aa r X i v : . [ c ond - m a t . s o f t ] M a y Free Energy of Twisted Semiflexible Polymers
Supurna Sinha
Raman Research Institute, Bangalore 560 080, India.
We investigate the role of fluctuations in single molecule measurements of torque-link ( t − lk )curves. For semiflexible polymers of finite persistence length ( i.e. polymers with contour length L comparable to the persistence length L P ), the torque versus link curve in the constant torque(isotorque) ensemble is distinct from the one in the constant link (isolink) ensemble. Thus, oneencounters the conceptually interesting issue of a “free energy of transition” in switching ensembleswhile making torque-link measurements. We predict the dependence on the semiflexibility parameter β = L/L P of this extra contribution to the free energy which shows up as an area in the torque-link plane. This can be tested against future torque-link experiments with single biopolymers. Webring out the inequivalence of torque-link curves for a stiff polymer and present explicit analyticalexpressions for the distinct torque-link relations in the two ensembles and the free energy differencein switching ensembles in this context. The predictions of our work can be tested against singlemolecule experiments on torsionally constrained biopolymers. PACS numbers: 87.15.-v,05.40.-a,36.20.-r
Statistical mechanics of semiflexible polymers is ofgreat current interest. Research in this area has beenmotivated by experiments[1, 2] on biopolymers in whichsingle molecules are stretched and twisted to measureelastic properties. These experiments are designed to un-derstand the role of semiflexible polymer elasticity[3, 4, 5]in, for instance, the packaging of these polymers in a cellnucleus. The process of DNA transcription can generatesupercoiling. It is also regulated by supercoiling[6]. Ina typical experiment[6] probing the twist elasticity of aDNA molecule, the ends of a single molecule of doublestranded DNA are attached to a glass plate and a mag-netic bead. Magnetic fields are used to rotate the beadand magnetic field gradients to apply forces on the bead.By such techniques the molecule is stretched and twistedand the extension of the molecule is monitored by the lo-cation of the bead. One can thus measure the extensionof the molecule as a result of the applied link and force[6] and also make a measurement of the torque versusapplied link [7] .In the context of force-extension measurements, an iso-metric setup is described by the Helmholtz free energy,whereas an isotensional setup is described by the Gibbsfree energy[8]. In statistical mechanics, these two ensem-bles are distinct[9, 10, 11]. In the thermodynamic limitthese two descriptions agree, but semiflexible polymers(those with contour length L comparable to the persis-tence length L P , i.e. β = L/L P ≃
1) are not at thethermodynamic limit.In the present paper we explore this issue in the contextof torque-link measurements. The two distinct statisti-cal mechanical ensembles here are the constant torqueensemble (isotorque ensemble) and the constant link en-semble (isolink ensemble). Here we focus on the role offluctuations in single molecule torque-link experiments.In order to correctly interpret such experiments one needs to understand the effect of fluctuations on the measuredquantities. For instance, it turns out, that an experi-ment in which the link applied to a polymer moleculeis fixed (isolink) and the torque fluctuates yields a dif-ferent result from one in which the torque is held fixed(isotorque) and the link fluctuates[12]. This differencecan be traced to large fluctuations about the mean valueof the torque or the link, depending on the experimentalsetup. Experimentally, both isolink and isotorque ensem-bles are realizable. Here is a schematic description of anexperimental setup for torque-link measurements of sin-gle biopolymer molecules. A polymer molecule attachedto a glass plate on one end is suspended in a suitablemedium with a magnetized bead attached to the otherend. The magnetized bead is kept in a magnetic trap.One can realize an isolink setup by using a “stiff mag-netic trap” and an isotorque setup by using a “soft mag-netic trap” which allows the applied link to fluctuate butapplies a fixed torque to the molecule[13, 14]. The fluc-tuations in torque-link measurements vanish only in thethermodynamic limit of very long polymers. In