADF/cofilin. First, from numerical simulations and mathematical analysis, we found that the average filament length,
ÆLæ, is controlled by the concentration of actin monomers (power law: 5/6) and ADF/cofilin (power law: ?2/3). We also showed
that the average subunit residence time inside the filament, ÆTæ, depends on the actin monomer (power law: ?1/6) and ADF/
cofilin (power law: ?2/3) concentrations. In addition, filament length fluctuations are ;20% of the average filament length.
Moreover, ADF/cofilin fragmentation while modulating filament length keeps filaments in a high molar ratio of ATP- or ADP-Pi
versus ADP-bound subunits. This latter property has a protective effect against a too high severing activity of ADF/cofilin. We
propose that the activity of ADF/cofilin in vivo is under the control of an affinity gradient that builds up dynamically along growing
actin filaments. Our analysis shows that ADF/cofilin regulation maintains actin filaments in a highly dynamical state compatible
with the cytoskeleton dynamics observed in vivo.
Actin dynamics (i.e., polymerization/depolymerization) powers a large number of cellular processes. However, a
Actin filaments, a major component of the cytoskeleton,
grow by polymerization of actinmonomers and organize into
dendritic networks or bundles in cell compartments (lamel-
lipodia or filipodia) (1). A long-standing challenge in cell
biophysics is to understand the molecular mechanisms con-
trolling the assembly and disassembly of actin cytoskeleton,
a dynamical process that generates forces and ultimately cell
movement (2–4). Indeed, depending on how actin filaments
are initiated by a nucleation-promoting factor (i.e., Arp2/3
complex, spire, formins), actin filaments will elongate be-
tween 11.6 mM?1s?1and 38 mM?1s?1(5–7). In the mean-
time, to avoid a depletion of the cellular concentration of
actin monomers, actin filaments need to be rapidly recycled
(8). Biomimetic systems helped to identify the minimal set
of actin binding proteins that are essential to maintain this
high turnover rate and induce actin-based motility (9,10).
Among these proteins, actin depolymerizing factor (ADF)/
cofilin stimulates actin cytoskeleton dynamics by severing
actin filaments (11–13) and increasing filament turnover in
vitro (14) or in biomimetic systems (9,10). Recently ADF/
cofilin has been shown to control the filament length in
parallel with a reduction of the subunit residence time in
filaments (6). Because these new facts change our under-
standing of actin dynamics, we present a model for the po-
lymerization of actin filaments in the presence of ADF/
cofilin. We base our approach on accepted mechanisms for
the polymerization of actin monomers and the interactions
between ADF/cofilin and actin subunits in a filament.
First, we assumed that each polymerized ATP-actin sub-
unit hydrolyzes its ATP independently in a first-order reac-
tion that is not influenced by surrounding subunits (15).
Second, ADF/cofilin accelerates phosphate dissociation (16).
Third, ADF/cofilin exclusivelybinds to actin subunits loaded
with ADP (16). Fourth, ADF/cofilin binds cooperatively to
subunits in the filament (17). In addition, we assumed that
subunits only. A recent study questioned the acceleration of
depolymerizationat thepointed end,showing thatitisalmost
independent of the presence of ADF/cofilin (12). Therefore,
we assumed that the pointed- and barbed-end depolymer-
ization rates are unaffected by ADF/cofilin. Finally, we sim-
ulated the set of chemical reactions in the presence of a large
excess of actin monomers, an assumption relevant to the
conditions in cells and to the experimental data used to val-
idate our approach.
We combined a stochastic molecule-based model, in
which single actin monomers or subunits inside the filament
and ADF/cofilin are the modeling units, and a continuous
approach to analyze the statistical properties of the control
Submitted September 13, 2007, and accepted for publication November 19,
Jeremy Roland and Julien Berro contributed equally to this work.
Address reprint requests to Jean-Louis Martiel, TIMC-IMAG Laboratory,
Taillefer Building, Faculty of Medicine, F-38706 La Tronche, France. Tel.:
33-456-520-069; E-mail: firstname.lastname@example.org.
Editor: Alexander Mogilner.
? 2008 by the Biophysical Society
2082 Biophysical JournalVolume 94March 2008 2082–2094
exerted by ADF/cofilin on filament dynamics. The Monte
Carlo simulation of the stochastic model illustrates how
ADF/cofilin controls the emergence of a stable dynamical
regime for actin dynamics and stimulates actin subunit
filament population, we analytically determined the average
filament length and the residence time of subunits in fila-
ments, with respect to the rate constants for the reactions and
the concentrations of actin monomers and ADF/cofilin. Our
study offers a satisfactory and coherent understanding of the
experiments in biomimetic assays (6) and presents a useful
tool to analyze in vivo mechanisms for cytoskeleton dy-
namics, in particular its fast actin turnover.
Model and simulation methods
Kinetic model for filament polymerization and severing
We developeda kinetic modelto simulatethe dynamicsof polymerization of
ATP-actin monomers in the presence of ADF/cofilin. Since free actin
monomers and free ADF/cofilin are small molecules (respectively, 42 kD
and 15 kD) that diffuse rapidly, we assumed their spatial distribution is
homogeneous. In addition, we hypothesized that the compartment where
reactions take place exchanges molecules with a large reservoir so that
concentrations of actin monomers and free ADF/cofilin are constant. We
considered polymerization of actin at both filament ends (reaction rates vB
and vP, Fig. 1 A, Tables 1 and 2), ATP hydrolysis and inorganic phosphate
release (respectively, reaction rates r1and r2, Fig. 1 A, Tables 1 and 2). ATP
hydrolysis and phosphate release are assumed to be independent and affect
actin subunits randomly. ADF/cofilin binding to ADP-bound subunits in-
duces the acceleration of phosphate release from surrounding ADP-Pisub-
units and the cooperative binding of new ADF/cofilin molecules (16).
