ARTICLE

S. Portet Æ J. A. TuszynskiC. W. V. Hogue Æ J. M. Dixon

Elastic vibrations in seamless microtubules

Received: 14 July 2004 / Revised: 18 December 2004 / Accepted: 18 December 2004 / Published online: 11 May 2005� EBSA 2005

Abstract Parameters characterizing elastic properties ofmicrotubules, measured in several recent experiments,reflect an anisotropic character. We describe the micro-scopic dynamical properties of microtubules using adiscrete model based on an appropriate lattice of dimers.Adopting a harmonic approximation for the dimer–di-mer interactions and estimating the lattice elastic con-stants, we make predictions regarding vibrationdispersion relations and vibration propagation velo-cities. Vibration frequencies and velocities are expressedas functions of the elastic constants and of the geome-trical characteristics of the microtubules. We show thatvibrations which propagate along the protofilament doso significantly faster than those along the helix.

Keywords Microtubule Æ Microtubule structure ÆMicrotubule dynamics Æ Lattice Æ Elastic vibrations

Introduction

Microtubules (MTs) are proteins organized in a networkthat is interconnected with microfilaments and inter-mediate filaments to form the cytoskeleton. Each of theprotein fibers has specific physical properties andstructures suitable for its role in the cell. MTs are long,hollow, cylindrical objects made up of protofilamentsinteracting laterally, and consisting of longitudinally

stacked a,b-tubulin heterodimers; longitudinal and lat-eral bonds exhibit different strengths (VanBuren et al.2002). In vivo MTs have typically 13 protofilaments but,for example, 15 protofilament MTs are found in Cae-norhabditis elegans. Unger et al. (1990) observed in vitroMTs with a number of protofilaments ranging from 8 to16 depending on the assembly conditions. The MTstructure can be represented by a helical surface latticeinterrupted by mismatches which are called seams(Chretien and Wade 1991). To accommodate differentnumbers of protofilaments the whole lattice is rotated tocompensate the extra protofilaments and to absorb themismatch (Ray et al. 1993). In the cell MTs are normallyorganized in an aster radiating near the nucleus towardsthe cell periphery. MTs are involved in a number offunctions of the cell, such as cell shape maintenance,mitosis and play an important role in intracellulartransport. They form the mitotic spindles for the segre-gation of chromosomes during cell division, and they aresupports for directional transport driven by motorproteins. MTs are highly dynamic polymers, character-ized, for example, by dynamic instabilities, successiveperiods of growth and shrinkage. Under specific condi-tions treadmilling may occur, where addition of het-erodimers at one end of an MT and release at the otherend take place at the same rate.

There have been a number of experimental studies inrecent years dealing with the various aspects of theelasticity of MTs. By biological standards, MTs are rigidpolymers with a large persistence length of 6 mm (Boal2002). From Janmey’s experiments (Janmey et al. 1991),MTs suffer a larger strain for a small stress comparedwith either microfilaments or intermediate filaments.The rupture stress for MTs is very small and typically isonly about 0.4–0.5 N/m2 (Janmey et al. 1991). From theliterature, we can observe that the shear and Young’smoduli are significantly different, which reflects struc-tural anisotropy of the material (Kis et al. 2002). Ananisotropic material has physical characteristics withdifferent values when measurements are made in severaldirections within the same material: e.g., values of the

S. Portet (&) Æ C. W. V. HogueThe Samuel Lunenfeld Research Institute,Room 1060, Mount Sinai Hospital,600 Univercity Avenue, Toronto,ON, Canada, M5G 1X5E-mail: sportet@mshri.on.ca

J. A. TuszynskiDepartment of Physics, University of Alberta,Edmonton, AB, Canada, T6G 2J1

J. M. DixonDepartment of Physics, University of Warwick,Coventry, CV4 7AL, UK

Eur Biophys J (2005) 34: 912–920DOI 10.1007/s00249-005-0461-4

Young’s modulus ranging from 106 to 109 N/m2 can befound in the literature (Kis et al. 2002; Tolomeo andHolley 1997).

A few already available dynamical elastic propertiesof MTs are listed in Table 1 although many more arestill to be determined. In connection with this it is worthnoting that there are several papers in the literaturedealing with static and elastic properties (Janosi et al.1998) and the vibrational properties of cytoskeleton fi-laments and, in particular, MT vibrations. Elastic vi-brations of subunits around their equilibrium positionsin a lattice have been called phonons in the literature(Pokorny et al. 1997; Sirenko et al. 1996b).

