actin mechanical properties

1. IntroductionThe cytoskeleton is an intricate network of bio-macromolecules which pervade the cytoplasm ofcells. The structural integrity of a cell depends uponits cytoskeleton, and for small deformations theelasticity of a cell depends on its actin network, amajor constituent of the cytoskeleton. Actin net-works are dynamic and consist of growing andshrinking filaments, and the cross-linking of thesefilaments in to a dynamic network. The purpose ofthis study is to capture the elasticity of actin net-works as a function of their dynamic behavior. Thecorrelation between actin dynamics and cellularelasticity can not only reveal interesting insightsinto the physics of cell mechanics, but also indicatethe presence of disease [1]. For example, cancerouscells have been found to exhibit an increase indeformability commensurate with disease progres-sion [2]. Understanding the interplay between

cytoskeletal structure and stiffness, therefore, isimportant for both the diagnosis and treatment ofthis disease. In this study a computer model of actindynamics yields network structures which can bedirectly fed into simulations of network elasticity.In other words, we use computer simulations todirectly relate the elastic properties of an actin net-work to the actin dynamics responsible for the for-mation of this network.Living cells have the ability to move and changeshape in response to environmental stimuli; a con-sequence of the dynamic nature of the cytoskeletonand its actin network [3]. In particular, actin net-works are formed from actin filaments which aretransiently cross-linked by actin-binding proteins(proteins with an affinity for actin). These actin fil-aments continually polymerize and depolymerizeto ensure a recycling of actin monomers, as theytreadmill through the filaments, with a net polymer-

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*Corresponding author, e-mail: [email protected] BME-PT

Actin dynamics and the elasticity of cytoskeletal networks

G. A. Buxton1*, N. Clarke2,3, P. J. Hussey2,4

1Department of Physics, Case Western Reserve University, Cleveland, OH 44106, USA2Biophysical Sciences Institute, Durham University, Durham, DH1 3LE, UK3Department of Chemistry, Durham University, Durham, DH1 3LE, UK4School of Biological and Biomedical Sciences, Durham University, Durham, DH1 3LE, UK

Received 15 April 2009; accepted in revised form 26 June 2009

Abstract. The structural integrity of a cell depends on its cytoskeleton, which includes an actin network. This network istransient and depends upon the continual polymerization and depolymerization of actin. The degradation of an actin net-work, and a corresponding reduction in cell stiffness, can indicate the presence of disease. Numerical simulations will beinvaluable for understanding the physics of these systems and the correlation between actin dynamics and elasticity. Herewe develop a model that is capable of generating actin network structures. In particular, we develop a model of actindynamics which considers the polymerization, depolymerization, nucleation, severing, and capping of actin filaments. Thestructures obtained are then fed directly into a mechanical model. This allows us to qualitatively assess the effects of chang-ing various parameters associated with actin dynamics on the elasticity of the material.

Keywords: modeling and simulation, biopolymers, actin networks, cytoskeleton

eXPRESS Polymer Letters Vol.3, No.9 (2009) 579587Available online at www.expresspolymlett.comDOI: 10.3144/expresspolymlett.2009.72

ization at the barbed end typically balancing a netdepolymerization at the pointed end [4]. Thebarbed (+) and pointed () ends are structurally dif-ferent and possess different polymerization kinet-ics. These dynamics are further regulated byadenosine triphosphate (ATP) hydrolysis; the ATPin free actin monomers that, once polymerized,hydrolyzes to adenosine diphosphate (ADP) andfacilitates depolymerization [5]. Furthermore, cell-specific regulatory proteins not only cross-linkthese dynamic filaments but also cap, sever andhelp nucleate them [6, 7]. In order to understand themechanical properties of actin networks we mustfirst elucidate their structure [8]. Therefore, wedevelop a model which computationally capturesthese complex dynamics and generates networkstructures which serve as the input to an elasticmodel.There have been many mathematical models ofactin polymerization kinetics which yield averagequantities [911], or simulations of individual fila-ments [12], however to directly simulate networkelasticity, large-scale discrete models are required.For example, Huber et al. [13] have recently pre-sented two-dimensional simulations of actin poly-merization which included the diffusion andpolymerization of individual actin monomers.Haviv et al. [14] adopted a similar kinetic MonteCarlo approach to model the self-assembly of astersand their transition into stars. Mogilner and Rubin-stein [15] have also used stochastic models to simu-late actin dynamics during filopedial protrusion.We adopt a similar, but more computationally effi-cient, approach and consider the filaments to bediscretized into 100 nm segments. Other investiga-tions of actin network elasticity, such as the studiesof Huisman et al. [16] or DiDonna and Levine [17],have considered randomly generated networkstructures.For many years the physics of elastic networkshave attracted attention in relation to the deforma-tion and fracture of heterogeneous materials andstructures [18, 19]. However, recently this empha-sis has shifted toward biological networks. Wil-helm and Frey [20] and Head et al. [21] investi-gated the deformation of two-dimensional net-works and found that more densely packedstructures deform more affinely (i.e., with a moreuniform strain field) and that the individual fila-ments increasingly deformed more through stretch-