the nextsection we describe the setup in more detail.The paper is organized as follows. In Sec. II we dis-cuss the isolink and the isotorque ensembles. In Sec. III we illustrate the phenomenon of inequivalence of ensem-bles in torque-link measurements by explicitly presentinganalytical expressions in the context of a stiff polymer.In Sec. IV we draw attention to the notion of the freeenergy of transition in going from one ensemble to an-other and its dependence on semiflexibility which can betested against future single molecule experiments on tor-sionally constrained polymers and simulations. Finally,we conclude the paper in Sec. V .Consider an experiment, as described in the Introduc-tion, in which one end of a biopolymer molecule is at-tached to a glass plate and the other end is attached toa magnetized bead (of magnetic moment ~µ ) kept in amagnetic field ~B which is used to rotate the bead. Wesuppose both ~B and ~µ are parallel to the glass plate. Theenergy of the bead is given by E = − ~µ. ~B = − µB cos( θ − θ ) (1) θ describes the direction of the magnetic field and θ thedirection of the magnetic moment ~µ . The variables lk and lk , which keep track of the number n of turns ofthe bead are related to θ and θ as follows: lk = θ + 2 nπ and lk = θ + 2 nπ . Consider P ( lk ) dlk , the numberof configurations (counted with Boltzmann weight) for apolymer of contour length L and bend persistence length L P , characterized by a semiflexibility parameter β = LL P ,in a link interval dlk of lk . The free energy defined byΦ( lk ) = − β lnP ( lk ) is the free energy pertaining to afixed link. The partition function[11, 15] for the com-bined system consisting of the polymer molecule and themagnetized bead in the magnetic field is given by: Z ( lk , β ) = Z + ∞−∞ dlke − β Φ( lk ) e βµB cos( lk − lk ) (2)Constant Link Ensemble: The Limit of a Stiff TrapIn the limit of a stiff trap ( µB → ∞ ), ~µ follows ~B closely and lk ≈ lk . Thus cos( lk − lk ) ≈ − ( lk − lk ) . Inthis limit the partition function for the combined system(molecule+trap) reduces to: Z ( lk , β ) = e βµB Z + ∞−∞ dlke − βφ ( lk ) e − βµB ( lk − lk (3)Clearly, in this limit the Gaussian factor pertaining tothe magnetic trap approaches a delta function and wehave: Z ( lk, β ) ≈ e − β Φ( lk ) (4)Here we have switched notation to write lk in place of lk . Thus a stiff trap realizes the Constant Link (isolink)ensemble by constraining fluctuations in lk . In order tochange the applied link from lk to lk + dlk one applies atorque < t > = ∂ Φ ∂lk . Thus one gets a torque-link ( < t >, lk ) curve by plotting < t > versus lk .Constant Torque Ensemble: The Limit of a Soft TrapIn the opposite limit of a soft trap µB is small butlarge enough that the polymer does not get untwisted. Insuch a situation the link fluctuates. One can adjust lk such that lk − lk ≈ π/
2. The magnitude of the torque | ~t | = | ~µ × ~B | = t = µB is held fixed for a particularmeasurement. In this limit the torque t is the controlparameter which can be changed from one reading to thenext by changing the magnitude B of the magnetic field.A feedback loop is used to ensure that < lk − lk > ismaintained at π/
2. Clearly, the potential energy for the trap, on expanding around lk − lk = π/ lk − lk takes the form: E = − ~µ. ~B = µB ( lk − lk ) = t ( lk − lk ) (5)Thus, in this limit Eq. (2) gives the following expres-sion for the partition function ˜ Z ( t, β ) = Z ( lk , β ) e βtlk (where lk is determined by the condition < lk − lk > = π ) for the combined system consisting of the polymermolecule and the trap [11, 15]:˜ Z ( t, β ) = Z + ∞−∞ dlke − β Φ( lk ) e βtlk (6)Thus in a soft trap link lk fluctuates but torque fluctu-ations are constrained[16]. One thus realizes the Con-stant Torque (isotorque) ensemble. ˜ Z ( t )[17] is the gener-ating function for the lk distribution. Given the constanttorque free energy Γ( t ) = − β ln ˜ Z ( t ) one gets the meanlink < lk > = − ∂ Γ ∂t and the ( t, < lk > ) torque-link rela-tion.Notice that ˜ Z ( t ) is the Laplace transform of Z ( lk ).In the thermodynamic limit of long polymers ( β → ∞ )the Laplace transform integral Eq.