Curvature Distribution of Worm-like Chains in Two and Three Dimensions

0.24 MB·English

by  S. M. Rappaport

Curvature Distribution of Worm-like Chains in Two and
Three Dimensions
Shay M. Rappaport, Shlomi Medalion and Yitzhak Rabin∗
Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
February 4, 2008
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n Bending of worm-like polymers carries an en- polymers involves generating a representative
a
ergy penalty which results in the appearance statistical ensemble of different conformations
J
of a persistence length l such that the poly- of space curves and weighting them by appro-
1 p
2 mer is straight on length scales smaller than l priate Boltzmann factors. Since these statisti-
p

and bends only on length scales larger than this cal weights are Gaussian functions of the local
]
t
f length. Intuitively, this leads us to expect that curvature, one expects that the most probable
o
themostprobablevalueofthelocalcurvatureof valueofthecurvatureiszero, i.e., thattheprob-
s
. a worm-like polymer undergoing thermal fluc- ability distribution of curvature is peaked at the
t
a
tuations in a solvent, is zero. We use simple origin. This expectation agrees with our intu-
m
geometric arguments and Monte Carlo simula- ition about semi-flexible polymers: since they
-
d
tions to show that while this expectation is in- deviate from straight lines only on length scales
n
o deed true for polymers on surfaces (in two di- exceedingtheirpersistencelength, thelocalcur-
c mensions), in three dimensions the probability vature of a typical configuration is expected to
[
of observing zero curvature anywhere along the vanish almost everywhere. As we will show in

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worm-like chain, vanishes. the following using both analytical arguments
v
3 and Monte Carlo simulations, the above con-
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clusions hold only in two dimensions; surpris-
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1 Introduction
3 ingly, the most probable value of the curvature
.
1 in three dimensions is not zero!
0 Semi-flexible polymers are usually modeled as
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space curves (either continuous or discrete) that
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2 Curves, curvature and
: possess bending rigidity. Following the linear
v
i theory of elasticity of slender rods, the bending measure
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energy of such worm-like polymers is taken to
r
a be a quadratic functional of the local curvature Consider a space curve defined by a vector (cid:126)r(s)
[1]. The curvature is a local geometrical prop- (with respect to some space-fixed coordinate
erty of a continuous curve - the reciprocal of system), wheretheparametersisthearclength.
the radius of the tangential circle at any point The shape and the orientation of the curve are
along the curve. In discrete models the curva- completely determined (up to unifrom transla-
ture is the angle (per unit length) between ad- tion)bythedirectionofthetangenttothecurve
jacent segments, in the limit when the length of at every point along its contour, tˆ(s) = d(cid:126)r/ds:
these segments goes to zero (see below). The
(cid:90) s
calculation of the statistical properties of such (cid:126)r(s) = ds(cid:48)tˆ(s(cid:48)) (1)
0
∗Nano-materials Research Center, Institute ofNan-
Since the tangent is a unit vector, |tˆ(s)| = 1, it
otechnology and Advanced Materials, Bar-Ilan Univer-
sity, Ramat-Gan 52900, canberepresentedasapointonasphereofunit
1
radius; thecoordinatesofthispointaregivenby where we used the equalities (tˆ )2 = (tˆ )2 =
n+1 n
the two spherical angles θ and φ. Finally, one 1 and tˆ · tˆ = cos(∆θ). This definition of
n+1 n
can introduce a local property of the curve, the the curvature is consistent with the one used in
curvature, defined as : the Frenet-Serret theory only if κ∆s << 1, in
which case κ∆s ≈ |∆θ|. Using Eqs.4 and 5 it is
(cid:12)(cid:12)d2(cid:126)r(cid:12)(cid:12) (cid:12)(cid:12)dtˆ(cid:12)(cid:12)
κ(s) = (cid:12) (cid:12) = (cid:12) (cid:12) (2) straightforward to show that:
(cid:12)ds2(cid:12) (cid:12)ds(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
dtˆ(s) = d(∆ϕ)(s)∆s2κ(s)dκ(s) (6)
Let us introduce a discrete version of the con-
tinuous curve of length L, by dividing it into N The above discussion applies to curves in 3 di-
segments of length ∆s (N∆s = L), such that mensions. In 2 dimensions the measure is rep-
the orientation of the n’th segment connecting resented by a single angle θ so that in the dis-
points s and s+∆s is defined by the direction crete representation of a curve, the angle be-
of the tangent at point s. One can define the tween successive segments, ∆θ, varies in the in-
orientation of the n’th tangent vector (i.e., the terval [−π,π]. Unlike the 3d case in which the
n’th segment) in terms of the (n−1)’th one curvature is a positive definite quantity (see Eq.
