j.a. tuszynski, t. luchko, s. portet and j.m. dixon: anisotropic elastic properties of microtubules

8/3/2019 J.A. Tuszynski, T. Luchko, S. Portet and J.M. Dixon: Anisotropic elastic properties of microtubules.

1/7

DOI 10.1140/epje/i2004-10102-5

Eur. Phys. J. E 17, 2935 (2005)THE EUROPEAN

PHYSICAL JOURNAL E

Anisotropic elastic properties of microtubules

J.A. Tuszynski1,a, T. Luchko1, S. Portet2, and J.M. Dixon3

1 Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2J12 Room 1060, The Samuel Lunenfeld Research Institute, Mount Sinai Hospital, 600 University Avenue, Toronto, Ontario,

Canada M5G 1X53 Department of Physics, University of Warwick, Coventry, CV4 7AL, United Kingdom

Received 23 July 2004 and Received in final form 20 December 2004 /Published online: 6 April 2005 c EDP Sciences / Societa Italiana di Fisica / Springer-Verlag 2005

Abstract. We review and model the experimental parameters which characterize elastic properties of mi-crotubules. Three macroscopic estimates are made of the anisotropic elastic moduli, accounting for themolecular forces between tubulin dimers: for a longitudinal compression of a microtubule, for a lateralforce and for a shearing force. These estimates reflect the anisotropies in these parameters observed inseveral recent experiments.

PACS. 36.20.-r Macromolecules and p olymer molecules 87.16.Ka Filaments, microtubules, their net-works, and supramolecular assemblies

1 Introduction

The cytoskeleton is composed of three different types of

filaments organized in networks: microfilaments (MF), in-termediate filaments (IF) and microtubules (MT). Eachof them has specific physical properties and structuressuitable for their role in the cell. For example, the two-dimensional arrangement of MFs in contractile fibers, so-called stress fibers, appears to form cable-like structuresinvolved in the maintenance of the cell shape and trans-duction pathways. F-actin can support large stresses with-out a great deal of deformation and it ruptures at approxi-

matively 3.5 N/m2 [1]. IFs have a rope-like structure com-posed of fibrous proteins consisting of two coiled coils andare mainly involved in the maintenance of cell shape andintegrity. Ma et al. [2] shows that IFs resist high applied

pressures by increasing their stiffness. They can withstandhigher stresses than the other two components withoutdamage [1].

MTs are long hollow cylindrical objects made up of12 to 17 protofilaments under in vitro conditions, andtypically of 13 protofilaments in vivo. Protofilaments con-sist of-, -tubulin heterodimers longitudinally arranged.Each protofilament is shifted lengthwise with respect to itsneighbor describing left-handed helical pathways aroundthe MT. MTs are involved in a number of functions of thecell such as cell shape maintenance, mitosis and they playan important role in intracellular transport. By biologicalstandards, MTs are rigid polymers with a large persis-

tence length of 6 mm [3]. From Jamneys experiments [1],a e-mail: [email protected]

MTs suffer a larger strain for a small stress compared toeither MFs or IFs. The rupture stress for MTs is very

small and typically is about 0.40.5 N/m

2

[1]. The lateralcontacts between tubulin dimers in neighboring protofil-aments have a decisive role for MT stability, rigidity andarchitecture [4]. Tubulin dimers are relatively stronglybound in the longitudinal direction (along protofilaments),while the lateral interaction between protofilaments ismuch weaker [5].

There have been a number of experimental studies inrecent years dealing with the various aspects of the elas-ticity of MTs. Section 2 summarizes in detail the resultsof experimental investigations. On the other hand, theo-retical effort in this area has been much more limited. Forexample Janosi et al. [6] have studied the elastic prop-erties of MT tips and walls and the various shapes ob-served from electron micrographs have been shown to beconsistent with their simple mechanical model. However,their model deals chiefly with the geometrical character-istics of MTs and not with their anisotropic propertieswhich is the focus of our effort. The limited flexibility ofinter-protofilament bonds in MTs, assembled from puretubulin, has also been investigated by Chretien et al. [7]via moire patterns in cytoskeletal micrographs. The posi-tion of tubulin subunits and their arrangement on the MTsurface enables the moire period to be predicted.

