Table Of ContentBending elasticity of macromolecules:
analytic predictions from the wormlike chain model
Anirban Polley,1 Joseph Samuel,1 and Supurna Sinha1
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1 1Raman Research Institute, Bangalore, India 560 080
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2 We present a studyof thebend angle distribution of semiflexible polymers of short and interme-
diate lengths within the wormlike chain model. This enables us to calculate the elastic response
n
of a stiff molecule to a bending moment. Our results go beyond the Hookean regime and explore
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the nonlinear elastic behaviour of a single molecule. We present analytical formulae for the bend
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angledistributionandforthemoment-anglerelation. Ouranalyticalstudyiscomparedagainst nu-
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merical Monte Carlo simulations. The functional forms derived herecan be applied to fluorescence
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microscopic studies on actin and DNA. Our results are relevant to recent studies in “kinks” and
cyclization in short and intermediate length DNAstrands.
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o PACSnumbers: 61.41.+e,64.70.qd,82.37.Rs,45.20.da
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a INTRODUCTION ply the general theory to a special case to illustrate its
m use: we treat the pure bending elasticity of a semiflexi-
- A classic study in elasticity is the bending of beams ble polymer not subject to stretching forces or twisting
d
n or rods subject to forces and moments[1]. At the cellu- torques.
o lar level there are many macromolecularstructures like Twodifferentexperimentaltechniquesarepossibleto
c actin and cytoskeletal filaments, which are like beams probe the elastic properties.
[ in giving rigidity and structure to the cell. Unlike the Distribution of tangent vectors: Onecantagtheends
1 beams studied by civilengineers,these macromolecular of the molecule with flourescent dye[12] to determine
v beamsaresubjecttothermalfluctuations. Thepurpose the direction of the initial and final tangent vectors.
3 of this study is to look at the role of thermal fluctua- Byfluorescencevideomicroscopy,onefindsthe angular
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tions in shaping the elastic properties of macromolecu- distribution P(θ) of the bending angle θ , where θ is
3
4 lar beams. defined by cosθ = tˆi.tˆf. This experimental technique
. Weworkwithinthewormlikechainmodel[2,3],which hasbeenusedtostudyactinintwodimensionalstudies
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has been known to describe double stranded DNA [4] in Ref.[12]. The angular distribution of θ gives us the
0
3 as well as actin filaments[5]. For clarity we consider an elasticpropertiesofthe moleculeandonecancompute,
1 experiment in which one end of the molecule is fixed for instance, the probability for a given bending angle
: at the origin and its tangent vector at the same end θ and compare with the theoretical expectation.
v
i is constrained to lie along the zˆ direction. We wish Measuring moment angle relations: A more inva-
X
to know the number of configurations (counted with sive experimental technique is to tether one end of the
r Boltzmann weight) that will result in the final tangent molecule, attach a magnetic bead to the other and ap-
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vector tˆ . In this paper we present an approximate ply bending moments to the molecule by varying the
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analytical study of this statistical mechanical problem. direction of an applied magnetic field. Note that we
Some earlier treatments of this problem[6, 7] restrict do notconstrainthe final positionofthe molecule x(L)
to small bending angles, so that the polymer is essen- butonlyitsfinaltangentvectortˆf. Auniformmagnetic
tially straight. One can then replace the sphere of tan- field will result in a pure bending moment without ap-
gentdirectionsbythetangentplanetothissphere. This plying any stretching force. Plotting the bending mo-
givesagoodaccountofsmallbending angles. However, ment vs the bending angle gives another experimental
thereisconsiderableexperimentalandtheoreticalinter- probe of the elastic properties. In this paper we derive
est in largebending angles to understand cyclizationof the predictionsofthe wormlikechainforboththese ex-
DNA [8–10]. Ref. [11] developed a new approximation perimental situations.
technique that worksevenfor largebending angles,say Wefirstderiveanapproximateanalyticalformulafor
of order π/2. The treatment of [11] is general and in- the free energy allowing for thermal fluctuations of a
cludes applied forces and torques. In this paper we ap- wormlike chain polymer. We then plot the expected
2
theoreticaldistribution ofbending anglesandthe theo-
retically expected moment-angle relation. We conclude
with a discussion.
