# 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:**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 - [show abstract] [hide abstract]

**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 - [show abstract] [hide abstract]

**ABSTRACT:**Mathematical modeling has established its value for investigating the interplay of biochemical and mechanical mechanisms underlying actin-based motility. Because of the complex nature of actin dynamics and its regulation, many of these models are phenomenological or conceptual, providing a general understanding of the physics at play. But the wealth of carefully measured kinetic data on the interactions of many of the players in actin biochemistry cries out for the creation of more detailed and accurate models that could permit investigators to dissect interdependent roles of individual molecular components. Moreover, no human mind can assimilate all of the mechanisms underlying complex protein networks; so an additional benefit of a detailed kinetic model is that the numerous binding proteins, signaling mechanisms, and biochemical reactions can be computationally organized in a fully explicit, accessible, visualizable, and reusable structure. In this review, we will focus on how comprehensive and adaptable modeling allows investigators to explain experimental observations and develop testable hypotheses on the intracellular dynamics of the actin cytoskeleton.Biophysical Journal 02/2013; 104(3):520-32. · 3.67 Impact Factor

Page 1

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

ABSTRACT

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

INTRODUCTION

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

ADF/cofilinseversfilaments betweentwoadjacentdecorated

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

doi: 10.1529/biophysj.107.121988

Submitted September 13, 2007, and accepted for publication November 19,

2007.

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: jean-louis.martiel@imag.fr.

Editor: Alexander Mogilner.

? 2008 by the Biophysical Society

0006-3495/08/03/2082/13$2.00

2082 Biophysical JournalVolume 94March 2008 2082–2094

Page 2

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

turnoverinfilaments.Then,fromthestatisticaldistributionof

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.

METHODS

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

specificroleof ADF/cofilininstimulatinglargefilamentfluctuationsandfast

monomer turnover.

Simulation methods

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

nucleotideboundtothe subunit,andthesubunitresidencetime.Allaveraged

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.

RESULTS

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

actinfilaments(Fig.2B).Afteraninitialperiodofcontinuous

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

activityofADF/cofilin(Fig.2B).Despitethehighlyirregular

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

Page 3

FIGURE 1

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

Page 4

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):

ÆLæ}ð½Actin?Þ5=6ð½ADF=cofilin?Þ?1=3;

ffiffiffiffiffiffiffiffiffiffiffi

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)

ffiffiffiffiffiffiffiffiffiffiffi

ÆDL2æ

p

), de-

ÆDL2æ

q

? 0:2ÆLæ:

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

study.

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

VariableReactionExpression

vB

Elongation rate at the

barbed end

Elongation at the pointed end

Total elongation

ATP hydrolysis

Phosphate release

Binding/unbinding of

ADF/cofilin to F-ADP

subunits

Cooperative binding/unbinding

of ADF/cofilin to F-ADP

subunits

Severing

kon,B[Actin]-koff,B

vP

v

r1

r2

r3

kon,P[Actin]-koff,P

v ¼ vB1 vP

kATP-hydrolysis

kPi-release

kon,ADF[ADF/cofilin]-koff,ADF

r4

kcoop,ADF[ADF/cofilin]-koff,ADF

r5

ksevering

TABLE 1Chemical rates constant

Chemical rate Numerical valueReference

kon,B

koff,B

kon,P

koff,P

kATP-hydrolysis

kPi-release

kPi-release(in the presence

of ADF/cofilin)

kon,ADF

koff,ADF

kcoop,ADF

ksevering

11.6 mM?1s?1(GATP)

1.4 s?1(GATP)

1.3 mM?1s?1(GATP)

0.8 s?1(GATP) 0.27 s?1(GADP)

0.35 s?1

0.0019 s?1

0.035 s?1

(5)

(5)

(5)

(5)

(15)

(32)

(16)

0.0085 mM?1s?1

0.005 s?1

0.075 mM?1s?1

0.012 s?1

(16)

(16)

(16)

(5,16)

GATP, ATP-loaded monomer.

GADP, ADP-loaded monomer.

ADF Drives Actin Filament Fluctuations2085

Biophysical Journal 94(6) 2082–2094

Page 5

KD;App¼½ADF=cofilin?Æ#FADPæ

Æ#FADP-ADFæ

;

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

FIGURE 2

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

Page 6

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

wherexisthepositionofanactinsubunitincorporatedintothe

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

resultsuggeststhatchemicaltransformationofATPintoADP-

Piand ADP, and the subsequent binding of ADF/cofilin to

F-ADP subunits, provides the timer necessary to control the

elongation/severing cycle.

