Elastic Anisotropy Scenario for Cooperative ... - ACS Publications

Subscriber access provided by UNIV OF CALIFORNIA SAN DIEGO LIBRARIES

Article

Elastic Anisotropy Scenario for Cooperative Binding of Kinesin Coated Beads on Microtubules Ken Sekimoto, and Jacques Prost J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b01627 • Publication Date (Web): 30 Mar 2016 Downloaded from http://pubs.acs.org on April 2, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry B is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Elastic Anisotropy Scenario for Cooperative Binding of Kinesin Coated Beads on Microtubules Ken Sekimoto∗,†,‡ and Jacques Prost∗,¶,§ †Matières et Systèmes Complexes, CNRS-UMR7057, Université Paris-Diderot, 75205 Paris, France ‡Gulliver, CNRS-UMR7083, ESPCI, 75231 Paris, France ¶Physico-chimie Curie, CNRS-UMR168, Institut Curie, 75231 Paris, France §Mechanobiology Institute, National University of Singapore, 5A engineering drive 1, 117411, Singapore E-mail: [email protected]; [email protected]

1 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract E. Muto, et al. reported in 2005 an observation called cooperative binding, according to which the initial binding of a bead covered with active kinesins on a microtubule filament was capable of favoring the subsequent binding of similar beads on the same filament up to distances of the order of a few microns. This positive bias is stronger ahead of the initially bound bead than behind. We explain this effect by combining the recently proposed notion of shear screening length with the notion of localized tubulin conformational transition induced by motor binding. Elastic terms linked to the polarity of protofilaments, up to now ignored, provide adequate description to the long range elastic shear generated by motor binding. The subsequent binding is favored when and where the shear displacement of protofilaments meets the requirement for specific strong binding. We propose experimental tests of our model, which open the way to a new type of spectroscopy for biomolecular processes.

Introduction In 2005, E. Muto, H Sakai and K. Kaseda 1 reported a very intriguing observation according to which the initial binding of a bead covered with active kinesins on a microtubule filament was capable of favoring the subsequent binding of similar beads on the same filament up to distances of the order of a few microns. This positive bias required ATP hydrolysis by kinesins. Moreover the bias was stronger ahead of the bead than behind. In view of the fact that motor activity results from short-range effects such as conformational transitions due to ATP hydrolysis and short-range interactions, this observation is unexpected and its interpretation is challenging. Even electrostatic interactions, which are considered to play an important role in the motor binding process, are short ranged since they are screened beyond one nanometer under normal physiological conditions. 1 Unscreened interactions such as Van der Waals and thermal Casimir interactions are completely negligible at those distances. The questions are three-fold : How can nanometer scale interactions bias binding probabilities at the µm scale? How is the binding of the second bead enhanced? And how does the front-rear asymmetry of the binding arise ? 2 ACS Paragon Plus Environment

Page 2 of 20

Page 3 of 20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


A long-range effect, stemming from short-range interactions is not a new phenomenon in condensed matter science, but it requires specific conditions. The most obvious case is that of a system with broken symmetry of a continuous group such as rotation groups. For instance, by changing boundary conditions at an interface, one can change the orientation of a nematic at an infinite distance from that interface even though all interactions are short-ranged. 2 Two more realistic scenarios for the emergence of a µm order length-scale have recently been proposed. One is the elastic anisotropy scenario 3 for explaining the unconventional thermal fluctuation spectrum of microtubules. 3,4 The authors recognized the importance of the characteristic length ℓc called “shear-screening length” to interpret their data: ℓc ≃ (E/µ)1/2 ∆, where E and µ are, respectively, the Young modulus of protofilaments (PFs) against longitudinal compression and the shear modulus associated to the relative sliding of the PFs and ∆ is a length of the order of tubulin size 3,5 . The other scenario for the µm order length-scale is based on the hypothetical existence of two conformational states for taxol stabilized tubulin monomers. 6,7 Assuming a cooperative transition of the states among neighboring monomers along the PF, the polymorphic order may extend to a length-scale beyond the monomer size. Whether this assumption can account for Muto et al.’s observations remains open. In the present paper, we propose answers to the two remaining questions. First we show that the elastic anisotropy model 3,4 needs to be supplemented by a new energy term which reflects the polar nature of the PFs. We then show that this term comes into play if we assume that the strong binding of a kinesin motor head to a tubulin induces a localized conformational transition in the tubulin. Next the enhancement of the binding rate of a second bead is ascribed to the active shear deformations of the PFs driven by the on-and-off switches of the conformational transition at the first binding site. We further discuss similar observations in the case of the actin-myosin system. 8,9 We propose possible experimental tests of our theory, which open the way to a new type of spectroscopy for biomolecular processes.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

i=0 i=−1 u (z) −1 i=1

u0(z) = 0

Page 4 of 20

u (z) 0

u−1(z) = 0 u1(z)

u1(z) = 0

i=0 u0(z) = 0

u (z) 0

z

Figure 1: Schematic representation of microtubule with three protofilaments (PF) i = 0, ±1. PFs are represented as continuous compressible rods (thick horizontal lines). Longitudinal displacements ui (z) with i = 0, ±1 (filled discs) from their respective reference positions ui (z) = 0 (open discs) are also defined, where the magnitude of displacements are exaggerated. The uppermost PF is the copy image of the lowermost one.