Recently, Prochniewicz et al. (18) established that the binding of a single
ADF/cofilin facilitates two distinct structural changes on actin filament that
may explain ADF/cofilin effects. First, we assumed that the binding of a
single ADF/cofilin to one ADP subunit accelerates the release of inorganic
phosphate and enhances the production of F-ADP for the whole filament
(modification of r2, Fig. 1 A). To justify this drastic hypothesis, we investi-
gated different models in which phosphate release acceleration is limited to
the R (R is an integer) subunits on both sides of a bound ADF/cofilin. Nu-
merical simulations proved that infinite cooperativity (i.e., a bound ADF/
cofilin affects the phosphate release of the whole filament) is an excellent
approximation of the filament dynamics. Second, we modeled ADF/cofilin
binding to actin subunits in filaments as a two-step process. Initially, a single
ADF/cofilin binds to a subunit bound to the nucleotide ADP (F-ADP) whose
two neighbors are free from ADF/cofilin (reaction rate r3, Fig. 1 B, Tables
1 and 2). Subsequently, the binding of a second ADF/cofilin to an F-ADP
subunit is facilitated by the neighboring decorated subunits (reaction rate r4,
Fig. 1 C, Tables 1 and 2).
We also assumed that filament severing occurs between two adjacent
F-ADP-ADF subunits (reaction rate r5, Fig. 1 D, Tables 1 and 2). The two
new pieces generated by severing have different fates (Fig. 1 D). Because of
the large amount of capping proteins in vivo (8), we assumed that the piece
associated with the new barbed end (i.e., fragment, Fig. 1 D) is immediately
capped and cannot elongate. Therefore, to simplify simulations and the
mathematical analysis in the Appendix, the piece associated with the old
barbed end, referred to as the ‘‘filament’’ (Fig. 1 D), remains under inves-
tigation. The other piece, associated with the old pointed end, referred to as
the ‘‘fragment’’ (Fig. 1 D), is discarded from simulations, except in Fig. 5 B.
Models for actin filament dynamics predicted the existence of a diffusive
length (;30–34 monomers2s?1) at the barbed end in conditions close to
chemical equilibrium (19,20). This result agrees with experimental work
(21,22) but represents only minor fluctuations of the filament length. Here,
although we used the same set of chemical reactions, we addressed the
We used the Gillespie algorithm to determine the evolution of the filament
and the chemical transformation of subunits (23,24). This molecule-based
approach provides precise information on the dynamics of actin filaments. In
particular, we could determine the spatial and temporal distribution of actin
subunits along the filament, the nature (i.e., ATP, ADP-Pi, or ADP) of the
variables (e.g., filament length or subunit residence time) were determined
from the sampling of time-dependent simulations (typically, simulations
during 10,000 s were sampled every 20 s). The analytical distribution of
filament length and subunit residence time in filaments is presented and
analyzed in the Appendix section.
ADF/cofilin induces large amplitude fluctuations
in growing actin filaments
Initially, we addressed the question of how actin filament
length reaches a steady dynamical regime by balancing as-
sembly and disassembly of subunits at both ends, indepen-
dent of the biochemical conditions in cells or in biomimetic
assays. We investigated the key issue of actin filament length
control by the severing activity of ADF/cofilin. First, we
assumed a control of the length of actin filaments based only
on an increase in the rate of depolymerization and in the
absence of ADF/cofilin-severing activity (ksevering¼ 0) (Fig.
2 A). Simulations show that a steady dynamical regime is
achieved for only a single value of the actin monomer con-
centration, somewhere between 0.8 and 0.9 mM (Fig. 2 A).
For different concentrations of actin monomer, actin fila-
ments will grow (above 0.9 mM) or shrink to zero length
(below 0.8 mM). Addition of the severing activity of ADF/
cofilin to this model substantially modified the behavior of
growth for ;150–300 s, actin filaments follow periods of
sustained polymerization and sudden shrinkage mediated by
ADF/cofilin severing (Fig. 2 B). Although the rate of po-
lymerization of actin subunits is constant, severing prevents
unrestricted filament growth and induces large-amplitude
fluctuations that follow a well-defined distribution (Fig. 2 C).
The succession of elongation and shortening periods for
actin filaments depends on the efficiency of the severing
behavior (see Fig. 2 B), the length distribution is bell-shaped,
with a marked peak sharper than in a Gaussian distribution
(Fig. 2 C and Eq. A9). Conversely, the average and standard
deviation of the filament length increase with the concen-
tration of actin monomers (Fig. 2 C, inset). Wederiveda very
simple relation between the average (respectively the standard
deviation) of the filament length and the rates of reactions for
ADF Drives Actin Filament Fluctuations2083
Biophysical Journal 94(6) 2082–2094
tin filament elongation and severing. (A) Actin filament
elongation by addition of actin monomers bound to
ATP at filament ends. vBand vPare, respectively, the
barbed- and pointed-end polymerization rates. We
assumed that random ATP hydrolysis (rate r1) is
followed by the release of the inorganic phosphate Pi
(rate r2). (B) ADF/cofilinbindsreversibly to an isolated
ADP-actin subunit in the filament. (C) Cooperative
binding of ADF/cofilin molecules facilitates the for-
mation of decorated subunit pairs along one of the two
actin filament strands. (D) Severing occurs between
two consecutive F-ADP-ADF subunits, creating a
piece containing the ‘‘old’’ pointed end and ‘‘new’’
barbed end (fragment) and a piece containing the
‘‘old’’ barbed end and a new pointed end (filament).