ing than through bending. Using a three-dimen-sional model of network elasticity we recentlyobserved a similar bending-to-stretching transitionas a function of network connectivity, which like-wise corresponded with a transition to more affinedeformations [22]. The ability to computationallycapture the mechanics of network structures has ledto recent computational investigations of actin net-works. Didonna and Levine looked at the effects ofprotein unfolding in randomly generated two-dimensional networks [17]. Huisman et al. [16]looked at the deformation of three-dimensional ran-dom network structures as a method of elucidatingthe mechanics of actin networks. Recently, Broed-ersz et al. have theoretically looked at networkswhere the cross-links are assigned stiffnesses muchlower than that of the filaments [23]. Here, weexpand on these studies by considering networkstructures which are created directly from a simula-tion of the underlying actin dynamics. In otherwords, the output from a model of actin dynamicsserves as the input to a mechanical model. The ben-efits of such a two-step methodology is that themechanical properties of these networks can bedirectly correlated with the actin dynamics whichled to network formation. We give details of thismethodology in the following section, before pre-senting results and drawing relevant conclusions insubsequent sections.

2. Methodology

2.1. Generation of network

We employ the first reaction method to simulateactin dynamics, where actin filaments are dis-cretized into regularly spaced nodes along the fila-ment path [24]. Dynamic events are assigned ratesat which they are likely to occur; for example, thepolymerization and depolymerization at both fila-ment ends, the nucleation of a filament, or the sev-ering, capping and uncapping of existing filaments.Upon capping the polymerization kinetics at thebarbed end are inhibited. Polymerization rates aretaken to be proportional to the availability of freeactin (i.e., w+ = k+C, where C is the actin concen-tration). Nucleation can occur either spontaneously(through the formation of relatively stable trimers)leading to a concentration dependence of the formwnew = knewC3 [25], or nucleation can be initiatedby actin binding proteins, with a rate given by

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Buxton et al. eXPRESS Polymer Letters Vol.3, No.9 (2009) 579587

wnew = knewC. The concentration of free actin isdepleted as filaments are created or polymerize,and increased when filaments depolymerize. Due tothe high diffusion coefficient of actin, and rela-tively slow growth of filaments, we assume spatialvariations of actin can be neglected. The rate atwhich a filament is likely to be severed is propor-tional to its length (i.e., wsever = kseverl). Although itis worth noting that the hydrolysis of ATP-actinmay effect the severing rate. For computationalefficiency we do not include ATP hydrolysis as oneof the kinetic events in the first reaction method,but rather assign a probability of hydrolysis occur-ring during the time step t of the form Phydro =1 exp(khydrot), where khydro is the rate of ATPhydrolysis on the actin filament. A filament seg-ment with ATP-actin undergoes hydrolysis if a ran-dom number between zero and one is less thanPhydro. Upon polymerization a new node at the endof a filament is created with a position ri+1 =ri + r0(ri ri1 + )/(|ri ri1 + |), where ri+1 is theposition of the new node, r0 = 100 nm is the spatialdiscretization of the filament and is a sphericallysymmetric random vector whose magnitude is cho-sen to coincide with the persistence length of actinfilaments. In particular, the orientational correla-tion function = exp(r0/lp), where lp =17 m is the persistence length [26, 27]. The rateequations, therefore, do not describe the addition ofactual molecular subunits but, rather, the sequentialaddition of several actin subunits 100 nm in length.Periodic boundary conditions are enforced on asystem of size L3 = (20 m)3 and filament lengthsare limited to 20 m and, therefore, do not span thelength of the simulation box; we limit the size ofthe filaments to ensure that the elasticity of the net-work is due to the cross-linked network structureand not as a consequence of a single filament span-ning the system. While this simulation size is com-parable to the average filament length found in thisstudy (typically less than 10 m), it is still adequateto determine the constitutive response of these sys-tems. In particular, Ostoja-Starzewski and Stahl[28] have found that displacement-based boundaryconditions (as used here) can accurately capture theelastic properties of random networks even for rela-tively small system sizes.A time is associated with each event of the formi = ln([0:1])/wi, where wi is the rate of the ith

