Polymorphic Dynamics of Microtubules
aa r X i v : . [ q - b i o . B M ] M a y Polymorphic Dynamics of Microtubules
Herv´e Mohrbach , Albert Johner and Igor M. Kuli´c ∗ Groupe BioPhysStat, Universit´e Paul Verlaine, 57078 Metz, France CNRS, Institut Charles Sadron, 23 rue du Loess BP 84047, 67034 Strasbourg, France (Dated: November 4, 2018)Starting from the hypothesis that the tubulin dimer is a conformationally bistable molecule -fluctuating between a curved and a straight configuration at room temperature - we develop a modelfor polymorphic dynamics of the microtubule lattice. We show that tubulin bistability consistentlyexplains unusual dynamic fluctuations, the apparent length-stiffness relation of grafted microtubulesand the curved-helical appearance of microtubules in general. Analyzing experimental data weconclude that taxol stabilized microtubules exist in highly cooperative yet strongly fluctuating helicalstates. When clamped by the end the microtubule undergoes an unusual zero energy motion - in itseffect reminiscent of a limited rotational hinge.
PACS numbers: 87.16.Ka, 82.35.Pq, 87.15.-v
Microtubules are the stiffest cytoskeletal componentand play versatile and indispensable roles in living cells.They act as cellular bones, transport roads [1] and cy-toplasmic stirring rods [2]. Microtubules consist of el-ementary building blocks - the tubulin dimers - thatpolymerize head to tail into linear protofilaments (PFs).PFs themselves associate side by side to form the hol-low tube structure known as the microtubule (MT). De-spite a long history of their biophysical study a deeperunderstanding of MT’s elastic and dynamic propertiesremains elusive to this date. Besides the unusual poly-merization related non-equilibrium features like ”tread-milling” and the dynamic instability there are a numberof other experimental mysteries - in thermal equilibrium-that presently defy coherent explanations, most notably:(i) The presence of high ”intrinsic curvature” [3]-[6] ofunclear origin, also identified as a long wave-length he-licity [3]. (ii) In various active bending [7] or thermalfluctuation experiments [4][5][8] MTs display length de-pendent, even non-monotonic apparent stiffness [5]. (iii)They exhibit unusually slow thermal dynamics in com-parison with standard semiflexible filaments [5][6].The most bizarre and controversial feature (ii) has beenthe subject of much debate and some theoretical expla-nation attempts based on low shear stiffness modulushave been put forward [9]. However a careful reanal-ysis of clamped MT experiments, Figs 2 , ∼ L while the relaxation timesscale as ∼ L . This exotic behavior naively suggests thepresence of a limited angular hinge at the MT clampingpoint. On the other hand artifacts that could triviallylead to a ”hinged behavior” (like loose MT attachmentand punctual MT damage) were specifically excluded inexperiments [4][5]. We will outline here a model basedon internal MT dynamics explaining phenomena (i)-(iii).It leads us to the origin of MT helicity (i) implying (ii)-(iii) as most natural corollaries [10]. The two centralassumptions of our model are as follows: (I) The tubulin
FIG. 1: Polymorphic Tube Model: (a) The tubulin dimerfluctuates between two states σ = 0 , P with phase angle φ describes the distribution of tubulinstates in the cross section. c) Polymorphic wobbling - thezero-energy motion of the phase angle at each cross-section.It is responsible for the rotation on a cone with opening angle α when clamped by the end. d) Defects in polymorphic order:Single defects (SD) have a cost proportional to their length.Double defects (DD) give only local energy contribution. dimer is a conformationally multistable entity and fluc-tuates between at least 2 states on experimental timescales. (II) There is a nearest-neighbor cooperative in-teraction of tubulin states along the PF axis. We arelead to assumptions I-II from several independent direc-tions: First , the experimentally observed MT helicity [3]implies that there is a symmetry breaking mechanism ofindividual PF’s conformational properties. In analogy tothe classic case of bacterial flagellum the existence of he-lices in azimuthaly symmetric bundles also necessitates acooperative longitudinal interaction along protofilaments[11][12].
Second , investigations of single protofilamentconformations by Elie-Caille et al [13] reveal that a singletaxol- PF can coexist in at least 2 states with comparablefree energy: a straight state κ P F ≈ κ P F ≈ / nm . Theseauthors also point out the apparent cooperative natureof straight to curved transition within single PFs. Third ,when mechanically buckled by AFM tips tubulin dimersoccasionally switch back to the initial straight conforma-tion [14].
