Cooperative lattice dynamics and anomalous fluctuations of microtubules

Abstract

Microtubules have been in the focus of biophysical research for several decades. However, the confusing and mutually contradictory results regarding their elasticity and fluctuations have cast doubt on their present understanding. In this paper, we present the empirical evidence for the existence of discrete guanosine diphosphate (GDP)–tubulin fluctuations between a curved and a straight configuration at room temperature as well as for conformational tubulin cooperativity. Guided by a number of experimental findings, we build the case for a novel microtubule model, with the principal result that microtubules can spontaneously form micron-sized cooperative helical states with unique elastic and dynamic features. The polymorphic dynamics of the microtubule lattice resulting from the tubulin bistability quantitatively explains several experimental puzzles, including anomalous scaling of dynamic fluctuations of grafted microtubules, their apparent length–stiffness relation, and their remarkable curved–helical appearance in general. We point out that the multistability and cooperative switching of tubulin dimers could participate in important cellular processes, and could in particular lead to efficient mechanochemical signaling along single microtubules.

Appendix

A. Polymorphic phase coherence length

In this section, we derive the formula $l_{\phi}=\frac{N^{2}b}{8\pi^{2}}\left(2+{\rm e}^{2J/k_{\rm B}T}\right)$ for the polymorphic phase coherence length. To this end we want to calculate the distribution of double junctions that leads to angular orientation change $\Updelta\Upphi$ on a scale l much larger than the tubulin dimer b, yet still much smaller than the total length, i.e., b ≪ l ≪ L. In this domain, at each cross-section we have three possibilities:

• 1. State j = 0 with no double defect. The rotation angle $\Updelta\Upphi$ is attached to the internal lattice rotation, $\frac{\Updelta\Upphi}{b}-q_{0}=0.$

• 2. State j = −1 for a left-handed double defect; $\Updelta\Upphi$ deviates from the internal twist: $\frac{\Updelta\Upphi}{b}-q_{0}=-\frac{1}{b}\frac{2\pi}{N}.$

• 3. State j = +1 for a right-handed double defect with $\frac{\Updelta\Upphi}{b}-q_{0}=+\frac{1}{b}\frac{2\pi}{N}.$

On a length l we are throwing a three-sided dice l/b times and the total rotation of $\Updelta\Upphi$ away from the optimal twist is $\Updelta\Upphi-q_{0}l=\frac{2\pi}{N}\sum\nolimits_{n=1}^{l/b}j_{n}.$ The variation of the polymorphic phase with respect to the internal twist is then

$$ \Updelta\phi=\frac{\Updelta\Upphi}{l}-q_{0}=\frac{1}{l}\frac{2\pi}{N}\sum _{n=1}^{l/b}j_{n}. $$

For l/a ≫ 1 the law of large numbers implies that the random variable $\Updelta\phi=\frac{1}{l}\frac{2\pi}{N}\sum\nolimits_{n=1}^{l/b}j_{n}$ becomes Gaussian distributed

$$ p\left(\Updelta\phi\right) \propto {\rm e}^{-\frac{\Updelta\phi^{2}}{2\left\langle \Updelta\phi^{2}\right\rangle}} $$

with mean $\left\langle \Updelta\phi\right\rangle =0$ and $\left\langle \Updelta\phi^{2}\right\rangle =\left(\frac{1}{l}\frac{2\pi}{N}\right) ^{2}\left(\frac{l}{b}\right) \left\langle j^{2}\right\rangle$ (as $\left\langle j_{n}j_{m}\right\rangle =\delta_{nm}\left\langle j^{2}\right\rangle$). The average $\left\langle j^{2}\right\rangle$ is given from the Boltzmann factors of the three different states $p_{0}={\frac{1}{1+2{\rm e}^{-2\beta J}}}$ and $p_{\pm1}=\frac{{\rm e}^{-2\beta J}}{1+2{\rm e}^{-2\beta J}},$ so that $\left\langle j^{2}\right\rangle =\frac{2{\rm e}^{-2\beta J}}{1+2{\rm e}^{-2\beta J}}.$ We can now interpret the quantity $1/\left(2\left\langle \Updelta \phi^{2}\right\rangle \right)$ as coming from an effective elastic energy over the interval l by writing $\frac{\Updelta\phi^{2}}{2\left\langle \Updelta\phi^{2}\right\rangle }=\frac{1}{2}\beta C_{\phi}l\left(\Updelta \phi\right)^{2}$, which allows us to identify the effective stiffness