2), thecurvatureofaplanarcurvecanbeeither
tˆ = cos(∆θ)tˆ +cos(∆ϕ)sin(∆θ)nˆ
n n−1 n−1 positive or negative (the torsion is zero every-
ˆ
+sin(∆ϕ)sin(∆θ)b
n−1 where). The direction of the tangent vector can
(3)
be represented as a point on a unit circle and,
As shown in Fig. 1, ∆θ ∈ [0..π] is the angle
therefore, the measure is given by
between the n’th and the (n − 1)’th segments
and∆ϕ ∈ [−π..π]isthepolaranglebetweenthe dtˆ(s) = d(∆θ)(s) ≈ ∆sdκ(s), in 2d (7)
two planes defined by tˆ ×tˆ and tˆ ×tˆ .
n−2 n−1 n−1 n
In the continuum limit ∆ϕ becomes the angle We conclude that in summing over the con-
of rotation of the normal nˆ and the binormal figurations of 3d curves, one can replace the
ˆb, with respect to the tangent tˆ to the curve. measure in the laboratory frame, D{(cid:126)r(s)}, by
In differential geometry this triad tˆ, nˆ and ˆb is that in terms of the intrinsic coordinates of
referred to as the Frenet frame and its rotation the deformed line, which is proportional to
as one moves along the curve is governed by the κ(s)D{κ(s)}. This differs from the correspond-
Frenet-Serret equations. As we will show in the ing measure for 2d curve which is proportional
following, in the limit of a continuous curve the to ∝ D{κ(s)}.
angle ∆θ is proportional to the curvature (one
can also show that the torsion is proportional
3 Curvature distribution of
to the angle ∆ϕ [2]).
The calculation of statistical averages involves worm-like polymers
integration over all possible configurations of
the polymer with respect to the measure In the worm-like chain model (WLC) a polymer
D{(cid:126)r(s)} or, alternatively, over D{tˆ(s)}. Dis- is modeled as semi-flexible rod whose elastic en-
cretization allows us to replace this measure by ergy is governed only by the curvature [3]:
the product of contributions of its segments,
1 (cid:90) 1
D{tˆ(s)} → Π (dtˆ ), where the measure of a E = b dsκ2(s) → b(cid:88)∆sκ2 (8)
n n WLC 2 2 n
single segment is given by n
where b is the bending rigidity. This energy
dtˆ= sin(∆θ)d(∆θ)d(∆ϕ) (4)
is invariant under polar rotations and conse-
The local curvature of a discrete curve can be quently the torsion angle ∆ϕ is uniformly dis-
defined by replacing the differentials by differ- tributed in the range [−π,π]. In 3 dimensions,
ences in Eq. 2: the probability p(κ)dκ of observing a value of
the curvature in the interval between κ(s) and
(cid:113) (cid:113)
∆sκ = (tˆ −tˆ )2 = 2(1−cos(∆θ)) (5) κ(s)+dκ at a point s along the contour of the
n n+1 n
2
curve, is proportional to the product of corre- between two neighboring segments is picked
sponding Boltzmann factor by the measure κdκ at random and a randomly oriented axis that
(see Eq. 6), passes through this point is defined. Then, the
entire part of the polymer that lies on one side
p(κ)dκ = lp∆se−12lp∆sκ2κdκ (9) of the chosen point, is rotated about the axis by
an angle ψ which is uniformly distributed in the
where l = b/k T is the persistence length
p B range [−π,π] (see figure 2). Since in the WLC
(for L >> l the polymer length does not af-
p model the total energy (in dimensionless units,
fect the statistics of the curvature). Inspec-
k T = 1) of a discrete polymer is:
B
tion of Eq.9 shows that if one rescales the cur-
(cid:113) (cid:88)
vature as κ → κ˜ = κ l ∆s, the probability E = b [1−cos∆θ ], (12)
p WLC n
distribution functions of polymers with differ- n
ent bending rigidities can be represented by a
the change of energy in a move is given by
single master curve κ˜exp[−(1/2)κ˜2]. This scal-
ing may lead to the conclusion that changing ∆E = b[cos(∆θ ) −cos(∆θ ) ] (13)
WLC n old n new
the persistence length is equivalent to chang-
ing the discretization. Indeed, calculation of A move is accepted or rejected according to
properties such as the mean curvature of a Metropolis rule and statistics is collected fol-
curve depends on the discretization length ∆s lowing equilibration determined by convergence
and diverges in the continuum limit, (cid:104)κ(cid:105) = of the total energy (after at least 106 moves).
