Single-polymer ‘flyfishing’ effect for nanoscale motors and machines:
an exact worm-like-chain model study
Nan Wang,1, 2 Ruizheng Hou,3, 2 Weizhu Bao,1, 2 and Zhisong Wang3, 2, 4, a)
1)
Department of Mathematics, National University of Singapore, Singapore 119076
Center for Computational Science and Engineering, National University of Singapore,
Singapore 119076
3)
Department of Physics, National University of Singapore, Singapore 119076
4)
NUS Graduate School for Integrative Sciences and Engineering, National University of Singapore,
Singapore 119076
2)
Single-polymer control effects are abundant in biological nanomotors and replicating these effects in innovatively designed molecular systems leads to artificial motors/machines for nanotechnology. Understanding
motor-relevant polymer effects in a general and conceptual clear way is important for the study of many
biological nanomotors and for the development of nanotechnological counterparts from scratch out of diverse
molecular building blocks. Here we study single-polymer control effects on the basis of the exact solution to
the worm-like chain (WLC) model in three dimensions. The results suggest the possibility of a surprisingly
accurate flyfishing-like control in which tilting one end of a flexible polymer enables positioning of the other
diffusing end to a remote location within an error of ∼ 1 nm. This single-polymer flyfishing not only provides
a directional bias, but also is a molecular-scale power stroke in the literal sense as a real force is transmitted
along a flexible polymer from the tilted end to the freely diffusing end. The flyfishing effect likely plays a
role in biological nanomotors, and may be used in artificial nanomotors to improve their directional fidelity,
efficiency and speed. We also obtain a new force-extension formula that is valid for both stretched and
compressed polymers. The formula provides a convenient tool to estimate direction and magnitude of the
intra-chain force, which are critical for the flyfishing and also for another directional bias called site-selective
dissociation in nanomotors.
I.
INTRODUCTION
Polymer mechanics play an important role in nanomotors from biology and nanotechnology, especially those
motors that walk along a linear track.
Virtually
all the track-walking biomotors1 and the man-made
counterparts2–4 reported to date use polymer linkers
to connect the leg-like track-binding parts. The polymer linkers are often semi-flexible oligomers that consist of 10—100 monomer units and exhibit a rich interplay between bending energy and conformational freedom (i.e., entropy). The polymer linkers can be peptides
in the form of soft random coils5 or relatively rigid alpha
helices6 , or be nucleotides2–4 in the form of single strands
or double-stranded helices. The length and intra-chain
force of a polymer linker largely decide how fast a diffusing leg binds to the track and how much force can be
transmitted into the motor’s forward motion against a
load. After an event of leg binding, the free energy of the
polymer linker can be high enough14–16 to affect the overall stability of the motor-track state, which in turn pertains to the leg dissociation for the motor’s continual motion. Therefore, controlling the polymer’s length, force
and free energy is a common mechanism by which biological and artificial nanomotors regulate their leg binding/dissociation for directional motion. Indeed, biological motors exhibit a wide variety7,8 of sophisticated poly-
a) Author
to whom correspondence should be addressed. Electronic
mail:[email protected]
mer control. As a prominent example, a bipedal biological motor called kinesin aligns5 a part of its soft peptide linker at one end in a zipper-like manner along the
track to throw forward the diffusing leg at the linker’s
other end. This polymer ’zippering’ provides a forward
bias for leg binding, and also is suggested to transmit
an intra-chain force like a power stroke9 . This force is
largely associated with the linkers entropy change10 . A
biological bipedal motor called myosin V generates a similar forward bias but by forward tilting11 its rigid helical
linker at one end. Interestingly, tilting the linker at its
one end can also transmit a force to selectively dissociate
a leg at the other end from the track even though the leg
binds the track stronger than the leg at the tilted end, as
found in another biological bipedal motor12,13 . For both
kinesin and myosin V, the free energy associated with the
polymer linker is comparable14,15 to the leg-track binding