Sirenko et al. (1996b) studied theoretically the vi-brational properties of MTs by adopting a continuummedium approximation in cylindrical geometry for anisotropic material. By describing filaments as elasticshells immersed in water, they identified several modesof vibration—radial, torsional and longitudinal—withthe attendant dispersion relations. Their predicted pho-non velocities ranged from approximately 200 to600 m/s. In a sequel paper (Sirenko et al. 1996a) theseauthors arrived at a higher estimate of the phonon ve-locities, namely 800–1,300 m/s. They also predicted theexistence of an infinite set of helical waves with parabolicdispersion relations. While these results are very usefulin identifying the most likely modes of vibrational pro-pagation, they still await experimental confirmation.The structure of MTs is highly heterogeneous, and bothexperimental measurements and theoretical analyses in-dicate a highly anisotropic dynamical picture (Kis et al.2002). Thus, we believe that a discrete approach is notonly adequate, but it also allows the expression of MTstructural anisotropy in a natural way.

Pokorny et al. (1997) attempted to find dispersionrelations based on a monatomic linear chain model re-presenting individual protofilaments made up ofmonomers, and by using additivity properties they de-scribed an MT and concluded that phonon frequenciesin the range 107–1010 1/s can be expected. They alsoshowed that the energy can be supplied by the hydrolysisof GTP.

Our approach will go further than the work of Po-korny et al. (1997) by modeling the structure of MTs bya cylinder unfolded into a planar lattice of tubulin dimerssatisfying appropriate boundary conditions in ac-cordance with the essential geometrical properties ofMTs. The structural transformation to accommodatedifferent numbers of protofilaments is the rotation of thewhole lattice (Chretien and Wade 1991), without mod-ifying the interprotofilament interactions. Consequentlythe local geometry of the lattice, (i.e., longitudinal and

lateral interactions between dimers) is preserved. Hence,from the canonical lattice describing a 13-protofilamentMT, we deduce the general dimensions of the lattice ofdimers representing an MT independently of the numberof protofilaments. From this lattice, we develop a mi-croscopic model of classical vibration modes to studythe dynamics of dimers within the MT surface (exclud-ing radial oscillations). We estimate dispersion relationsby using heterogeneous elastic coefficients that reflect theanisotropic character of MTs. The elastic coefficientvalues are obtained from molecular dynamics simula-tions of tubulin–tubulin interactions (Sept et al. 2003)and from experimental data (de Pablo et al. 2003). Ourpredicted vibration frequencies and corresponding ve-locities, which are expressed as functions of elasticconstants and structural properties of MTs, are in thesame range as those estimated by Sirenko et al. (1996a,1996b), and slightly larger than those estimated by Po-korny et al. (1997). Although no experiments have yetbeen performed to validate these predictions, they areclearly within today’s experimental capabilities. Thus, byusing recently published information regarding elasticproperties and the molecular geometry of MTs we de-scribe at a microscopic level the dynamical elasticproperties of MTs by accounting for their anisotropy.

Modeling

In the literature, theoretical and experimental data canbe found for an MT description using a lattice ofmonomers (Amos 1995; Chretien and Wade 1991), butrarely by a lattice of dimers (Metoz and Wade 1997).Chretien and coworkers (1991, 1998) have given precisedescriptions of MT structures in terms of a surface lat-tice composed of tubulin monomers in the lattice ac-commodation model. In an MT with 13 protofilaments,the protofilaments are parallel to the MT axis forming ahollow cylinder (Chretien and Wade 1991). Tubulinmonomers are longitudinally aligned to form the pro-tofilaments; the monomer spacing along the protofila-ment is b=4.05 nm (Chretien et al. 1998). Eachprotofilament is shifted lengthwise with respect to itsneighbour describing left-handed helical pathways run-ning around the MT surface. The longitudinal shift be-tween monomers of adjacent protofilaments ise=0.935 nm (Chretien et al. 1998). The accumulatedcircumferential shift is 13·0.935 nm, which is equal to ahelix pitch of 3·4.05 nm, indicating that three helices ofmonomers are necessary to describe the whole MTsurface (Chretien et al. 1998). The separation betweenprotofilaments is d=5.13 nm (Chretien et al. 1998). The13-protofilament and 3-start helix MT is the so-calledcanonical MT.