In this paper we discuss the elastic properties of MTs.In particular we try to utilize the experimental data foundin the literature for our theoretical estimations. Our start-ing point is a review of the information found in the liter-ature regarding the results of experiment and theoretical


2/7

30 The European Physical Journal E

models of elastic properties of MTs. Using recently pub-lished information regarding dimer-dimer interactions andthe molecular geometry of the MT, we provide estimatesof the elastic moduli accounting for the anisotropy of the

MT filament.

2 Elastic parameter review

The response of a cylinder of length L and cross-sectionalarea A, to an extensional force (stretching), F, is describedby Youngs modulus, Y, and the relationship between thesephysical quantities is given by Hookes law,

F

A= Y

L

L, (1)

where L is the small length change and the relative ex-

tension,LL , is the unidirectional strain along the direc-

tion of the force applied to the cylinder.Another elastic property is characterized by the shear

or twisting modulus, G, that is expressed, for a homoge-neous isotropic material, by [5]

G =Y

2(1 + ), (2)

where is Poissons ratio, representing the relative mag-nitude of transverse-to-longitudinal strain and its valuetypically lies in the range 0 < < 0.5.

Another indicator of the elastic property of a solid isthe parameter called flexural rigidity, f, that determinesthe resistance of a filament to a bending force. The higherthe flexural rigidity, the greater the resistance to bending.The flexural rigidity is completely determined by the prop-erties of the bonds between the atoms within each proteinsubunit and the properties of the bonds that hold thesubunits together in MTs. For isotropic and homogeneousmaterials, the flexural rigidity, f, can be represented asthe product of two terms (characterizing material proper-ties and their shape), by

f = YIy , (3)

where Iy is a shape-dependent parameter called the sec-

ond moment of inertia. For a hollow cylinder [8,3] it isgiven by Iy = (R

4o R

4

i )/4, where Ro and Ri are theouter and the inner radii, respectively. However, for ahollow cylinder made up of n cylindrical protofilaments,where each protofilament has radius r, Iy is given byIy =

2

2n3 + n

4

r4 [8].The persistence length, p, is another index which de-

scribes the filaments resistance to thermal forces. Roughlyspeaking this length is the distance over which a filamentappears straight and can be expressed by

p =f

kBT=YIy

kBT, (4)

where kB is Boltzmanns constant and T is the tem-perature in Kelvin. The more flexible the filament, the

Table 1. Experimental data for elastic properties of MTs (andother proteins) from the literature.

Youngsmodulus, Y

(1.22.7) 109 N/m2 [811]

(15) 108 N/m2 [5,12]

Shear mod-ulus, G

1.4 N/m2 [13]

103 N/m2 between MTs [10]

34 N/m2 gel (concentr.-dependent)

[1]

1.4106 N/m2 at 25 C(temp.-dependent)

[5]

Flexuralrigidity, f

(1645) 1024 N m2 [1417,8,1820]

(2.95.1) 1024 N m2 [21]

Poissonsratio,

0.3 for macromolecules [22,12]

0.4 for MFs [11]

Persistencelength, p

(16.3) 103 m [3,15,16]

smaller the persistence length. Polymers for which per-sistence length and contour length are similar are calledsemiflexible polymers.

In Table 1 we list the different values found in theliterature for the above-defined parameters for MTs. FromTable 1 we observe that the shear and Youngs moduliare significantly different. This observation describes ananisotropic material. Hence, MTs are inhomogeneous and

anisotropic. The bending deformation and stretching offilaments are governed by Youngs modulus and the slidingbetween filaments is governed by the shear modulus [5].Note that the values of the shear modulus, G, seem to bestrongly dependent on the experimental conditions andshow little consistency, as we will attempt to elucidate inSection 4.

3 Estimation of tubulin-tubulin interactions

Tubulin was first imaged to atomic resolution in 1998 [23],

following 30 years of difficult work by a number of labswith this protein. Nogales et al. published the structureof - and -tubulin which were co-crystallized in the het-erodimeric form [23]. Electron diffraction was performedfor tubulin assemblies in the form of zinc sheets. The workestablishes that the structures of - and -tubulin arenearly identical and confirms the consensus speculation. Adetailed examination shows that each monomer is formedby a core of two -sheets that are surrounded by -helices.The monomer structure is very compact, but can be di-vided into three functional domains: the amino-terminaldomain containing the nucleotide-binding region, an in-termediate domain containing the taxol-binding site, andthe flexible carboxy-terminal domain.

