Motor driven microtubule shape fluctuations - force from within the
  lattice
Motor driven microtubule shape fluctuations - force from within the lattice
Hervé Mohrbach1 and Igor M. Kulić2
1Laboratoire de Physique Moléculaire et des Collisions, Université Paul Verlaine - 57012 Metz, France
2School of Engineering and Applied Sciences, Harvard University, Massachusetts 02138, USA
(Dated: October 31, 2018)
We develop a general theory of microtubule (MT) deformations by molecular motors generating
internal force doublets within the MT lattice. We describe two basic internal excitations, the S
and V shape, and compare them with experimental observations from literature. We explain the
special role of tubulin vacancies and the dramatic deformation amplifying effect observed for katanin
acting at positions of defects. Experimentally observed shapes are used to determine the ratio of
MT shear and stretch moduli (≈ 6 × 10−5) and to estimate the forces induced in the MT lattice
by katanin (10’s of pN). For many motors acting on a single MT we derive expressions for the
end-to-end distance reduction and provide criteria for dominance of this new effect over thermal
fluctuations. We conclude that molecular motors if acting cooperatively can ”animate” MTs from
within the lattice and induce slack even without cross-bridging to other structures, a scenario very
much reminiscent of the motor driven axoneme.
PACS numbers: 87.15.-v 87.16.Ka 87.16.Nn
Microtubules are the stiffest cytoskelletal component
and constitute the main routes for motor mediated in-
tracellular cargo transport in higher organisms [1]. Un-
derstanding their physical properties is at the heart of
many biological problems from cellular mechanics to in-
formation and material trafficking in the cell. Since
the discovery of their high elastic anisotropy [2] it be-
came increasingly clear that MTs are mechanically more
complex than other semiflexible biofilaments. The high
anisotropy has been impressively confirmed by thermal
fluctuation analysis of beads attached to MTs of different
lengths[3]. The emerging picture of the MT is that of an
anisotropic fiber reinforced material [2, 3] with the tubu-
lin protofilaments (PF) acting as strong fibers weakly
linked with easily shearable lateral bonds. Remarkably
this type of design is also found in higher structures like
axonemes (constituting the backbone of flagella and cilia)
where relatively inextensible MTs are held together with
highly stretchable nexin connections [1]. This remark-
able structural self-similarity of the two nested structures
(MT and axoneme) indicates further analogies in the way
they respond to external and internal forces. We explore
here important consequences of MT geometry and elas-
tic properties and show that motors acting on the MT
surface can generate internal lattice strains sufficient to
induce observable lateral and longitudinal deformations
of the MT backbone.
In the following we describe a twist-free MT of length
L consisting of N identical PFa with constant dis-
tance a and a circular cross section, Fig 1. Each PF,
parametrized by the MT backbone arc length s has a
position dependent displacement uk (s) from its equilib-
rium position, k = 1, ...N . The backbone shape is de-
scribed by a curvature vector ~κ (s) = d
~t (s) with ~t (s)
the bundle centerline tangent. The elastic properties
of the MT are characterized by a PF bending stiffness
B = 1
πa4Y and compressional modulus Kc =
with Y ≈ 0.1−1.5GPa [2, 3] being the PF Young’s mod-
ulus. Additionally there are shear elastic forces restoring
the longitudinal displacement between the PFs governed
by a very soft elastic shear modulus Ks ≈ 10
−3 − 1MPa
[2, 3]. The elastic energy is given by
EMT =
∫ L/2
B~κ2 +Kcu
k +Ks∆
ds (1)
with the first term being the bending energy, the second
the PF compression and the third describing the relative
shear energy between the neighboring PFs. The shear
displacement ∆k is related to the difference of PF dis-
placements uk−uk−1 and a curvature induced additional
displacement via
∆k (s) = uk (s)− uk−1 (s) +
~κ (s′) ·∆~rkds
′ (2)
With ∆~rk = ~rk − ~rk−1 and ~rk = RMT
cos 2πk
, sin 2πk
the vector pointing from the MT center to the k-th PF,
cf. Fig 1. Equations 1-2 are 3-D analogues of the previ-
ously proposed stretchable railway-track [4] or wormlike-
bundle [5] model for the case of a hollow circular bundle.