There exists an important variability in the number ofprotofilaments (Unger et al. 1990). MTs accommodatedifferent numbers of protofilaments by skewing theirprotofilaments (Langford 1980), without modifiyingtheir lateral interactions. The structural alteration is

Table 1 Dynamical elastic properties of microtubules (MTs)

Phonon frequencies 107–1010 1/s (Pokorny et al. (1997)Phonon velocities 200–600 m/s (Sirenko et al. 1996b)

100–1,000 m/s (Pokorny et al. 1997)800–1,300 m/s (Sirenko et al. 1996a)

913

equivalent to the rotation of the whole protofilamentsheet by an angle h that can be expressed as a function ofthe number of protofilaments N, the number of helices Sconstituting the MT, the separation between protofila-ments d, the longitudinal shift between adjacent proto-filaments e, and the spacing between subunits b(Chretien et al. 1998):

tan h ¼ 1

dSbN� e

� �: ð1Þ

Thus, the interactions between protofilaments arelocally preserved in MTs. In other words, the localgeometry, i.e., the subunit neighbourhood geometry, isindependent of the protofilament numbers (Ray et al.1993). The number of protofilaments only influences theMT radius r (see Fig. 7 in Chretien et al. 1998) and theskew angle h. Hence, the local geometry of the subunitneighbourhood can be deduced from the geometry of thecanonical MT.

To study the dimer vibrations inside the MT wall, weuse a lattice model of dimers which is considered to be aseries of coupled oscillators whose equilibrium positionsrepresent the geometry of the MT.

Lattice of dimers

The longitudinal interdimer interactions are very similarto those between monomers within a dimer. However,an extra electrostatic interaction exists between mono-mers that does not occur between dimers. This compo-nent affects the strength of longitudinal intradimerinteractions (Nogales 1999), which are stronger thanlongitudinal interdimer interactions (Nogales et al.1999). Furthermore, the basic subunit for the MT as-sembly is not the monomer but the dimer. Consequently,we assume longitudinal intradimer interactions to bestructurally stable inside the MT wall, and we investigatethe vibration dynamics of dimers inside the MT surface.

A seamless MT can be modeled as a three-dimen-sional helical lattice of dimers (Fig. 1a) (Metoz andWade 1997). We consider dimers as points at theircenters of mass (Fig. 1b). The pitch of helices that runaround the MT wall is proportional to the equilibriumdistance, b, between adjacent dimers along a protofila-

ment. Hence pitch is equal to Sb, where S represents thenumber of helices necessary to describe a whole MTsurface (Fig. 1b). Helices are well defined by a pitch, Sb,and a radius, r, and are characterized by the radius ofcurvature expressed as r 1þ Sb=2prð Þ2

h i. From helical

geometry, we can calculate the dimensions of the dimerneighbourhood in the lattice.

We open the three-dimensional lattice along thelength of a protofilament, parallel to the MT axis as weconsider a 13-protofilament MT, and unroll it to obtaina two-dimensional planar lattice of dimers. The char-acteristic dimensions of the two-dimensional planarlattice arise directly from the three-dimensional latticegeometry; hence, all equilibrium angles and distancesbetween neighbouring dimers are preserved (Figs. 1c,2a). By construction and from the radius of curvature ofhelices, we obtain the equilibrium distance, a, separatingadjacent dimers along a given helix or layer (Fig. 2a):

a ¼ 2r 1þ Sb2pr

� �2" #

sinpN: ð2Þ

As we consider the geometry of the canonical MT,the number of protofilaments is N=N0=13 and its ra-dius is r=r0. Furthermore, the canonical MT has 3-starthelices of monomers (Chretien and Wade 1991), whichcorresponds to S=S0=2 helices of dimers. From thehelix angle (in the helical structure, the angle between arow of dimers and the horizontal), arctan(Sb/2pr), wecalculate the equilibrium distance, d, between adjacentprotofilaments, and the equilibrium longitudinal shift, e,between adjacent protofilaments (Fig. 2a):

d ¼ a 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þðSb=2prÞ2p ¼ 2r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ Sb

2pr

� �2qsin p

N ;

e ¼ a Sb=2prffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ Sb=2prð Þ2p ¼ Sb

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ Sb

2pr

� �2qsin p

N :ð3Þ

The notation is as before. Protofilaments interactlaterally with a lateral spacing d. Each protofilament isshifted lengthwise by e with respect to its neighbour.Dimensions of the dimer neighbourhood in the latticeare listed in Table 2.