Using the refined tubulin crystal structure producedby Lowe et al. [24], Sept et al. [25, 26] calculated the


3/7

J.A. Tuszynski et al.: Anisotropic elastic properties of microtubules 31

Fig. 1. MT protofilament. and monomers are colored gray and white, respectively. Image created with VMD [27].

protofilament-protofilament energy as a function of theshift along the protofilament axis. Two protofilamentswere constructed similar to Figure 1 and translated rel-

ative to each other along the MT axis. Using Poisson-Boltzmann calculations and a surface area term the in-teraction energy was computed over a 80 A translation in2 A steps. The data published in this work indicate thepresence of two stable equilibrium positions that clearlycorrespond to the known MT lattice types A and B.

In fact, a recent paper by VanBuren et al. [28] esti-mated the values of lateral and longitudinal bond energiesin an MT structure using a stochastic model of assemblydynamics. Their analysis predicts the lateral bond energyto be in the range from 2.2 to 5.7 kT (which translatesto 9.1 1021 to 2.4 1020 J) while the longitudinalbond energy is given from 6.8 to 9.4 kT (2.8 1020

to 3.91020 J). However, without the knowledge of thepotentials positional dependence these values cannot pro-vide information on the corresponding elastic coefficients.

Using this potential map (Sept et al., Fig. 1 [26]) wehave evaluated the corresponding elastic coefficient in aharmonic approximation around the potential minimum.We found the value k 4 N/m which will be used inthe next section. Note that an early phenomenological es-timate of the spring constant k [19] was approximately40 times smaller and predicted incorrectly that the rigid-ity of MTs is 100 times lower than that of actin. Unfor-tunately, still no results are available at present whichwould give the values of the elastic coefficients along differ-ent directions and not just as the result of protofilament-protofilament shifting along the MT axis as demonstratedin Sept et al. [25, 26]. While this calculation imposes acomputational challenge, it is only a matter of time be-fore these values are obtained in a subsequent calculationof MTs anisotropic elastic properties.

Recently, De Pablo et al. have estimated experimen-tally by radial indentation of MTs with a scanning forcemicroscope tip a spring constant value related to radialinteractions between neighboring protofilaments [29]. In-dentations, induced by the nanometer-sized tip positionedon the top of an MT, result in a linear elastic responsewith the spring constant k = 0.1 N/m. Clearly, k k.Tubulin dimers are relatively strongly bound in the longi-tudinal direction (along protofilaments), while the lateralinteraction between protofilaments is much weaker [5,30,

0

F

F

F

F

r

Fig. 2. A cubic lattice of particles separated by r0 when inequilibrium, connected by elastic springs with spring constants and subjected to a force F [8].

26]. Hence, if we define p as the elastic constant betweenthe two dimers along a protofilament, and l as the elas-tic constant between the two dimers belonging to adjacentprotofilaments, then p > l = k.

An order of magnitude estimate of this particular elas-tic constant, l, can be found in [19] which, when units

are converted, results in l = 0.1 N/m. This compares verywell with the value given above based on the experimentreported in [29].

4 Estimate of anisotropic elastic propertiesfrom molecular forces

In this section we will make estimates of the anisotropicelastic moduli measured for individual MTs based on theinternal geometry of the MT and the molecular forces act-ing between individual dimers. To give a simple conceptualexample we follow [8] and imagine a perfect cubic latticeconnected by springs, see Figure 2.

Suppose a force F is applied along the line of springsas indicated in Figure 2. Each bond experiences this force


4/7


which is perpendicular to one side of the lattice. Eachspring will be extended by a small distance r, fromits equilibrium state r0, according to the application ofHookes law to a spring i.e. F = r. Dividing this rela-

tion through by r

2

0 we obtainF

r20

=

r0

r

r0. (5)

The left-hand side of equation (5) represents the force perunit area or stress whereas r

r0on the right-hand side, de-

formation per unit length, describes the strain involved.From equation (1), we therefore obtain Youngs modulusof this material as Y=

r0. In essence the value of Youngs

modulus is proportional to the spring constant which de-pends on the strength of the intermolecular forces. It isinversely proportional to the equilibrium separation be-tween neighboring molecules.