lim (l ∆s)−1/2 = ∞. This, however, is not In order to double check our method, we calcu-
∆s→0 p
true in general and calculation of physical ob- lated the tangent-tangent correlation function
servables such as the tangent-tangent correla- and the mean square end to end distance as
tion function, yields well-behaved results: well and confirmed that they agree with well-
known analytical results for worm-like chains
(cid:68) (cid:69) (cid:68) (cid:69)
tˆ(0)·tˆ(s) = tˆ ·tˆ = (cid:104)cos(∆θ)(cid:105)n (10)
0 n (not shown). Inspection of Fig. 3 shows that
there is perfect agreement between our simula-
From Eqs.9 and 5 one can easily conclude that:
tion results and the analytical expression, Eq.
(cid:104)cos(∆θ)(cid:105) = 1−∆s/l ∼ exp[−∆s/l ]. Insert-
p p 9. The two main features of the distribution
ing the above expression into Eq. 10 one gets
are the existence of a peak at a finite value of
the WLC relation:
the curvature and the fact that the probabil-
(cid:68) (cid:69) ity to observe zero curvature vanishes identi-
tˆ(0)·tˆ(s) = e−s/lp (11)
cally. Atfirstsight,thepredictionthatanylocal
measurement on a worm-like chain will yield a
This correlation function and all properties that
non-vanishing curvature, contradicts our intu-
can be derived from it, such as the mean square
ition about stiff polymers. Note, however, that
end to end distance, depend only on the per-
experiments (e.g., by AFM) that monitor the
sistence length and are independent of the dis-
configurations of semi-flexible polymers such as
cretization.
dsDNA, do not directly measure the curvature
butratherthelocalbendingangle. Eventhough
4 Simulations and analyti- (cid:113)
(cid:104)κ(cid:105) = 1/ l ∆s does not vanish, the average
p
cal results local angle (measured on length scales smaller
(cid:113)
than the persistence length) (cid:104)δθ(cid:105) = ∆s/l is
p
In order to study the curvature distribution always small.
function of a worm-like polymer numerically, The above discussion applies to worm-like poly-
we performed Monte-Carlo simulations of a dis- mers that do not have intrinsic curvature (their
crete polymer made of N equal segments, in 3d. curvature is generated by thermal fluctuations
To generate the conformations of a polymer we only). Although the WLC model has been ap-
used the following moves (see ref. [4]): a point plied [5] to interpret mechanical experiments
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on individual double stranded DNA (dsDNA) experiments. To the best of our knowledge, all
molecules [6], it has been suggested[7] that ds- experiments that probed the local curvature
DNA has sequence-dependent intrinsic curva- of DNA (by electron microscopy [9] and by
ture. In this case, the experimentally observed SFM [10, 8]) were done on dsDNA molecules
local curvature of dsDNA contains both the deposited on a surface. It has been argued
contribution of intrinsic curvature and that of that in this case one has to distinguish between
thermal fluctuations [8]. The generalization to strong adsorption in which case dsDNA attains
the case of spontaneous curvature is straightfor- a purely 2d conformation, and weak adsorption
ward: even though the expression for the bend- in which the 3d character of the adsorbed
ing energy Eq. 8, is replaced by macromolecule is maintained, at least in part
1 (cid:90) 1 (see ref. [11]). While in the former case the
E = b dsδκ2(s) → b(cid:88)∆s(κ −κ )2
WLC n 0,n measure of Eq. 7 applies and the most probable
2 2
n
curvature vanishes [12], in the latter case one
(14)
should use the 3d measure of Eq. 6 for which
the measure is not affected and depends only
the most probable curvature is finite and the
on the dimensionality of the problem. Using
probability to observe zero curvature vanishes.
the discrete form of curvature Eq. 5 the distri-
New experiments that could probe the 3d
butionfunctionofcurvatureforagivenintrinsic
conformations of DNA and other worm-like
curvature κ˜ is:
0
polymers are clearly necessary to test our
p(κ˜|κ˜0)dκ˜ = A(κ˜0)κ˜e−12l(κ˜−κ˜0)2dκ˜ (15) predictions.
where
Acknowledgments We would like to acknowl-
(cid:20)(cid:90) 1 (cid:21)−1/2
A(κ˜ ) = dκ˜κ˜exp(− l(κ˜ −κ˜ )2) (16) edge correspondence with A.Yu. Grosberg Y.
