Table Of ContentMolecular Elasticity and the Geometric Phase
Joseph Samuel and Supurna Sinha
Raman Research Institute, Bangalore 560080,India
(February 1, 2008)
We present a method for solving the Worm Like Chain (WLC) model for twisting semiflexible
polymers to any desired accuracy. Weshow that theWLC free energy is a periodic function of the
applied twist with period 4π. We develop an analogy between WLC elasticity and the geometric
phase of a spin 1 system. These analogies are used to predict elastic properties of twist-storing
2
polymers. We graphically display the elastic response of a single molecule to an applied torque.
This studyis relevant to mechanical properties of biopolymers likeDNA.
PACS numbers: 82.37.-j,03.65.Vf,87.14.-G,87.15.-V
3
0
Some long molecules are as stiff as needles, others explore all configurations which it can reach by contin-
0
as flexible as thread. The elasticity of a stiff molecule uous deformation. As we have shown, a configuration
2
is dominated by energy, while the elasticity of a flexi- with a 4π twist can be continuously deformed to a con-
n
ble molecule is dominatedby its configurationalentropy. figuration with 0 twist. It follows that the free energies
a
J It has lately become feasible [1] to stretch, bend and forthesetwosituationsarethesame. Ifoneweretomea-
3 twist single molecules to study their elastic properties. sure the free energy G(ψ) of the molecule as a function
2 The subject of this paper is the elasticity of semiflexible ofappliedtwistψ (bysaymeasuringthe torque-twistre-
polymers in which there is competition between ener- lation), one would find that G(ψ) is a periodic function
4 getic and entropic effects. We consider a polymer which of ψ with period 4π. From the 4π periodicity of the free
v
can bend as well as twist [2]. The flexibility of such a energy,itfollowsthatthetorque-twistrelationandother
1
molecule is characterized by two dimensionless param- measurable elastic properties have the same periodicity.
9
4 eters α = LBP/LTP and β = L/LBP, where L is the The above discussion is even more relevant to theo-
3 length of the molecule and L and L are the bend retical models of polymers which do not incorporate self
BP TP
0 and twist persistence lengths. For example DNA has a avoidance. Such a polymer is a ‘phantom chain’ [4] and
2
bend persistence length of about 53nm and a twist per- can pass through itself. In the absence of self avoidance,
0
sistence length of about 70nm. a polymer does not even need to pass around its end: it
/
at The main purpose of this letter is to draw attention canpassthroughitselfandsoreleasetwisttwo turns at a
m to an experimentally relevant topological subtlety which time. Indeed,sucheffects havebeenseeninrecentsingle
hasnotbeendiscussedinprevioustheoreticaltreatments moleculeexperiments[5]andevenearlier[6]: theenzyme
-
d of the elasticity of twisting polymers. To appreciate the topoisomerase II converts a real DNA into a ‘phantom
n
point, take a strip of paper (a ribbon or belt will do as chain’ [5,7] and in the presence of this enzyme (which
o
well) and tape one end of the strip to a table. Pull the playsacrucialroleinreplication),the DNAmoleculere-
c
: striptautby its otherendandtwistit byfour half-turns leasestwisttwoturnsat atime[5,6]. Thereleaseoftwist
v
(4π rotation). If you now slacken the strip, you will find through bending modes (geometric untwisting) has also
i
X that it is possible, keeping the end fixed, to pass the been discussed in [8].