(E) Code for the different symbols in A–D. The
elongation and reactions rates (vB, vP, and r1-5) repre-
sent molecule fluxes and are basically obtained as the
product of an intrinsic reaction rate by a concentration
(Tables 1 and 2).
Molecule-based stochastic model for ac-
2084 Roland et al.
Biophysical Journal 94(6) 2082–2094
the polymerization and ADF/cofilin-dependent severing
(Eqs. A10, A12, and A13). Basically, the average filament
length, ÆLæ (respectively standard deviation,
pends almost linearly on the actin monomer concentration
(power law: 5/6) and is inversely proportional to the ADF/
cofilin concentration (power law: ?1/3):
This analytical result shows that the average filament length
and the size of the fluctuations, determined by the standard
deviation of the distribution, are reduced in the presence of a
high ADF/cofilin concentration, in agreement with the nu-
merical simulations presented in Fig. 2 C (and inset). This
reduction of fluctuations is also visible in Fig. 2 B, with a
marked correlation between the severing activity of ADF/
cofilin and the fluctuation amplitude. The actin fragment
average length (Fig. 2 D) decreases to a value below 0.5 mm
for ADF/cofilin concentration above 0.2 mM. This empha-
sizes that high ADF/cofilin concentrations will generate actin
filaments too small to be detected in light microscopy.
Inversely, for a constant concentration of ADF/cofilin (1 mM)
an increase in actin monomer concentration induces an al-
most linear increase in actin filament mean lengths (Fig.
2 D, inset, and Eq. A14).
We also analyzed models where R actin subunits (R is an
integer) on both sides of a bound ADF/cofilin have their rate
of phosphate release increased. In the case of finite cooper-
ativity, the average phosphate release rate for a filament
slows the transformation of F-ADP-Piinto F-ADP and the
subsequent binding to ADF/cofilin. Therefore, the average
filament length is increased in reference to the model with
infinite cooperativity (compare black and red curves in
Supplementary Material Fig. S1, inset). However, the devi-
ation from this last model, which is maximal for R in the
range10–90actinsubunits, becomespractically undetectable
for R larger than 125 subunits. Prochniewicz et al. (18)
measured that the increased torsional flexibility after the
binding of a single ADF/cofilin affects 427 6 355 subunits,
to which a parameter R in the range 40–390 corresponds in
our modeling approach. Hence, because these numbers are
highly variable and because model outputs are practically
indistinguishable for R $ 125 subunits, the infinite cooper-
ativity hypothesis is excellent; we used it throughout this
Chemical composition of the actin filament
ADF/cofilin controls the emergence of a steady dynamical
regime, with a well-defined average length and fluctuation
amplitude (Fig. 2 C). Since ADF/cofilin preferentially binds
to ADP-actin subunits, the severed fragments are principally
made ofsubunits bound toADP,whereas the remainingactin
filament is composed of younger subunits bound to ATP or
ADP-Pi. Consequently, in the steady regime, the molar
fraction of the different nucleotide on a filament is highly
dependent on the severing activity, as shown in Fig. 3 A.
Although ATP or ADP-Pirepresent only transient chemical
states for the nucleotide bound to subunits (half-time lives
are, respectively, 2 s and 6 min), their molar fraction in the
actin filament increases regularly with ADF/cofilin concen-
trations (Fig. 3 A). At ADF/cofilin concentrations above
0.1 mM, ATP/ADP-Pi-bound subunits represent .50% of
the total subunits in a filament (Fig. 3 A). Conversely, most
of ADP-bound subunits are removed from the filament by
severing, and their molar fraction drops to only 20% for
ADF/cofilin above 1 mM (Fig. 3 A). Therefore, ADF/cofilin
directly controls the age of thefilament by removing subunits
bound to ADP (Fig. 3 B).
Although the ADF/cofilin concentration used in the sim-
ulations lies in the range 0.001–10 mM, the number of dec-
orated pairs remains globally low and plateaus at ;25 pairs
per filament (Fig. 3 C), which represents only a small per-
centage of the total number of actin subunits. To measure the
apparent drop of binding efficacy of ADF/cofilin, we defined
an apparent ‘‘dissociation equilibrium constant’’ of cofilin
bound to actin filament by
TABLE 2 Reaction rates
Elongation rate at the
Elongation at the pointed end
ADF/cofilin to F-ADP
of ADF/cofilin to F-ADP
v ¼ vB1 vP
TABLE 1Chemical rates constant
Chemical rate Numerical valueReference
kPi-release(in the presence
0.8 s?1(GATP) 0.27 s?1(GADP)
GATP, ATP-loaded monomer.
GADP, ADP-loaded monomer.