event and [0:1] is a random number between

0 and 1 [24]. The event with the lowest i is exe-cuted at each iteration and this time is removedfrom the times associated with the remainingevents. New is, for the recently performed (andany new) events, are calculated and, again, theevent with the lowest i is executed. The process isreiterated until steady state occurs. We start froman initial condition consisting of one hundred 1 mlong filaments, however, the final steady-statestructures have evolved sufficiently that they arenot sensitive to this choice. We run the simulationsfor 1108 iterations (time steps are on the order ofmilliseconds). The systems considered here areobserved to reach steady state before 1107 itera-tions (defined as when the average quantitiesdescribing the system reach relatively constant val-ues). Once the steady-state filament structures areestablished we determine a probability of cross-linking between neighboring segments.From the tube model of actin filaments the fluctua-tions of a filament, perpendicular to its length, arebelieved to be limited to a distance of 400 nm [29].Therefore, if the separation distance between twosegments is greater than this distance we assumethe probability of cross-linking to be zero. If dis-tances are less than 400 nm the probability of cross-linking is taken to be of the form Pcross =1 exp(kcrosst), where t refers to the time thesegments have coexisted and kcross is a rate of cross-linking. We take kcross = kABPAB, where PAB is theprobability of contact between two sites, defined aswhen two filaments (confined within tubes of400 nm diameters) come within 40 nm of eachother (the size of a cross-linking protein [30]).Therefore, the probability of contact is simplyPAB = 0.01 and we can vary kAB to generate cross-linked structures with variable cross-link density.

2.2. Network elasticity

Once the network structures are obtained we canfeed these structures directly into an elastic modeland correlate network geometry with elastic prop-erties. The elastic energy of a filament is given byEquation (1) [31]:

(1)

where the first term accounts for filament stretch-ing and the second term filament bending. E is the

sR

EIss

uEaA

+

= 22 12121

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Youngs modulus, a is the cross-sectional area, I isthe area moment of inertia, R is the radius of curva-ture and u is the displacement along the filamentcurvature, s. Similar elastic models have proven tobe highly successful at simulating the complexdeformations of actin networks [16, 17, 32], andelastic networks in general [20, 21].We take the flexural rigidity of the actin filamentsto be EI = 7.31026 Nm2 [33] and the filamentstiffness to be 34.5103 Nm1 [34]. Cross-linkingproteins are described by linear springs with a stiff-ness much lower than that of the actin filaments. Inparticular, the stiffness is taken to be 5105 [35]and the equilibrium length is taken to be 40 nm[30]. However, other more complicated elementscould be considered; for example, DiDonna andLevine [17] have considered unfolding cross-linkswhich reversibly unfold at a critical pulling force.Maintaining periodic boundary conditions we canevolve the network to equilibrium using the follow-ing Langevin Equation (2):

(2)

where is the friction coefficient and is a Gauss-ian noise term which satisfies the fluctuation dissi-pation theorem [36]. The filament is discretizedinto a series of points, or nodes, 100 nm apart. Thevelocity of a node is dr/dt and the elastic forces act-ing on this node is F. For numerical stability, /t istaken to be 10106 Nm1 and where t is the time step, kb is the Boltzmann con-stant, T = 300 K is the temperature, and G is aGaussian distributed random number with zeromean and unit standard deviation. Incorporatingthermal noise into the relaxation dynamics ensuresthat we capture entropic effects. The shear moduluscan then be obtained from the elastic energy densitystored in a network as a consequence of the appliedshear. This requires us to contrast the elastic energyin a given network subject to an applied shear(using Lee-Edwards boundary conditions) with theelastic energy in the same network without appliedshear (but still in a non-equilibrium state due tothermal noise). In other words, the difference inelastic energy is related to shear modulus byAshear Aundeform = G2/2, where Ashear and Aundeformare the elastic energy densities in the sheared andundeformed networks, respectively [31]. Note thatthe elastic energy density is the total energy stored

within all the elastic filaments in the simulation,divided by the volume of the simulation. G is theshear modulus and is the applied shear. Thestretching and bending of actin filaments, orstretching of cross-link proteins, results in forcesacting on the discrete nodes which characterize thefilament backbone. These forces are used to evolvethe Langevin equation, until the system relaxes toequilibrium. The difference in elastic energy den-sity between equilibrated systems with and withoutapplied shear can then be used to calculate the shearmodulus. Furthermore, not only can we calculatethe shear modulus but also isolate the contributionfrom different structural elements (stretching andbending energy of actin filaments and stretchingenergy of cross-linking proteins). The averages andstandard deviations are presented from three inde-pendent simulation (with each simulation takingapproximately 100 hours of cpu time on a standardlinux processor).