Although the above estimate appears to be rathercrude at first sight, it compares quite well to earlier workon polymer elasticity [31] which gives a formula for theYoungs modulus of a filamentous polymer

Y= r0

a2(6)

with r0 denoting the centre-to-centre distance betweenneighboring monomers and a representing the radius ofthe filament.

Using the same method, we estimate below the effec-tive elastic moduli for a longitudinal, a lateral and a sheardeformation (Fig. 3).

4.1 Effective Youngs modulus due to longitudinalcompression (directed parallel to the protofilament ofan MT)

When a compressive force, F, is applied as shown in 1) ofFigure 3, there will be a small displacement, rp, of thedistance between two dimers of a particular protofilamentgiven by

F = rp , (7)

where is the elastic constant between the two dimersalong a protofilament (this was referred to earlier as p).

Dividing both sides of equation (7) by the area, A, overwhich the force is applied we obtain

F

A=

Arp. (8)

Suppose N is the number of protofilaments around theMT, then

A =1

N(R2o R

2

i ) (9)

where Ro and Ri are the outer and the inner radii of theMT, respectively.

Hence, from equations (79) we obtain

F

A

=

N l

(R2o R2

i )

rp

l

. (10)

F

F

F

F

F

F

F

1)Longitudinal

compressive force

2)Lateral compressive

force

4)Lateral shearing

force

3)Longitudinal

shearing force

Fig. 3. Different types of forces applied to a cylinder and re-sulting deformations. 1) A compressive force applied longitudi-nally to the tip of the filament. 2) A compressive force appliedlaterally to the wall of a filament. 3) A longitudinal shearingforce. 4) A lateral shearing force.

The left-hand side of equation (10) is the stress appliedand the term in the brackets on the right, assuming l is theequilibrium distance between dimers, is the strain or incre-mental displacement per unit length. Hence, from equa-tion (1) the effective Youngs modulus, Y, is given by

Y =N l

(R2o R2

i ). (11)

We estimate the value of to be 4 N/m which is obtainedfrom the harmonic approximation of the interaction en-ergy profile published by Sept et al. (see Fig. 1 in [26]).Further, taking N = 13, l = 8nm, Ro = 12.5nm, andRi = 7.5nm, and from equation (11) we find that our

estimate of Y is 1.32 109 N/m2 [26] which is in very

good agreement with the experimental data found in theliterature (see Tab. 1).

4.2 Effective elastic modulus due to a lateral force(perpendicular to the protofilament)

In this case (namely, entry 2) of Fig. 3)

F = r, (12)

where r is the displacement of dimers in two adjacentprotofilaments and is the elastic constant for the com-pressive force (see Fig. 4(b)), the experimental value anal-ogous to has been measured by De Pablo et al. [29].

Since there are two adjacent dimers and we need toproject the force along the line joining the two dimers, we


5/7


R0

F

F

R o

(a) (b)

Fig. 4. (a) Side elevation of an MT showing the area in greyto which the force F is applied. (b) Cross-section of an MTshowing the direction of the applied force.

multiply both sides of equation (12) by 2 sin , where is

the angle subtended by 2 adjacent dimers at the center ofthe MT (see Fig. 4(b)). We then divide the result by anappropriate area (see Fig. 4(a)), A, given by

A = Rol, (13)

l being the distance between two neighboring layers(Fig. 4(a)) calculated as a center-to-center distance. Thus,

2F sin

A=

2 sin

l

r

Ro

. (14)

The part on the left-hand side of equation (14), 2F/A,represents the applied stress, and putting the left-hand

sin to the right-hand side gives (r/Ro sin ) as the re-sultant strain, where Ro sin is the equilibrium separationof one pair of dimers. Hence, the effective lateral elasticmodulus, Y, is estimated to be given by

Y =2 sin

l. (15)

If we use the value of as the currently available es-timate for the elastic coefficient [29], i.e. = 0.1 N/m, =

6, and l = 8 nm then the corresponding value of the

Youngs modulus is found to be Y = 4 106 N/m2. At

any rate, due to the presence of the sin factor in equa-

tion (15) and a small value of ( =

2

N1 , where N is thenumber of protofilaments) we expect the Youngs modulusto be anisotropic for the parallel and perpendicular direc-tions along which it may differ by as much as an order ofmagnitude.