energy in magnitude. This makes a symmetry breaking
possible14,16 by which a local asymmetry in the control of
one end of the polymer linker may be amplified into larger
directional steps of the motor as a whole. This polymer
mechanics-mediated symmetry breaking has been identified in biological nanomotors14,15 and later implemented
in artificial DNA nanomotors3,17 .
The motor-enabling polymer effects identified so far
are likely a ‘tip of iceberg’, implying a remarkable wealth
of single-polymer control mechanisms that are likely a
common mechanistic basis for nanoscale motors and machines. Understanding the rich single-polymer control
effects is important not only for the study of biological nanomotors, but also for the development of arti-
2
ficial counterparts for nanotechnology. While biological nanomotors are largely limited to peptides, artificial
nanomotors may be made of diverse building blocks ranging from peptides, nucleotides to synthetic polymers. Besides, any effective polymer control may be attributed to
a fine interplay among the conformational entropy and
bending energy of the polymer, and the energy associated with its binding with other molecular partners (e.g.,
tracks). Therefore, the study of motor-relevant polymer controls calls for a general and rigorous polymermechanical framework that counts the conformational
entropy and bending energy accurately and allows the
study of polymer controls in a generic, conceptually clear
way. Such a general framework will be particularly useful
in providing guidelines for developing artificial nanomotors from scratch. Previous theoretical studies13–16,18 of
motor-relevant polymer effects focus on specific molecular systems (mostly biological motors and associated
peptide systems) or are based on approximate solution
of polymer mechanics. A good candidate for developing the general framework is the worm-like chain (WLC)
model for semi-flexible polymers. On one hand, the exact solution to the WLC model in two dimensions19,20 becomes available recently to allow a precise counting of the
conformational entropy, which remains difficult for other
methods like atomic simulations. On the other hand, the
WLC model has only two parameters: contour length
as a measure of the maximum stretch of a polymer and
persistence length as a measure of the polymer’s bending
rigidity. Both parameters can be extracted by fitting experimental data, potentially making the WLC model a
realistic working model for any polymeric building blocks
of nanomotors.
In this study, we extend the method in19,20 for solving the WLC model in two dimensions to three dimensions and then apply the exact WLC model to study
several single-polymer control effects concerning remote
positioning and force transmission, which are pertinent
to biological and artificial nanomotors. The results suggest feasibility of these single-polymer controls up to a
surprising accuracy even for a rather soft polymer, which
rationalizes high optimality previously found for some biological nanomotors and reveals new mechanistic regimes
to improve performance of artificial nanomotors. This
study demonstrates the capacity of the exact WLC model
in three dimensions to serve as a general working framework to study motor-relevant polymer effects. A new
force-extension formula is also obtained for the purpose
from the exact solution of the WLC model in three spatial
dimensions. The new formula relation has an improved
accuracy over the widely used approximate formula21 for
stretched polymers, and also is applicable to compressed
polymers.
II.
A COMPUTATIONAL METHOD
The simplest model for describing semiflexible polymers without self-avoidance is the so-called worm-likechain (WLC) model22 . In this model, the polymer is
modeled as a continuous curve — a configuration C —
that can be specified by a d-dimensional (d > 1) curve
~x(s) (0 ≤ s ≤ lc ), with s the arc-length parameterization
of the curve and lc is the contour length of the polymer.
The unit tangent vector to the chain at s is denoted as
~t(s) = d~x(s) (|~t(s)|2 = ~t(s) · ~t(s) = 1) and the curvature
ds
~(s) of the polymer at s is given by κ(s) = dtds
. For the
simplicity of notations, we suppose that one end of the
polymer is tethered to the origin, i.e. ~x(0) = 0, and the
other end ~x(lc ) = ~r is tagged. As the polymer configuration changes with thermal agitation, the location ~r of
its tagged end fluctuates. The quantity we want to compute is Q(~r), which is the probability distribution for the
location ~r of the tagged end, and the free energy F (~r)
which is defined as the change in Q(~r) if the tagged end