MATERIALS AND METHODS Elastic model of MT as bundle of non-polar rods The measurements of thermal fluctuations of end-grafted microtubules (MT) 3 were interpreted by assuming a strong elastic anisotropy of MT: while the longitudinal modulus E was E ≃ 1.5 GPa along the MT axis, the shear modulus µ associated to the mutual sliding of neighboring PFs was µ ≃ 1.5 kPa. A similar estimate was also obtained in, 5 where the ratio E/µ was 6 × 105 instead of 106 . (An other approach 10 reports a more moderate contrast between E and µ.) The high contrast of elastic moduli E/µ . 106 guarantees that the longitudinal displacement u j (z) along the j-th PF of a MT from z to z + u j (z) is a slowly varying variable. We can thus use a gradient expansion to discuss its elastic properties. In 1 we show a simplified model of MT with three PFs, i = 0, ±1. We adopt the anisotropic elasticity model of MT used in the literature, 5 which contains a microscopic length parameter ∆ (see below). The elastic energy H reads !2 p Z p Z X X E du j µ u j+1 − u j 2 2 2 H= ∆ dz + ∆ dz, 2 dz 2 ∆ j=−p j=−p

(1)

where u j (z) ( j = −p, . . . , p) is the longitudinal displacement of the j-th PF as function of z, and we have defined u p+1 (z) ≡ u−p (z). Merely for the simplicity of notations, we will describe only the case of odd number (2p + 1) of PFs. For the typical thirteen PFs we take p = 6. Or, by 4 ACS Paragon Plus Environment

Page 5 of 20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1

introducing the discrete Fourier amplitudes, Uk (z) = (2p + 1)− 2 where ζ ≡ exp (2πi/(2p + 1)), we can simplify Eq.(1) as follows:

Pp

j=−p

ζ −k j u j (k = −p, . . . , p),

# !2 Z Z " p X E dU0 E dUk 2 |Uk (z)|2 2 2 + (k) H= ∆ dz, ∆ dz + 2 2 dz 2 dz (ℓ ) c k=−p(,0) ℓc(k)

≡

2 sin

1

kπ 2p+1

ℓc ,

E ℓc ≡ µ

! 12

∆.

(2)

(3)

The set of Uk ’s contains the same degrees of freedom as u′j ’s because Uk and U−k with k , 0 are related by the identity U−k (z) = Uk∗ (z), where the asterisk means the complex conjugate. In the above ∆2 corresponds, a priori, to the sectional area of each PF and appears effectively only in the longitudinal deformational energy. In reality the PF is not a smooth cylinder and ∆2 requires reinterpretation. The shear deformational energy may include a dimensionless factor associated to the aspect ratio of a tubulin monomer in the radial vs azimuthal directions. In brief we will deal with ∆ as an adjustable molecular length, rather than the real size of a tubulin. We will decide later the value of ∆ so that the model be consistent with the experiments. In Eq.(3) ℓc(k) is the shear screening length 3,4 for the k-th mode. If we take E ≃ 1.5GPa, µ ≃ 1.5KPa, therefore, 1

(E/µ) 2 = 103 , 3 we have ℓc ≃ 103 ∆. In the original model 5 the choice of ∆ = 6nm yields ℓc ∼ 1µm with some geometrical factor being adjusted. Later on we will assume a larger value for ∆ about 50nm. It is in agreement with the data of Muto et al. 1 since in that data it is rather ≃ ℓc /4 that should be compared with the experimental decay length. See 4. Eq.(2) does not include the bending energy of PFs. This is legitimate as the following argument shows. For bending modes the relevant displacement fields vi are orthogonal to the microtubule axis. Subtle couplings between shear and bending modes 11 lead to a scale dependent bending modulus, but do not change this scaling argument. The bending terms involve second order derivatives with respect to z. If we put in Eq.(1) the terms (∆ × d2 vi /dz2 )2 in place of (dui /dz)2 terms (as well as ((v j+1 − v j )/∆)2 in place of ((u j+1 − u j )/∆)2 ), then we obtain a characteristic length 1

(E/µ) 4 ∆ ∼ 400nm, for ∆ ∼ 50nm in place of ℓc ∼ 10 µm. This curvature related length is too 5 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 20

small to be relevant to the observed cooperative binding. Furthermore, in the set-up of 1 the MT axis, or the neutral curve in the terminology of elasticity, 12 is straight because the MT is adsorbed on the surface. We do not discuss chiral effects 13 since they are small in MTs, and our primary interest is in the polarity.