While in general all the N + 3 fields , i.e. the 3 compo-
nents of ~κ (s) and the PF displacements {uk (s)}k=1,..,N
enter the eqs 1-2 in the limit of small MT deviations
from a straight line the problem can be drastically sim-
plified. We first expand the tangent ~t ≈ (θx, θy, 1) and
θ′x, θ
in terms of two angular projections θx
and θy of ~t in x and y direction respectively. Exploit-
ing the circular geometry of the PF arrangement and the
Fourier representation uk (s) =
ûq (s) e
2πikq
N over k we
quickly realize that only the longest wavelength mode
û1 (s) couples to overall MT backbone shape given by
the curvature ~κ. This leads to total energy decoupling
EMT = E
MT + E
MT + E
MT into a shape- independent
http://arxiv.org/abs/0704.1911v1
FIG. 1: (Color online) The basic geometry of motors induc-
ing internal force doublets along the MT backbone: between
two PFs (S-let) and along the same PF (V-let). The red and
blue ovals represent two coupled motors or two motor sub-
units (legs) of the same motor. Tubulin lattice vacancies at
the motor position strongly amplify the MT backbone defor-
mation (V gap -let).
component E0MT and two shape dependent contributions
(in x and y direction) given by:
EiMT =
∫ L/2
B̂∆θ′2i + K̂cU
2 + K̂s∆̂
ds (3)
With ∆̂i (s) = a (θi (s)− θi (−L/2)) − Ui (s) , a =
|∆~rk| the inter-protofilament distance , i = x, y and
~U (s) = (Ux, Uy) = (Reχû1, Imχû1) with χ = 1 −
e−2πi/N and renormalized constants B̂ = NB, K̂c =
NKc/ (4− 4 cos (2π/N)) and K̂s = NKs. Visually the
new variable ~U (s) is a x-y vector at each MT -crosscut
and can be interpreted as the (vectorial) mean over rel-
ative PF displacements of neighboring PFs. With this
enormous simplification at hand we can consider now ba-
sic motor induced MT excitations (Fig 1). There are two
elementary configurations in which motors can induce in-
ternal MT strains: 1) A motor (or a complex of several
motors) acting between two (not necessarily neighboring)
PFs and 2) A motor (or a complex of several motors) act-
ing at two points within the same PF. For reasons that
will soon become clear we call the excitation 1 S-type or
simply an ”S-let” and excitation 2 we call an V-type ex-
citation or ”V-let”. Both excitations are ”internal” in the
sense that there is no net torque or force on the system
motor+MT similarly to the case of a beating flagellum[1].
Elementary internal MT excitations. In the following
we want to understand the properties of the two basic
types of excitations from Fig 1 and focus on the S-type
first. We assume a single motor (or a complex of two
motors) at position s = s0 bridging between two PF
with index k1 and k2 and exerting opposing forces F
and −F onto them respectively, Fig 1 a) (left) + b).