To further investigate the effect of subunit ageing on the

binding of ADF/cofilin along growing actin filaments, we

analyzedthefractionofsubunitsboundtoADF/cofilinandthe

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

withrespecttothebarbedend.)The0interceptwiththexandy

axes corresponds to the position of the growing barbed end

whereactinsubunitsarealwaysintheATP-boundstate(Fig.4

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

FIGURE 3

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

Page 7

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

actin filament

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

assumption,weplottedthenetbalanceofactinsubunitsinthe

filament (i.e., the difference between the rates of addition and

FIGURE 4

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

F-ATPandF-ADP-Pi(respectivelywithactinADP).KD0¼20mMandKD1¼

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/

FIGURE 5

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

generatedbythesamefilament(dashedblackcurve).Theaveragedurationofa

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

Page 8

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

thesimulations,thislatterphenomenonrepresented;80%of

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

distribution(Eqs.A6andA7;comparealsoFigs.2CandS5).

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-

omer,wealsodeterminedtheresidencetimeofonesubunitin

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

not shown).

Fragments generated by

ADF/cofilin severing

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).

DISCUSSION

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

theforminfamily.Here,wedevelopedakineticmodelforthe

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).

FIGURE 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

Page 9

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-

proaches.

Filaments are in a stable dynamical regime

that is independent from the

chemical conditions

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

twicetheaveragetimespentinthemotherfilament (Fig.5B).

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

furthertimereductionishardlyseenatconcentrationsabove

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

fragments.

Nucleation of new filaments and inhibition

of severing

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

Page 10

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

ADF/cofilinforgrowingactinfilaments)showamarkeddrop

athighADF/cofilin concentration(Figs.2D,3B,4,and5B).

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

filaments.

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

andconcentrationstocontrolactinfilamentslength,chemical

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

(27,28).

APPENDIX

Distribution of filament length and subunit

residence time

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

indicatesthedistanceittraveledsinceitsadditiontothefilamentatthebarbed

end. The number of filaments of length L at time t, denoted F(L,t), is the

solution of an integrodifferential equation (29):

@FðL;tÞ

@t

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

1r5PðLÞ

L

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Biophysical Journal 94(6) 2082–2094

¼ nðFðL ? d;tÞ ? FðL;tÞÞ

ð1Þ

ZN

Fðs;tÞds

ð2Þ

?r5FðL;tÞ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ZL

ð3Þ

0

PðsÞds:

Parameter

or variableDimension Definition

x

L Monomer position along the filament

(origin at the barbed end)

Time

Filament length

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

filament

Global (de)polymerization rate

t

L

T

[Actin]

T

L

T

mM

[ADF/cofilin]

mM

F(L)

G(T)

L?1

T?1

v ¼ ðkon;B1kon;PÞ

½Actin? ? ðkoff;B

1koff;PÞ

d

T?1

LStep change in filament length

associated with polymerization of

one monomer

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)

r1

r2

r3

T?1

T?1

T?1

r4

T?1

r5

P(L)

T?1

L?1

ADF Drives Actin Filament Fluctuations2091

Page 11

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

@FðL;tÞ

@t

¼ ?nd@FðL;tÞ

@L

1r5PðLÞ

ZL

ZN

L

Fðs;tÞds

? r5FðL;tÞ

0

PðsÞds:

At steady state, we have

0 ¼ ?nd@FðLÞ

@L

1r5PðLÞ

ZN

L

FðsÞds ? r5FðLÞ

ZL

0

PðsÞds:

Introducing an auxiliaryvariable ZðLÞ ¼RN

nd@2ZðLÞ

@L2

ZL

By direct integration, and using the conditions ZðNÞ ¼ 0 and Zð0Þ ¼ 1; one

gets

ZL

from which we obtain the number of filaments of length L at steady state

ZL

LFðsÞds; the final equationreads

¼ r5PðLÞZðLÞ ? r5@ZðLÞ

@L

ZL

3

0

PðsÞds ¼ ?r5

@

@L

ZðLÞ

0

PðsÞds

??

:

ZðLÞ ¼ exp ?r5

nd

0

ZS

0

PðsÞds

??

dS

??

;

FðLÞ ¼r5

nd

0

PðsÞds

??

exp ?r5

nd

ZL

0

ZS

0

PðsÞds

??

dS

??

:

(A1)

This result is analogous to Eq. 8 in Edelstein-Keshet and Ermentrout (29).

Model for the severing probability function P

Let

fS1ðtÞ; S2ðtÞ; S3ðtÞ; S4ðtÞ; S5ðtÞg

betheprobabilitydistributionthatagivensubunitisinoneofthestateslisted

below

fF-ATP; F-ADP-Pi; F-ADP;

F-ADP-ADF;F-ADP-ðADFÞ2g

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

dS2

dt¼ r1? ðr11r2ÞS2? r1S3? r1S4? r1S5

dS3

dt¼ r2S2? r3S3

dS4

dt¼ r3S3? r4S4

dS5

dt¼ r4S4

(A2)

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 +

4

i¼1

Kiexpð?ritÞ;

(A3)

with

Ki¼ ?