$$ C_{\phi}=kT\frac{N^{2}}{8\pi^{2}}\left(2+{\rm e}^{2J\beta}\right) b. $$

Note that this expression is valid for large enough J suppressing higher-order defects, i.e., in the limit when multiple double defects sitting on a single lattice site (i.e., | j| > 1) can be ignored.

B. The variation of the polymorphic modulus

In this appendix we compute the energy variation due to a deviation of the polymorphic modulus |P| away from its optimal value | P ^*| minimizing the energy, i.e., the change of the number of switched PFs. We start with the energy density of a MT cross-section

$$ e=\frac{B}{2}\left(\left(\kappa-\kappa_{\rm pol}(p)\right) ^{2}+\kappa _{1}^{2}\left(\gamma\frac{\pi}{N}p-\sin^{2}\left(\frac{\pi}{N}p\right) \right) \right), $$

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whose minimum is reached for $p^{\ast}=\frac{N}{2}-\frac{N}{2\pi }\arcsin\gamma.$ Assuming a continuous number of PFs, the energy of a state with $p=p^{\ast}+\Updelta p$ switched PFs reads to quadratic order

$$ \begin{aligned} e(p^{\ast}+\Updelta p) & \approx e(p^{\ast})-\frac{\pi^{2}B}{N^{2}}\kappa _{1}^{2}\cos\left(\frac{2\pi}{N}p^{\ast}\right) \Updelta p^{2} \\ & =e(p^{\ast})+\frac{\pi^{2}B}{N^{2}}\kappa_{1}^{2}\sqrt{1-\gamma^{2}}\Updelta p^{2}, \end{aligned} $$

where we used $\cos(\pi-\arcsin\gamma)=-\sqrt{1-\gamma^{2}}.$ Therefore, the energy variation of a segment of length l reads

$$ \Updelta E\approx\frac{\pi^{2}B}{N^{2}}\kappa_{1}^{2}\sqrt{1-\gamma^{2}} {\textstyle\int_{0}^{l}} {\rm d}s(p(s)-p^{\ast}(s))^{2}. $$

Now using $\left\vert P\left(s\right) \right\vert =\left\vert \sin\left(\frac{\pi}{N}p\right) \right\vert /\sin(\pi/N)$ we can write the energy variation to the same (quadratic) order as

$$ \Updelta E\approx B\kappa_{1}^{2}\sin^{2}(\pi/N)\sqrt{1-\gamma^{2}} {\textstyle\int_{0}^{l}} {\rm d}s((\left\vert P\left(s\right) \right\vert -\left\vert P^{\ast}\right\vert))^{2}. $$

Therefore, any deviation of P from its optimum state P ^* is associated with an energy cost proportional to the length l of the region in the unfavorable state.