ADF Drives Actin Filament Fluctuations2085
Biophysical Journal 94(6) 2082–2094
where Æ#FADPæðrespectivelyÆ#FADP-ADFæÞ is the aver-
age number of actin subunits bound to ADP in the filament
(respectively the average number of ADF/cofilin molecules
bound to the actin filament). Simulations demonstrated that
KD,Appincreases with the concentration of free ADF/cofilin
(Fig. 3 C, inset). This apparent dissociation equilibrium
constant remained low at ADF/cofilin concentrations below
depolymerization at the pointed end is multiplied by a factor of up to 25 and that ADF/cofilin has no severing activity (14). The actin monomer concentrations
used in the simulations are (from bottom to top) 0.63 (s), 0.71 (h), 0.81 (d), 0.92 (n), and 1.05 ( ) mM. (B) ADF/cofilin-mediated large amplitude
fluctuations. Increasing concentrations of ADF/cofilin reduce the average filament length and the amplitude of the fluctuations. We used a single actin
monomer concentration of 1 mM; no qualitative changes have been observed for different actin monomer concentrations. Black curves 0.1 mM ADF/cofilin,
dark gray 1 mM ADF/cofilin, and light gray 10 mM ADF/cofilin. (C) Increasing severing activity decreases both the average and the variance of the
distribution of filament length for different values of ADF/cofilin (light gray 10 mM; gray 1 mM, and black 0.1 mM) and [Actin] ¼ 1 mM. The model
prediction from the balance equation (Eq. A5, solid curve) matches the empirical distribution based on the molecule-based model (points). (Inset) Model
prediction (solid curve) and empirical (points) distributions for the filament length with different concentrations of actin monomers (light gray curve 0.5 mM,
gray curve 1 mM, and black curve 2 mM). The ADF/cofilin is fixed at 1 mM. (D) ADF/cofilin controls the average filament length (solid curve) and the
average fragment size (dashed curve); [Actin] ¼ 1 mM. Note that for ADF/cofilin above 0.2 mM, the fragment size is ,0.5 mM, the resolution limit in light
microscopy, as indicated by the shaded area. (Inset) The average filament length increases with the concentration of actin monomers. ADF/cofilin is held
constant at 1 mM. Parameters for A are vB¼ 11.6[Actin] s?1, vP¼ 1.3[Actin]-6.75 s?1, r1¼ 0.3 s?1, r2¼ 0.0019s?1, r3¼ 0 s?1, r4¼ 0 s?1, and r5¼ 0 s?1.
Parameters for B–D are vB¼ 11.6[Actin] s?1, vP¼ 1.3[Actin]-0.27 s?1, r1¼ 0.3 s?1, r2¼ 0.035 s?1, r3¼ (0.0085[ADF/cofilin]-0.005) s?1, r4¼
(0.075[ADF/cofilin]-0.005) s?1, and r5¼ 0.012 s?1.
Steady-state dynamics of actin filaments. (A) Filament length time course for different actin monomer concentrations. We assumed that
2086Roland et al.
Biophysical Journal 94(6) 2082–2094
0.1 mM. For ADF/cofilin concentration in the range 0.1–10
mM, the KD,Appincreased, implying a drop in the available
binding sites for ADF/cofilin on the filament.
To test the role of subunit ageing, we determined the aver-
age spatial distribution of actin subunits, given the state of the
associated nucleotide (ATP, ADP-Pi, or ADP, Figs. S2 and
S3). It turns out that a long simulation (10,000 s) is sufficient
for the spatial distribution of actin subunit to stabilize, except
large fluctuations at the pointed end (Fig. S2). Using the time-
dependent solution of the system of Eq. A2, which expresses
the time course of the chemical transformation of the nucleo-
tide, and from the conversion between time and space x ¼ vdt
filament t ago (d and n are, respectively, the size of an actin
subunit and the polymerization rate), we can match the time-
dependent curves (Fig. S3) with the spatial distribution ob-
tained from a long run of the stochastic model (Fig. S2). This
Piand ADP, and the subsequent binding of ADF/cofilin to
F-ADP subunits, provides the timer necessary to control the
To further investigate the effect of subunit ageing on the
binding of ADF/cofilin along growing actin filaments, we
spatial variation of the local dissociation constant of the actin
subunit-ADF/cofilin complex formation, denoted KD,Spatialas
a function of the position along the actin filament (Fig. 4).
(Note that KD,Appis the average of the dissociation constant
determined from the whole filament, discarding the infor-
mation coming from the spatial position of actin subunits
axes corresponds to the position of the growing barbed end
A). At this position, KD,Spatialis ;20 mM (Fig. 4 B) and the
molar fraction of bound ADF/cofilin is nearly 0 (Fig. 4 A).
KD,Spatialdecreases sharply to reach the value of the actual
ADF/cofilin concentration controls the molar fraction of the nucleotide bound
Chemical composition of actin filaments at steady state. (A)
to subunits in filament: F-ATP (solid thick curve), F-ADP-Pi(long dashed
curve), F-ADP (dot-dashed curve), and F-ADP-ADF/cofilin (dotted curve).
The removal of a large piece made of F-ADP subunits favors the molar ratio
ATP or ADP-Piversus ADP and ADP-ADF/cofilin-bound subunits. (B) The
age of the filament decreases with ADF/cofilin activity. The age of a particular
subunit in an actin filament at time t is the time spent by this subunit since its
polymerization in the filament before time t. The filament ageis determinedby
averaging the subunit ages in a filament. (C) The number of ADF/cofilin
decorated subunit pairs depends on the ADF/cofilin concentration. The
number of ADF/cofilin-decorated subunit pairs is a sigmoidal function of
the ADF/cofilin concentration, which plateaus at high ADF/cofilin level.
(Inset) Variation of the apparent dissociation equilibrium constant, KD,Appfor
the binding of ADF/cofilin to actin filaments. For concentrations of ADF/
cofilin below 0.01 mM, the KD,Appis low and almost constant. Because high
ADF/cofilin levels favor low F-ADP molar ratio in filaments, KD,Appincreases
linearly with ADF/cofilin concentration. At concentrations of ADF/cofilin
higher than 50 mM, KD,Appplateaus at a value of 20 mM (data not shown).
Parameters for simulations are listed in the legend of Fig. 2 B.
ADF Drives Actin Filament Fluctuations2087
Biophysical Journal 94(6) 2082–2094
dissociation constant of ADF/cofilin for ADP-bound actin
filaments (Fig. 4 B) at 2 mm from the growing barbed end,
corresponding toa molar fractionofbound ADF/cofilin of;1
(Fig. 4 A).