MECHANICS AND FLUCTUATIONS
In the simplest wormlike chain model, we model the
polymerbyaspacecurve~x(s). ~x(s)describesthecurve,
and tˆ(s) = d~x, its tangent vector. s is the arc length
ds
parameter along the curve ranging from 0 to L, the
contour length of the curve. ~x(0) = 0 since one end is
fixed at the origin. The tangent vectors at both ends
tˆ(0),tˆ(L) are fixed to tˆ and tˆ respectively.
i f
Themathematicalproblemwefaceistocomputethe
partition function
( )
Q(tˆi,tˆf)= exp E C . (1) FIG. 1. Two configurations of a molecule with fixed end
XC −hkBTi tangent vectors tˆi and tˆf. (left) Minimum energy configu-
ration and (right) the same configuration is shown slightly
In Eq.(1), the sum is over all allowed configurations of
perturbed bythermal fluctuations.
the polymer, those which satisfy the boundary condi-
tionsforthetangentvectoratthetwoends: tˆ =tˆ(0)=
i
zˆ,tˆ =tˆ(L). The energy functional is given by minimumtoquadraticorder. Thiscalculationwasdone
f
in general form in Eq.(53) of Ref. [11] for a single
A L dtˆ dtˆ molecule subject to forces and twisting torques. The
( )= ( . )ds, (2)
E C 2 Z ds ds determinant of the fluctuation operator is calculated
0 O
in [11]. In the present case, this fluctuation determi-
whereAisanelasticconstantwithdimensionsofenergy
nant can be calculated either by specialising Eq.(53) of
times length. The quantity L = A/kT is the persis-
p [11] to the case of zero force and torque of [11] or by
tencelength. Forexampleactinhasapersistencelength
the method described in [13] in the context of Brown-
ofabout16µmandanelasticconstantof6.7 10−27 in
ian motion on the sphere. As is to be expected, both
×
units of Nm2. Lengths of actin filaments can go up to
methods give the same answer
hundreds of microns, so the molecule is of intermediate
flexibility. Det =L2sinθ/θ. (4)
O
Wewillcomputethepartitionfunctionassumingthat
near the stiff limit (L not much larger than L ), the Performing the Gaussian integration over the
p
sum over curves is dominated by configurations near fluctuations we find the distribution function
the minimum of the energy. 1/ (DetO)exp−EkmTin and we arrive at the ap-
The minimum energy configurations of the polymer propximate formula
are those where the tangent vector tˆ(s) describes the
shortest geodesic connecting tˆi to tˆf (Fig. 1). The Q(θ)= N(L) θ exp[ Aθ2 ] (5)
minimum energy is easily computed to be L rsinθ −2LkT
forthepartitionfunctionasafunctionofthefinalangle
E =Aθ2/(2L), (3)
min θ,where isanormalizationconstanttobedetermined
N
by the normalization condition
which is a purely mechanical contribution to the
energy[1, 9]. π
To go beyond mechanics and include the effect of Q(θ)sinθdθ =1 (6)
Z
thermal fluctuations (Fig. 1), we use an approximation 0
described in [11]. The approximation consists of using which includes the measure sinθ on the sphere.
an expansion of the energy about the minimum energy The result (5) implies that the number of configura-
configurationandretainingfluctuationtermsaboutthe tions with a final tangent vector tˆ making an angle θ
f
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defined by arccos(tˆ.tˆ ) is given by P(θ) = Q(θ)sinθ keeping L = Nl and L = 2l fixed, one recovers
i f p (∆θ)2
leading to the simple analytic approximate formula for the wormlike chain model. A frequency distribution of
the bend angle distribution the number of configurations with angle θ between the
initial and final tangent vectors was generated.
Aθ2
P(θ)= N√θsinθexp[ ] (7) Figure 2 shows a comparison between our predicted
L −2LkT analytical form, the planar formula and the results of
for θ. Note that this is a closed analytic form rather computersimulation. Inthestiffregime(β =L/Lp less
than an infinite series[13]. It is thus suitable and con- than 1) eq. (7) is virtually exact, in the semiflexible
venient for experimental comparison. regime (β around 5) it remains a reasonable approxi-
In some earlier works [6, 7] stiff polymers were con- mation. Thus, in an experiment which takes repeated
sidered as perturbations about the straightline. This snapshotsoftheanglebetweenthe finalandinitialtan-
effectively describes the tangent vector as varying on gent vectors of a molecule, we expect to find a distri-
a tangent plane to the unit sphere. This leads to the bution of angles given by Eq. (7). This is a prediction
planar formula of the wormlike chain model and it goes beyond earlier
studies based on classical elasticity[9].