New energy associated to the polarity of MT Even though equation (2) is commonly used to discuss elastic properties of microtubules, it misses terms related to the polar symmetry of PFs which are important for describing the long range consequences of local effects. More precisely the polar symmetry of PFs allows for the following additional terms: Hp =

Z X p

dm

m=−p

p X

j=−p

uj

∂u j+m ∆dz, ∂z

(4)

where dm with m = −p, . . . , p are the coupling parameters which characterize polarity and the suffix j for u j is to be understood modulo (2p + 1), i.e., u j±(2p+1) ≡ u j . The independence of H p on the choice of the z > 0 direction can be satisfied if we assume that dm is transformed to (−dm ) under the inversion, z 7→ −z. The discrete translational symmetry is also taken into account with respect to the shift of the indices of the PFs, j 7→ j + m with an integer m. Finally the translational symmetry u j 7→ u j + a with a being constant independent of j is also observed. 2 gives an intuitive explanation of the origin of a typical term in Eq.(4) with m = 1. In practice, only those terms with 2, suggests that dm can be neither as large as E nor as small as µ,

m = ±1 will be important.

because of partial steric interference (see below). Using the variables Uk we find

Hp =

where we introduced Dk ≡

Pp

m=−p

Z X p

∂Uk∗ ∆ dz, Dk Uk ∂z k=−p

(5)

dm ζ km , with ζ being defined as before. If we write dm as the sum

(s) (a) of the symmetric part dm(s) and the anti-symmetric part dm(a) (i.e., dm(s) = d−m and dm(a) = −d−m ), the (a) (s) (s) corresponding decomposition of Dk as Dk = D(s) k + Dk inherits similar properties, i.e. Dk = D−k


Page 7 of 20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


z−∆

z

z+∆ PF−(j+1) PF−j

PF−(j+1) PF−j

Figure 2: A schematic explanation for the origin of the free energy term with m = 1 in Eq.(4). A raw of triangles represents a PF, and the triangular elements signifies its polarity. The thick dots represent the positions by which the displacements of tubulin dimers are referred. The upper figure displaying six triangles shows the reference state without displacements, u j = u j+1 = 0. In the lower figure the elongation is introduced only on PF-( j + 1) around the element at z, that is, u j+1 (z + ∆0 ) > u j+1 (z) = 0 > u j+1 (z − ∆0 ), leading to ∂u j+1 (z)/∂z > 0 in the continuum approximation. Here ∆0 is the length of the unit along the z axis, i.e. 4nm for a tubulin monomer or 8nm for a tubulin dimer. Physically, the molecular field at z in the j-th PF created by the elongated ( j+1)-th PF must be asymmetric because of the head-tail asymmetry of the triangles. Such effect can be reflected in the free energy term, and u j (z) [∂u j+1 (z)/∂z] is the simplest one. (a) and D(a) k = −D−k . We then have the following neat expression for H p :

! Z X p  (s) ∗  Dk ∂|Uk |2 D(a)  ∂U ∂U k    ∆ dz. Hp = + k Uk k − Uk∗ 2 ∂z 2 ∂z ∂z  k=−p

(6)

In view of our previous remark concerning the weakness of chirality, we shall ignore the antisym(s) metric part (∝ dm(a) or D(a) k ). Then there remains the symmetric terms including Dk , which are

integrable along z and, therefore, depend on the limits of the integral along z. If the only limits are the ends of MT, the energy due to the polarity, H p , is completely ignorable because it simply redefines the edge energy, with different values for the plus and minus extremities. However, the action of active motor on a tubulin or tubulin dimers can also contribute to the above energy (see below).