To calculate the position vector, Rij, in the two-di-mensional plane, of a dimer that belongs to the proto-filament i¢ and to the layer of dimers j¢ in a three-dimensional structure representing the canonical MT,we use the mapping

f : 1::Nf g � N! R2

ði0; j0Þ7!f ði0; j0Þ ¼ Rij ¼ðj0 � 1Þbþ Frac i0�1

N

� �Ne

Frac i0�1N

� �Nd

" #: ð4Þ

In the three-dimensional helical structure, i¢ re-presents the protofilament (1 £ i¢ £ N), and j¢ is theindex of the dimer layer in the MT (1 £ j¢). In thetwo-dimensional planar lattice, i is related to the y-po-sition, j to the x-position: the x-axis is parallel to the

Table 2 Parameter values which, used in Eq. 4, give the two-di-mensional planar lattices of dimers in accordance with the MT’sgeometry and with no mismatch as shown in Figs. 1c and 2a

N0 r0 S0 b Pitch a d e

13 12 2 8 16 6.01 5.88 1.24

N0 and S0 represent the number of protofilaments and helices in thecanonical MT. Its radius r0 and the pitch (S0b) characterize thehelices defining the canonical MT. b is the distance between twoadjacent subunits along the same protofilament. a, d and e arecalculated by Eqs. 2 and 3. Values of r0, b, pitch, a, d and e areexpressed in nanometers

914

protofilament axis (Fig. 2a). In Eq. 4 the function Frac[Æ] is the fractional value function; in the first coordinate,it is used to identify the layer, while in the second itdesignates which protofilament of the MT the particulardimer belongs to. The neighbourhood of a dimer in thethree-dimensional helical lattice (Fig. 1b) is preserved inthe two-dimensional planar geometry owing to the per-iodicities implied by the use of the function Frac [Æ].Hence, in the two-dimensional planar lattice, for twoadjacent dimers Di,j and Di,j+1 along the same proto-filament, we have Ri,j+1=Ri,j+b, and for two adjacentdimers Di,j and Di+1,j along the same layer or helix,Ri+1,j=Ri,j+a. Subsequently, Ri+1,j+1=Ri,j+a+b.Vectors a and b are the lattice vectors, jjajj ¼ a andjjbjj ¼ b (Fig. 2a).

For noncanonical structures, the vector position, Ri,j,must be rotated by an angle h, given in Eq. 1 with theparameters of the dimer lattice to resolve the mismatchinduced by extra protofilaments.

Governing equations

By connecting adjacent dimers of the planar lattice bysprings with heterogeneous elastic constants (Fig. 2), we

model an MT by a lattice with anisotropic elasticproperties. This provides a framework to study the vi-brational dynamics of dimers within the MT surface.

To study the internal MT dynamics, we model thesmall displacements of dimers, inside the MT wall, froman equilibrium position. Any stretching or compressionof any bond between nearest-neighbour dimers (Fig. 2a)in the lattice will result in a restoring force obeyingHooke’s law. We denote the vector displacement, ui,j, ofthe dimer Di,j in the orthogonal coordinate system (x,y)(Fig. 2a) by

ui;j tð Þ ¼ xi;j tð Þ; yi;j tð Þ� �T

; ð5Þ

where xi,j(t) and yi,j(t) denote small displacements fromequilibrium of the dimer, Di,j, in the directions along theprotofilament (x) and orthogonal to it (y), respectively(Fig. 2a). In the three-dimensional helical structure,xi,j(t) corresponds to longitudinal displacements fromequilibrium, and yi,j(t) are analogous to angular dis-placements (Sirenko et al. 1996a).

As the deviations from equilibrium are small, weapproximate the elastic potential energy, /i,j, of a di-mer, Di,j, as a harmonic potential. Then, /i,j, has theform

Fig. 1 Geometrical model of amicrotubule (MT). Here thecanonical MT is used as anexample. a An MT is a long,hollow cylinder with a wallmade up of N protofilamentsconsisting of cylindrical tubulindimers longitudinally stacked.b An MT is a three-dimensionalhelical lattice made up of thecenters of mass of the dimers.Centers of mass of the dimersare aligned with a spacing b toform the protofilaments thatrun lengthwise along the wall.Each protofilament is shiftedlengthwise with respect to itsneighbour describing helicalpathways around the MT. Eachhelix has a radius r and a pitchSb, where S is the number ofhelices necessary to describe thewhole MT surface. The greyand black helices depict twohelices running around the MTwall. c The two-dimensionallattice of dimers: the three-dimensional lattice islongitudinally opened andflattened. Arrows represent theprotofilament axis