(6) is dominated bythe saddle point value and therefore Φ( lk ) and Γ( t ) arerelated by a Legendre transform:Φ( lk ) = Γ( t ) + tlk. (7)For finite β i.e. for a polymer of finite extent, the saddlepoint approximation no longer holds true and fluctua-tions about the saddle point value of the free energy be-come important. Thus one finds that for a finite β , Φ( lk )and Γ( t ) are not Legendre transforms of each other butare related via a Laplace transform [Eq. (6)].We illustrate the issue of inequivalence of ensemblesin the context of torque-link measurements explicitly byconsidering an analytically tractable and instructive spe-cial case, the torque-link relation for a stiff polymer.Our starting point is the WLC Hamiltonian with bendand twist degrees of freedom in the presence of a stretch-ing force f and a torque t [12]: H = p θ / p φ − A φ ) / θ − f cos θ − αt / p θ and p φ are momenta conjugate to the Euler an-gles θ and φ . The momentum conjugate to the Euler an-gle ψ is p ψ = it , a constant of motion, which contributesa term − αt / α is the ratio of the bend persistence length L P tothe twist persistence length L T . The ‘vector potential’ A φ = it (1 − cos θ )[4, 12]. For a stiff polymer with oneend clamped along the ˆ z direction, we can approximatethe sphere of directions by a tangent plane at the northpole of the sphere as the angular coordinate θ always re-mains small. In this limit (the PWLC model [4]) wherethe tangent vector never wanders too far away fromthe north pole of the sphere of directions, the polymerHamiltonian[3, 4, 12, 18] reduces to : H P W LC = p θ p φ − A φ ) θ − αt − f (1 − θ H P − f − αt H P is the Hamiltonian of interest in the paraxiallimit after we take out a constant piece. In this limit A φ = itθ . We introduce Cartesian coordinates ξ = θ cos φ and ξ = θ sin φ on the tangent plane R at thenorth pole.The PWLC model has been applied earlier in the con-text of flexible polymers at high tension[4]. Here we ap-ply it in the context of stiff polymers at f = 0. In thestiff limit, the tangent vector to the polymer points es-sentially along a fixed direction (the north pole) even at f = 0. As in Ref. [4] we restrict to the case of α = 0[19].In Ref. [20] we have derived the partition function Z ( f ) of a stiff polymer with both end tangent vectorspointing along a fixed direction ˆ z at a force f . In thepresence of a force f and a torque t this expression goesover to:˜ Z f ( t ) = p f − t / βf ) / (2 π sinh (cid:2) β p f − t / (cid:1) . (9)Notice that the “effective force” p ( f − t /
4) replaces f when there is a competition between a force f and atorque t . This is simply due to the fact that the Hamil-tonian H P in the presence of a force f and a torque t pertains to that of a two dimensional harmonic oscillatorwith a frequency ω = p ( f − t /
4) rather than ω = √ f which is the expression for the frequency of a two dimen-sional harmonic oscillator in the presence of a force f and no torque.At zero force, the expression simplifies and reduces to:˜ Z ( t ) = t/ (4 π sin (cid:2) βt/ (cid:1) . (10)¿From the partition function ˜ Z ( t ), we get the followinganalytical expression for the torque-link ( t, < lk > ) rela-tion in the isotorque ensemble: < lk > = (cid:0) βt −
12 cot (cid:2) βt (cid:3)(cid:1) . (11)Given ˜ Z ( t ) one can get the partition function Z ( lk )in the conjugate domain of an isolink ensemble (See Sec. II ) [20, 21]: Z ( lk ) = ( π/ β ) / (cid:0) cosh (cid:2) πlk ] (cid:1) . (12)This leads to the following torque-link ( < t >, lk ) rela-tion: < t > = (2 π/β ) tanh (cid:2) πlk ] . (13) LINKTORQUE T O R Q U E LINK
FIG. 1: Torque-Link Curve in the isolink (upper curve) andisotorque (lower curve) ensembles for β = 1 . Thus, the torque-link ( t, < lk > ) relation obtained inthe isotorque ensemble (Eq.[11]) is distinct from the one( < t >, lk ) obtained in the isolink ensemble (Eq.[13]) [SeeFig. 1].In the next section we derive explicit expressions forthe extra contribution to the free energy due to changesin ensemble.In an isotorque setup, torque is the control parame-ter and one measures the mean link to plot the torque-link ( t, < lk > ) curve. In an isolink setup the roles oflink and torque are interchanged. Consider going from asmall torque ( lk , t ) to a large torque ( lk , t ) configu-ration via an isotorque setup and returning from a largelink ( lk , t ) to a small link ( lk , t ) configuration via anisolink setup. Since the torque-link relation depends onthe chosen ensemble, in general there will be two dis-tinct curves joining the points ( lk , t ) and ( lk , t ) inthe torque-link plane, describing the two processes. Sucha transformation could lead to a net area being enclosedin the torque-link plane. This appears paradoxical sinceit seems to suggest that one can extract work from thesystem via a cyclic process. The resolution of this para-dox is as follows. In completing the cycle and returningto the initial state one is in fact changing ensembles twiceat the two end points. Such ensemble changes in a cyclictransformation involve finite changes in free energy whichneed to be taken into consideration. In particular, we no-tice for the special case of a stiff polymer the torque-linkrelations (Eq.