C. Persistence length(s)

A definition of the persistence length, often used in single-molecule experiments, is expressed in terms of the lateral deviation $\overrightarrow {\rho}=(x(s),y\left(s\right))$ of a MT clamped at s = 0 from its attachment axis as $l_{\rm p}^{\ast}\left(s\right) =(2/3)s^{3}/\left\langle \left(\overrightarrow{\rho}\left(s\right) -\left\langle \overrightarrow {\rho}\left(s\right) \right\rangle \right) ^{2}\right\rangle,$ where $\left\langle \cdot \right\rangle$ is the statistical average. The equivalence of the x and y directions implies that $l_{\rm p}^{\ast}\left(s\right) =1/3s^{3}/\left\langle \left(y\left(s\right) -\left\langle y\left(s\right) \right\rangle \right) ^{2}\right\rangle.$ The second often used alternative but more standard definition of the persistence length—the tangent persistence length—is related to the angular correlation $l_{\rm p}\left(s-s^{\prime}\right) =\left\vert s-s^{\prime}\right\vert /V(s-s^{\prime})$ with variance $V=\left\langle \left(\theta_{y}\left(s\right) -\theta_{y}\left(s^{\prime}\right) \right) ^{2}\right\rangle$ (by symmetry we have the same expression with θ _x ). Whereas for an ideal WLC $l_{\rm p}^{\ast}=l_{\rm p}=l_{\rm B}$ is position and definition independent, this is not the case for a polymorphic chain (see Fig. 13). For small angular deformations, the decoupling of the chain’s fluctuations into polymorphic and purely elastic contributions allows one to decompose the persistence length as l ⁻¹_p  = l ⁻¹_pol  + l ⁻¹_B , this result being valid for both definitions of the persistence length.

Fig. 13

Different definitions of the persistence length can deviate from each other for a polymorphic chain: the “clamped persistence length” $l_{\rm p}^{\ast}$ (thick line) and the “tangent persistence length” l _p (thin line) (for l _B = 10 mm, l _ϕ  = 50 μm, κ ₀ = 0.03 μm⁻¹, and q ₀ = 0.7 μm⁻¹)

We first focus on the first definition, for the clamped persistence length. In this case the polymorphic persistence length $l_{\rm pol}^{\ast}(s)$ is given by

$$ l_{\rm pol}^{\ast}(s)=1/3s^{3}/\left\langle \left(y_{\rm pol}\left(s\right) -\left\langle y_{\rm pol}\left(s\right) \right\rangle \right) ^{2} \right\rangle, $$

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where y _pol(s) is the lateral polymorphic displacement in the y direction. Integrating over the rotational zero mode readily implies $\left\langle y_{\rm pol}\left(s\right) \right\rangle =0$ (see Eq. 18). From Eq. 19 one can write

$$ \left\langle y_{\rm pol}^{2}\left(s\right) \right\rangle =\int\limits_{0}^{s}\int\limits _{0}^{s}G(s_{1},s_{2}){\rm d}s_{1}{\rm d}s_{2} $$

with the angular correlation function $G(s_{1},s_{2})=\left\langle \theta_{y,{\rm pol}}(s_{1})\theta_{y,{\rm pol}}(s_{2})\right\rangle$ given by the integration over the zero mode

$$ G(s_{1},s_{2})=\int\limits_{0}^{2\pi}\frac{{\rm d}\phi_{0}}{2\pi}G_{0}(s_{1},s_{2},\phi _{0}) $$

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of the angular correlation function at fixed value of ϕ ₀, i.e., $G_{0}(s_{1},s_{2},\phi_{0})=\left\langle \theta_{y,{\rm pol}}(s_{1})\theta _{y,{\rm pol}}(s_{2})\right\rangle |_{\phi_{0}}.$ This last expression, from the relation $\theta_{y,{\rm pol}}(s)=\kappa_{0}\int_{0}^{s}\sin\left(\widetilde{\phi }\left(s^{\prime}\right) +q_{0}s^{\prime}+\phi_{0}\right) {\rm d}s^{\prime}$ (cf. Eq. 17), is explicitly given by

$$ \begin{aligned} G_{0}(s_{1},s_{2},\phi_{0}) & =\kappa_{0}^{2}\int\limits_{0}^{s_{1}}\int\limits_{0} ^{s_{2}}\left\langle \sin\left(\widetilde{\phi}\left(s\right) +q_{0}s+\phi_{0}\right) \right. \\ & \left. \sin\left(\widetilde{\phi}\left(s^{\prime}\right) +q_{0}s^{\prime}+\phi_{0}\right) \right\rangle |_{\phi_{0}}{\rm d}s{\rm d}s^{\prime}. \\ \end{aligned} $$