Residence time of actin subunits in the
We next examined the role of ADF/cofilin on the time spent
bysubunitsin the filament and on the global turnover of actin
monomers. The restricted filament lengthvariation, as shown
in Fig. 2 B, suggests that gain and loss of actin subunits
should be balanced over long periods. To further test this
filament (i.e., the difference between the rates of addition and
cofilin. (A) The binding of ADF/cofilin to ADP-bound actin subunits
increases with the distance to the growing barbed end, symbolized by the
x, y axis 0 position. We assumed a constant level (time and space) for the
actin monomer concentration ([Actin] ¼ 1 mM) and ADF/cofilin ([ADF/
cofilin] ¼ 1 mM). We simulated Eq. A2 with r1¼ 0.3 s?1, r2¼ 0.035 s?1,
r3¼ 0.0085[ADF/cofilin]-0.005 s?1, r4¼ 0.075[ADF/cofilin]-0.005 s?1,
and r5¼ 0.012 s?1. Note that the spatial distribution was obtained by
converting time into space (Eq. A4). The growing barbed end is assumed to
beat position0. (B)ADF/cofilin-actinsubunitdissociation constantatsteady
state. From the spatial distribution of the different nucleotide states bound to
actin subunit (ATP, ADP-Pi, or ADP) found by solving Eq. A2 (the
simulation parameters are given in A), we determined the local apparent
dissociation constant of the complex ADF/cofilin-actin subunit given by
KD;Spatial¼ ðfATP1fADP?PiÞKD01ð1 ? fATP? fADP?PiÞKD1; where KD0(re-
spectively KD1) is the dissociation constant of the complex ADF/cofilin with
0.58mM (16); fATP, fADP-Piare, respectively, the average fraction of actin
subunits bound to ATP, with ADP-Piat position x from the barbed end.
Effect of the ageing of actin filament on the binding of ADF/
unit gain/loss balance rate. After an initial transient decrease, the net balance
between gain and loss of actin subunits rapidly converges to zero, regardless
of the ADF/cofilin concentration used in simulations (black curve corresponds
to 0.1 mM; dark and light gray are obtained with 1 and 10 mM, respectively).
(Inset) Net balance fluctuation histogram determined over the last 2000 s of
simulations. (B) ADF/cofilin-severing activity controls the average time spent
by asubunitinactinfilaments(solid black curve) andinthedifferentfragments
life cycle (monomer to filament to monomer) is given by the dotted black
curve. Parameters for simulations are listed in the legend of Fig. 2 B.
Subunit dynamics in a single filament. (A) Variation of the sub-
2088Roland et al.
Biophysical Journal 94(6) 2082–2094
loss of actin monomers) for three ADF/cofilin concentrations
(Fig. 5 A).
After the first initial transient phase, due to the lag between
actin filament elongation and ATP hydrolysis of subunits
(Fig. 5 A), the balance between subunit gains and losses
presents zero-centered fluctuations (Fig. 5 A and inset) in-
dicating that, on average, the number of subunits in the fil-
ament will practically remain constant. Note that the actin
subunit loss includes contributions from actin monomer de-
polymerization at both ends and the sudden removal of a
large amount of subunits in the case of filament severing. In
the total subunit loss (Fig. S4).
The dynamics of actin filament length regulation directly
affects the residence time of actin subunits in the filament.
We analyzed the average time spent by a particular subunit in
the filament, between its incorporation and its release, either
by depolymerization or by severing (Fig. 5 B). ADF/cofilin
drastically reduces this average time at concentrations below
1 mM. However, further reductions are hardly seen for con-
centrations above 1 mM, and the minimal average time re-
mains ;25 s. Both ADF/cofilin and actin monomer control
the average residence time negatively (Supplementary Ma-
terial Figs. S5 and S6), in agreement with the analytical
In addition, note that actin concentration increases the aver-
age length (Fig. 2 C, inset) whereas it has an opposite effect
on the subunit residence time (Fig. S6).
To address the question of the global turnover of a mon-
the fragments obtained after filament severing. Assuming
that fragments are immediately capped, as in vivo, the time
spent by a particular subunit in the successive fragments is
about twice the time spent in the filament (Fig. 5 B, dashed
curve). Finally, we also determined the average actin mon-
omer life cycle duration (i.e., from monomer to filament to
monomer).As shown bythedotted curveinFig.5B,thetotal
time spent is substantially reduced (down to ;50 s) in the
presence of high ADF/cofilin concentrations, above 1 mM,
and the global monomer turnover is accelerated by a factor of
.100, when compared to the situation without cofilin (data
Fragments generated by
The severing of filaments produces fragments of different
sizes. To analyze the fragmentation process, we determined
the distribution of the fragment lengths generated from a
single filament (Fig. S7). Although ADF/cofilin favors fila-
ment severing, the proportion of large filaments, above 0.5
mm, diminishes abruptly with the severing activity (compare
red, blue, and black curves in Fig. S7). This observation
suggests that most of the severing activity above 1 mM ADF/
cofilin will be undetectable by light microscopy. This con-
clusion is valid over four orders of magnitude for the ADF/
cofilin concentration and is consistent with the statistical
distribution of filaments (Fig. 2 C). At steady state, the
fragmentation rate increased with the severingactivity before
plateauing at high ADF/cofilin concentration (Fig. 6, solid
curve). If we considered fragments larger than 0.5 mm only
(that are observable experimentally by light microscopy), the
apparent fragmentation rate is optimal for [ADF/cofilin] of
;0.2 mM (Fig. 6, dashed curve). However, fragments ,0.5
mm are preferentially produced at higher concentrations of
ADF/cofilin (Fig. 6, dotted curve; see also Fig. S7).