?Y

4

j¼1j6¼i

rj

??Y

4

j¼1j6¼i

ðrj? riÞ

??1

:

To determine the severing probability, P, we need a connection between the

actinsubunitage,t,andits positioninthe filament,x.If weneglect stochastic

fluctuationsoftheactual(de)polymerizationrate,thedistancetravelledbyan

actin subunit during time t is

x ¼ ndt:

(A4)

In consequence, the probability that a filament is severed in the interval

[x,x1d] is

PðxÞ ¼1

dS5

x

nd

??

¼1

d

11 +

4

i¼1

Kiexp ?ri

ndx

??

??

:

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

FðLÞ ¼

r5

nd

??

P1ðLÞexp ?

r5

nd

??

P2ðLÞ

??

(A5)

with

P1ðLÞ ¼

ZL

0

PðxÞdx ¼1

nd

ri

d

L1 +

4

i¼1

Ki

??

nd

ri

???

? +

4

i¼1

Ki

??

exp ?ri

ndL

?

;

and

P2ðLÞ ¼

ZL

0

P1ðxÞdx ¼1

?

d

L2

21L +

?

4

i¼1

?2

Ki

nd

ri

?

???

nd

ri

??

? +

4

i¼1

Ki

nd

ri

?2

1 +

4

i¼1

Ki

exp ?ri

ndL

?!

:

2092Roland et al.

Biophysical Journal 94(6) 2082–2094

Page 12

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

equations,including1),thepolymerizationstep,v(firstequation),and2),the

outflow after severing (last equation, variable S6(t))

dS1

dt

dS2

dt

dS3

dt

dS4

dt

dS5

dt¼ r4S4? r5S5

dS6

dt¼ r5S5

¼ n ? r1S1

¼ r1S1? r2S2

¼ r2S2? r3S3

¼ r3S3? r4S4

(A6)

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

filamentattimeT.Thedistributionoftheresidencetimeofasinglesubunitin

the filament is obtained by differentiating the above expression with respect

to T

GðTÞ ¼ r5S5expð?S6ðTÞÞ:

(A7)

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

theseparametersispossibleinthegeneralcase.However,inthelimitoflarge

polymerization rate, i.e., if

?

holds, the Taylor expansion of P (Eq. A5) reads

?

Therefore, we get the filament length distribution

?

3exp ?1

6!

maxi¼1?4

riL

nd

?

?

? 1(A8)

PðLÞ ¼r1r2r3r4

4!

L

vd

?4

1O

maxi¼1?4

riL

vd

???5

!

:

FðLÞ¼1

5!

1

d

r1r2r3r4r5

v5

?

L

d

? ?5

r1r2r3r4r5

v5

??

L

d

? ?6

!

;

(A9)

to which corresponds an average filament length

ZN

ÆLæ¼

0

LFðLÞdL¼c1d

v5

r1r2r3r4r5

!

??1=6

;

c1¼pð2Þ2=3ð3Þ?2=351=6G

5

6

? ??1

?2:78...;

(A10)

and variance

ÆDL2æ¼

ZN

0

L2FðLÞdL?ÆLæ2¼c2d2

? ??4

?:290...

v5

r1r2r3r4r5

??1=3

;

c2¼4p

27ð10Þ1=3G

2

3

ð2Þ1=3ð3Þ2=3p2?ð3Þ13=6G

2

3

? ?3

!

(A11)

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:

ÆTæ¼c1ðr1r2r3r4r5vÞ?1=6;

ÆDT2æ¼c2ðr1r2r3r4r5vÞ?1=3:

(A12)

where c1and c2are given by Eqs. A10 and A11.

Equations A9–A12 have important consequences that characterize the

dynamicsofactinfilamentssubjectedtoADF/cofilinsevering.First,theratio

standard deviation to average length (respectively subunit residence time) is

independent of the kinetic parameters r1-5, v, or d:

ffiffiffiffiffiffiffiffiffiffiffi

c1

ÆDL2æ

ÆLæ

p

¼

ffiffiffiffiffiffiffiffiffiffiffi

ÆDT2æ

ÆTæ

p

¼

ffiffiffiffi

c2

p

?

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

:290...

2:78...

p

?:194...?20%:

(A13)

Second, the control of the average filament length by actin scales as

ÆLæ

ÆL0æ¼

v

v0

? ?5=6

:

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

scaling:

?

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

ÆLæ

ÆL0æ¼

½Actin?

½Actin?0

?5=6

:

(A14)

ÆLæ

ÆL0æ¼

ðr3r4Þ0

ðr3r4Þ

?1=6

¼

ADF=cofilin

ADF=cofilin

½

½?0

?

??1=3

;

(A15)

ÆTæ

ÆT0æ¼

v0

v

? ?1=6

¼

Actin

Actin

½

½?0

?

??1=6

:

(A16)

Conversely, two different levels of ADF/cofilin (at constant actin monomer

concentration) give an equation similar to Eq. A15:

?

ÆTæ

ÆT0æ¼

ðr3r4Þ0

ðr3r4Þ

?1=6

¼

ADF=cofilin

ADF=cofilin

½

½?0

?

??1=3

:

(A17)

SUPPLEMENTARY MATERIAL

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

Page 13

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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