4.3 Effective elastic modulus due to a lateral shearingforce (see entry 4) in Fig. 3)

The cross-sectional area of the MT is R2o, whereas thelength of the section of an MT (see Fig. 5(a)) is L = nlwhere n is the number of layers of dimers and l the dis-tance between two neighboring layers. The net macro-scopic displacement, x (see Fig. 5(a)) is obviously givenby x = nx, where x (see Fig. 5(b)) is the microscopic

x

L

F

F

After

f

f

r

rh

x

A

Before

l l

C

B

(a) (b)

Fig. 5. (a) Section of an MT of length L, subjected to a lateralshearing force, F. (b) Movement of a single dimer as a resultof the force F indicating the undisturbed p osition (before, B )and the position following displacement (after, A).

lateral displacement due to the shearing force. With frdenoting the inter-dimer force (see Fig. 5(b)) between adisplaced dimer, A, and a relatively stationary one, C,we have

fr = s(l l) , (16)

where s is the microscopic elastic constant and l l

denotes the change in the interdimer distance (i.e. thedifference in length between BC = l and AC = l).

We denote the restoring force across the section of theMT by frh (see Fig. 5(b)) which, by vectorial addition, isrelated to the restoring force between dimers A and C,

fr, byfrh = fr sin . (17)

IfN is the number of protofilaments around the perimeterof the MT we have

F = N frh = N fr sin , from equation (17)= N s(l

l)sin , from equation (16)= N sl(

1

cos 1) sin ,

(18)since it follows from Figure 5b that l = l cos . Rearrang-ing equation (18),

F = N sl(1 cos )tan (19)

and, using the fact that the cross-sectional area is A =R2o and tan =

xl

= xL

, we have

F

A=

N

R2osl(1 cos )

x

L= G

x

L

, (20)

where G is the shearing modulus. Hence, we deduce thatthe shearing modulus can be finally expressed by

G =N

R2osl(1 cos )

N

R2osl

2

2, (21)

where is the angle of deformation in radians. We haveused the fact that is small by expanding the cosine. The

angle can lie between the following limits:0.001 in radians 0.1 . (22)


6/7


From equation (21) we see that the shear modulus, G,varies greatly with possible values of and hence also withthe size and magnitude of the displacement. Taking typicalvalues, for example N = 13, l = 8nm, Ro = 12.5 nm and

s

0.5 N/m, we find G may vary between the followingestimated limits:

5 3 N m2 G 0.5 106 N m2 . (23)

Note that the large variation in the value of G found inequation (23) reflects the large differences seen in the var-ious experimental data (see Tab. 1) and may provide anexplanation for this diversity. The crucial point to makehere is that the deformation angle, , enters into the shearmodulus formula. Combining equations (20) and (21) al-lows us to calculate the force required for the shearingaction as a function of the angle, . For example, withthe use of typical structural parameters and for = 0.1

the shearing force amounts to more than 20 pN which isquite significant. While the computer simulations of Septet al. [26] addressed the problem of longitudinal shearingforces (see entry 3) in Fig. 3) and the resulting strain, thecurrently available experimental techniques are unable tomeasure these effects. On the other hand, lateral shearingforces and their effects can be tested experimentally withno major difficulties.

5 Discussion

The objective of this paper was to analyze, describe and

theoretically estimate the anisotropic elastic properties ofMTs. This has been possible due to a recently obtained ex-perimental body of data coming from several groups anddescribing a number of physical and structural propertiesof MTs. In order to synthesize the data we reviewed theliterature taking into account the geometry of MTs, exper-imental data, and their extrapolation. The most markedaspect of MTs observed was the anisotropy of the elas-tic coefficients. This led us to propose a molecular modelthat explains the bulk values in terms of the molecularprotein-protein interactions. By and large the agreementbetween theory and experiment is satisfactory. An inter-esting by-product of this calculation was that the shear