The total energy is Etot = EMT + ES−mot with EMT
given by eqs 1-2 and the potential energy of the motor
ES−mot = −F
∑k=k2
∫ L/2
δ (s− s0)∆k (s) ds . As we
had for EMT before ES−mot also decouples into inde-
pendent modes in the Fourier representation over k and
Emot,S = E
S−mot+ E
S−mot+E
S−mot with E
S−mot a cur-
vature independent term and the two shape dependent
contributions E
S−mot (in x and y direction) given by:
EiS−mot = −F
∫ L/2
δ (s− s0) ∆̂i (s) ds with i = x, y
and ~µ being the vector connecting the two attachment
points of the motor (or motor complex) with components
µi = ~µ · ~ei , Fig 1 b. The equilibrium solution is given
by the Euler Lagrange equations: δEitot/δUi = 0 and
δEitot/δθi = 0 with E
tot = E
S,mot + E
MT and bound-
ary conditions θ′i (±L/2) = U
i (±L/2) = 0 (vanishing
bending and shearing stresses at the ends). A short cal-
culation leads to ∆̂′′i − λ
−2∆̂i = (1 + α)
δ (s− s0)
with the shear decay length λ =
λ−2c + λ
)−1/2
Here the two important length scales λc =
K̂c/K̂s
and λB =
B̂/a2K̂shave the physical meaning of a
pure compression- / pure bending- induced shear screen-
ing length respectively, with their squared ratio α =
(λc/λB)
≈ 35 (N = 13 PF). The remaining equations
lead to conservation laws (aθi + αUi)
= 0 which com-
bined with the equation for ∆̂i give for the simplest sym-
metric case s0 = 0 the tangent angles (up to an arbitrary
constant)
θi (s) = Φi
cosh (|s| /λ− L/2λ)
sinh (L/2λ)
With Φi = Φ
i = ~µ · ~ei
. The resulting MT backbone
curvature has a jump at s = 0 and attains its maximal
modulus there. The resulting MT shape is planar (con-
tained in the plane spanned by ~µ and ~t at any position)
and S-shape-like with initial and final angle coinciding
θi (−L/2) = θi (+L/2) which explains our nomenclature
”S-type excitation” or ”S-let”. Interestingly the length
scale λ over which θi (s) declines allows us to indepen-
dently estimate the ratio of stretch and shear moduli
from the observation of S-let deformations coming from
katanin action[7], Y/Ks ≈ 64π
−1 (λ/a)
≈ 6 × 105 for
λ ≈ 1µm (cf, Fig 2b) and a ≈ 6nm. This value is close to
the result obtained by Pampaloni et al. [3] (Y/Ks ≈ 10
Further the maximal deflection angle of θmax ≈ 38
(= 0.66) measured in Fig 2a gives via 4 an estimate
for the involved motor forces F ≈ 2B̂θmax/(λ |µ|) =
πa4Y/(λ |µ|) = 20−250pN (for |µ| = a ≈ 6nm and the
range of values for Y from literature [2, 3]). This indi-
FIG. 2: a: The shapes of V and S-type excitations as given
by Eq. 4 and their experimental observations for katanin op-
erating on MTs (b,c). b: Adapted and edge enhanced image
from [7]. c: Adapted from [6], black bars highlight the MT de-
formation. The upper and lower images show the MT before
and after kink generation respectively.
cates that many katanin motors might act cooperatively
to generate the observed shape change in Fig 2b. Another
interesting possibility is that katanin might be different
from dynein/kinesin by generating only small contractile
displacement ”powerstrokes” . 1nm with a more effi-
cient chemical-mechanical ATP-energy conversion lead-
ing to larger contractile forces F & 15kBT/nm = 60 pN .
The second fundamental internal MT excitation ap-
pears when a motor (or motor complex) acts along
a single PF with index k compressing or stretching
it. The motor energy in this case can be written
as EV,mot = F (uk (s0 + |µ| /2)− uk (s0 − |µ| /2)) ≈
F |µ|
δ (s− s0)u
k (s) ds where |µ| is the size of the
motor step. Like in the previous case it is sufficient
to keep the energy contribution of the mode q = 1
as the others decouple from each other and from the
curvature term. Along very similar line of derivation
as in the S-type excitation case we obtain a solution