r strip around the end. Pulling the strip taut will reveal Itisthusclearthatintheoreticalmodelswhichdonot
a
that the 4π twist has been released. This demonstration have self avoidance, the free energy is a periodic func-
shows that a polymer can release twist, two turns at a tion of applied twist with period 4π. Bearing this in
timebygoingaroundits end. Atwistof2π cannotbe so mind, we will now explore the Worm Like Chain (WLC)
released but can be transformed to −2π. This of course, [9]modelfortwistingpolymers,whichignoresselfavoid-
merelyillustratesthe wellknownmathematicalfactthat ance. As was emphasized recently [10] in a review of
the rotation group is doubly connected [3]. We will now single molecule experiments, “the precision of control
see that this mathematical fact has concrete experimen- andquantitativemeasurementandsimple interpretation
talconsequencesfortheelasticpropertiesoftwist-storing of these experiments make detailed theoretical analyses
polymers. Mentally replace the table with a translation appropriate”. While there has been some theoretical
stage, the strip with DNA molecule with a micron sized progress [4,11] on the WLC for twisting polymers, in-
magneticbeadatitsfreeendandletthetwistingbedone terest has been confined primarily to the high tension
with a magnetic field. If the DNA molecule is about16µ regime,whichistheoreticallymoretractablebecausethe
long,andnotpulledtaut,itcanrelease4πworthoftwist moleculehasonlysmallperturbationsaboutalinearcon-
by passing around the bead. One will find that twisting figuration. Experiments of Bensimon [12] have also ex-
thebeadby4π isequivalenttonottwistingitatall! Un- plored this regime. In contrast the present letter theo-
der the influence of thermal agitation, the molecule will retically explores the non linear regime of small forces,
1
where the perturbative methods described in [4,11] are frame index i) and R(s)∈SO(3) is a 3×3 rotation ma-
inapplicable. In this regime we find that the WLC free trix. There is a clear analogy between the elastic prop-
energy has a 4π periodicity in contrast to the aperiodic erties of the WLC and the motion of a top. Indeed as
free energy of the high tension regime. MezardandBouchiat[11]andMorozandNelson[4]point
We first describe the WLC model and summarize the out, the WLC problem can be mapped to the quantum
existingtheoreticalanalysesofthismodel. Wethenpoint mechanics of a symmetric top. From this mapping, one
out a physically relevant mathematical subtlety without maynaivelyconclude [13]thatthe periodicity ofthe free
which the solution of the WLC model is incorrect. We energy is 2π. We now show that a careful treatment of
thencorrectlysolvetheWLCmodelbyacombinationof the path integral (3) gives the correct 4π periodicity.
analytical and numerical techniques and elucidate some EachconfigurationC in(3) ischaracterizedby acurve
of the elastic properties that emerge from the model in {R(s)} in the rotation group with fixed end points R(0)
graphical form. A concluding discussion interprets the and R(L). The sum is over all configurations which are
results. sampledundertheinfluenceofthermalagitation. Itisev-
WLC Model: The WLC model ignores self avoid- identthatweshouldonlysumoverconfigurations{R(s)}
ance and views the polymer as a framed space curve inasinglehomotopyclass: thermalagitationonlycauses
C ={~x(s),eˆ(s)},i=1,2,3, 0≤s≤L of contour length continuous deformations and therefore cannot knock the
i
L, with an energy cost E(C) for bending and twisting. polymer out of its homotopy class. Q thus depends not
We suppose that one end of the polymer is tethered to only on (~r,R(0),R(L)) , but also on the homotopy class
the origin ~x(0) = 0 and the other end at ~x(L) = ~r is [{R(s)}] of the paths {R(s)} being summed over. This
tagged. The unittangentvectoreˆ =d~x/dstothe curve information is exactly captured by going to the cover-
3
describes the bending of the polymer while the twisting ing space SU(2) of the rotation group SO(3). This step
is captured by a unit vector eˆ normal to eˆ . eˆ is then is essential to correctly describe the elastic properties of
1 3 2
fixed by eˆ =eˆ ×eˆ to complete the right handed mov- the twisting polymer. (If we do not take this step but
2 3 1
ing frame eˆ(s),i = 1,2,3. The rate of change of the remain on SO(3), we are effectively summing over both
i
moving frame eˆ(s) along the curve can be measured by homotopy classes, which is a physically incorrect proce-
i
its “angular velocity vector” Ω~ defined by dure.) The result is that while the WLC Hamiltonian
is the same as that of the top, the WLC configuration
d eˆ(s)=Ω~ ×eˆ(s). (1) space is not the configuration space SO(3) of the top,
i i
ds but its double cover SU(2) [3]. As we will see below,
thisresultsina4π periodicityforthefreeenergy. SU(2)
The components of Ω~ in the moving frame are Ω =Ω~.eˆ
i i is the same as S3, the four dimensional sphere defined
and the energy E(C) of a configuration C is given by
by {xα,α = 1,2,3,4,},Σ (xα)2 = 1. In fact, xα are
α
the Cayley-Klein parameters [14] traditionally used in
L
E[C]=1/2 [A((Ω )2+(Ω )2)+C(Ω )2]ds, (2) describing tops.