3. Results and discussion

The polymerization kinetics are taken from Pollardet al. [7] (Table 1.). Note that due to our discretiza-tion the polymerization rates used in the simulationdiffer from those quoted and correspond to thepolymerization of 100 nm pieces, rather thanmolecular subunits. The rate of hydrolysis is takento be 0.3 s1 [37] and the capping rate at the barbedend is taken to be 1 s1 (although this will dependon the concentration of capping proteins). We varythe nucleation rate (for systems where the nucle-ation rate is either proportional to concentration, orthe concentration cubed), the severing rate, theuncapping rate and the concentration of actin. Therange of these parameters are given in Table 2.Figure 1 shows a snapshot of a simulation withknew = 0.1 M1s1m3, ksever = 1106 s1m1,kuncap = 0.1 s1 and = 1 M. The network is con-nected with a rate of cross-linking kAB = 1 s1. The

/2 GtTkb =

t

rF

d

d=+

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Table 1. Polymerization kinetics [7]

Polymerization at barbed end with ATP-actin 11.6 s1 M1

Depolymerization at barbed end with ATP-actin 01.4 s1

Polymerization at pointed end with ADP-actin 00.16 s1 M1

Depolymerization at pointed end with ADP-actin 00.27 s1

Polymerization at barbed end with ADP-actin 03.8 s1 M1

Depolymerization at barbed end with ADP-actin 07.2 s1

Polymerization at pointed end with ATP-actin 01.3 s1 M1

Depolymerization at pointed end with ATP-actin 00.8 s1

filaments are shown as red cylinders and the cross-linking proteins are shown as black cylinders. Forthis value of kAB we find 6320 8 cross-linkingproteins (connecting 609 filaments with an averagelength of 8.2 m). For smaller values of kAB = 0.01and 0.1, we find 2304 39 and 5568 25 proteins,respectively. In this study we maintain kAB = 1which corresponds to a relatively high cross-linkdensity and note that smaller values of kAB willincrease the compliance of the networks.The effects of varying the nucleation rate is shownin Figure 2. We vary the nucleation rate from 0.1 to10 M1s1m3 (or M3s1m3 for wnew =knewC3). We contrast this difference in power anddepict the average length of the filaments and thenumber of filaments. For power = 1 (wnew = knewC)we find the average number of filaments increasesfrom 500 to 3500 and the length decreases from9 to 2 m. For power = 3 (wnew = knewC3) theeffects of varying knew are less severe. The concen-tration of free actin in the systems considered hereis roughly 0.6 m and so the effects of increasingknew are expected to be 3 times less for systemswhere power = 3.

Figure 3 shows the effects of varying ksever from1108 to 1104. As expected the number of fila-ments increases and the average length decrease. Inparticular, the number of filaments increases from500 to 2200 and the average length decreases from10 to 2 m. This range of filament lengths is com-parable to those found in experimental studies [38].Figure 4 shows the effects of increasing the uncap-ping rate from 0.001 to 1 s1. The number of fila-ments initially decreases before increasing, whilethe average filament length doubles from 6 to12 m. The percentage of filaments that are cappedas a function of uncapping rate is shown in theinset. As filaments become increasingly uncapped

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Figure 1. Snapshot of a simulation with knew =0.1 M1s1m3, ksever = 1106 s1m1,kuncap = 0.1 s1 and = 1 M. The network isconnected with a rate of cross-linking kAB =1 s1. The filaments are shown as red cylindersand the cross-linking proteins are shown asblack cylinders.

Figure 2. The effects of varying the nucleation rate from0.1 to 10 M1s1m3 (or M3s1m3 forwnew = knewC3) on the number and averagelength of actin filaments. ksever = 1106 s1m1,kuncap = 0.1 s1 and = 1 M. Nucleation rates ofthe form wnew = knewC are contrasted withnucleation rates of the form wnew = knewC3 tocompare spontaneous with protein mediatednucleation.