Dynamic organization of actin filaments into highly ordered
arrays (actin cables or a dendritic network) that produce the
forces necessary to deform or move cells requires a coordi-
nation of actin-binding protein activity together with the
transduction of chemical energy into force (1). Recently
a biomimetic system, comprising a minimal set of actin-
interacting proteins (including formin, ADF/cofilin, and
profilin), was able to reproduce actin filament dynamics at a
rate compatible with in vivo actin filament turnover (6). This
study demonstrated that ADF/cofilin was the only actin-
binding protein necessary to rapidly disassemble growing
actin filaments generated by an actin-promoting factor from
control of ADF/cofilin on single actin filament dynamics. We
showed that ADF/cofilin regulates the actin filament length
(Fig. 2), resembling the fast elongation periods followed by
abrupt shrinkage events observed in biomimetic assays (6).
actin filament increases with ADF/cofilin activity (solid black curve). Large
filaments, above 0.5 mm, are optimally produced for low severing activity
(dashed black curve); conversely, pieces below 0.5 mm are predominant at
large ADF/cofilin concentration (dotted black curve). Parameters for sim-
ulations are listed in the legend of Fig. 2 B.
ADF/cofilin severing at steady state. The production of new
ADF Drives Actin Filament Fluctuations2089
Biophysical Journal 94(6) 2082–2094
Model simulations (Fig. 2) and mathematical analysis (Ap-
pendix) suggestthattheconjunctionoftheageing ofsubunits
in the actin filament and the binding of ADF/cofilin to actin
subunits loaded with ADP followed by severing are essential
for actin filament dynamics. Under these conditions, actin
filament length distribution reaches a stable stationary re-
gime. This is an emergent property of the actin system that
constitutes a building block for future investigations of the
ADF/cofilin-driven control over actin dynamics in more
complex systems, both experimentally and in modeling ap-
Filaments are in a stable dynamical regime
that is independent from the
The presence of actin-interacting proteins produces different
biochemical conditions that can affect actin filament po-
lymerization quite dramatically. Therefore, we addressed
whether a stable regime for actin dynamics (i.e., a balance
between assembly and disassembly) ispossible, whatever the
biochemical conditions in cells or in biomimetic assays. In
the presence of ADF/cofilin, simulations suggest that fila-
ment length and chemical composition, though highly vari-
able, have a perfectly defined average and standard deviation
(Eqs. A10–A12). Additionally, both the average and the
amplitude of actin filament length fluctuations depend on the
actin monomer or ADF/cofilin concentrations only, with a
constant fluctuation/average ratio (;20%, Eq. A13). The
existence of a stable dynamical regime, as shown in Fig. 2,
implies that the contribution of actin filament elongation is
balanced by subunit loss (combining depolymerization and
severing). The match between gain and loss of actin subunits
emerges from the combination of constant ageing of actin
subunits in the filament and from the specific higher affinity
for binding of ADF/cofilin to F-ADP subunits (16). Since
only a few ADF/cofilin-actin subunit pairs are necessary to
fragment an actin filament, the balance between gain and
loss of subunits becomes almost independent of the actual
ADF/cofilin concentration, except at very low ADF/cofilin
(;0.1 nM). This resolves the apparent contradiction between
the drop of the apparent binding affinity of ADF/cofilin at
large concentration (Fig. 3 C, inset) and the severing efficacy
illustrated in Figs. 2 B and 5 B.
This result has important consequences for in vivo or
in vitro conditions, where nonequilibrium conditions often
prevail. A previous report (6) and this study highlight that
a stable dynamical regime is achieved for a whole set of
ADF/cofilin and actin monomers (Fig. 2, C and D). This is
possible because ADF/cofilin cannot bind to F-ATP or
F-ADP-Piactin subunits (16) and, consequently, the fila-
ment region close to the elongated barbed end is never
severed. This has the further consequence of preventing
total disassembly of a filament at its growing end due to
ADF/cofilin activity that is too high.
Subunit residence time in filaments and global
turnover of actin monomers
By severing the oldest part of the filament, ADF/cofilin lar-
gely contributes to the active turnover of subunits (Fig. 5 B),
simultaneously enriching the molar ratio of the remaining
actin filament with subunits bound to ATP or ADP-Pi(Fig.
3A).Experiments(6)andsimulations provethat themaximal
efficiency of ADF/cofilin is obtained at concentrations below
1 mM, in agreement with the evolution of KD,App.
We also investigated the dynamics of fragments, assuming
that they were immediately capped by capping proteins be-
fore further severing by ADF/cofilin. We found that the av-
erage subunit residence time in such daughter fragments,
originating from the same mother filament, happens to be
Similarly, we examined the global monomer turnover by
looking atthe time spent by a particularmonomer throughout
its complete life cycle. All residence times decrease rapidly
at low ADF/cofilin level (below 1 mM, Fig. 5 B), whereas
1 mM, in agreement with experimental data (see Fig. 3 E in
Michelot et al. (6)). This is a consequence of the protection
provided by the F-ATP and F-ADP-Pipopulation of subu-
nits against severing. The residual turnover observed at
large ADF/cofilin concentrations represents the time delay
necessary for ATP hydrolysis and phosphate release (Figs.
S2 and S3). More interestingly, Fig. 5 B gives the correct
order of magnitude for actin filament turnover (;50 s) in
vivo (3) or in biomimetic assays (9). As suggested by the
model and in conjunction with experimental data, ADF/
cofilin-driven filament fragmentation is likely the most
important factor that determines actin turnover through the
acceleration of the monomer life cycle in filaments and/or
Nucleation of new filaments and inhibition
Each fragment generated by severing is a potential seed for
the generation of a new actin filament (Fig. 6), unless rapidly
capped with capping protein. To reconcile this model-driven
analysis with recent results showing that ADF/cofilin severs
filaments, with optimal activity ;0.01 mM (whereas higher
levels, above 0.1 mM, stabilize the filaments (12,25,26)), one
has to consider the initial composition of the actin filament.