Thermal effects and the dynamics of protofilaments We can calculate the thermal amplitude of the relative sliding displacement of the adjacent PF by using standard methods of statistical physics. Below, however, we give a physical argument that reproduces the same results. This estimation is important both to find the reasonable range of the 7 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 20

adjustable parameter ∆ and to know the thermal and active environment for the kinesin motor on the MT. Since the longitudinal stiffness of the PFs, E, serves to correlate the displacement over a long distance along PF it also works to reduce the longitudinal thermal fluctuations of tubulins. In the absence of longitudinal stiffness, the elastic restoring force would arise only from the shear deformation between the laterally neighboring tubulin monomer. The associated energy cost would be µ(δu/∆)2 × ∆3 , where δu(z) ≡ u j+1 (z) − u j (z) is the sliding displacement between the neighboring 1

PF. Therefore, the equipartition law would lead to hµ(δu/∆)2 × ∆3 i ≃ kB T , or h(δu[nm])2 i 2 ∼ 1

100/(∆[nm]) 2 , where we took µ = 2.5kPa (so that (E/µ) = 6 × 105 with E = 1.5GPa 5 ) and both δu and ∆ are expressed in unit of nm. This model without longitudinal stiffness is unrealistic because 1

the thermal amplitude h(δu)2 i 2 exceeds the tubulin monomer size (∼ 4nm) unless ∆ ≥ 100nm. In reality, the longitudinal stiffness E of each PF imposes a collective displacement over a distance 1

1

of (E/µ) 2 times ∆, that is, over ∼ ℓc along a PF. The amplitude of thermal fluctuations h(δu)2 i 2 is reduced by (E/µ)1/4 ≃ 30 times with respect to the previous case because the total restoring shear force on this block increases by a factor (E/µ)1/2 while the random thermal force increases by a 1

1

factor (E/µ)1/4 . We, therefore, have h((δu)2 i ≃ kB T /(∆µ) × (µ/E) 2 , or, δu[nm] ∼ 1.44(∆[nm])− 2 with the above mentioned parameter values for K and µ. The relative thermal displacement δu thus obtained is subject to two physical constraints. Within the resolution of modern cryo-electron microscopy data, 14 on the one hand, that is, about 1

0.2nm, no displacements are reported. This gives the lower bound for ∆ so that 1.44(∆[nm])− 2 < 0.2[nm]. The empirical stability criterion against melting, the so-called Lindemann’s criterion 15 asserts, on the other hand, that the thermal fluctuations of an atomic position is no more than around 10% of the inter-atomic distance in any inorganic solid and also in proteins, see for example the simulation study in. 16 Since melting is dominated by steric interactions, we may apply this criterion to the mesoscopic scale, that is, we regard the tubulin as a solid particle which itself is well stable. Then the criterion of δu for non-melting is 0.1∆0 = 0.4nm for ∆0 = 4nm, the tubulin monomer size. Since the former criterion from the structural data is more stringent, we adopt for ∆ ≃ (1.44/0.2)2 ≃ 52nm. 8 ACS Paragon Plus Environment

Page 9 of 20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


. To investigate further the dynamics of PF, we introduce a phenomenological model of the displacement fields ui , based on the energy functional H + H p discussed above and the dissipation ˙ associated to the frictional forces between adjacent PFs: functional W X ˙ =η W 2 j

Z

! ∂u j+1 (z) ∂u j (z) 2 − dz, ∂t ∂t

where η is a phenomenological friction constant. The Langevin equation for u j (z) can be constructed by adding a white Gaussian random noise with zero mean, ξi (z, t), to the equation of linear non-equilibrium thermodynamics: o ˙ δW δ n H + Hp + ξ j (z, t), =− δ[∂u j (z)/∂t] δu j (z)

(7)

where δ/δ[ f (z)] denotes the functional derivative with respect to the function f at the argument z. We argue later that we can neglect the dissipation due to the fluid viscosity around the PF. In terms of the mode amplitudes, {Uk }, the above equation Eq.(7) becomes " #) ( 2 1 ∂ (k) 2 ∂ − 1 Uk (z, t) + (2ηk kB T ) 2 Θk (z, t) = 0 −ηk + µk ℓc 2 ∂t ∂z

(8)

for k , 0, where ηk ≡ 2 sin2 [kπ/(2p+1)]η, µk ≡ 2 sin2 [kπ/(2p+1)]µ, and {Θk (z, t)} are independent white Gaussian random noises with zero mean and standardized variance; hΘk (z, t)Θk′ (z′ , t′ )i = δk,k′ δ(z − z′ )δ(t − t′ ). The pre-factor of Θk (z, t) in Eq.(8) has been determined so that the canonical equilibrium is attained in the absence of active forces. Also, in this way, the local translational invariance of the system is assured.