modulus depends on the magnitude of the deformationwhereas Youngs moduli due to a longitudinal or lateralforce are only dependent on the structure of MTs. Hencethe diversity of experimental data may be explained asarising due to different magnitudes of the forces applied.It should be noted that the calculation of the shear mod-ulus, G, defined in equation (21), refers to a completelydifferent physical situation than the one involved in flexu-ral rigidity which is commonly measured experimentally,e.g., using optical tweezers [32]. In the latter case a can-tilevered beam is subjected to a force, F, applied to its freetip while the other end is clamped. Take the length of therod to be L and the free tip to be deflected from the rest-ing configuration by a distance y(L). The basic formulafor small-angle deflection is given by F = y(L), wherethe cantilever spring constant is given by = 3f/L

3

and f, defined in equation (3), is the flexural rigidityof the rod [8]. With the flexural rigidity of an MT esti-mated as f = 30 10

24 N m2, Gittes et al. [17] foundthat the spring constant of an MT with L = 10 m is

approximately

10

7

N/m. Consequently a force ofonly 1 pN is able to deflect an MT by up to 1 m which issupported by measurements using laser tweezers [8]. How-ever, such large deflections are not produced by shearingforces as calculated above. The difference in the physi-cal mechanisms is that shearing forces lead to stackingdisplacements at right angles to the stack while flexuralforces cause a small-angle bending between dimers in ad-jacent stacks. Clearly, it is easier to bend a filament thanto exert a shearing force.

All the theoretical estimates of the molecular springconstants required a knowledge of the tubulin dimer-dimerinteraction potential. In Section 3 we discussed in some

detail how the estimate of the spring constant was madebased on molecular-dynamics simulations or experiments.This is still a major computational challenge that mer-its an independent study in its own right. Fortunately,the published atomic resolution structure of tubulin af-forded us with an opportunity to carry out preliminarymolecular-dynamics simulations for this problem that pro-vided only order of magnitude estimates with some degreeof confidence. With the knowledge of one particular inter-action strength between two neighboring dimers [25,26] asguidance, we have obtained rough estimates of the molec-ular spring constants along three different directions onthe MT surface.

An objection may be raised that due to the fact thatan MT is an anisotropic elastic structure, it posesses manymore deformation tensor components than those intro-duced in the present article. In particular elastic coef-ficients corresponding to the various off-diagonal effectslinking forces applied in one direction and deformations inanother have not been exhaustively analysed here. This istrue and the deformation tensor would in general contain21 components [33,34]. However, both experimentally andtheoretically we are not yet at the stage of being able toestimate many of these off-diagonal components. An MTsstructure is highly anisotropic elastically and hence we ex-pect the direct effects of the application of compressionaland tensional forces to be dominant. In our estimates ofthe continuum elastic coefficients the link was made to themolecular structure and protein-protein interactions. Wehave used, in addition to geometrical properties of MTs,approximate values for the handful of spring constantsdescribing the dimer-dimer interactions in an MT. Oncemore detailed molecular-dynamics simulations reveal ad-ditional interaction constants we will be able to extend themodel to include the remaining tensor components. Alter-natively, experimental techniques may be utilized to domeasurements of the dispersion relations for the variousphonon modes, e.g. through Brillouin scattering, whichwould provide the required information indirectly as isdone in the studies of ferroelasticity.

In conclusion, while the gross elastic anisotropic fea-tures of MTs are quite well explained with the existing


7/7


models and parameter values, still more work is requiredto obtain a higher level of confidence and excellent quan-titative agreement.

This project has been supported by grants from NSERC andMITACS. J.M. Dixon would like to thank the staff and mem-bers of the Physics Department of the University of Alberta, forall their kindness and thoughtfulness during his stay. S. Portetwas supported by Genome Canada, the Ontario R&D Chal-lenge Fund and the Canadian Institutes of Health Research.

References

1. P.A. Janmey, U. Euteneuer, P. Traub, M. Schliwa, J. CellBiol. 113, 155 (1991).

2. L. Ma, J. Xu, P.A. Coulombe, D. Wirtz, J. Biol. Chem.274, 19145 (1999).

3. D. Boal, Mechanics of the Cell (Cambridge UniversityPress, 2002).4. P. Meurer-Grob, J. Kasparian, R.H. Wade, Biochemistry

40, 8000 (2001).5. A. Kis, S. Kasas, B. Babic, A.J. Kulik, W. Benoit, G.A.D.