Fourth , tubulin multistability was inferred fromthe formation of stable circular MT arcs in kinesin drivengliding assays by Amos & Amos [15]. Unfortunately theirclear, seminal observations were subsequently forgottenfor decades leading to much of the confusion about MTswe are witnessing today.
Polymorphic MT Model . Starting from assumptions I − II we model the tubulin dimer state by a twostate variable σ n ( s ) = 0 , n = 1 , ...N ( N = 11 − s . The total elastic + conformational energy canbe written as E MT = R L ( e el + e trans + e inter ) ds with e el = Y Z Z ( ε − ε pol ) rdrdα (1) e trans = − ∆ Gb X Nn =1 σ n ( s ), (2) e inter = − Jb X Nn =1 (2 σ n ( s ) −
1) (2 σ n ( s + b ) −
1) (3)where the integration in e el goes over the annular MTcross-section with r ≈ . nm, r ≈ . nm the innerand outer MT radii, with ∆ G > b ≈ nm the monomerlength, J the ”Ising” cooperative coupling term alongthe PF contour and with the polymorphism induced pre-strain ε pol ∝ ε P F σ n ( s ) [16] where ε P F is the strain gen-erated in the curved state. The latter can be estimatedfrom the switched PF curvature κ P F ≈ (250 nm ) − [13]to be ε P F = d P F κ P F / ≈ − . For an isotropic Euler-Kirchhoff beam, the actual material deformations are re-lated to the centerline curvature via ε = − ~κ · ~r with ~r theradial vector in the cross-section . Upon inspection it becomes clear that the phase be-havior (straight or curved state stability) is containedin the interplay of the first two terms e el and e trans while the thermal dynamics is governed by the 3rd e inter which rules over defect behavior (cf. Fig. 1d). Tounderstand the basic behavior we first consider a shortMT section along which the PFs are in a uniform state σ n ( s ) = σ n ( s + b ) ( e inter =const. can be dropped). Fur-thermore we resort to the single block ansatz, i.e. ateach cross-section there is only one continuous block ofswitched PFs of length p . This ansatz was successfullyused by Calladine in modelling bacterial flagellin poly-morphic states [11]. In this approximation the energydensity becomes e = B (cid:16) ( κ − κ pol ( p )) + κ (cid:16) γ πN p − sin (cid:16) πN p (cid:17)(cid:17)(cid:17) (4) with the bending modulus B = Y π (cid:0) r − r (cid:1) and thepolymorphic curvature κ pol ( p ) = κ sin (cid:0) πN p (cid:1) with κ = κ FP ( r − r ) π ( r + r ) . The MT phase behavior depends on thepolymorphic-elastic competition parameter γ = κ PF κ − N ∆ GbBκ . Physically, γ measures the ratio between poly-morphic energy of tubulin switching and the elastic costof the transition. For γ < − σ = 1 while for γ > σ = 0 - both corresponding toa straight MT. For − < γ < p = 0 or p = N )and curved state with p >
0. For − γ < γ < γ with γ ≈ .
72 the curved state is the absolute energy mini-mum and the straight state is only metastable. There-fore in this regime, the ground state of a microtubulebearing natural lattice twist will be helical (cf. Fig. 1b).Assuming a stable helical state as observed in [3] wehave p/N ∈ [1 / , /
2] giving us an estimate for the ra-dius of curvature κ − pol ≈ − µm . This compares favor-ably with an estimate of observed helices κ − ≈ µm from [3]. The helical stability and the magnitude of theprotofilament curvature κ P F ≈ / nm [13] with a typi-cal protein Young modulus Y ≈ − GP a, allows us alsoa simple estimate of the transition energy per monomer∆ G ≈ +1 . kT. In general, the energy in Eqs.1-3gives rise to a complex behavior and we focus on basicphenomena. It turns out that a most remarkable devia-tion from standard wormlike chain behavior comes fromthe change of polymorphic phase that we consider in thefollowing.
Polymorphic Phase Dynamics.