Eq.(8) shows that the bundle of tubulin works as a spatio-temporal low-pass filter characterized by the length and time scales ∼ ℓc(k) (≃ ℓc ) and ηk /µk (≃ η/µ), respectively. In other words, the



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 20

spectral response Uˆ k (q, ω) to the force ξˆk (q, ω) is described as Uˆ k (q, ω) ≃

1 ξˆk (q, ω), iηω + µ(1 + ℓc2 q2 )

(9)

apart from the numerical factors of order 1 which depend on k. Experiments 3 show that the longest decay time of MT fluctuations is ∼ 50ms. These results strongly suggest that η/µ . 50ms, which implies η . 102 Pa.s. This viscosity is much higher than water viscosity (≃ 0.9 × 10−3 Pa.s), and corresponds to the PF-PF direct friction. 4,17–19 This order of magnitude is very reasonable, corresponding to the monolayer-monolayer friction coefficients. 20 Eq.(9) also shows that thermal fluctuations on length scales smaller than ∼ ℓc are suppressed, which in turn tells us that shear propagates essentially uniformly over that length scale. We thus expect that the impact of a localized discontinuity in the displacement field of a PF spreads over a spatial range of order µm. This discontinuity may be generated by the binding of a motor. Before arguing its feasibility, we note that the walking of the motors at the speed ∼ 1µm/s does not play a role. In the following we, therefore, neglect the motor’s walking along the MT.

Binding of the first kinesin coated bead It was supposed that only a single dimeric motor is involved when the bead is bound to MT. 1 We, therefore, consider the binding of motor on a single PF. We discuss in the Supporting Information SI.3 the case where two PFs are involved. 3 symbolically shows the binding of wild type motor heads WT/WT (3(a)), or heterodimeric half-deficient motor WT/E236A (3(b)) to a PF. When an

(a)

(b)

Figure 3: The binding of a dimeric kinesin motor to a single protofilament (PF) with (a) homodimeric active wild-type heads (filled discs) (b) heterodimeric half-deficient heads. The inactive mutant head (cross) is permanently fixed to a PF. In both cases two heads of a dimer interact with the same PF and cause a localized conformational transition of tubulins of MT. This change causes a discontinuity in the longitudinal displacements. 10 ACS Paragon Plus Environment

Page 11 of 20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


active dimeric motor, either of wild type WT/WT or of mutant WT/E236A, binds strongly to a tubulin monomer, we suppose that a conformational transition is induced in that tubulin monomer or the tubulin dimer including the former. Such conformational transition may be related to the two state model of PFs proposed recently 6,7 and reported experimentally 21 14 . In our description using slowly varying displacements u j (z), the localized conformational transition of tubulin implies a “jump” δ j in this displacement, where j specifies the PF where the motor is bound. If the discontinuity occurs at z = z0 with the following form,

δ j = u j (z0 + 0) − u j (z0 − 0),

(10)

or equivalently, 1

Uk (z0 + 0) − Uk (z0 − 0) = (2p + 1)− 2 ζ −k j δ j .

(11)

with δ j a constant, then H p becomes, p X Dk(s) |Uk (z0 − 0)|2 − |Uk (z0 + 0)|2 ∆. Hp = 2 k=−p

(12)

We note that, for any energy functional which is quadratic in its dynamical variables (Uk here), the thermal average of those variables at finite temperature coincide with the result at T = 0K. Therefore, the individual values of u j (z0 +0) and u j (z0 −0) can be determined by further minimizing the total energy, H + H p , with respect to these variables under the constraint Eq.(10) or (11). From this minimization procedure a long range polar asymmetry results. The latter fact can be understood easily by looking at Eq.(12). For example, if D(s) k > 0, the term including this factor evidently acts to reduce |Uk (z0 − 0)| and to enhance |Uk (z0 + 0)|. As a result, the amplitude of displacement |Uk (z)| in the front region (z > z0 ) is larger than that in the back region (z < z0 ). The result of the minimization is (see the Supporting Information SI.1 for the derivation),    −(z−z0 )/ℓc(k)  for z > z0   Uk (z0 + 0)e Uk (z) =   (k)    Uk (z0 − 0)e(z−z0 )/ℓc for z < z0 11

ACS Paragon Plus Environment

(13)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 20

with    1 D(s) ℓc(k)  ζ − jk k  p Uk (z0 ± 0) = ± ± δ j, 2 E∆  2p + 1

(14)

where the ± signs should be chosen coherently. The front-rear asymmetry in the amplitude |Uk (z0 ± (k) 0)| is borne by the factor |D(s) k ℓc /(E∆)|.

In order not to be contradictory to the existing cryo-electron microscopy data the magnitude of δ j should be at most about 0.2nm. The mutant head of kinesin (E236A), which is inactive and permanently attached to a tubulin, may or may not induce the conformational transition of tubulin. As will be seen below our theory does not depend on this choice. In any case, once a “jump” of δ j is generated, Eq.(13) predicts an asymmetric displacements of PFs. In Eq.(14), the parameter (k) (s) √ characterizing the asymmetry, D(s) k ℓc /(E∆) ∼ Dk / Eµ can be appreciable as compared with unity when the coupling constants D(s) k (or dm ), which has the dimension of elastic modulus, is much larger than the shear modulus µ while it is definitely smaller than the compressive modulus E, i.e. µ ≪ |D(s) k | ≪ E. It may well be the case for the conformational transition which generically involves transversal steric interactions between the tubulin monomers that belong to neighboring PFs. As we will see below, comparison with the experimental data of 1 supports this hypothesis.