915

/i;j ¼1

2

X1l¼�1

X1m¼�1

kiþl;jþmjjui;jðtÞ � uiþl;jþmðtÞjj2 ð6Þ

where ki+l,j+m is the (l,m) element of the contact matrix,Ki,j, for the dimer, Di,j, defined by the neighbourhoodsystem shown in Fig. 2b. Hence, the matrix, Ki,j, has theform

Ki;j ¼0 kh kakp 0 kpka kh 0

24

35; ð7Þ

where kh is the elastic constant along the main helix, kpis the elastic constant along a protofilament, and ka isthe elastic constant along the antihelix (Fig. 2b). Thevalues of the elastic constants kh, kp and ka are assumedto be different from each other. This assumption directlyresults in an elastic lattice anisotropy that is well docu-mented in the literature (Kis et al. 2002), although pre-cise numerical magnitudes of these constants are yetunknown.

Using Newton’s second law of motion and the elasticpotential energy Eq. 6, we find the two Cartesian com-ponents of the equation of motion for the dimer Di,j tobe

m@2xi;j

@t2¼ kh xi�1;j þ xiþ1;j � 2xi;j

� �þ kp xi;j�1 þ xi;jþ1 � 2xi;j

� �þ ka xi�1;jþ1 þ xiþ1;j�1 � 2xi;j

� �;

ð8Þ

and

m@2yi;j

@t2¼ kh yi�1;j þ yiþ1;j � 2yi;j

� �þ kp yi;j�1 þ yi;jþ1 � 2yi;j

� �þ ka yi�1;jþ1 þ yiþ1;j�1 � 2yi;j

� �;

ð9Þ

where m is the mass of a dimer. From Eqs. 5, 8 and 9 weobtain

m@2ui;j@t2¼ kh ui�1;j þ uiþ1;j � 2ui;j

� �þ kp ui;j�1 þ ui;jþ1 � 2ui;j

� �þ ka ui�1;jþ1 þ uiþ1;j�1 � 2ui;j

� �:

ð10Þ

The boundary conditions are uN,j=u0,j+S, because di-mers DN,j and D1,j+S are adjacent along a given helix topreserve the helical pathways running around MTs.

Accordingly we look for wavelike solutions of Eq. 10of the form

ui;j ¼ ~cei k�Rij�xtð Þ; ð11Þ

where ~c is an arbitrary vector amplitude, x a fre-quency, k a two-dimensional wave-vector with com-ponents kx and ky, and Rij ¼ j� 1ð Þb cos hþ i� 1ð Þ:½e cos h� d sin hð Þ; j� 1ð Þb sin hþ i� 1ð Þ e sin hþð

d cos h�T is the position vector of the dimer Di,j. Herethe position vector, Ri,j, results from a rotation of thecorresponding canonical structure position vector byangle h. We substitute Eq. 11 into Eq. 10 to find theassociated dispersion relation between x and k givenby

Fig. 2 Geometrical characteristics of the planar lattice of dimers.The x-axis is the protofilament axis represented by arrows in Fig. 1.a Lattice representing the canonical MT, where a=(e,d) andb=(b,0). The longitudinal shift between two adjacent protofila-ments is e. Here d represents the distance between two adjacentprotofilaments, and b is the distance between two adjacent dimersalong a given protofilament. Parameter values are given in Table 2.Bonds in the lattice describe the nearest-neighbour dimers ininteraction with a given dimer. b The neighbourhood system of agiven dimer Di,j. c The canonical MT lattice is shown in black andthe lattice for an MT made up of 15 protofilaments and 3 helices(corresponding to the 15-protofilament and 5-start helix MT in themonomer lattice) is in gray. To accommodate the two extraprotofilaments the lattice is rotated by an angle h

916

x2¼ 2

mkh 1�cos a½kxcosðaþhÞþky sinðaþbÞ�

� �� ��

þkp 1�cos b½kx coshþky sinh�� �� �

þka 1�cos kxðbcosh�acosðaþhÞÞð½

þkyðbsinh�asinðaþbÞÞ���

: ð12Þ

For an M·N orthogonal two-dimensional latticerepresenting the canonical structure composed of Mdimers along the x-direction, and N dimers along the y-direction, the wave-vectors, k, may be defined by (Kittel1956)

kx ¼qp

b M þ 1ð Þ ; ky ¼pp

a N þ 1ð Þ ; ð13Þ

where p=1...N and q=1...M. To consider MTs with adifferent number of protofilaments we must rotate thewave-vector by an angle h. So the general expression ofthe vector, k, is defined by

kx ¼ qpb Mþ1ð Þ cos h� pp

a Nþ1ð Þ sin h;

ky ¼ qpb Mþ1ð Þ sin hþ pp

a Nþ1ð Þ cos h;ð14Þ

where h (see Eq. 1) is the skew angle from the axis of theMT.