To better under-stand the central phenomenon, we define at each MTcross-section the complex polymorphic order parameter P ( s ) = P Nn =1 e πin/N σ n ( s ) = | P ( s ) | e iφ ( s ) where | P ( s ) | denotes the ”polymorphic modulus” and φ the ”polymor-phic phase” (cf. Fig. 1b). The polymorphic state canthen be described by the local (complex) centerline cur-vature ˆ κ pol ( s ) = κ e iq s P ( s ) with κ = κ sin π/N and q the natural lattice twist that varies with PF number [17].This gives rise to a helical MT shape described by the cur-vature | ˆ κ pol | = κ pol and torsion τ ≈ φ ′ + q . For large act-ing forces both the polymorphic phase φ and amplitude | P | will vary along the contour, however for small (ther-mal) perturbations the phase fluctuations will be domi-nant [18]. Based on this and on the observation of stablehelical states [3] we will now assume | P | = const, andwrite the total energy of the MT whose centerline deflec-tion is described by a complex angle θ ( s ) = θ x ( s )+ iθ y ( s )(deflection angles in x/y direction) as follows: E tot = E el ( θ, φ ) + E pol ( φ ) (5)The first energy term is the ”wormlike-chain” bendingcontribution E el ( θ, φ ) = B R | θ ′ − ˆ κ pol | ds . The sec-ond term is the polymorphic phase energy E pol ( φ ) = L p ∗ [ µ m ] L [ µ m] Pampaloni et al.Taute et al.Polymorphic Tube Model FIG. 2: Effective persistence length l ∗ p ( s ) as a function ofthe position from the attachment point along the MT con-tour. The experimental and theoretical prediction with l B =25 mm, λ = 7 . µm, κ − = 18 µm, q l φ >> C φ R L φ ′ ds with the polymorphic phase stiffness C φ = k B T N b π (cid:0) e J/k B T (cid:1) which can be related to the den-sity of double defects with energy 2 J (cf. Fig. 1d), givingrise to a new length scale - the polymorphic phase coher-ence length l φ = C φ /k B T. The most unusual property of a polymorphic chain isreflected in the rotational invariance of E pol ( φ ). Thebroken cylindrical to helical symmetry of the straightstate is restored by the presence of a ”Goldstone mode” φ → φ + φ [19] consisting of a rotation of P by an arbi-trary angle φ in the material frame (cf. Fig. 1c). Thismode that we will call the ”wobbling mode” is a fun-damental property of a helically polymorphic filament.The wobbling mode leads to dramatic effects on chain’sfluctuations and is the clue to the resolution of myster-ies (i)-(iii). To see this we will first investigate the staticproperties resulting in length dependent variations of thepersistence length. Persistence Length Anomalies.
Among several defini-tions of the persistence length [20] we consider for directcomparison with experiments [4][5], the lateral fluctua-tion persistence length l ∗ p ( s ) = (2 / s / D | ρ ( s ) | E with ρ ( s ) = x ( s ) + iy ( s ) the transverse displacement at po-sition s of a MT clamped at s = 0 and h .. i the sta-tistical average. It is easy to see from Eq. 5 that forsmall deflections, ρ decouples into independent elasticand polymorphic displacements ρ ( s ) = ρ el + ρ pol , suchthat l ∗ p = (cid:16) l ∗− pol + l − B (cid:17) − with l ∗ pol = (2 / s / D | ρ pol | E where ρ pol = κ R s R s e iq ˜ s + iφ (˜ s ) d e sds . The coherenthelix nature of the MT observed in [3] and the ab-sence of a plateau in l ∗ p imply [21] that l φ >> λ =2 πq − (the helix wave length). In that limit we ob-tain D | ρ pol | E ≈ κ q − [ q + s + s l φ − q e − s lφ ((1 + s l φ ) cos q s + sq sin q s )]. Whereas for an ideal WLC l ∗ p = l B ≡ B/kT is position and definition independent, −2 −1 τ [ s e c ] L [ µ m] Taute et al.Wobbling model prediction
FIG. 3: Longest relaxation time of a microtubule of lengthL, experimental data from [5] and theoretical prediction fromthe wobbling mode approximation and the wobbling angle α = 2 κ /q ≈ . ◦ extracted from static data [4][5], Fig. 2. the polymorphic fluctuations induce a strong position /distance dependence - a behavior that could be inter-preted as ”length dependent persistence length”. Indeedfor l φ >> s >> q − the persistence length displays anon-monotonic oscillatory behavior around a nearly lin-early growing average value l ∗ p ( s ) ≈ q κ s + q κ sin ( q s ).This oscillation is related to the helical ground state whilethe linear growth l ∗ p ( s ) ∝ α s is associated to the conicalrotation of the clamped chain (wobbling mode), cf. Fig.1c, with an angle α = 2 κ q − . For s >> l φ the satura-tion regime with a renormalized l ∗ p ( ∞ ) = 1 / (cid:16) l − pol + l − B (cid:17) with l pol = 2 l φ q κ − is reached. The theory can nowbe compared with the experimental data [4][5] (cf. Fig.2) that reveal several interesting characteristics in agree-ment with predictions. In particular the mean lineargrowth of l ∗ p ( L ) (single parameter fit l ∗ p ∼ L δ gives δ = 1 .