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After integration over ϕ ₀ and using the known result $\left\langle \cos\left(\widetilde{\phi}\left(s_{1}\right) -\widetilde{\phi}\left(s_{2}\right) \right) \right\rangle ={\rm e}^{-\left\vert s_{1}-s_{2}\right\vert /2l_{\phi}}$ which results from the WLC-type probability distribution of the field $\widetilde{\phi},$ i.e., $P[\widetilde{\phi}]\sim\exp(-\frac{l_{\phi}}{2} \int_{0}^{L}{\rm d}s\widetilde{\phi}^{\prime2})$, one obtains the rotational invariant correlation function in the form

$$ G(s_{1},s_{2})=\frac{\kappa_{0}^{2}}{2}\int\limits_{0}^{s_{1}}\int\limits_{0}^{s_{2} }{\rm e}^{-\frac{\left\vert s-s^{\prime}\right\vert }{2l_{\phi}}}\cos\left(q_{0}(s-s^{\prime})\right) {\rm d}s{\rm d}s^{\prime}. $$

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Computation of the integrals in Eq. 38 gives finally the following expression for the polymorphic contribution to the transverse displacement:

$$ \begin{aligned} \left\langle y_{\rm pol}(s)^{2}\right\rangle & =\frac{2\kappa_{0}^{2}l_{\phi} }{3(1+4l_{\phi}^{2}q_{0}^{2})^{4}}\left\{ P_{1}-{\rm e}^{-\frac{s}{2l_{\phi}}} P_{2}\cos\left(q_{0}s\right) \right.\\ &\quad \left. +{\rm e}^{-\frac{s}{2l_{\phi}}}P_{3}\sin(q_{0}s)\right\} \\ \end{aligned} $$

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with ${P}_{1}(s){ =24l}_{\phi}^{3}\left(1-6x+x^{2}\right) { -3l}_{\phi}\left(1+x-x^{2}-x^{3}\right) { s}^{2}$ ${ +}\left(1+3x+3x^{2}+x^{3}\right) { s}^{3}, { P}_{2}(s){ =24l}_{\phi}^{3}\left(1-6x+x^{2}\right)$ ${ +12l}_{\phi}^{2}\left(1-2x-3x^{2}\right) { s}$, and P ₃(s) = 192l ⁴ _ϕ q ₀(1 − x) + 24l ³ _ϕ q ₀(3 + 2x − x ²)s, where we have introduced the notation x = 4l ² _ϕ q ²₀ .

From Eq. 39, we get the polymorphic persistence length $l_{\rm pol}^{\ast}(s)$ defined in Eq. 35, and in turn the global persistence length $l_{\rm p}^{\ast}(s)$ depicted in Fig. 13. Its physical interpretation is discussed in the main text.

We now consider the second definition of the persistence length $l_{\rm p}\left(s-s^{\prime}\right) =\left\vert s-s^{\prime}\right\vert /V(s-s^{\prime}).$ From Eq. 38, the angular variance V _pol can easily be evaluated as

$$ \begin{aligned} V_{\rm pol}(s) & =\frac{2\kappa_{0}^{2}l_{\phi}}{1+4l_{\phi}^{2}q_{0}^{2} }\left(s-\frac{2l_{\phi}\left(1-4q_{0}^{2}l_{\phi}^{2}\right) }{1+4l_{\phi}^{2}q_{0}^{2}}\right) \\ &\quad+\frac{4\kappa_{0}^{2}l_{\phi}^{2}{\rm e}^{-\frac{s}{2l_{\phi}}}}{\left(1+4l_{\phi}^{2}q_{0}^{2}\right) ^{2}}\left(\left(1-4q_{0}^{2}l_{\phi} ^{2}\right) \cos\left(q_{0}s\right) \right.\\ &\quad\left. -4q_{0}l_{\phi}\sin\left(q_{0}s\right) \right) \\ \end{aligned} $$