0 0
2
Rabin would like to acknowledge support by
is a normalization factor and l is the persistence
a grant from the US-Israel Binational Science
length. An experiment measuring the curva-
Foundation.
ture distribution of an ensemble of conforma-
tions of a polymer with a specific primary se-
quence {κ (s)}, would measure the distribution
0
Eq. 15, from which the intrinsic curvature can
be calculated as a fit parameter. If one moni-
torsthecurvaturedistributionofanensembleof
random copolymers whose sequences are char-
acterized by a distribution of intrinsic curvature
p(κ˜ ), the corresponding curvature distribution
0
function will be:
(cid:90)
p(κ˜) = dκ˜ p(κ˜|κ˜ )p(κ˜ ) (17)
0 0 0
In order to check this result we performed
Monte Carlo simulations with different distri-
butions of κ˜ . As can be observed in Fig:4, the
0
simulations are in perfect agreement with the
analytical results for the curvature distribution
calculated using Eq. 17
5 Discussion
Finally, we would like to comment on the ap-
plicability of our results to present and future
4
References
[1] LDLandau, EMLifshitzTheory of Elastic-
ity , 1970, Pergamon Press, Oxford, New
York
[2] S.M.Rappaport, Y.Rabin J. Phys. A:
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[3] M.Rubinstein, R.H.Colby Polymer Physics
Oxford University Press 2003 ch.2;
[4] Hongmei Jian, Alexander V. Vologodskii
and Tamar Schlick J.Comp.phys, 1997,
136, 168
[5] J.F.MarkoandEDSiggiaMacromolecules,
1995, 28, 8759
[6] C.Bustamante,S.B.Smith,J.Liphardt,D.
Smith, Curr.Op.Struct.Biol 2000 10, 279;
T. Strick, J.F. Allemand, V. Croquette,
D. Bensimon, Progress in Biophysics and
Molecular Biology, 2000, 74, 115.
[7] EN Trifonov Trends Biochem Sci, 1991,
16(12) 467.
[8] A.Scipioni, C.Anselmi, G.Zuccheri,
B.Samori and P.De Sentis Biophys. J,
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[9] G.Muzard, B.Theveny and
B.Revet EMBO.J, 1990, 9, 1289.;
J.Bednar,P.Furrer, V.Katritch,
A.Z.Stasiak, J.Dubochet and A.Stasiak
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[10] C.Riveti, M.Guthold and C.Bustamente
J.Mol.Biol, 1996, 264 919.
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285, 997.
5
Fig. 1 3 successive segments are shown by
black arrows. ∆θ is the angle be-
tween the (n−1)’th segment (the ra-
dius of the unit sphere) and the n’th
segment. ∆ϕ is the polar angle de-
fined by the projection of tˆ (dashed
n
arrow) on the plane defined by the
ˆ
normal nˆ and the binormal b
n−1 n−1
(red arrows)
Fig. 2 Sketch of a Monte Carlo move. The
right hand part of the curve rotates
with random angle ψ with respect to
arandomaxis(dashedredline)pass-
ing through a random point m
Fig. 3 Distribution function of scaled cur-
vature κ˜ for a polymer of N = 104
segments of length ∆s = 1 each.
The empty symbols are the simu-
lation results for l = 25 (black
p
squares), l = 50 (red circles) and
p
l = 100 (green triangles). The rigid
p
line is the analytical scaled distribu-
tion function κ˜exp[−(1/2)κ˜2] where
(cid:113)
κ˜ = κ l ∆s. There are no fit pa-
p
rameters.
Fig. 4 In all figures the blue squares and
green diamonds are the histograms
of curvature and of intrinsic curva-
ture, respectively. Theredlineisthe
analyticaldistributionfunctioncom-
puted using Eq. 17.
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Figure 2: Caption of Fig. 2.
Figure 1: Caption of Fig. 1.
0.7
0.6
0.5
0.4
)
~
(
P
0.3
0.2
0.1
0.0
0 1 2 3 4 5 6 7 8 9 10
~
Figure 3: Caption of Fig. 3.
7
0.6 5
0.5
4
P P
0.4
3
0.3
2
0.2
0.1 1
0 0
(a) 0 0.5 1 1.5 2 (b) 0 0.5 1 1.5 2
kappa kappa
5 5
4 4
P
3 3
2 2
1 1
0 0
(c) 0 0.5 1 1.5 2 (d) 0 0.5 1 1.5 2
kappa kappa
Figure 4: Caption of Fig. 4.
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