05) and the non-monotonic l ∗ p ( L ) dependence [5]are well captured by the theory. The linearly growingexperimental spread of l ∗ p with L is likely linked to thespread of q in the MT lattice populations [17]. Thelarge length plateau s >> l φ is not reached even forlongest MTs ( ∼ µm ) in agreement with coherent he-lices [3]. Our best comparison between theory and ex-periments (cf. Fig. 2) gives l B = 25 mm correspond-ing to Y ≈ GP a (proteins with Y up to 19 GP a ex-ist [22]) and a helix wave length λ ≈ . µm. This isclose to the expected 6 µm corresponding to the twist [17]of the predominant 14 PF MTs fraction in the in-vitroMTs preparation of [4][5]. It turns out that l B is largerthan in previous studies l B ∼ − mm where howeverpolymorphic fluctuations were neglected. The absence ofthe plateau also allows a lower estimate of the coherencelength l φ > µm and the coupling constant J > k B T . Polymorphic Phase Dynamics.
To describe the MTfluctuation dynamics we consider the total dissipa-tion functional P diss = P ext + P int which is com-posed of an internal dissipation contribution P int = ξ int R ˙ φ ds and an external hydrodynamic dissipation P ext = ξ ⊥ R | ˙ ρ | ds with ξ ⊥ = 4 πη/ (ln (2 L/r ) − / η the solvent viscosity, r and L the MT radius and length. The time evolu-tion equation of the phase variable φ ( s, t ) and elasticdisplacement ρ el ( s, t ) is given by the coupled Langevinequations δEδφ = − δP diss δ ˙ φ + Γ φ and δEδρ el = − δP diss δ ˙ ρ el + Γ ρ with Γ φ/ρ the thermal noise term. In general this dy-namics is highly non-linear however in the experimen-tally relevant regime where the behavior is dominatedby the wobbling mode the equations simplify greatlyand we end up with a simple diffusive behavior of thewobbling mode ddt φ ( t ) = ξ tot L − R L Γ φ ( s, t ) ds witha friction constant given by ξ tot = ξ int + ξ ext where ξ ext = 2 ξ ⊥ κ q − ((1 + cos Lq ) − Lq + q L / . Forcomparison with the experiment we compute the timecorrelation of the y deflection. A short calculation gives h y pol ( L, t ) y pol ( L, t ′ ) i ∝ e − | t − t ′ | /τ ( L ) with the relaxationtime τ ( L ) ≈ Lξ tot /k B T . For small lengths, τ ( L ) ≈ Lξ int /k B T is dominated by internal dissipation while forlarge lengths τ ( L ) ≈ ξ ⊥ k B T ( κ /q ) L . A careful analy-sis of the experimental data [5] reveals in fact the latterscaling. An independent single exponent fit gives τ ∝ L α with α = 2 .
9. Using the value ( κ /q ) ≈ . × − from Fig. 2 and ξ ⊥ ≈ η with η = 10 − P a · s [23] we findthe theoretical value τ th /L = 7 . × s/m that canbe compared with the fit of experimental data (Fig. 3) τ fit /L = 6 . × s/m . The excellent agreement ofboth the exponent and the prefactor leads us again to thestrong conclusion that in these experiments the clampedMT is an almost rigid helical polymorphic rotor whosebehavior is dominated by the zero energy (”wobbling”)mode and hydrodynamic dissipation. For very short MTsthe linearly scaling internal dissipation dominates andwe could measure ξ int from the limit value of τ th /L, for L →
0. For the available data
L > µm [5] this plateau-regime is not yet fully developed and we can only providean upper estimate from the data ξ int . × − N s . Conclusion . The MT fluctuations are well described -both dynamically and statically - by the bistable tubu-lin model and the reason for appearance of MT helicesbecomes obvious. The otherwise mysterious lateral fluc-tuations reflected in l ∗ p ( L ) ∼ L and τ p ( L ) ∼ L scalingare mere consequences of the ”wobbling motion” of apolymorphic cooperatively switching helical lattice. Wespeculate that the implied conformational multistablityof tubulin and the allosteric interaction are not