Z 1 2 3
0 Let{g(s)}beacontinuouscurveinSU(2)whichmaps
down to the curve {R(s)}. g(s) satisfies the differential
where A is the bending modulus and C the twist modu-
lus. equation dg(s) = i/2(Ω~.~σ)g(s), whose solution is a path
ds
Imagine that the ends of the polymer and the frames ordered exponential: g(s)=P[exp siΩ~(s′).~σ/2ds′]g(0).
0
at these ends (eˆ(0),eˆ(L)) are held fixed. We wish to C is nowdescribedby a curve{g(s}RinSU(2),with fixed
i i
compute the number of configurations end points g(0) and g(L). The standard Euler angles
(θ,φ,ψ)ontherotationgroupcanbeusedasco-ordinates
Q(~r,eˆi(0),eˆi(L))=ΣCexp(−E[C]/kT), (3) on SU(2) = S3 if the range of ψ is extended to 4π [3].
SU(2) acts on itself by right and left action generated
counted with Boltzmann weight exp(−E[C]/kT), which
[15]byspacefixed(J ,J ,J )andbodyfixed(J ,J ,J )
x y z 1 2 3
start at the origin with initial frame eˆ(0) and end at
i angular momenta.
~r with final frame eˆ(L). The function Q is related to
i We can now write (3) more correctly as Q(~r,q ,q )
0 L
the free energy of the molecule and its measurable elas-
where q =g(0) and q =g(L), to explicitly display the
0 L
ticpropertiesliketheforce-extensionrelation(FER)and
homotopy class dependence of Q. Q(~r,q ,q ) has the
0 L
the torque-twist relation (TTR). An overall multiplica-
path integral representation:
tive constant is not important in the calculation of Q.
This only leads to an additive constant in the free en-
N D[g(s)]e[−E(C)/kBT]δ(~x(L)−~r). (4)
ergy, which drops out on differentiation and does not Z
affect elastic properties.
Letusfixa“labframe”e˜b andwriteea(s)=Ra (s)e˜b, N is a normalisation constant and the path integral is
where a = 1,2,3 is a vectior index (asiopposed bto thie over all paths that go from q0 to qL on S3. We now
pass from Q(~r,q ,q ) to its Laplace transform defined
0 L
2
as Q˜(f,q ,q ) = d~rexp[f~.~r/L ]Q(~r,q ,q ), where which is normalised (ξ†ξ = 1). We can write ξ1 =
0 L BP 0 L
L = A/kT. PerRforming the elementary integrations cosθ/2exp−iφ/2expiψ/2,ξ2 =sinθ/2expiφ/2expiψ/2and
BP
andchangingvariablestoτ =s/L and~ω =Ω~L ,we thus introduce co-ordinates (θ,φ,ψ) ranging from 0 to
BP BP
seethatQ˜(f,q ,q )canberepresentedasNZ(f,q ,q ), (π,2π,4π) respectively. These are similar to Euler an-
0 L 0 L
where Z has the path integral representation gles on the rotation group and differ only in the range
of ψ. The frame eˆ can be expressed as eˆ = ξ†~σξ,
Z D[g(τ)]e−[R0βdτ12(ω12+ω22+α−1ω32)−f~.eˆ3], (5) eˆ1+ieˆ2 = ξT(iσ2)~σξi, where the σs are the u3sual Pauli
matrices. Notice that altering ψ by 2π flips only the
where α = A/C = L /L . This is clearly the quan- sign of ξ and therefore does not affect the frame. Using
BP TP
tum amplitude < q |exp[−βH ]|q > for a particle on this mapping between 2 component spinors and frames,
L f 0
the surfaceofa 3-sphereto gofromaninitialpositionq we can import ideas from the geometric phase to under-
0
on S3 to a final position q in imaginary time β in the stand WLC elasticity. The information in the spinor ξ
L
presence of an external force field. The Hamiltonian is can be decomposed into an overallphase ψ/2 describing
H = H −fcosθ, where H = 1/2(J 2+J 2+αJ 2), twist and a ray eˆ describing bend. Fix a configuration
f 0 0 1 2 3 3