which is planar and contained in the plane spanned by
the vector ~rk and ~t. The resulting tangent angles for
an excitation in the middle of the MT (s0 = 0) are
given by eq 4 with Φi = Φ
i (s) =
(~ei ·∆~rk)
γiF |µ|
2(1+α)B̂
which is now s dependent and changes sign at
s = 0. Here (γx, γy) = sin (π/N) (1− cos (2π/N))
(sin (2π (k + 1/2)/N) ,− cos (2π (k + 1/2) /N)) . The re-
sulting shape, that we call a ”V-let”, is a smooth V-
shaped planar kink in the MT backbone with continuous
curvature which relaxes on the length-scale λ. While su-
perficially similar the S-let and V-let solutions are physi-
cally very different for two reasons. First, the s/ |s| factor
in the V-let solution changes the symmetry with respect
to the S-let which leads to dramatic effects on the end-
end distance as we see below. Second difference lies in
the different scaling of the numerical prefactors which in
the case of a V-let do not contain the screening length
λ and involve additionally a very large reduction factor
1/ (1 + α) . In practice this suppresses significantly the
involved deformations (θ ≈ 10−5 − 10−6), orders of mag-
nitude below that of a S-let corresponding to the same
force (θ ≈ 10−1). However the situation changes dramat-
ically if the motor is operating at a position of a vacancy
in the tubulin lattice, cf. Fig 1. In this case the mo-
tor is not hindered by the large rigidity of the short PF
portion that the motor acts on. Formally there is no re-
quirement of continuity for uk (s) of the involved PF k
at the position of the gap. For such a combined defect
+ motor excitation which we call V gap-let, the motor en-
ergy is given by EV gap,mot = F (uk (s0 + 0)− uk (s0 − 0))
and its Ux/y dependent component becomes E
V gap,mot
= Fγi (Ui (s0 + 0)− Ui (s− 0)) . After a short calcula-
tion in direct analogy with the previous cases we obtain
the same form as in eq 4 but with Φi = Φ
V gap
i (s) =
(~ei ·∆~rk)
.While its functional form and symme-
try coincide with that of the V-let, the prefactor of a
V gap-let solution is more similar in magnitude to that
of a S-let. It is intuitive to think of a V gap - let as a
V-let with an effectively renormalized motor force F̃ =
2a−1λ (1 + α)F. The distribution of angles of observed
V-shapes measured by Davies et al [6] of θmax ≈ 15
30◦(cf.Fig 2c) would suggest for the defect free case (V-
let) a very large required force of F = 6×104−6×105pN .
However , the same estimate for the V gap-let case gives
much more moderate 10 − 100pN (few tens of motors),
showing the prime importance of lattice defects in the
case of V type excitations. Interestingly Davies et al. [6]
also suggested a crucial role of lattice defects based on the
pattern and kinetics of MT decomposition by katanin.
Statistical mechanics of multiple excitations. It is
particularly interesting to understand the collective
contribution of a large number of internal MT ex-
citations acting at random positions sj and orien-
tations −→µ j in addition to the MT thermal fluctu-
ations. The motor energy for S- and V-lets be-
comes now respectively EiS,mot = −
j∆̂i (sj) and
EiV,mot = −F
j |µ| γ
i (sj). It is convenient to in-
troduce the vector −→µ S/V,j such that µ
S,j ≡ µ
j and
µiV,j ≡ |µ| γ
j (i = x, y). For a fixed (but arbitrary)
distribution of motors the partition functions Zi =
DUiDθi exp
EiMT + E
and the correla-
tion functions
θi,qθi,p
(q, p 6= 0) can be obtained. Here
〈...〉 denotes the average over the random motore distri-
bution and (...) is the average over the thermal noise.
θi,qθi,p
decomposes into the sum of a thermal contri-
bution 〈θi,qθi,p〉T = 2kBTL
−1G(q)δp,q with the propa-
gator G(q) =
B̂q2 + q
2K̂sa
q2+K̂s/K̂c
and a motor con-
tribution
θi,qθi,p
= ΨS/V (p)ΨS/V (q)CS/V (q, p) .