For our slanted lattice, the wave-vectors are such that

~kx ¼ kx;~ky ¼ ky sin aþ kx cos a;

ð15Þ

where the angle a is the complementary angle of thehelix angle, a ¼ p=2ð Þ � arctan Sb=2prð Þ. The component~kx is related to the protofilament direction, and ~ky isrelated to the orientation of helices. The parameter M isproportional to the number of helices, S, runningaround the MT wall.

Substituting the previous expressions from Eqs. 14and 15 into Eq. 12 gives the dispersion relation betweenx and ~k:

For small p,q we note that xp,q fi 0 in Eq. 16. Theytherefore represent acoustic vibrational modes.

By estimating the gradient of xp,q in specific direc-tions, we obtain the group velocity of vibrations pro-pagating along the protofilament and along the helix.The velocity of longitudinal vibrations along protofila-ments is

vx ¼ffiffiffiffi1

m

rkha2 cosðaþ hÞ � cot a sinðaþ hÞð Þ2�

þ kpb2 cos h� cot a sin hð Þ2

þ ka b cos h� a cosðaþ hÞð

� cot a b sin h� a sinðaþ hÞð ÞÞ21=2

: ð17Þ

The velocity of vibrations along the helix is

vy ¼ ad

ffiffiffi1m

qkha2 sin2 aþ hð Þ þ kpb2 sin2 h

þka½b sin h� a sinðaþ hÞ�2o1=2

:ð18Þ

The velocities of vibrations (along protofilaments andalong helices) depend on the elastic constants, kp, kh andka, and the mass, m, of tubulin dimers. They also dependon the structural properties of MTs determined by theskew angle h and the helix angle a. In the three-dimen-sional helical structure, at a macroscopic level, thelongitudinal vibrations along protofilaments correspondto longitudinal vibrations of the MT, and vibrationsalong helices to torsional vibrations of the MT.

Results

Elastic constants

Longitudinal interactions between dimers have polar,hydrophobic and electrostatic components (Nogaleset al. 1998). The contact interface is highly com-plementary in shape; hence, van der Waals interactionsare important. Lateral interactions (between a–a and b–

b) have more important electrostatic character thanlongitudinal interactions (Nogales et al. 1999). The lat-

x2p;q ¼

2

mkh 1� cos a

qpbðM þ 1Þ cos h� pp

aðN þ 1Þ sin h

� �����cosðaþ hÞ

þ qpbðM þ 1Þ sin hþ pp

aðN þ 1Þ cos h

� �sinðaþ hÞ

���

þ kp 1� cos bqp

bðM þ 1Þ cos h� ppaðN þ 1Þ sin h

� ����cos h

þ qpbðM þ 1Þ sin hþ pp

aðN þ 1Þ cos h

� �sin h

���

þ ka 1� cosqp

bðM þ 1Þ cos h� ppaðN þ 1Þ sin h

� ���b cos h� a cosðaþ hÞð Þ

þ qpbðM þ 1Þ sin hþ pp

aðN þ 1Þ cos h

� �b sin h� a sinðaþ hÞð Þ

���: ð16Þ

917

eral contacts between tubulin dimers in neighbouringprotofilaments have a decisive role for MT dynamics,stability, rigidity and architecture (Meurer-Grob et al.2001; Nogales et al. 1999). Tubulin dimers are relativelystrongly bound in the longitudinal direction (alongprotofilaments), while the lateral interaction betweenprotofilaments is much weaker (Kis et al. 2002; Nogales1999; Sept et al. 2003); hence, kp>kh, ka. While all thelongitudinal intraprotofilament interactions are assumedto be the same, the lateral interactions between theprotofilaments are known to be different (Li et al. 2002)depending on direction; hence, kh „ ka.