whichistheHamiltonianofasymmetrictop. Iftheexact C and note that ξ(s)=g(s)ξ(0) satisfies the Schr¨odinger
eigenstates of Hf were known, we could write: differential equation idξ(s) = hˆξ(s), where hˆ = −Ω~.~σ/2
ds
Z =Σ exp[−βE ]u∗(q )u (q ), (6) isthe“Hamiltonian”ofaspinhalfparticleinanexternal
n n n 0 n L magnetic field Ω~. We can now decompose the difference
where{un(q)}isacompletesetofnormalisedeigenstates ψ/2 = ψ(L)/2 − ψ(0)/2 between the final and initial
of the Hamiltonian Hf and En are the corresponding phases into a geometric phase and a dynamical phase.
eigenvalues. EventhoughHf cannotbediagonalisedan- The dynamical phase is given by the integral of the ex-
alytically, we can exploit its symmetries to reduce the pectation value of the “Hamiltonian” Ldsξ†hˆξ, which,
0
problemtoanumericallytractableform. Jz andJ3 com- using the definition of eˆ above is seenRto be LdsΩ /2,
mute with the Hamiltonian H , reflecting the symme- 3 0 3
f half the twist Ω integrated along the polymeRr. The ge-
tryunderspace-fixedandbodyfixedrotationsaboutthe 3
ometric phase is given by half the solid angle swept out
third axis. As a result, Z depends only on the differ-
by the ray eˆ (s). If the initial and final rays are distinct
ences φ = φ − φ and ψ = ψ − ψ and we write 3
L 0 L 0 (but not antipodal), one can join them by the unique
Z(f,θ ,θ ,φ,ψ). Consider the dependence of Z on ψ.
0 L shorter geodesic [18] to enclose a solid angle. The total
Sinceallthewavefunctions in(6)aresinglevaluedfunc-
twistdifferenceψisthesumofthe“dynamicaltwist”-the
tions on S3, it follows that Z(ψ) is periodic in ψ with
integratedlocaltwist- Ω dsandthe“geometrictwist”-
period 4π: Z(ψ+4π)=Z(ψ). This means that the free 3
the solid angle swept Rout by the tangent vector- which
energy G = −1/βLog[Z] and all elastic properties have
dependsonthebending ofthepolymer. Intheliterature
the same period. This is the first main result of this let-
[4,11,19],the distributionofappliedtwistbetweentwist-
ter. Thefactthattheperiodicityis4π andnot2πcanbe
ing and bending is compared with the decomposition of
traced to the fact the sum in (6) extends not only over
link into twist and writhe. This result is referred to as
tensorial states but also over spinorial ones. The vari-
White’stheorem[20](thoughanearlierreferenceis[21]).
ation of G with respect to the variables (f,θ ,θ ,φ,ψ)
0 L The discussion [4,11,19] applies to closed, self-avoiding
gives the elastic response to stretch (f), bend (θ,φ) and
polymers which havebeen twisted anintegralnumber of
twist (ψ). G can be computed numerically for any value
times. In contrast, our treatment applies also to open
of its arguments using mathematica programs [16] that
polymers which have been twisted a fractional number
run for a few minutes on a PC, using methods similar to
of times. However, since the WLC model does not take
those of [17]. From G we can extract all possible infor-
into account self avoidance, a twist of 4π is equivalent
mation regarding the elasticity of a polymer with bend
to no twist and the integral part of the twist is only
and twist degrees of freedom. For instance one can pre-
measuredmodulo two. Our treatment captures the frac-
dict the formof extensionversustwistcurves for various
tionalpartoftheappliedtwist(whichisgeometrical)and
values of the stretching force.