is pulled from ~r to ~r + d~r. Thus Q(~r) is directly related
to the force-extension relation of the polymer, i.e. intrachain force f (~r).
A.
Probability distribution function Q(~r)
Different analytical, asymptotic and numerical methods have been proposed in the literature19,20 for solving
the WLC model. Here we extend the method for solving
the WLC model for a two-dimensional case19 to a full
three-dimensional case. For a given three-dimensional
curve C: ~x(s) (0 ≤ s ≤ lc ) of a polymer, we have
Rl
~r = 0 c ~t(s)ds and the bending energy of the polymer is
R lc
2
~
E(C) = A
2 0 |dt(s)/ds| ds with A the bending modulus.
To compute Q(~r), we need to sum over all polymer configurations C which end at ~r, with a Boltzmann weight or
the partition function according to the Boltzmann’s Law
X
Z=
e−E(C)/kB T ,
(1)
C
where kB is the Boltzmann constant and T is temperature. This is a standard counting problem in statistical mechanics and can be naturally addressed in the
language of path integration by considering the Brownian motion in the space of the tangent vectors ~t(s)
(0 ≤ s ≤ lc ). The advantage of this approach is that the
tangent vectors form a unit sphere and thus the problem reduces to studying Brownian motion on the unit
sphere, which can be handled by the standard operator
techniques familiar from quantum mechanics.
Denote ~tA = ~t(0) and ~tB = ~t(lc ) as the two unit tangent vectors at the two ends of the configuration C, respectively, of the polymer, and let lp = kBAT be the persistence length of the polymer and denote the dimensionless
parameter β = llpc . Then Q(~r) with ~tA and ~tB two given
3
unit vectors for the configuration C can be expressed by
the path integral representation19,20
Q(~r) = N
Z
~
tB
lp
D[~t(s)]e− 2
R lc
0
|d~
t(s)/ds|2 ds
Let Ylm := Ylm (θ, ϕ) (l ≥ 0 and |m| ≤ l) be the standard spherical harmonic functions, then we have
~
tA
×δ
3
~r −
Z
lc
!
~t(s)ds ,
0
~r ∈ R3 ,
(2)
where δ 3 (·) is the Dirac function in three dimensions and
N is a normalization constant such that all polymer configurations C which end at ~r with different ~tA and ~tB are
counted. We remark here that Q(~r) = 0 when |~r| > lc .
In order to find Q(~r), we define its generating function
P̃ (~k) through the Laplace transform
Z
1
~
~
~k ∈ C3 .
P̃ (k) =
d~r ek·~r/lp Q(~r),
(3)
N R3
From (2) and (3) and the inverse Laplace transform, us√
ing the change of variable ~k = − iβu~ with i = −1, we
get
Z
N
Q(~r) =
P̃ (−i~u/β) ei~u·~r/lc d~u, ~r ∈ R3 . (4)
(2πlc )3 R3
Plugging (2) into (3), using the change of variable s =
ilp τ and denoting ~t̃(τ ) = ~t(s) = ~t(ilp τ ), we obtain
P̃ (~k) =
Z
~
tB
~
tA
D[~t̃(τ )] ei
R −iβ
0
~ )
2 ~ ~
[ 12 | dt̃(τ
dτ | +k·t̃(τ )]dτ
.
(5)
~ 2
dt̃ In this equation, if τ and 12 dτ
+ ~k · ~t̃ are viewed as
time and Lagrangian, respectively, then it can be interpreted as the path integral representation for the kernel of a quantum particle on the unit sphere at inverse temperature β. Thus we can express P̃ (~k) for
and ~k = (k1 , k2 , k3 )T ∈ C3 as the quantum amplitude to go from an initial tangent vector state ~tB to
the final tangent vector state ~tA in the imaginary time
with length β in the presence of an external potential
−k1 sin θ cos ϕ − k2 sin θ sin ϕ − k3 cos θ with (θ, ϕ) the
spherical coordinates of the unit sphere. Therefore (5)
can be written as a “vacuum persistence amplitude”
P̃ (~k) = hgA |e−i(−iβ) H~k |gB i = hgA |e−β H~k |gB i,
(6)
P̃ (~k) =
∞
X
l
X
l,l′ =0 m,m′ =−l
≈
L
X
l
X
l,l′ =0 m,m′ =−l
′
′
′
′
hgA |Ylm
i hYlm
|e−βH~k |Ylm i hYlm |gB i
′
′
hgA |Ylm
i hYlm
|e−βH~k |Ylm i hYlm |gB i
′
′
:= U T e−βG V,
(9)
where L is an integer chosen numerically for the truncation, U and VR are
two column vectors with components
π R 2π
m
hgA |Yl i := 0 0 gA (θ, ϕ)Ylm (θ, ϕ) sin θdϕdθ (|m| ≤ l
and 0 ≤ l ≤ L) and hYlm |gB i (|m| ≤ l and 0 ≤ l ≤ L),
respectively. Here For example, if the end A of the
polymer is aligned at a fixed unit direction, i.e. ~tA =
(sin θA cos ϕA , sin θA sin ϕA , cos θA ) with θA and ϕA two
fixed angles, then gA (θ, ϕ) = δ(cos θ − cos θA )δ(ϕ − ϕA );
if the end B of the polymer is free, i.e. ~tB is uniformly
1
distributed on the sphere, then gB (θ, ϕ) = 4π
. G is a
m′
m
m′
matrix with entries hYl′ |H~k |Yl i = hYl′ |H0 |Ylm i −
′
′
′
k1 hYlm
|H1 |Ylm i − k2 hYlm
|H2 |Ylm i − k3 hYlm
|H3 |Ylm i
′
′
′
(|m|, |m′ | ≤ l and 0 ≤ l, l′ ≤ L). By using the properties of the standard spherical harmonic Ylm , we can
find for |m|, |m′ | ≤ l and l, l′ ≥ 0 that
( l(l+1)
′
′
m′
m
2 , l = l & m = m,
hYl′ |H0 |Yl i =
0,
otherwise,
′
hYlm
|H1 |Ylm i
′
 1
√ ,