Tuszynski et al. (2005) have done molecular dynamicscomputations to study the electrostatic properties oftubulin heterodimers. Sept et al. (2003) performed acomputation of the protofilament–protofilament bind-ing free energy as a function of the longitudinal shiftalong the protofilament axis. The data published in thislatter work indicate the presence of two stable equili-brium positions that correspond to MT lattice types Aand B. They also discriminate the polar and electrostaticcontribution to the free binding energy. We have pro-cessed the potential diagram published by Sept et al.(2003) in their Fig. 1 in order to evaluate the corre-sponding elastic coefficient in a harmonic approximationaround the potential minimum. We found the value tobe k.4 N/m.

De Pablo et al. (2003) estimated experimentally byradial indentation of MTs with a scanning force mi-croscope tip an elastic constant value related to lateralinteractions between protofilaments. Indentations, in-duced by the nanometer-sized tip positioned on the top

of an MT, result in a linear elastic response with theelastic constant k=0.1 N/m.

The estimate of Sept et al. (2003) corresponds to in-teractions between two protofilaments when one is slidalong its length against the other that is kept fixed, theprotofilament–protofilament shift along the protofila-ment axis. However, the estimates of de Pablo et al.(2003) correspond to a radial dependence of the force onthe distance to the centre of the MT.

Vibration dynamics

While we have no direct experimental evidence at pre-sent to claim the anisotropy in the elastic coefficients ka,kh and kp, it is more prudent to allow these parametersto differ from each other. To approximately representthe anisotropy, we judiciously chose kp=4.5 N/m,kh=0.1 N/m and ka=0.01 N/m. The mass of the dimeris taken to be m=1.83·10�22 kg.

Dispersion relations along the protofilament andhelix directions are represented in Fig. 3. Frequenciesreach 3·1011 1/s along the protofilament direction, andonly 5·1010 1/s along the helix direction. Both long-itudinal and transverse modes are acoustical (Fig. 3).The corresponding propagation velocities range be-tween 145 and 1,260 m/s, where the smallest velocity isrelated to the helix direction and the largest to theprotofilament direction. The range of propagationvelocities is in accordance with the previously pre-dicted values (Pokorny et al. 1997; Sirenko et al.1996a, 1996b). Our frequencies are slightly higher thanthe predicted frequencies of Pokorny et al. (1997)(Table 1). These results, however, are still subject toexperimental determination. It is noteworthy that thevibrations which propagate along the protofilament doso significantly faster than those along the helix,vx>vy (Table 3).

Table 3 Estimates of frequencies x, velocities v and wavelengths kalong the protofilament direction (labelled x) and along the helicalpathway (labelled y) for four MTs

S=2A N=13 xx=3.15·1011 1/s xy=4.8·1010 1/s

vx=1,256 m/s vy=147 m/skx=3.9·10�9 m ky=3·10�9 m

B N=15 xx=3.15·1011 1/s xy=5·1010 1/svx=1,237 m/s vy=151 m/skx=3.9·10�9 m ky=3·10�9 m

S=3C N=13 xx=3.15·1011 1/s xy=4.8·1010 1/s

vx=1,222 (m/s) vy=197 (m/s)kx=3.8·10�9 m ky=4.1·10�9 m

D N=15 xx=3.15·1011 1/s xy=5·1010 1/svx=1,263 m/s vy=167 m/skx=4·10�9 m ky=3.3·10�9 m

A the canonical MT (N=13 and S=2), B the MT with 15 proto-filaments and 2 helices, C the MT with 13 protofilaments and 3helices [3 helices in the dimer lattice corresponds to 5-start helices inthe monomer lattice (Chretien et al. 1998)] and D the MT with 15protofilaments and 3 helices

Fig. 3 Dispersion relations for the canonical MT. The curvelabeled Longitudinal represents the dispersion relation forvibrations propagating along protofilaments. Frequencies can yield3· 1011 1/s. The curve labeled Transverse is the dispersion relationfor vibrations propagating along helices. Frequencies can reach5·1010 1/s

918

In Fig. 4 we represent the effects of the structuralproperties of MTs (number of protofilaments and he-lices) on the vibration velocities. The vibration velocitiesalong the protofilament and helix directions are verysensitive to the number of protofilaments and to thenumber of helices. The number of helices, S, has a largerinfluence than the number of protofilaments. Further-more S has a major influence when the number of pro-tofilaments is very small, i.e., N=8.

In Table 3 we give our estimates of the frequencies,velocities and wavelengths along the direction of theprotofilaments and of the helices.