For S- and V-let case we have respectively ΨS (p) =
2FK̂cp
L(K̂cp2+K̂s)
G(p) and ΨV (p) =
2FaK̂sp
L(K̂cp2+K̂s)
G(p),
with the motor position and orientation correla-
tor CS/V (q, p) =
j,l µ
S/V,jµ
S/V,l cos (qsj) cos (psl)
CS/V (q, p) is easily computed for the simplest choice
of a uniform motor position and orientation distribu-
tion: P (si) = 1/L ,
−→µ S/V,j =
∣µS/V
∣ (cosφj , sinφj)
with an random angle φj with a probability distribution
P (φj) = 1/2π. In this case we obtain the length reduc-
/L ≈ 1
θ2x,q
θ2y,q
which naturally
decomposes in a sum of a thermal fluctuation term and
a motor term
. In the relevant
limiting case L/λ ≫ 1 we obtain for the thermal part
a2K̂sl0p
with the large and small scale persistence lengths
given by l∞p =
B̂ + a2K̂c
/kBT and l
(α/ (1 + α))
B̂/kBT . Interestingly the term a
can be formally understood as an intrinsic self-tension
straightening the MT at small scales. Similar formulas
appear in different geometries for the railway-track[4] and
wormlike-bundle model[5]. The motor dependent length
reduction for the S-, V- and V gap-let excitations with line
density ρ is given by:
ρF 2µ2
a4K̂2sλ
ρF 2µ2L
a2K̂2sλ
V gap
= cV gap
ρF 2µ2L
a4K̂2sλ
with cS = α
2 (1 + α)
/16 ≈ 6 × 10−2, cV
= 0.18α2 (1 + α)
≈ 1.3 × 10−4 and cV gap =
0.73α2 (1 + α)
≈ 0.7. Remarkably the S- and V/V gap-
lets show different scaling. In particular
V/V gap
grows with L (in analogy to the first term in the ther-
mal contribution 5) while
/L stays length inde-
pendent. The physical reason for this difference becomes
obvious from Fig 2, as the relative slack
/L induced
by a single S-let scales with λ/L, while for an V/V gap-
let it is essentially length independent. For longer MTs
this effect leads to a strong dominance of V gap-lets over
S-lets
V gap
∼ (L/λ) ≫ 1. Although hav-
ing the same L scaling the minute prefactor of defect
free V-lets renders their contribution relatively insignif-
icant
∼ a2λ−3L ≈ 10−6 even for very
long MTs (L = 100µm), underlining the importance
of lattice vacancies transforming a V-let into a V gap-
let. Another interesting observation is that in all three
cases
S/V/V gap
∝ ρF 2. Fixing the number of mo-
tors Nmot but regrouping them into Nmot/M clusters of
size M we have ρ → M−1ρ, F → MF and therefore
S/V/V gap
∝ M, i.e. the slack grows linearly with
the cluster size. This indicates that cooperativity (po-
sitional correlation) of motor action can lead to strong
enhancement of the slack length.
From Eqs.5-7 we can derive criteria for the dominance
of motor slack over the thermal slack. For instance us-
ing the values estimated from Fig 2 b,c for katanin for
elastic constants from [3] (F = 20pN , λ = 1µm) and
µ = 8nm, L = 20µm we obtain
/L = ρ/ρc with
ρc,S ≈ 0.25nm
−1, ρc,V gap ≈ 1.2 × 10
−3nm−1. For large
enough motor densities the katanin action easily domi-
nates over the thermal slack
/L ≈ 6× 10−4.
Being evolutionary specialized for MT deformation and
degradation katanin is likely to be among the strongest
slack generating motors. We suspect however that classi-
cal motors like dynein and kinesin might cause less pro-
nounced but observable effects as well. While dynein
is known to walk between several PFs, kinesin is very
strictly following a single one[1]. Our theory suggest that
dynein should induce moving S-lets, yet with quickly fluc-
tuating signs which would diminish the effect consider-
ably. A battery of many kinesins, however, walking over a
MT region with many tubulin vacancies, would give rise
to spatially stationary V gap-lets blinking between ”on”
and ”off” states. The theoretical and experimental ex-
ploration of these issues is an interesting future direction.
The authors acknowledge fruitful discussions with E.
Frey, C. Heussinger, M.Bathe, O. Campas, J.F. Joanny
and P.C. Nelson. I.M.K. acknowledges support by the
Max-Planck Society.
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[7] J.J. Hartmann et. al, Cell, Vol. 93, 277287; Movie at
http://valelab.ucsf.edu/images/mov-rhomtsvkat.mov
with kind permission by R. Vale
http://arxiv.org/abs/cond-mat/0702097
http://valelab.ucsf.edu/images/mov-rhomtsvkat.mov