6


q


(l+m+2)(l+m+1)



4(2l+1)(2l+3) ,


q



− (l−m)(l−m−1)
4(2l−1)(2l+1) ,
=
q


− (l−m+2)(l−m+1)


4(2l+1)(2l+3) ,
q



(l+m)(l+m−1)



4(2l−1)(2l+1) ,


0,
l = m = 0 & l′ = m′ = 1,
l′ = l + 1 & m′ = m + 1 & l ≥ 1,
l′ = l − 1 & m′ = m + 1 & l ≥ 1,
l′ = l + 1 & m′ = m − 1 & l ≥ 1,
l′ = l − 1 & m′ = m − 1 & l ≥ 1,
otherwise,
′
where
1
H~k = − ∇2 − k1 sin θ cos ϕ − k2 sin θ sin ϕ − k3 cos θ
2
:= H0 − k1 H1 − k2 H2 − k3 H3 ,
(7)
is the corresponding Hamiltonian with
1
H0 = − ∇2 ,
H1 = sin θ cos ϕ,
2
H2 = sin θ sin ϕ,
H3 = cos θ.
and gA := gA (θ, ϕ), gB := gB (θ, ϕ) are the distribution
functions of the unit vectors ~tA and ~tB over the unit
sphere, respectively.
(8)
hYlm
|H2 |Ylm i
′

− √i6 ,



q


(l+m+2)(l+m+1)


−i

4(2l+1)(2l+3) ,


q



i (l−m)(l−m−1)
4(2l−1)(2l+1) ,
=
q



−i (l−m+2)(l−m+1)

4(2l+1)(2l+3) ,
 q




i (l+m)(l+m−1)


4(2l−1)(2l+1) ,


0,
l = m = 0 & l′ = m′ = 1,
l′ = l + 1 & m′ = m + 1 & l ≥ 1,
l′ = l − 1 & m′ = m + 1 & l ≥ 1,
l′ = l + 1 & m′ = m − 1 & l ≥ 1,
l′ = l − 1 & m′ = m − 1 & l ≥ 1,
otherwise,
4
′
hYlm
|H3 |Ylm i
′
 1
√ ,

 q3



 (l+1+m)(l+1−m) ,
= q (2l+1)(2l+3)

(l−m)(l+m)