Figure 3. The effects of varying the severing rate overorders of magnitude (from 1108 to 1104) onthe number and average length of actin fila-ments. knew = 0.1 M1s1m3, kuncap = 0.1 s1

and = 1 M

Table 2. Polymerization rates

Nucleation rate knew = 0.1 10 M1s1m3

Severing rate ksever = 1108 1104 s1m1

Uncapping rate kuncap = 0 1 s1

Actin concentration = 0.5 1.5 M

the length increases, as the polymerization kineticsare less inhibited, and the reduction in free actinreduces the nucleation rate. However, newly cre-ated filaments have a greater chance of survival asthey are less likely to be capped. This reduction infilament nucleation, but increasing likelihood of fil-ament survival, results in the number of filamentsinitially decreasing before increasing.The effects of increasing actin concentration areshown in Figure 5. For an actin concentration of0.5 M there are relatively few filaments of verysmall length. As the concentration is increased to1.5 M the number of filaments increases linearlyto 1200 and the average filament length increases to9 m. Newly created filaments consist of ATP-

actin and the critical concentration at the pointedend with ATP-actin is 0.61. Therefore, the newlycreated filaments with concentrations less than thisdepolymerize from the pointed end and have lesschance of survival, whereas for concentrationsgreater than 0.61 the filaments initially grow fromboth ends before hydrolysis ensures the preferentialdepolymerization of the pointed end. We, therefore,see a dramatic increase in filament length around aconcentration of 0.61.We now turn our attention to the mechanical prop-erties of these networks. Figure 6 shows the effectsof varying actin concentration on the shear modu-lus of the network. For the simulations shown inFigure 5 we feed the structures in to the mechanicalmodel and calculate the shear modulus. The errorbars correspond to the standard deviation of threeruns. In particular, we plot the shear modulus as afunction of F-actin concentration. As the total con-centration of actin is increased to 1.5 M the con-centration of F-actin increases to 0.9 M. Thisresults in an exponential increase in the mechanicalstiffness of the material. The shear modulusincreases to 7 Pa as not only the density of the fila-ments increases, but also the number of cross-link-ing proteins increases. This range of shear modulusis consistent with experimental studies [3941]although the shear modulus varies as the concentra-tion to the power of 4.7 (rather than 2.5 as predictedtheoretically and found in Gardel et al. [41]). Thismight be a consequence of the elastic energy beingprimarily stored within the cross-linking proteinsand not the actin filaments.

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Figure 4. The effects of varying the uncapping rate (at thebarbed end) from 0.001 to 1 s1 on the numberand average length of actin filaments. Note thecapping rate is kept at 1 s1. knew =0.1 M1s1m3, ksever = 1106 s1m1 and = 1 M. The percentage of filaments which arecapped is shown within the inset.

Figure 5. The effects of varying the total actin concentra-tion from 0.5 to 1.5 M on the number and aver-age length of actin filaments. knew =0.1 M1s1m3, ksever = 1106 s1m1 andkuncap = 0.1 s1

Figure 6. Plot of shear modulus, G, as a function of F-actin concentration. The variation in F-actin isobtained through varying the total actin concen-tration (see Figure 5)

The effects of increasing the severing rate in Fig-ure 3 predictably resulted in an increase in the num-ber of the filaments and a reduction in filamentlength. Figure 7 shows the effect of this on themechanical properties of the actin networks. Whilethe concentration of F-actin also shows a slightvariation as a function of severing rate (not shown)the variations are on the order of a percent. Thevariation in mechanical stiffness is, therefore, aconsequence of network geometry. In particular,neither the networks consisting of many small fila-ments, nor the networks consisting of just a fewhundred large filaments, appear to provide the bestmechanical properties. Interestingly, the optimummechanical performance occurred in systems inbetween these two extremes.

4. Conclusions

To summarize, we have coupled a model of actindynamics with a mechanical model to directly cor-relate network formation and elasticity. We modelthe network formation through a model which con-siders the rates at which various events occur (poly-merization, depolymerization, nucleation, severing,capping, uncapping and hydrolysis). This is thenfed directly into an elastic model which allows us toobtain a shear modulus for the structure. These pre-liminary results offer interesting physical insightsinto these systems, and provide a platform fordeveloping specific biological models which couldbe directly compared to commensurate experimen-tal studies.

Future work will further explore the parameterspace and analyze the statistics of these heteroge-neous networks. In particular, the severing ratemight depend on the hydrolysis of ATP-actin andvary along the length of a filament (with ADP-actinmore likely to sever [10]) or the heterogeneity ofactin concentration could be important in modelingmore realistic cellular environments. In terms of themechanical simulations, the incorporation of non-linear protein deformations as a consequence ofprotein unfolding could improve the dynamics ofthe model and future work could explore the effectsof varying cross-link density.The focus in the current study, however, was thedevelopment of a two-step methodology capable ofcorrelating actin dynamics with actin networkmechanics. Our new methodology could provideinsights into how variations in actin dynamicsbetween different systems, or in response to a dis-ease [2], can ultimately effect the mechanical prop-erties of the cell.

AcknowledgementsWe thank Dr Junli Liu and Dr Bernard Piette (Durham Uni-versity) for their critical reading of the manuscript, and thereferees for their useful and insightful comments.

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