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The resulting persistence length l _p (depicted in Fig. 13) shows a rich behavior similar to the persistence length $l_{\rm p}^{\ast}\left(s\right)$ but displays a functional form distinct from the latter. However, as expected, both curves reach the same asymptotic value at very short and very long MT lengths.

D. Zero-mode dynamics

The evolution of the zero mode $\phi_{0}\left(t\right)$ is given by Eq. 28 as

$$ \frac{{\rm d}}{{\rm d}t}\phi_{0}\left(t\right) =\frac{1}{\xi_{\rm tot}}L^{-1}\int\limits_{0} ^{L}\Upgamma_{\phi}\left(s,t\right) {\rm d}s $$

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with a friction constant ξ _tot = ξ _int + ξ _ext, where ξ _ext is given by Eq. 27. The correlation function of the thermal white noise $\Upgamma_{\phi}\left(s,t\right)$ is $\left\langle \Upgamma_{\phi }\left(s,t\right) \Upgamma_{\phi}\left(s^{\prime},t^{\prime}\right) \right\rangle =D\delta(s-s^{\prime})\delta(t-t^{\prime})$ with a coefficient D that can be determined in the following manner. Notice first that ϕ ₀ performs free Brownian motion and that its quadratic fluctuations necessarily satisfy the relation $\left\langle \left(\phi_{0}\left(t\right) -\phi_{0}\left(0\right) \right) ^{2}\right\rangle ={ \frac{2k_{\rm B}T}{L\xi_{\rm tot}}}t.$ On the other hand, integrating Eq. 41 yields

$$ \phi_{0}\left(t\right) -\phi_{0}\left(0\right) =\xi_{\rm tot}^{-1} {\displaystyle\int\limits_{0}^{t}} \left(\frac{1}{L}\int\limits_{0}^{L}\Upgamma\left(s,t^{\prime}\right) {\rm d}s\right) {\rm d}t^{\prime}, $$

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and exploiting the white noise type auto-correlation of $\Gamma$ one obtains $\left\langle \left(\phi_{0}\left(t\right) -\phi_{0}\left(0\right) \right) ^{2}\right\rangle =\frac{D}{\xi_{\rm tot}^{2}L}t,$ from which we readily deduce D = 2ξ _tot k _B T, as expected from the fluctuation–dissipation theorem.

The relaxation time is generally given from the time correlation function <y _pol(s, 0)y _pol(s, t)> with the lateral position $y_{\rm pol}(s,t)=\frac{\kappa_{0}}{q_{0}^{2}}(sq_{0}\cos\left(\phi_{0}\left(t\right) +\alpha\right) +\sin\left(\phi_{0}\left(t\right) +\alpha\right) -\sin\left(q_{0}s+\phi_{0}\left(t\right) +\alpha\right))$ obtained from Eq. 14 with l _ϕ  ≫ s. The average must first take into account all statistically equivalent values of angular orientations $\alpha\in\left[ 0,2\pi\right] ,$ such that $\left\langle y_{\rm pol}(s,0)y_{\rm pol}(s,t)\right\rangle =\int\nolimits_{0}^{2\pi}\left\langle y_{\rm pol}(s,0)y_{\rm pol}(s,t)\right\rangle _{\alpha}\frac{{\rm d}\alpha}{2\pi},$ and we obtain

$$ \left\langle y_{\rm pol}(s,0)y_{\rm pol}(s,t)\right\rangle =\left\langle y_{\rm pol} ^{2}(s)\right\rangle \left\langle \cos(\phi_{0}\left(t\right) -\phi _{0}\left(0\right))\right\rangle $$