(2l−1)(2l+1) ,


0,
the free energy of the ensemble F (~r) can be computed as
l = m = m′ = 0 & l′ = 1,
l′ = l + 1 & m′ = m & l ≥ 1,
′
′
l = l − 1 & m = m & l ≥ 1,
otherwise.
In practical computations, we choose the integer L
large enough such that the truncation error in (9) can be
negligible and choose the distribution functions gA (θ, ϕ)
and gB (θ, ϕ) for the two unit vectors ~tA and ~tB , respectively, based on the assumption of the chain, e.g. both
end free, or one end free and the other end fixed, etc.
Then we can compute the matrix G and the two vectors
U and L in (9). Then the matrix exponential e−βG is
computed numerically via the Páde approximation and
P̃ (~k) can be evaluated by matrix multiplication. Finally
Q(~r) is obtained from P̃ (~k) by a three-dimensional inverse fast Fourier transform (FFT) via (4). To check the
method and our code as well as to compare it with existing results, we consider a WLC polymer with one end
located at the origin and aligned towards the z direction,
i.e. ~tA = (0, 0, 1)T and the other end free. After we compute Q(~r), we display the distribution in the x-z plane
for different y values (Fig. 1). The results agree very well
with those reported in20 by using a different method.
FIG. 1. The probability distribution Q(x, y, z) taken at different y planes for different values of β with a fixed lc = 1.
B.
Free energy F (~r) and intra-chain force f (~r)
Based on the probability distribution function Q(~r),
we can derive the thermodynamic quantities such as the
free energy F (~r) and the intra-chain force f (~r). In fact,
F (~r) = −kB T ln Q(~r),
|~r| < lc .
(10)
In order to keep the free end staying at the position ~r, a
force must be applied, i.e. intra-chain force f (~r) which
can be computed as
f~(~r) = ∇F (~r),
III.
|~r| < lc .
(11)
RESULTS AND DISCUSSION
A. Single-polymer ‘flyfishing’ by a local alignment at one
end
FIG. 2. Mechanics of a worm-like-chain polymer with its one
end located at the origin (x = 0, y = 0, z = 0) and the other
end diffusing to reach a location on the z axis. Shown are
the probability distribution of the free end along the z axis,
Q(z) = Q(0, 0, z), and the free energy F (z) = F (0, 0, z), the
intra-chain force f (z) = F ′ (z) derived. The results are for
two scenarios: The end at the origin either has free orientation, or is aligned at a fixed angle θ with respect to the z
direction. The polymer has a contour length lc = 10 nm and
a persistence length lp = 4 nm.
In the study of biological nanomotors, it is long believed that a zippering-like control of a soft polymer
5
linker at one end biases a motor leg at the other end of the
linker to bind the track forward towards the zippering direction over opposite direction. As we shall show below,
the exact WLC dynamics in three dimensions suggests
that more effects pertinent to a motor’s position-selective
binding can occur generally. Fig. 2 presents the probability distribution of one free end of a WLC polymer along
the z axis when the other end is located at the origin
and has different orientations for its tangent: the end at
the origin either has free orientation or is aligned towards
an angle θ = 90, 45, 0 degree with reference to the positive z direction to mimic the zippering effect (following
a treatment of Spakowitz and Wang in20 ). For the free
orientation and the vertical orientation (θ = 90 degree),
the free end has two symmetric most probable positions
at a positive and a negative z value. When the fixed end
is tilted from 90 to 45 degree and further 0 degree, the
probability peak at negative z values virtually vanishes
but the peak at the positive z value rises, in consistence
with the well-known zippering-induced bias. However,
an examination of the extent of accuracy by which aligning the end can position the other free end to a certain
location reveals several new, distinct patterns.
FIG. 3. The most probable end location and the full width
at half maximum probability (FWHM) versus the angle of
alignment at the other end for the polymer of Fig. 2. The