the earlier treatment captures its integral part (which is
The 4π periodicity of the free energy strongly moti-
topological). In this sense, the two discussions are com-
vates the use of spinorial methods. In fact, the WLC
plementary. The analogy between twist elasticity and
configuration space S3 is the same as the set of nor-
the geometric phase is the second main result of this let-
malised states of a spin-1/2 quantum system. We will
ter. The analogy has also been noted in Ref [19], which,
now show that there is a mapping between a configu-
however, uses a vectorial correspondence rather than a
ration of a twisting polymer and the quantum evolu-
spinorial one. The decomposition of applied twist into
tion of a spin-1/2 system. This brings out an inter-
a geometrical and a dynamical part leads to a coupling
esting connection between WLC elasticity and the Ge-
between the bend and the twist degrees of freedom and
ometric Phase. Let us introduce a 2 component com-
hasadirectbearingontheelasticpropertiesoftheWLC
plex vector (a spinor) ξ1 = x1 + ix2,ξ2 = x3 + ix4,
3
polymer. As a specific illustrationwe givethe results for
the special case of pure twist elasticity.
[1] J.M. Schurr and S.B. Smith, Biopolymers 29, 1161
Pure twist elasticity: We suppose that the tagged end
(1990); S.B.Smith and A.J. Bendich, Biopolymers 29,
is not constrained in position, but only in orientation.
1167 (1990); S.B.Smith et al, Science, 258, 1122 (1992).
Integrating Q(~r,q ,q ) over ~r, we see that the applied
0 L J. Marko and E. D. Siggia, Macromolecules 28, 8759
force f vanishes. We also suppose that the initial eˆ3(0) (1995). C. Bouchiat et al, Biophysical Journal 76, 409
and final eˆ (L) tangent vectors are both in the same di-
3 (1999). C. Bustamante et al, Current Opinion in Struc-
rection (which we take to be the z direction). We com- tural Biology 10, 279 (2000).
pute the distribution Z(ψ) of ψ. In this case only states [2] S. Panyukov and Y. Rabin, Europhys. Lett. 57, 512
for which m = g contribute [22] and Z takes the form (2002); Y.Kats,D.A.KesslerandY.Rabin,Phys. Rev.
Z = Σ Z . Using standard techniques from angular E 65, 020801 (2002); S. Panyukov and Y. Rabin, Phys.
g gg
momentum theory, we can express Z as Z = Σ eigψQ Rev. E 62, 7135 (2000).
g g
whereQ =Σ∞ (2j+1)exp[−β/2(j(j+1)+(α−1)g2)], [3] J. J. Sakurai Modern Quantum Mechanics The Ben-
g j=|g| jamin/CummingsPublishingCompany,Inc.MenloPark,
wherejrunsinintegersteps. ByaninverseFouriertrans-
California, pp 163-171; R. W. Wald General Relativity
form, we compute P(ψ) and the free energy G(ψ) =
The University of Chicago Press, p 346.
−1/βLog[P(ψ)]. By differentiating with respect to ψ,
[4] J.D. Moroz and P. Nelson, Macromolecules 31, 6333
we compute the torque Θ = ∂F/∂ψ needed to twist the
(1998).
moleculebyanangleψ. Thetorque-twistrelationisplot- [5] T.R. Strick et al, Nature 404 901 (2000).
ted in Fig.1. This graph, which is the third main result [6] P.O.BrownandN.R.Cozarelli, Science2061081(1979);
of this letter, describes the pure twist elastic properties [7] B.Albertsetal.MolecularBiologyoftheCellThirdEdi-
of a molecule in the WLC model. These graphs are eas- tion Garland Publishing, Taylor and Francis Group, pp
ily interpreted in terms of the geometric phase ideas de- 260-262.
scribed earlier. For large α (α is the ratio of the bend [8] R. E. Goldstein et al, Phys. Rev. Lett. 80, 5232 (1998);
to the twist persistence length), twist costs very little N. H.Mendelson et al, J. Bacteriol. 177, 7060 (1995).