Briggs, C. Schonenberger, S. Catsicas, L. Forro, Phys. Rev.Lett. 89, 248101 (2002).

6. I.M. Janosi, D. Chretien, H. Flyvbjerg, Eur. Biophys. J.27, 501 (1998).

7. D. Chretien, H. Flyvbjerg, S.D. Fuller, Eur. Biophys. J.27, 490 (1998).

8. J. Howard, Mechanics of Motor Proteins and the Cy-closkeleton (Sinauer Associated Inc., 2001).

9. Dimitrije Stamenovic, Srboljub M. Mijailovich, Iva Marija

Tolic-Norrelykke, Jianxin Chen, Ning Wang, Am. J. Phys-iol. Cell Physiol. 282, 617 (2002).10. J.A. Tolomeo, M.C. Holley, Biophys. J. 73, 2241 (1997).11. O. Wagner, J. Zinke, P. Dancker, W. Grill, J. Bereiter-

Hahn, Biophys. J. 76, 2784 (1999).12. Y.M. Sirenko, M.A. Stroscio, K.W. Kim, Phys. Rev. E 53,

1003 (1996).13. M. Sato, W.H. Schwartz, S.C. Selden, T.D. Pollard, J. Cell

Biol. 106, 1205 (1988).

14. L. Cassimeris, D. Gard, P.T. Tran, H.P. Erickson, J. CellSci. 114, 3025 (2001).

15. M. Elbaum, D.K. Fygenson, A. Libchaber, Phys. Rev.Lett. 76, 4078 (1996).

16. D. Kuchnir Fygenson, M. Elbaum, B. Shraiman, A. Libch-aber, Phys. Rev. E 55, 850 (1997).

17. F. Gittes, B. Mickey, J. Nettleton, J. Howard, J. Cell Biol.120, 923 (1993).

18. B. Mickey, J. Howard, J. Cell Biol. 130, 909 (1995).19. J. Mizushima-Sugano, T. Maeda, T. Miki-Noumura,

Biochim. Biophys. Acta 755, 257 (1983).20. Francois Nedelec, J. Cell Biol. 158, 1005 (2002).21. H. Felgner, R. Frank, M. Schliwa, J. Cell Sci. 109, 509

(1996).22. Y.M. Sirenko, M.A. Stroscio, K.W. Kim, Phys. Rev. E 54,

1816 (1996).23. E. Nogales, S. Wolf, K. Dowling, Nature 391, 199 (1998).24. J. Lowe, H. Li, K. Dowling, E. Nogales, J. Mol. Biol. 313,

1045 (2001).25. David Sept, Nathan A. Baker, J. Andrew McCammon, in

46th Annual Meeting of the Biophysical Society, San Fran-

sico, CA, 2002, Book of Abstracts.26. David Sept, Nathan A. Baker, J. Andrew McCammon,

Protein Sci. 12, 2257 (2003).27. W. Humphrey, A. Dalke, K. Schulten, J. Mol. Graph. 14,

33 (1996) http://www.ks.uiuc.edu/Research/vmd/.28. Vincent VanBuren, David J. Odde, Lynne Cassimeris,

Proc. Natl. Acad. Sci. U.S.A. 99, 6035 (2002).29. P.J. de Pablo, I.A.T. Schaap, F.C. MacKintosh, C.F.

Schmidt, Phys. Rev. Lett. 91, 098101 (2003).30. E. Nogales, Cell. Mol. Life Sci. 56, 133 (1999).31. F. Ooshawa, S. Asakura (Editors), The Thermodynamics

of the Polymerization of Protein(Academic Press, London,1975).

32. M. Kurashi, M. Hoshi, H. Tashiro, Cell Motil. Cytoskeleton30, 221 (1995).

33. L.D. Landau, E.M. Lifshitz, L.P. Pitaevshii, Theory ofelasticity, 3rd edition (Butterworth Heinamann, 1995).

34. M.P. Marder, Condensed Matter Physics (John Wiley andSons, New York, 2000).

j.a. tuszynski, t. luchko, s. portet and j.m. dixon: anisotropic elastic properties of microtubules

Documents