results (symbols) are extracted from the Q(z) = Q(0, 0, z)
distributions in Fig. 2.
First, the full width at half maximum (FWHM) of the
probability peaks becomes as small as ∼ 2 nm for the
zero-degree alignment (Fig. 3). This suggests the possibility of positioning the free end to the most probable
location (zc ) within an error of ∼ 1 nm for a polymer
with a contour length of 10 nm. Second, such a precise site-selective positioning may occur for rather flexible
polymers with lp /lc = 0.3 − 0.4, not necessarily require
a rigid molecular rod (lp /lc > 1). Indeed the probability
for locations around zc peaks at intermediate values of
persistence length (lp ) for a certain contour length (lc )
(Fig. 4). And the FWHM becomes largely flat when
lp /lc changes from 0.3 to 1 (Fig. 5). Hence a flexible
polymer of lp /lc ∼ 0.3 − 0.4 already accesses the regime
of precise positioning. Third, the precise positioning can
be modulated by adjusting the end alignment and polymer rigidity. Changing the end alignment from 90 degree
to 0 degree shrinks the FWHM of the probability peaks
monotonically (Fig. 3), hence improves the positioning
accuracy. The most probable location of positioning (zc )
may be modulated too by the alignment (Fig. 3), or by
changing the polymer rigidity (Fig. 5), which may be
done by adjusting the solution conditions like ion concentrations, pH values, temperature etc.
FIG. 4. Dependence of the end probability, free energy and
intra-chain force on persistence length for WLC polymers as
for Fig. 2 for the alignment angle θ = 0. The polymers have
the same contour length lc = 10 nm but different persistence
length (lp ). The exact solutions are shown by symbols with
the lines to guide eyes.
FIG. 5. The most probable end location and the full width at
half maximum probability (FWHM) versus persistence length
for WLC polymers as for Fig. 2. Shown are the results for
the scenarios of the end at the origin having free orientation
or being aligned parallel to the z axis (θ = 0). The polymers
have the same contour length lc = 10 nm but different persistence length (lp ). The results (symbols) are extracted from
the Q(z) distributions as in Fig. 2.
Hence aligning a flexible polymer of lp /lc = 0.3 − 0.4
and lc = 10 nm at one end positions the other free end
to a most probable position of zc ∼ 8 nm within an error
of ∼ 1 nm along the path parallel to the alignment. Such
a surprisingly precise positioning to a unique, remote lo-
6
cation (relative to the length of the polymer) is achieved
by controlling the flexible polymer at one end, essentially
resembling the art of flyfishing but at molecular level.
The direction and speed of a bipedal nanomotor is
largely decided by the position-selective binding of its
legs, which is in turn affected by the mechanics of the
inter-leg polymer linker. Fine-tuning a nanomotor into
the regime of molecular ‘flyfishing’ may improve the motor’s directional fidelity23,24 , which is important25 for
making motors of high energy efficiency. The flyfishing
effect also helps improve a motor’s speed by accelerating
forward binding of its legs: From the free or vertical orientation to zero-degree orientation, the FWHM drops by
a factor of ∼ 2 (Fig. 3) and two most-probable positions
merge into one. This amounts to a drop of a factor of
∼ 4 for the search volume of a diffusing leg at the linker’s
free end. A rough estimation for the first-passage time26
suggests more than two-fold increase in the leg’s forward
binding rate by the flyfishing. Hence the single-polymer
flyfishing likely plays a role in biological nanomotors that
possess high directional fidelity, energy efficiency or high
speed. This is consistent with the observation of zippering in biomotor kinesin, which is the smallest bipedal
motor found in biology but has a directional fidelity24
of more than 90%, an efficiency27 of 60% - 80% and a
speed28 of a few micrometers per second.
B. Single-polymer power stroke and intra-chain force
transmission