All previous studies use F-ADP actin filaments, which be-
come decorated on each subunit very rapidly in the presence
of excess ADF/cofilin. This rapid and huge change of the
composition stabilizes the filament and prevents its severing.
In our model, we started from short filaments made of ATP-
bound subunits which become decorated by ADF/cofilin
after the hydrolysis and the release of the g-phosphate bound
to the nucleotide. However, since ADF/cofilin-decorated
subunits are scattered, severing occurs before the complete
2090 Roland et al.
Biophysical Journal 94(6) 2082–2094
stabilization of the structure, giving rise to a new filament
made of ATP or ADP-Pi-bound subunits. Therefore, the
initial composition keeps the severing of growing actin fila-
ments on, avoiding the stabilization of growing actin fila-
ments at high ADF/cofilin concentration.
Apparent non-mass-action kinetics
Most of the parameters analyzed so far (average filament
length, subunit residence time, fraction of bound ADF/cofilin
to filaments, or apparent equilibrium dissociation constant of
Although the binding of ADF/cofilin to actin subunits has a
constant affinity (Table 1), the apparent equilibrium disso-
ciation constant, KD,App, increases from 1.66 mM at very
low ADF/cofilin concentrations to 8 mM at 10 mM of ADF/
cofilin (Fig. 3 C, inset). At low ADF/cofilin activity, long
and aged filaments (most of the subunits are bound to ADP,
Fig.3A)offer alarge number ofbindingsites,hence the low
value for KD,App. Conversely, if severing activity is high,
actin filaments are short and subunits are predominantly
bound to ATP or ADP-Pi(Fig. 3 A). As a consequence, the
number of potential binding sites for ADF/cofilin is low,
resulting in a low apparent affinity of ADF/cofilin for actin
Extension of the model to in vivo situations
This numerical study documents quantitatively the role of
ADF/cofilin severing on actin filament turnover and predicts
that growing filaments reach a stable dynamical regime, in-
dependent from the concentration of the different factors
modulating the reaction rates (formin, profilin, ADF/cofilin)
or the concentration of available actin monomers ready to
polymerize. This may explain how different cell types or
organisms use the same battery of proteins (i.e., formin,
profilin, ADF/cofilin) with similar but fluctuating activities
composition, and turnover.
In addition, we proposed that the activity of ADF/cofilin
in vivo is modulated by a gradient of spatial affinity. At the
growing barbed end of actin filaments, which is likely lo-
cated near a membrane structure, the high apparent disso-
ciation equilibrium constant KD,Spatiallimits ADF/cofilin
activity. As we moved along the growing actin filament
from the barbed end to the pointed end, the KD,Spatialde-
creased progressively to reach a low value ;2 mm away
from the growing barbed end (Fig. 4, A and B). Therefore,
the molar ratio of ADF/cofilin along growing actin fila-
ments derived from our analysis (Fig. 4 A) is an effective
way to predict the number of available sites for the fixation
of ADF/cofilin. This predicted gradient of ADF/cofilin
binding sites based on the variation of KD,Spatialagrees with
the observed localization of ADF/cofilin activity in vivo
Distribution of filament length and subunit
Model variables and parameters
Modeling hypothesis and equation
We combined the contribution of both barbed and pointed ends to filament
dynamics into a unique term, denoted v:
n ¼ ðkon;B1kon;PÞ½Actin? ? ðkoff;B1koff;PÞ
Note that v can be modulated by actin-binding proteins or the concentration
of available actin monomers. For example, in the presence of profilin, the
polymerization at the pointed end vanishes (i.e., kon;P¼ 0); another simpli-
fication occurs if we consider formin-driven polymerization, for which one
has kon;B½Actin? ? ðkoff;P1koff;BÞ: Here, we assume that a), free actin
monomers are continuously supplied to the reaction system, and b), the
polymerization rate at the pointed end is negligible.
Let x be the position of a subunit along the filament. By convention, the
barbed end is at x ¼ 0 so that the position of a subunit in the filament also
end. The number of filaments of length L at time t, denoted F(L,t), is the
solution of an integrodifferential equation (29):
Biophysical Journal 94(6) 2082–2094
¼ nðFðL ? d;tÞ ? FðL;tÞÞ
or variableDimension Definition
L Monomer position along the filament
(origin at the barbed end)
Subunit residence time in a filament
Concentration of actin monomers
(assumed constant and homogeneous)
Concentration of ADF/cofilin (assumed
constant and homogeneous)
Distribution of filaments of length L
Distribution of subunits of age T in the
Global (de)polymerization rate
v ¼ ðkon;B1kon;PÞ
½Actin? ? ðkoff;B
LStep change in filament length
associated with polymerization of
Pidissociation rate (F-ATP to F-ADP-Pi)
Pirelease rate (F-ADP-Pito F-ADP)
Fixation rate of the first ADF/cofilin
molecule on F-ADP
Cooperative-fixation rate of the second
ADF/cofilin molecule on F-ADP-ADF
Severing rate of F-ADP-(ADF)2
Filament-severing probability at a
distance L from position x ¼ 0
(position of the filament barbed end)
ADF Drives Actin Filament Fluctuations2091
The first term (1) represents filament elongation (or shortening, if v , 0) by
monomer addition at the filament barbed end (x ¼ 0). The second term (2)
represents the fragmentation by severing the filament at position L. The last
term (3) gives the removal of a filament of length L by cutting anywhere
between x ¼ 0 and L. Because of the modification of the nucleotide bound to
actin subunits (due to ATP hydrolysis and phosphate release), the function P
changes in time. However, Monte Carlo simulations showed that the spatial
distribution of subunit types along the filament is rapidly achieved at steady
state (see Supplementary Material Fig. S2); hence, we are justified in using a
time-independent expression for P.