[9] O. Kratky and G. Porod, Recl. Tral. Chim. 68, 1106
energy, the molecule twists without bending, and as it
(1949).
takes hardly any torque to twist the molecule, the TTR
[10] S. Cocco et al, cond-mat/0206238.
isalmostflat. Asαdecreases,theappliedtwistisshared
[11] C. Bouchiat and M. Mezard, Phys. Rev. Lett. 80, 1556
between the twist and the bend. When α is zero twist- (1998). C. Bouchiat and M. Mezard, Eur. Phys. J. E2,
ing is prohibitively expensive and the applied twist is all
377 (2000).
takenupbythebend. Thiscausesthemoleculetobuckle [12] T.R. Strick et al, Science 271 1835 (1996).
just as a towel does when it is wrung. When α = 0, [13] (seeeq.8of[4])Thisremarkdoesnothowever,invalidate
the “polarisationvector”eˆ is paralleltransportedalong themainconclusionsofRefs.[4,11].Theseworksarepri-
1
the polymer. The distribution Z(ψ) then reduces to the marilyinterestedinthelargeforcelimitoftheWLC.Re-
distribution of solid angles (Berry Phases) enclosed by lease of 4π twist is then exponentially suppressed by en-
closed Brownian paths on the (Poincare) sphere, which ergetics.Thepresentstudydoesnotaddressthisregime.
[14] H. Goldstein, Classical Mechanics, Addison-Wesley
was calculated in [23] in the context of depolarised light
(1980), USA.
scattering.
[15] L.D. Landau and E.M. Lifshitz Quantum Mechanics,
In this letter we have solved the WLC model with
Pergammon Press, Oxford (1977).
bend and twist degrees of freedom and noticed analo-
[16] Stephen Wolfram, The Mathematica Book,
gies to spin1/2 systems and the geometric phase. These
Third Edition (Wolfram Media/ Cambridge University
analogiesleadtoa descriptioninterms ofaparticle ona Press,1996).
sphereinexternalgravitationalandmagneticfields. Such [17] J. Samuel and S. Sinha Phys. Rev. E 66, 050801(R)
analogies,apartfromgivingusanalytictoolstosolvethe (2002).
problem virtually exactly for the first time, also provide [18] J. Samuel and R. Bhandari Phys. Rev. Lett. 60, 2339
simple physicalpictures: Imposing a twist on a molecule (1988).
is like applying a magnetic field. The helical shape of a [19] A.C.MaggsandV.Rosetto,Phys. Rev. Lett.87,253901
(2001); eprint cond-mat/0111011.
towelwhenitis wrungissimilartothe helicaltrajectory
[20] J.H. White, Amer. J. Math. 91, 693 (1969); F.B. Fuller,
of a particle in a magnetic field. We hope this letter will
Proc. Nat. Acad. Sci. USAbf 68, 815 (1971).
encourageexperimentalworkontwistingpolymersinthe
[21] G. Calugareanu, Rev. Math. Pures. Appl. 4, 5 (1959).
nonlinearlowtensionregimeandsetupadialogbetween
[22] A.R.EdmondsAngularMomentuminQuantumMechan-
the theory and experiments on molecular elasticity. ics Princeton University Press, Princeton (1960).
Acknowledgements: It is a pleasure to thank A. Dhar, [23] S.Sinhaand J. Samuel,Phys.Rev. B 50,13871 (1994);
Y. Hatwalne B. R. Iyer, V.A. Raghunathan and M. Rao M.M.G. Krishna et al, J. Phys. A. 33 5965 (2000).
for their comments and R. Goldsteinfor his constructive
suggestions which improved the paper and for drawing
attention to related work [8].
4
FIG. 1. The Torque-Twist relation for β = L/LBP = 1,
α=L/LTP =0,1,7.
5