The polymer zippering effect found in biomotors is often likened to the power stroke9 of a piston in macroscopic engines; but validity of the notation of power
stroke for a polymer linker is questioned29 since any effective transmission of force along a soft polymer is hard
to imagine. The molecular flyfishing effect lends a support to the single-polymer power stroke. Consider again
the flexible polymer of lp /lc = 0.3 − 0.4 and lc = 10 nm.
Keeping an alignment at one end requires application of
a force, which may be regarded as an input. The outcome
at the other end is a combination of two effects along the
direction of the alignment. First, the two most probable
positions before the alignment converge into one, resulting in a forward displacement of 2 × zc ≈ 16 nm for the
free end. Second, the alignment causes a free-energy difference of ∼ 11kB T over the displacement, amounting to
a force transmission of 2.75 pN (for room temperature
of 25◦ C) from the controlled end to the free end. The
directional displacement and the force transmission of a
few pN combine to suggest a piston-like power stroke in
real sense.
Furthermore, a finite force transmission is possible by
the flysfishing effect even for very soft polymers. The
force transmission becomes zero only for zero displacement, which is made clear below using the Gaussian
chain model and the freely joint chain model (14 and
16), and is confirmed by the exact solution (Fig. 6). For
the same displacement of 16 nm, the alignment causes a
free-energy difference of ∼ 7kB T for a polymer even with
lp /lc → 0. This implies a minimum force transmission of
1.75 pN by the flyfishing for arbitrary polymers. Interestingly, the minimum free-energy difference of ∼ 7kB T
is close to the observed entropy difference10 (∼ 6kB T )
associated with the directional bias of biomotor kinesin,
which possesses an inter-leg peptide linker9 of lc ∼ 10 nm
and lp < 1 nm30 and makes a forward leg binding over a
regular displacement of ∼ 16 nm.
FIG. 6. The free energy difference, ∆F (z) = F (−z) − F (z),
for WLC polymers as for Fig. 2. Shown are the results for the
scenario of the end at the origin being aligned parallel to the
z axis (θ = 0). The polymers have the same contour length
lc = 10 nm but different persistence length (lp ). In the left
panel the exact solution (symbols) are extrapolated to zero
persistence length to yield ∆F ∗ = ∆F at lp → 0, which are
shown as the solid line in the right panel. For small lp , the
probability can be well described by the Gaussian function
2
as Q(0, 0, z) = 103 σ1√2π exp(− (0.1z−µ)
) with µ = 0.1021lp
2σ 2
2
and σ = 0.0670lp . Taking lp → 0, the Gaussian fit yields
∆F ∗ (z) = 0.30478z (dashed line in the right panel).
The end alignment-induced free-energy change may be
derived analytically for Gaussian chains and freely joint
chains. Following reference22 , the end-to-end distance
distribution of a Gaussian chain under the approximation
of one-dimensional random walk with a total of N steps
is:
q2
1
PN (q) = √
e− 2N .
2πN
(12)
Each step is either in forward or backward direction and
q is the position after making N steps, so −N ≤ q ≤ N .
If the first step is toward the direction of the positive q
direction, the end-to-end distance distribution function
becomes P + as
PN++1 (q) = PN (q − 1),
PN++1 (−q) = PN (−q − 1). (13)
Therefore the free energy difference between q and −q in
the dimensionless variable 0 ≤ z := Nq ≤ 1 is
∆F (z) := kB T ln
= kB T ln
PN++1 (N z)
PN++1 (−N z)
= kB T ln
PN (q − 1)
= 2kB T z.
PN (−q − 1)
PN++1 (q)
PN++1 (−q)
(14)
7
For the two- and three-dimensional random walks, the
factor 2 becomes 4 and 6, respectively.
For a one-dimensional freely joint chain without the
Gaussian approximation, the end-to-end distance distribution is:
PN (q) =
1
N!
,
2N [(N + q)/2]! [(N − q)/2]!
q = −N, −N + 2, . . . , N − 2, N.
(15)
Following a derivation similar to the previous one for the
Gaussian approximation, the free energy difference can
still be written in a simple analytical form (with z :=
q
N +1 the dimensionless variable)
N +1+q
1+z
= kB T ln
N +1−q
1−z