A Aθ2
P(θ)= θexp , (8)
LkT −2LkT
for the distribution of final angles. Such a planar ap-
proachneglectsthecurvedgeometryofthesphere. One
also encounters a hybrid formula [9, 14]
A Aθ2
P(θ)= sinθexp , (9)
LkT −2LkT
in which one computes Q(θ) using the planar approxi-
mationandincludes the correctcurvedmeasuresinθdθ
on the sphere. The present treatment (unlike the ear-
lier planar ones) deals with geodesics on the sphere of
tangents and thus takes into account the curvature of
the sphere both in the computation of Q(θ) and in the
FIG.2. (Coloronline)ComparisonwithsimulationsofP(θ)
measure sinθ. In the limit that the bending angle θ is
vs θ for a range of flexibilities. Plotted are the simulation
small, Eq.(7) reduces to the earlier planar and hybrid
data(bluedots),theplanarformula(thickredline)andour
formulae (8,9). Our formula for the bend angle distri- analytic formula Eq. (7)) (thin black line). Notice that our
bution has a wider range of applicability compared to closed form formula gives a reasonable fit to the simulation
the earlier ones: it is valid for contour length compara- data even for β =L/Lp as large as 5.
ble to the persistence length (stiff and the semiflexible
regime) and also correctly describes the rare events in-
volving large bend angles.
MOMENT ANGLE RELATIONS
DISTRIBUTION OF BEND ANGLES
The partition function Q(θ) can be converted into
a free energy by the formula (we drop θ independent
We have performed Monte-Carlo simulations using terms as they areadditive constants in the free energy)
the Kratky-Porod model for comparison with the an-
alytical form (7). We randomly generated 106 config- Aθ2 kT θ
(θ)= kTlnQ(θ)= ln (10)
urations of the polymer chain assuming it to be built F − 2L − 2 sinθ
of N identical short straight segments each of length l.
andusedtocalculatethebendingmomentM neededto
Given one segment, the direction of the next segment
bend the final tangent vector through an angle θ. The
was assumed to be uniformly distributed on a cone of
approximate closed form predicted by our theory is
semivertical angle ∆θ = .1 radian around the preced-
ing segment. In the continuum limit when the number Aθ kT 1
of segments goes to infinity N ,l 0,∆θ 0 M = ( cotθ) (11)
→ ∞ → → L − 2 θ −
4
The first term in (11) represents the mechanical elastic elastic rod, such as a poker or a ball point pen refill,
energy and the second (proportional to kT) is due to one sees that if the bending angle exceeds π, the con-
thermal fluctuations. figurationbecomesunstablebecausethe tangentvector
no longer traces the shortest geodesic. The poker then
buckles into a new configuration with lower energy. As
Mechanical
10 WithThermalFluctuations showninFig. 3,thermalfluctuationscausethebuckling
tosetinatbending anglesconsiderablysmallerthanπ.
Lm 5 For small bending angles θ, one can Taylor expand
N
-21 the free energy (10) and find that the effect of thermal
0
H1 0 fluctuations is to effectively reduce the bending elas-
M
tic constant A by kTL/6. The fluctuations effectively
soften the elastic response of the polymer. For larger
-5
bending angles, one can no longer think of the fluc-
tuations as simply renormalising the bending modulus
0.0 0.5 1.0 1.5 2.0 2.5 3.0
ΘHradL A, since the form of the moment angle relation is non
Hookean.
FIG. 3. (Color online) Bending moment vs bending angle While we have used actin as a typical example of
theoretically expected. The thin blue line shows the me- a semiflexible polymer, our study is also relevant to
chanicalelasticresponse,andthethickredlineincludesthe DNA and efforts to understand the bending elasticity
effect of thermal fluctuations. The moment is plotted in
of small segments. Experiments and all atom simula-
units of 10−21 N-m. The values of A and kT are chosen to
tions [8, 9, 14] are performed to understand the forma-
reflect actin filamentsunderphysiological conditions.
tion of “kinks” and cyclization of short and intermedi-
ate length DNA strands. Our present approach is an-
As Fig. 3 shows, the mechanical contribution is
alytical and computes experimentally relevant quanti-
Hookean. Within a purely mechanical approach, when
ties characterisingthe bending elasticity of semiflexible
the bending angle reaches π, the geodesic connecting tˆ
i polymers within the wormlike chain model. We expect
totˆ isnolongertheshortestoneandtheconfiguration
f ourresultstointerestresearchersstudyingactinaswell
becomes unstable andbuckles to a lower energyconfig-
as DNA and other biopolymers.
uration. This picture is drastically altered by thermal
Oneofus(SS)acknowledgesdiscussionswithM.San-
fluctuations. As shown by the red curve in Fig. 3, the
tosh on all atom simulations of DNA.
bucklinghappensatananglesmallerthanπ. Thether-
malfluctuationshavetheeffectof“softening”thelinear
response to external bending.
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CONCLUSION
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(1995)
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http://link.aps.org/doi/10.1103/PhysRevE.77.041804
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5
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