Since a typical filament length, L, is much larger than d, one uses the
Taylor expansion of the first term to obtain
At steady state, we have
0 ¼ ?nd@FðLÞ
FðsÞds ? r5FðLÞ
Introducing an auxiliaryvariable ZðLÞ ¼RN
By direct integration, and using the conditions ZðNÞ ¼ 0 and Zð0Þ ¼ 1; one
from which we obtain the number of filaments of length L at steady state
LFðsÞds; the final equationreads
¼ r5PðLÞZðLÞ ? r5@ZðLÞ
PðsÞds ¼ ?r5
ZðLÞ ¼ exp ?r5
This result is analogous to Eq. 8 in Edelstein-Keshet and Ermentrout (29).
Model for the severing probability function P
fS1ðtÞ; S2ðtÞ; S3ðtÞ; S4ðtÞ; S5ðtÞg
fF-ATP; F-ADP-Pi; F-ADP;
at time t, given that it entered the filament, as F-ATP, at time t ¼ 0. Note that
the cooperative binding of a second ADF/cofilin molecule is coded into the
state F-ADP-(ADF)2, which actually represents pairs of actin subunits
decorated by ADF/cofilin. Because of probability conservation
S1ðtÞ1S2ðtÞ1S3ðtÞ1S4ðtÞ1S5ðtÞ ¼ 1;
we are left with a system of four linear differential equations
dt¼ r1? ðr11r2ÞS2? r1S3? r1S4? r1S5
dt¼ r2S2? r3S3
dt¼ r3S3? r4S4
with initial conditions
S2?5ð0Þ ¼ 0;
and r1¼ kATP-hydrolysis;r2¼ kPi-release;r3¼ ðkon;ADF½ADF?Þ;r4¼ ðkcoop;ADF
½ADF?Þ;r5¼ kserving. Note that in Eq. A2, we neglect ADF/cofilin disso-
ciation from actin filaments. The probability that a subunit is in the state
F-ADP-ADF2at time t is obtained as the solution of Eq. A2:
S5ðtÞ ¼ 11 +
To determine the severing probability, P, we need a connection between the
actinsubunitage,t,andits positioninthe filament,x.If weneglect stochastic
actin subunit during time t is
x ¼ ndt:
In consequence, the probability that a filament is severed in the interval
The prefactor d?1ensures normalization so that P is the probability density
for severing. Using this expression for P and Eq. A1, the steady-state
distribution for filament length reads
2092Roland et al.
Biophysical Journal 94(6) 2082–2094
Distribution of monomer lifetime in filaments
To gain further insight into the distribution of filament age or subunit res-
idence time in filament, we look at the subunit loss after severing. We
changed the previous analysis slightly and used a different set of differential
outflow after severing (last equation, variable S6(t))
dt¼ r4S4? r5S5
¼ n ? r1S1
¼ r1S1? r2S2
¼ r2S2? r3S3
¼ r3S3? r4S4
The other variables or parameters are those of Eq. A2. The probability that a
monomer incorporated at time t ¼ 0 (as F-ATP) leaves the filament at time
T . 0
1 ? expð?S6ðTÞÞ:
where S6(T) represents the probability that a subunit is severed from the
the filament is obtained by differentiating the above expression with respect
GðTÞ ¼ r5S5expð?S6ðTÞÞ:
Average and variance of the filament length at
low reaction rates
From Eqs. A5 or A7, one can obtain the average and variance of the filament
length or subunit residence time. Unfortunately, no closed expression for
polymerization rate, i.e., if
holds, the Taylor expansion of P (Eq. A5) reads
Therefore, we get the filament length distribution
to which corresponds an average filament length
Using Eq. A7, and the condition (Eq. A8) to simplify the expression of S6(t)
in Eq. A6, we get the average and variance of the subunit residence time:
where c1and c2are given by Eqs. A10 and A11.
Equations A9–A12 have important consequences that characterize the
standard deviation to average length (respectively subunit residence time) is
independent of the kinetic parameters r1-5, v, or d:
Second, the control of the average filament length by actin scales as
where v and v0correspond to two different polymerization rates (e.g., two
different actin monomer concentrations). In the presence of a large excess of
actin monomers or rapid polymerization (e.g., with formins (30,31)), v is
approximately proportional to the concentration of actin monomers. There-
fore, the average filament length can be expressed directly as
The control exerted by ADF/cofilin through rates r3,4gives a different
where ðr3;r4Þ and ðr3;0;r4;0Þ are associated to two different ADF/cofilin
concentrations. From Eq. A12, we see that the average residence time for a
single subunit in the filament scales as
Conversely, two different levels of ADF/cofilin (at constant actin monomer
concentration) give an equation similar to Eq. A15:
To view all of the supplemental files associated with this
article, visit www.biophysj.org.
ADF Drives Actin Filament Fluctuations2093
Biophysical Journal 94(6) 2082–2094
The authors thank Dr. Christopher J. Staiger and Dr. Rajaa Boujemaa-
Parterski for their help in handling the manuscript and fruitful discussions.
Financial support was provided by the Agence Nationale de la Recherche
(Programme physique et chimie du vivant, Mac-Mol-Actin project) and the
Rho ˆne-Alpes Institute of Complex Systems (IXXI), France.
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