∞
X
z 2j 
= 2kB T z 1 +
, 0 ≤ z ≤ 1. (16)
2j + 1
j=1
∆F (z) = kB T ln
in which the fishing rod is tilted to pull a fish out of water. Moreover, the intra-chain force on the two ends have
opposite directions, which leads to dissociation of one
leg but not the other depending on their positions if the
leg-track binding is responding to forward or backward
force in a symmetric way. The site-selective leg dissociation (often called gating7 ) and the asymmetric leg-track
binding32,33 are two major directional biases identified in
biological nanomotors. For an alignment with an intermediate θ angle between 0 and 90 degree, the intra-linker
force may be distributed on the two ends in two dimensions in a highly asymmetric way13 . As a consequence, a
strong leg may be dissociated by the linker alignment at
a weak leg as has been observed12 for biological motors.
Interestingly, a latest report of an artificial DNA bipedal
nanomotor4 demonstrates experimentally how leg dissociation can be induced purely by the force transmitted
within the inter-leg polymer linker and in a site-selective
way to ensure a unidirectional motor.
The free energy difference is higher than the Gaussian
chain result but decays to the latter for z ∼ 0. Both the
Gaussian chain and freely joint chain models predict a
finite ∆F (z) for non-zero z values.
C.
Site-selective dissociation by intra-chain force
Unlike the force transmission through a free diffusing
end, the force transmission along a polymer with both
ends restricted is well known. This type of force transmission occurs in the inter-leg linker of a bipedal nanomotor when its two legs are bound to the track, and has
been suggested to be a cause5,13 for leg dissociation in
biomotors. Indeed, leg dissociation induced purely by
an intra-polymer force transmission has been observed
for biomotors12 , and successfully implemented in a latest
artificial DNA nanomotor4. These experimental studies
are supported by our exact WLC solution in three dimensions.
Consider again the example of a flexible polymer of
lp /lc = 0.3 − 0.4 and lc = 10 nm for the inter-leg linker
of a bipedal nanomotor. Imagine the nanomotor on a
linear track along the z axis with one leg bound at the
origin z = 0 and the other leg near the most probable
positions allowed by the inter-leg linker, namely z = +zc
and −zc with zc > 0. Note that zc remains close to 8
nm regardless of end orientation as predicted by the exact WLC solution in three dimensions (Fig. 2). When
the second leg is bound near z = −zc , the intra-linker
force is increased when the linker’s end at the origin is
changed from free or vertical orientation to a forward orientation parallel to the track (i.e., θ = 0) (Fig. 7). The
force becomes more than 12 pN at θ = 0, which is rather
sufficient31,32 to dissociate a leg off the track for many
biological nanomotors. This is like the normal fishing
FIG. 7. The intra-chain force versus the angle of alignment at
the other end for the polymer of Fig. 2. The results (symbols)
are extracted from the f (z) distributions in Fig. 2.
D.
New force-extension formula
The intra-chain force dissociating a leg depends on the
end-to-end distance as well as a polymer’s persistence
length and contour length (Figs. 2&4). To facilitate
nanomotor studies, we obtain an analytical formula capturing these dependencies by fitting the force-extension
results from the exact WLC solution in three dimensions
for 0 ≤ z < lc
z
f (z)
β z
1
1 β5.1
0.3 lc
=
+
− e
kB T
lc lc
4(1 − z/lc)2
4
!
2
10 z
1
z
+ 0.94β 52.4
− 35.8
1 − e − β lc .
lc e
lc
This new formula differs from the previous proposed
force-extension formulas in two aspects. First, the
new fitting formula recovers
the empiricalMarko-Siggia
β
z
1
1
formula21 , kfB(z)
=
+
T
lc lc
4(1−z/lc )2 − 4 , at the long
chain limit β = llpc → ∞, but captures the exact WLC
solution better than the latter for short chains with small
β values (Fig. 8). We note that short chains with β < 30
are most relevant to nanomotors. As a matter of fact, the
8
mechanics of such short biopolymers are much less known
to date than long ones due to difficulty of single-molecule
measurement. Applicable to both long and short polymers, the new formula is useful not only for nanomotor
studies but also for interpreting single-molecule mechanical experiments on short polymers. Second, the MarkoSiggia formula and some later improvements34 apply only
to stretched polymers which have an end-to-end distance
larger than the thermodynamically most probable extension. The new formula applies not only to stretched polymers but also compressed polymers which have an end-toend distance smaller than the most probable extension.
As found in ref.4 , both the stretched and compressed
regimes of a polymer can be used to induce site-selective
dissociation for a nanomotor, though the direction of the
intra-chain force is revered for the two regimes. For a
polymer of a certain contour length, changing its end-toend distance and persistence length switches the polymer
between the stretched and compressed regimes, resulting
in different direction and magnitude of the intra-chain
force. Quantitative details of these patterns from the exact solution are well captured by the new formula (Fig.
8).
in biological nanomotors, especially those of high
directional fidelity, efficiency and speed.
• The flyfishing is a molecular-scale power stroke in
the literal sense as a real force is transmitted from
an input at the tilted end to the freely diffusing end
along a flexible polymer. This helps clarify the long
debated notation of power stroke in the study of
biological and artificial nanomotors. We note that
power stroke-like polymer effects are being actively
pursued in the study of artificial nanomotors3,4,35 .
• A new force-extension formula is obtained from the
exact solution to the WLC model. The new formula is obtained by fitting the exact results for
semiflexible oligomeric chains relevant to nanomotors, but also recovers the Marko-Siggia formula for
long, stretched polymers. The new formula applies
to both stretched and compressed polymers, and
provides a convenient estimation of the intra-chain
force (both direction and magnitude) critical for
flyfishing and site-selective dissociation in nanomotors.
These findings are all based on the exact solution to the
WLC model in three dimensions, suggesting the capacity of this model for the study of motor-relevant polymer
effects in a general and conceptual clear way. Hence the
three-dimensional WLC model may form a basis for a
general and rigorous polymer mechanics framework for
nanomotor research, especially for guiding the development of artificial nanomotors from scratch out of diverse
molecular building blocks. It will be interesting to see
how the single-polymer flyfishing effect predicted here
may find use in future artificial nanomotors.
ACKNOWLEDGMENT
FIG. 8. New approximate formula for the force-extension
relation of WLC polymers. Shown are the force-extension
relation from the exact solutions in three dimensions for a
lc = 10 nm polymer in comparison with the predictions by
the new formula and the Marko-Siggia formula. Both ends of
the polymer have free orientation.
IV.
CONCLUSION AND PERSPECTIVES
The findings of this study are summarized as follows.
• A precise, flyfishing-like control is possible in which
tilting one end of a flexible polymer enables positioning of the other diffusing end to a remote location within an error of ∼ 1nm. The location
and accuracy of this single-polymer flyfishing can
be modulated by adjusting the tilting angle and
the chain rigidity. The flyfishing likely plays a role
We acknowledge support from the Ministry of Education of Singapore grant R-146-000-196-112 (W. Bao) and
grants R-144-000-320-112 and R-144-000-325-112 (Z.S.
Wang).
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