RESULTS AND DISCUSSION Active shear fluctuations and cooperative binding After the binding of the first kinesin-coated bead, the active motor heads thereon repeat the active cycle of strong/weak binding on tubulin. This cyclic process should induce in the tubulin the transitions back and forth between the two conformations. Since the timescale of the ATP hydrolysis cycle on the kinesin head, τcyc (∼ 30ms, typically), is just within the range of the low-pass filter mentioned above (see the paragraph of Eq.(9)), the static result Eq.(13) holds to a good approximation. Even if the conformational transition of the tubulins is a spatially localized event, it induces the displacements of the PFs back and forth over a long distance with broken front-rear


Page 13 of 20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


symmetry, extending over ∼ µm along the MT, with the amplitude and timescale of displacements being . 0.2 nm and ∼ 30ms, respectively. We shall interpret these active displacements in terms of effective temperature: Recalling that in equilibrium thermal random forces on the tubulin network cause random displacements |δu| . 0.2 nm of PF with their mean square displace1

ments, h(δuth )2 i 2 ∝ T, and also assuming that the active cycle of motors can cause independent random displacements δuac which are of comparable amplitude (. 0.2nm), we conclude that the active cycle will raise the temperature of the shear mode from the ambient one to an effective one, T eff ∝ h(δuth + δuac )2 i = h(δuth )2 i + h(δuac )2 i, which attains up to about twice the room temperature, T . We remark that, while the additional temperature associated to δuac persists as long as the bound kinesin undergoes the ATPase cycle, this activity will not appreciably warm up the other modes of protein dynamics. If the inactivated binding frequency of the second kinesin-coated bead ν is written as ν = ν∗ e−∆E/kB T , where ν∗ is the pre-exponential factor, the active fluctuations of PF by the active kinesin will modify this frequency at position z to ν(z) = ν∗ e−∆E/kB Teff , where the effective temperature at position z is written as T eff (z) = T + δT (z) with δT (z)/T = h(δuac )2 i/h(δuth )2 i. For the thermal 1

fluctuation we use the previously discussed result, h(δuth )2 i ≃ kB T /(∆µ) × (µ/E) 2 , where ∆ is our microscopic parameter introduced in Eq.(1) and E and µ are, respectively the Young modulus and shear modulus of the MT also introduced in Eq.(1). For the active fluctuation h(δuac )2 i we use Eqs.(13) and (14). Leaving the detailed account of the approximation and calculation in the Supporting Informations (SI.2), the result for δT (z)/T reads as follows:  2  1 d1(s)  ∓2(z−z0 )/ℓc(k=p) δT (z) 2 = α  ± √  e T 2 µE

(15)

p where α is the dimensionless constant which is essentially δ j / h(δuth )2 i apart from a numerical factor of order one. It characterizes the metrical importance of the conformational transition rel-

ative to the thermal fluctuations. We recall that δ j introduced in Eq.(10) characterizes the “jump” in the tubulin displacement on the j-th PF, jump which is turned on and off by the active motor.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 4: The binding frequency (in µm−1 s−1 ) of the second kinesin-coated beads vs the distance from the reference bead (in µm). The joined thick dots represent the experimental data read off from Fig.3D of. 1 The upper and lower data show the binding of active and non-active kinesin motors, respectively. Theoretical model, Eqs. (1) and (3) in the Supporting Information SI.2 , can fit these data with reasonable values of the model parameters (see the text). We also recall that d1(s) characterizes the energy associated polar asymmetry of PF (see Eq.(4) and d1(s) = (d1 + d−1 )/2), whose physical origin is schematically explained in Figure. 2. We then expect that such elevated effective temperature of MT favors the subsequent binding of kinesin coated beads by a statistical mechanism : Binding of the (second) bead to MT requires a specific conformational matching between a kinesin motor head on the bead and a protofilament of the MT. The active shear displacements of PFs induced by the motors on the first bead should help the exploration of the specifically matched conformation of the binding.