[11] and Eq. [13]) lead to a free energy dif-ference of : ∆ stiff =2 β ln | cosh( πlk )cosh( πlk ) |− ( t lk − t lk )+ 1 β (cid:20) ln | t t |− ln | sin( βt )sin( βt ) | (cid:21) for a transformation between the states ( lk , t ) and( lk , t ). In the stiff regime (i.e. at small β ) the de-pendence of the free energy on β will be dominated bythe first term. In other words, the free energy difference∆ stiff ∼ β . This is a prediction of our analysis whichcan be tested against experiments with stiff biopolymerslike actin filaments.An analysis similar to the one in Ref. [9] applied to thecontext of torque-link measurement shows that one getsa contribution ∆ = β ln Φ ′′ ( lk ∗ ) to the free energy com-ing from fluctuations around the long polymer ( β → ∞ )limit by expanding ∆( lk ) = Φ( lk ) − tlk around the sad-dle point value lk = lk ∗ pertaining to the long polymerlimit. Here Φ( lk ) corresponds to the isolink free energy.This extra contribution ∆( lk ) vanishes in the limit of β → ∞ . For finite β , this nonzero contribution to thefree energy accounts for the transition between the con-stant link ensemble and the constant torque ensemble.Work is done on the bead by the trap in making thetrap stiffer, while in going from a stiff trap to a soft trapwork is extracted from the bead by the trap. The network done is the difference between the work done atthe two ends of the torque-link curves in switching en-sembles. This net work exactly cancels out the nonzeroarea enclosed in the torque-link plane. The net area en-closed in the torque-link plane pertaining to the “FreeEnergy of Transition” scales as 1 /β and therefore growswith the rigidity of the polymer. These predictions ofour study can be tested against future simulations andsingle molecule experiments.To summarize, in this paper we have studied inequiv-alence of ensembles for torque-link measurements. Wehave calculated the free energy difference between torque-link measurements in the isotorque and isolink ensemblesfor a stiff polymer. In addition, we have determined thecontribution to the “free energy of transition” in goingbetween the isolink and the isotorque ensembles by ex-panding the free energy difference ∆( lk ) around the longpolymer limit. We predict the dependence on the semi-flexibility parameter β of this extra contribution to thefree energy which shows up as an area in the torque-link plane. For the special case of a stiff polymer wefind explicit analytical expressions for the torque-link( t, < lk > ) relation obtained in the isotorque ensem-ble (Eq.[11]) and show that it is distinct from the one( < t >, lk ) obtained in the isolink ensemble (Eq.[13]). Wealso show that in this stiff regime the free energy differ-ence ∆ stiff ∼ β . All the predictions mentioned here canbe qualitatively and quantitatively tested against futuresingle molecule experiments on torsionally constrainedbiopolymers.The theoretical predictions presented in this study areexpected to generate interest in torsionally constrainedsingle molecule experiments which will eventually lead to a deeper understanding of the role of twist elasticityin biological processes involving gene regulation[22]. Acknowledgements:
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Statistical Physics, PartI , Pergamon Press, (1980).[9] Supurna Sinha and Joseph Samuel, Physical Review E , 021104 (2005).[10] A. Rosa, D. Marenduzzo and S. Kumar, Europhys. Lett., , 818 (2006).[11] D. Keller, D. Swigon and C. Bustamante, BiophysicalJournal , 733 (2003).[12] C. Bouchiat and M. Mezard, Eur. Phys. J. E , 377(2000).[13] A. La Porta and M. D. Wang, Physical Review Letters , 190801 (2004).[14] One way to apply torque could be to explore some in-trinsic birefringence property of the material by usingpolarized light. See for instance A. Ghosh et al., Phys.Biol. Phys. Rev.