Discussion

From the helical structure of MTs, we developed aplanar lattice of dimers that are considered to be coupledoscillators vibrating about equilibrium positions. Theuse of a discrete model was necessary to obtain vibrationdispersion relations accounting for the geometricalproperties of MTs and particulars of protein–proteininteractions. Our planar lattice of dimers without a seamrespects the geometrical properties of MTs. Equations ofmotion for dimers were derived and an ansatz of har-monic solutions found which satisfy the periodicboundary conditions. We obtained dispersion relationsfor a number of vibrational acoustic modes and thecorresponding propagation velocities. We give explicitexpressions for vibration frequencies and velocities,along both protofilament and helix directions. Thesequantities are functions of only elastic constants andstructural parameters of MTs.

Thus, from the planar lattice we are able to studyvibration modes of dimers along protofilament and helixdirections, which respectively correspond to thelongitudinal and torsional modes of vibrations of dimersinside the MT wall. It is noteworthy that the vibrationswhich propagate along the protofilament do so sig-nificantly faster than those along the helix. We alsoobserve a strong dependence of the microscopic vibra-

tional properties of MTs on the number of protofila-ments and of helices. Finally, our predicted vibrationalfrequencies and corresponding velocities are in the samerange as those estimated by Sirenko et al. (1996a,1996b), and slightly larger than those estimated byPokorny et al. (1997) (Table 1).

This work was possible owing to a recently obtainedexperimental body of data coming from several groupsand describing a number of physical and structuralproperties of MTs (de Pablo et al. 2003; Kis et al. 2002;Nogales 1999; Sept et al. 2003). The most marked aspectof MTs observed was the anisotropic character of MTelasticity that is reflected in the heterogeneity of elasticcoefficients. Theoretical estimates of the elastic constantsrequired knowledge of the tubulin dimer–dimer inter-action potential or relevant experimental data. With theknowledge of particular interaction strengths betweentwo neighbouring dimers, resulting from the protofila-ment–protofilament shifting along the MT axis (Septet al. 2003) or during the radial indentations of MTs (dePablo et al. 2003), as guidance, we obtained rough es-timates of the molecular elastic constants along threedifferent directions on the MT surface. These valueswere used for the modeling of the vibrational dynamics.

In this work we were guided by earlier works thatexplored both the continuum medium approach to acylindrical hollow structure (Sirenko et al. 1996a) and aone-dimensional approximation to the calculation ofphonon dispersion (Pokorny et al. 1997). Nevertheless,we believe that neither the continuum model nor theone-dimensional approximation is adequate owing tothe heterogeneity of the MT structure. However, theknowledge of several modes of vibration as well as thepredicted dispersion relations and phonon velocitieswere very useful in our modeling. The radial motions arenot described in our model (Sirenko and Dutta 2001). Interms of the continuum medium approximation, ourMT would be a cylinder with a fixed radius.

The viscosity of the surrounding solution (in vitro) orof cytoplasm (in vivo) may affect vibrations by dampingthem out (Foster and Baish 2000). Foster and Baish

618

900

516

1000

1100

14 4 S

1200

12N 31028

518 4.5

200

300

16 4

400

14 3.5

500

S12 3N

600

2.510

700

28

a b

Fig. 4 The effects of structuralparameters on vibrationvelocities along theprotofilament and helixdirections. Velocities ofvibrations, vx and vy, as afunction of the number ofprotofilaments and of helices,along the protofilament andhelix directions, respectively

919

calculated the relaxation time caused by viscous damp-ing to be in the nanosecond region. For our smallestpredicted frequency, we obtain a period 2 orders ofmagnitude smaller than the relaxation time. Further-more, Pokorny (2004) has shown that the viscousdamping effects due to the cytoplasm viscosity can beminimized by the ion condensation around the MT.Thus, in our model, the viscous damping effects areneglected.

Our lattice of dimers without mismatch can be clas-sified as a B-type lattice where lateral contacts are madebetween a–a and b–b subunits. This lattice type is pre-dominantly observed in MT structures; however, MTscan possess a seam in which lateral contacts exist be-tween heterologous subunits. In our dynamical model,the seam could be described by different elastic constantsfor the boundary nodes to model differences of energyoccurring at these contacts. But no change will occur inthe geometrical model. This aspect still requires furtherattention.

Acknowledgements S.P. and C.H. were supported by GenomeCanada through the Ontario, Genomics Institute and the OntarioR&D Challenge Fund. J.T. was supported by grants from NSERCand MITACS. J.M.D. would like to thank the staff and membersof the Physics Department of the University of Alberta for all theirkindness and thoughtfulness during his stay.

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