Comparison with the experimental data We can assess the parameters of our model by comparing our theoretical predictions with the experimental binding frequency of the second kinesin-coated beads. 1 4 shows how our model fits the experimental binding frequency as function of the distance from the first bound bead (Fig.3D of 1 ). Experimental data also show the absence of enhancement for the inactive kinesin motor heads. In our scheme it is because the kinetic temperature of the PF is in that case simply equal to room temperature. The fit of the experimental data gives, besides the decay distance of ≃ 10 µm, constraints on two key parameters. The first one is associated to the enhancement of the binding, α, characterizing the importance of the conformational transition. The other is the dimensionless 14 ACS Paragon Plus Environment

Page 14 of 20

Page 15 of 20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


√ coupling that leads to the asymmetry of the enhancement, d1 / µE in Eq.(4). The fit in 4 gives √ √ us the values, α = 3.5 ≃ 2 and d1 / µE = 0.12. The former (α) seems to be consistent with the picture that both δuth and δ j are of similar magnitude in the sub-nm range. Furthermore, the value √ √ of the latter (d1 / µE = 0.12, which means µ ≪ d1(s) . µE ≪ E), also seems to be reasonable (see the Supporting Information SI.2 for the details). In summary, the binding of the first bead with active motors enhances the binding rate of the second kinesin coated bead at large spatio-temporal scale ∼ µm and ∼ msec, and in a polar asymmetric manner.

CONCLUSION We show that a local tubulin conformation change can induce shear deformations over several microns. Such a change can be generated by kinesin strong biding to tubulin. The active transitions between strong and weak binding create favorable conditions for a second motor binding at distances over which the shear deformations are propagated. These deformations exhibit front-rear asymmetry. Our considerations provide a natural explanation for the experimental observations of E. Muto et al. 1 Obtaining an experimental test of these ideas is obviously very desirable. Changing physicochemical parameters such as salt concentrations, ATP, etc will have consequences but their signature will not be easy to interpret. We propose here a new type of spectroscopy which directly probes the importance of shear. More concretely, we propose to generate shear displacements among PFs by using a pair of optically trapped beads chemically bound to a MT. As these beads are most often bound to different PFs, the externally controlled displacement of the foci of two optical traps will induce either stochastic or systematic shear deformations among the PFs depending on the driving. By measuring the binding of motors as a function of shear amplitude and frequency this experimental set-up may work as a new type of spectroscopy probing the binding dynamics of



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 20

motors on their filament. Cooperative binding to biofilaments is a very general phenomenon , not specific to the microtubulekinesin system. We may expect similar phenomena in other systems. It is an important problem since many non-motile filament binding proteins that are relevant to development and disease could potentially exhibit similar behaviors. While observations of phase segregation by cryo-EM (e.g. Vilfan et al. 22 ) have been reported for some time, thus revealing cooperative binding of motors on filaments, more recent studies have examined filament mediated interactions, notably in Roos et al. 23 They show that energies of interactions of order a few kB T can favor proximity-dependent enhancements of motor-filament affinity. Uppulury et al. 24 also proposed that filament-mediated motor interactions, albeit at shorter distance, can lead to synergistic cooperation, and enhanced transport velocities under load. The originality of the cooperative binding of Muto et al. 1 stems from its long-range, asymmetric and active aspects.

Acknowledgement We thank Giovanni Cappello, Anne Houdusse, Pascal Martin and Etsuko Muto for helpful discussions. JP thanks also Christoph Schmidt for fruitful discussions. We also thank Erwin Frey for the careful reading the manuscript and very useful comments before submission.

Supporting Information Available § SI.1 Derivation of Eq.(13). § SI.2 Details of comparison with the experimental data. § SI.3 Alternative mechanism of active shearing of MT This material is available free of charge via the Internet at http://pubs.acs.org/.