just na-ture’s way to modulate the elastic properties of its mostimportant cytoskeletal mechano-element. It could alsobe a missing piece in the puzzle of dynamic instabil-ity. Another intriguing possibility of using this switchfor long range conformational signalling in vivo, couldhardly have been overlooked by evolution. I.M.K thanks Francesco Pampaloni for stimulating discussions. ∗ Email: [email protected][1] J. Howard,Mechanics of Motor Proteins and the Cy-toskeleton, Sinauer Press 2001; L. A. Amos & Amos W.G., Molecules of the Cytoskeleton , Guilford Press 1991[2] I.M.Kulic et al. Proc. Natl. Acad. Sci. USA 105, 10011(2008).[3] P. Venier et al. J. Biol. Chem. 269, 13353 (1994).[4] Pampaloni et al. Proc. Natl. Acad.Sci. USA. 103, 10248(2006).[5] K.M. Taute et al, Phys. Rev. Lett. 100, 028102 (2008).[6] M. Janson & M. Dogterom, Biophys. J. 87, 2723 (2004);C. Brangwynne et al. Biophys. J. 93, 346 (2007).[7] M. Kurachi, M. Hoshi, & H. Tashiro. Cell Motil. Cy-toskel. 30, 221 (1995); T. Takasone et al, Jpn. J. Appl.Phys.41, 30153019 (2002); A . Kis, et al. Phys. Rev. Lett.89, 248101 (2002); T. Kim et al. Biophys J. 94, 3880(2008).[8] Keller et al. Biophys. J. 95, 1474 (2008).[9] C. Heussinger, M. Bathe & E. Frey, Phys. Rev. Lett. 99,048101 (2007); H. Mohrbach & I.M. Kulic, Phys. Rev.Lett. 99, 218102 (2007).[10] We will focus on the simplest case of equilibrated, taxolstabilized MTs where dynamic instability is absent yetthe very rich thermal behavior i-iii is pronounced.[11] S. Asakura, Advan. Biophys. (Japan) 1, 99 (1970); C.R. Calladine, Nature (London) 255, 121 (1975).[12] S. V. Srigiriraju & T. R. Powers Phys. Rev. Lett. 94,248101 (2005); H. Wada and R. R. Netz, Europhys. Lett,82, 28001 (2008)[13] C. Elie-Caille et al., Curr. Biol. 17, 17651770 (2007).[14] I.A.T. Schaap et al. Biophys. J. 91, 15211531 (2006).[15] L. A. Amos and W. B. Amos , J. Cell. Sci. Suppl. 14,95101 (1991).[16] ε pol = ε PF σ n ( s ) (cid:2) I [ R − d PF ,R − d PF ] ( r ) − I [ R − d PF ,R ] ( r ) (cid:3) · I [ πN n + q s, πN ( n +1)+ q s ] ( α ) where I [ . ] ( x ) = 1 if x ∈ [ . ]and 0 otherwise, d PF the PF diameter and q the naturallattice twist.[17] D. Chr´etien et al, J. Cell. Bio. 117, 1031-1040 (1992);S.Ray, E.Meyhofer, R. A. Milligan, and J. Howard, J.Cell Biol., 121,1083, (1993).[18] Phase φ fluctuations are induced by double defects whichcarry a limited local energy cost ∆ E = 2 Jn ( n being thenumber of double defects). | P | variations are induced bysingle defects having in general larger energy cost thatgrows with their end distance, cf. Fig.1d.[19] This mode has a N fold symmetry however for large num-ber of PFs N = 11 −
15 it can be approximated as con-tiuuous.[20] Another more common definition, from angular correla-tion h cos ( θ ( s ) − θ ( s ′ )) i exhibiting a similarly rich be-havior as l ∗ p (yet a distinct functional form) will be dis-cussed elsewhere.[21] In the plateau region the helix looses its ”coherence” andthe collective rigid rotational conical motion softens untilan uncorrelated segment movement becomes dominant.[22] N. Kol et al. Nano Lett. 5, 1343 (2005).[23] In the experimental range [5] of 2 . µm < L < µm : ξ ⊥ ≈ . ηη
15 it can be approximated as con-tiuuous.[20] Another more common definition, from angular correla-tion h cos ( θ ( s ) − θ ( s ′ )) i exhibiting a similarly rich be-havior as l ∗ p (yet a distinct functional form) will be dis-cussed elsewhere.[21] In the plateau region the helix looses its ”coherence” andthe collective rigid rotational conical motion softens untilan uncorrelated segment movement becomes dominant.[22] N. Kol et al. Nano Lett. 5, 1343 (2005).[23] In the experimental range [5] of 2 . µm < L < µm : ξ ⊥ ≈ . ηη − . ηη