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with $<y_{\rm pol}^{2}(s)>=\frac{\kappa_{0}^{2}}{q_{0}^{2}}\left(\frac{s^{2}}{2}+\frac{1-\cos(q_{0}s)}{q_{0}^{2}}-\frac{s\sin(q_{0}s)}{q_{0}} \right),$ corresponding to the static result (Eq. 39) in the limit s/l _ϕ  ≪ 1. With 42 defining a simple Gaussian random walk process, one straightforwardly obtains

$$ \left\langle \cos(\phi_{0}\left(t\right) -\phi_{0}\left(0\right) )\right\rangle ={\rm e}^{-t/\tau} $$

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with the relaxation time given by

$$ \tau=L\frac{\xi_{\rm tot}}{k_{\rm B}T}. $$

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E. Comment on MT surface attachment and the robustness of “wobbling”

Throughout this work we have assumed that the free rearrangement of the polymorphic lattice states is not significantly hindered by the covalent surface attachment of the MT, as e.g. performed by Pampaloni et al. (2006) and Taute et al. (2008). This assumption is integral to the “wobbling” motion and in turn to understanding the static and dynamic data scaling. It therefore deserves closer consideration.

In the experiments by Pampaloni et al. (2006) and Taute et al. (2008), the adsorbed MT part is attached to a gold (electron microscopy grid) surface via thiol groups. It is likely that $\approx$1–2 protofilaments will establish localized chemical contacts with the gold microgrid. While substantial perturbation of the dimer such as denaturation appears unlikely, it is unclear to what extent this procedure will perturb the inner (polymorphic) dynamics of the entire tubulin dimer units. In principle, one can anticipate two plausible scenarios that would interfere to a varying degree with the free “wobbling” motion:

• S1. Due to high cooperativity (large coupling J) the polymorphic state transition can propagate within a certain penetration depth into the adsorbed (straight-planar) MT section.

• S2. The cooperativity is too weak to compete with the constraints imposed by the surface (including chemical perturbations), and the polymorphic transition does not propagate into the straight adsorbed MT section.

In both cases we have a nonvanishing deflection angle between a forced (adsorbed) planar section and the free helical section direction, effectively causing the characteristic MT “kink” at the surface interface. However, the rotational mobility of this “kink” (wobbling mode) which is integral to our theory will be affected in a slightly different manner.

If in case S1, in the adsorbed section, the polymorphic order parameter P can rearrange to some extent (by switching the monomer states without causing detectable deformation) except for possibly in the few surface-interacting dimers, then the effects of the “wobbling” motion will be hindered only mildly in the following sense. To retrieve the anomalous lateral fluctuations it is indeed enough for the wobbling angle ϕ ₀ to move freely in a certain nonvanishing angular interval. A single complete or multiple rotations of the order parameter $\vec{P}$ are not strictly necessary for the “hinge” effect, and they are in fact equivalent in lateral projection (as in experiment) to the motion of the wobbling angle ϕ ₀ in the smaller interval [−π/2, +π/2]. Note that even intervals smaller than this will lead to a similar phenomenology (in particular, dynamic and static variable scalings with length). Thus, the conical hinge-like motion is in a sense robust with respect to a limited local rotational hindrance perturbation in the adsorbed region.

In scenario S2 the situation is somehow simpler as the polymorphic dynamics of the adsorbed region is not involved in the process (the polymorphic order parameter vanishes there: $\vec{P}=0$). Wobbling is realized through a coherent rearrangement of the free MT section alone, without strong coupling to the adsorbed region.

Although both attachment scenarios S1 and S2 appear to some extent plausible, at present it is difficult to make reliable statements about their respective likelihood. In fact only a posteriori can we cautiously state that, based on the experimental static and dynamic measurement evidence, the chain “wobbles” to a high enough extent to display the effects that we observe.