Page 17 of 20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


References (1) Muto, E.; Sakai, H.; Kaseda, K. Long-range Cooperative Binding of Kinesin to a Microtubule in the Presence of ATP. J. Cell. Biol. 2005, 168, 691–696. (2) de Gennes, P.-G.; Prost, J. The Physics of Liquid Crystals; Clarendon Press, 1995. (3) Pampaloni, F.; Lattanzi, G.; Jonáš, A.; Surrey, T.; Frey, E.; Florin, E.-L. Thermal Fluctuations of Grafted Microtubules Provide Evidence of a Length-dependent Persistence Length. PNAS 2006, 103, 10248 –10253. (4) Janson, M. E.; Dogterom, M. A Bending Mode Analysis for Growing Microtubules: Evidence for a Velocity-dependent Rigidity. Biophysical J. 2004, 87, 2723–2736. (5) Mohrbach, H.; Kulić, I. M. Motor Driven Microtubule Shape Fluctuations: Force from Within the Lattice. Phys. Rev. Lett. 2007, 99, 218102. (6) Mohrbach, H.; Johner, A.; Kulić, I. M. Tubulin Bistability and Polymorphic Dynamics of Microtubules. Phys. Rev. Lett. 2010, 105, 268102. (7) Mohrbach, H.; Johner, A.; Kulić, I. M. Cooperative Lattice Dynamics and Anomalous Fluctuations of Microtubules. Eur Biophys J. 2012, 41, 217–239. (8) Tokuraku, K.; Kurogi, R.; Toya, R.; Uyeda, T. Q. Novel Mode of Cooperative Binding between Myosin and Mg2+ -actin Filaments in the Presence of Low Concentrations of ATP. J. Mol. Biol. 2009, 386, 149 – 162. (9) Uyeda, T.; Iwadate, Y.; Umeki, N.; Nagasaki, A.; Yumura, S. Stretching Actin Filaments within Cells Enhances Their Affinity for the Myosin II Motor Domain. PLoS One 2011, 6, e26200. (10) Van den Heuvel, M.; de Graaff, M.; Dekker, C. Microtubule Curvatures under Perpendicular Electric Forces Reveal a Low Persistence Length. PNAS 2008, 105, 7941–7946. 17 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(11) Heussinger, C.; Bathe, M.; Frey, E. Statistical Mechanics of Semiflexible Bundles of Wormlike Polymer Chains. Phys. Rev. Lett. 2007, 99, 048101. (12) Landau, L. D.; Lifshitz, E. M. Theory of Elasticity (Course of Theoretical Physics, Volume 7), 2nd ed.; Reed Educational and Professional Publishing Ltd, 2000. (13) Heussinger, C.; Schüller, F.; Frey, E. Statics and Dynamics of the Wormlike Bundle Model. Phys. Rev. E 2010, 81, 021904. (14) Morikawa, M.; Yajima, H.; Nitta, R.; Inoue, S.; Ogura, T.; Sato, C.; Hirokawa, N. X-ray and Cryo-EM Structures Reveal Mutual Conformational Changes of Kinesin and GTP-state microtubules upon binding. EMBO J. 2015, 34, 1270–1286. (15) Lindemann, F. A. The Calculation of Molecular Vibration Frequencies. Physik. Z. 1910, 11, 609–612. (16) Zhou, Y.; Vitkup, D.; Karplus, M. Native Proteins Are Surface-molten Solids: Application of the Lindemann Criterion for the Solid Versus Liquid State. J. Mol. Bio. 1999, 285, 1371– 1375. (17) Tawada, K.; Sekimoto, K. Protein Friction Exerted by Motor Enzymes Through a WeakBinding Interaction. J. Theor. Biol. 1991, 150, 193–200. (18) Brangwynne, C. P.; Koenderink, G. H.; Barry, E.; Dogic, Z.; MacKintosh, F. C.; Weitz, D. A. Bending Dynamics of Fluctuating Biopolymers Probed by Automated High-Resolution Filament Tracking. Biophysical J. 2007, 93, 346–359. (19) Taute, K. M.; Pampaloni, F.; Frey, E.; Florin, E.-L. Microtubule Dynamics Depart from the Wormlike Chain Model. Phys. Rev. Lett. 2008, 100, 028102. (20) Campillo, C.; Sens, P.; Köster, D.; Pontani, L.-L.; Lévy, D.; Bassereau, P.; Nassoy, P.; Sykes, C. Unexpected Membrane Dynamics Unveiled by Membrane Nanotube Extrusion. Biophysical J. 2013, 104, 1248–1256. 18 ACS Paragon Plus Environment

Page 18 of 20

Page 19 of 20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(21) Krebs, A.; Goldie, K.; Hoenger, A. Complex Formation with Kinesin Motor Domains Affects the Structure of Microtubules. J. Mol. Biol. 2004, 335, 139–153. (22) Vilfan, A.; Frey, E.; Schwabl, F.; Thormählen, M.; Song, Y.-H.; Mandelkow, E. Dynamics and cooperativity of microtubule decoration by the motor protein kinesin1. J. Mol. Biol. 2001, 312, 1011 – 1026. (23) Roos, W. H.; Campàs, O.; Montel, F.; Woehlke, G.; Spatz, J. P.; Bassereau, P.; Cappello, G. Dynamic kinesin-1 clustering on microtubules due to mutually attractive interactions. Phys. Biol. 2008, 5, 046004. (24) Uppulury, K.; Efremov, A. K.; Driver, J. W.; Jamison, D. K.; Diehl, M. R.; Kolomeisky, A. B. How the Interplay between Mechanical and Nonmechanical Interactions Affects Multiple Kinesin Dynamics. J. Phys. Chem. B 2012, 116, 8846–8855, PMID: 22724436.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Graphical TOC Entry


Page 20 of 20

Elastic Anisotropy Scenario for Cooperative ... - ACS Publications

Recommend Documents