thermal denaturation of a helicoidal dna modelpolymer.chph.ras.ru/asavin/dna/bl_03pre.pdfthermal...
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PHYSICAL REVIEW E 68, 061909 ~2003!
Thermal denaturation of a helicoidal DNA model
Maria Barbi*Laboratoire de Physique The´orique des Liquides, Universite´ Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris Cedex 05, France
Stefano Lepri†
Istituto Nazionale per la Fisica della Materia, UdR Firenze, via G. Sansone 1, 50019 Sesto Fiorentino, Italy
Michel Peyrard‡
Laboratoire de Physique, E´cole Normale Supe´rieure de Lyon, 46 Alle´e d’Italie, 69364 Lyon Cedex 07, France
Nikos Theodorakopoulos§
Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation, Vasileos Constantinou 48, Athens 11635,~Received 17 July 2003; published 23 December 2003!
We study the static and dynamical properties of DNA in the vicinity of its melting transition, i.e., theseparation of the two strands upon heating. The investigation is based on a simple mechanical model whichincludes the helicoidal geometry of the molecule and allows an exact numerical evaluation of its thermody-namical properties. Dynamical simulations of long-enough molecular segments allow the study of the structurefactors and of the properties of the denaturated regions. Simulations of finite chains display the hallmarks of afirst order transition for sufficiently long-ranged stacking forces although a study of the model’s ‘‘universalityclass’’ strongly suggests the presence of an ‘‘underlying’’ continuous transition.
DOI: 10.1103/PhysRevE.68.061909 PACS number~s!: 87.15.Aa, 05.20.2y
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I. INTRODUCTION
The study of simple dynamical models describing variofeatures of DNA dynamics has interested many authorsalmost 20 years@1#. This vivid interest arises of course fromthe biological relevance of DNA but also from its physicproperties which can now be probed through single-molecmicromanipulation experiments like stress-induced trantions @2# or strand separation@3#. This series of studies haclearly pointed out that DNA must be considered as anamical object, whose~nonlinear! distortions could play amajor role in its functions.
One feature of DNA that attracted a lot of attention frophysicists is its thermal denaturation, i.e., the transition frthe native double-helix B-DNA to its melted form where thtwo strands spontaneously separate upon heating@4#, be-cause it provides an example of aone-dimensional phastransition. Experiments show that this transition is vesharp, which suggested that it could be first order and thisto numerous investigations, first to justify the existence otransition in a one-dimensional system and second to demine its order@5–12#. In spite of these efforts the nature othe transition is still unclear and moreover, as discussedlow, the determination of its ‘‘true’’ order from experimenmay turn out to be virtually impossible.
In the theoretical approaches, the level of complexityreduced to the minimum by taking into account only t~classical! motion of large subunits rather than the full~quan-
*Electronic address: [email protected]†Electronic address: [email protected]‡Electronic address: [email protected]§Electronic address: [email protected]
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tum! many-body dynamics of all the atoms. Clearly, an apropriate choice of the relevant degrees of freedom, depeing on the specific problem at hand, is crucial. Models baon the theory of polymers use self-avoiding walks to dscribe the two strands@10–12#. They can be very successfuin studying the properties of the melting transition at tlargest scale but, as they do not describe DNA at the levethe base pairs, they cannot be used to investigate propethat depend on the sequence, or probe DNA at a microscscale such as some recent single-molecule experiments.of the simplest models that investigates DNA at the scalea base pair is the Peyrard-Bishop~PB! model @13–15#. Thecomplex double-stranded molecule is described by postuing some simple effective interaction among the bases wia pair and along the strands. The model has been succfully applied to analyze experiments on the melting of shDNA chains@16#. Furthermore, it allows to easily include theffect of heterogeneities@17# yielding a sharp staircase structure of the melting curve~number of open base pairs asfunction of the temperatureT) @4#. Beyond its original mo-tivation to explain the denaturation transition, the PB mohas an intrinsic theoretical interest as one of the simpone-dimensional systems displaying a genuine phase tration @8,9#.
The PB model has, however a serious shortcomingcause it does not take into account the helicoidal structurDNA. In the following, we consider an extended versionthe PB model that has been proposed to avoid this weak@18,19#. The introduction of DNA geometry induces an important coupling between base pair opening and helical twlargely substantiated for real DNA@18#. A modified versionof this helicoidal model has also been successfully appliedescribe the denaturation of the chain induced either thmally or mechanically by applying an external torque to t
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BARBI et al. PHYSICAL REVIEW E 68, 061909 ~2003!
chain ends@20#. Furthermore, its nonlinear excitations habeen studied: small amplitude breather-like solutions hbeen analytically determined@21# as well as large localizedbubbles@22#. Both types of excitations are of interest as thmay be thermally excited as precursors for the DNA straseparation. A first effect of the helicoidal structure, namelybring close to each other bases which are not consecutivthe sequence, was treated in@23#, by introducing an interac-tion between these bases. This first approach however didtake into account the important geometrical effect thatwant to examine here, the coupling between openingtwist.
The aim of this work is to give further insight on thmelting transition of the helicoidal model, both from the stistical and from the dynamical point of view. After havinrecalled the model and its state variables~Secs. II and III!,the first part~Secs. IV and V! is devoted to its exact thermodynamics and to a simulation study of its statistical propties. We find that the melting transition is extremely shabearing essentially all of the hallmarks of a first-order trasition, at all temperature sampling steps studied~down to0.01 K!. However, a study of the model’s ‘‘universalitclass,’’ using finite-size scaling techniques, allowing sovariation of the relevant physical parameters, and drawfrom analogies with a Schro¨dinger-like equation~the Appen-dix!, strongly suggests the presence of an ‘‘underlying’’ cotinuous transition of the Kosterlitz-Thouless type in the asence of nonlinear stacking. This should be contrasted wthe exact second-order transition obtained in the zestacking limit of the PB model@9#.
In the second part~Sec. VI! we investigate dynamicastructure factors that are of particular relevance to both ntron @24# and Raman@25# scattering experiments. The undelying idea is that, upon approaching the transition, the mecule should display precursor effects in the form of‘‘mode softening,’’ i.e., a slowing-down of the dynamicwith appearance of a low-frequency component in the sptrum. Slowing effects have been, to some extent, observethe structure factors of the PB model@15# thus encouragingthis type of investigation.
II. THE HELICOIDAL MODEL
Two versions of the model have been proposed in eastudies. The first one@18# describes an elastic backbone afixed base-pair planes while the second@20# considers a rigidbackbone and moving base-pair planes. The two modelsplay a very similar behavior with respect to denaturationthe potentials associated to the base-base interactionspair and along the strands are the same in both casesbecause both introduce a coupling between openingtwist that results from the helicoidal geometry. In the following, we will consider the fixed-planes model, sketchedFig. 1.
The helical structure of DNA is introduced essentiallythe competition between a stacking interaction that tendkeep the base-pairs close to each other~given by the fixeddistanceh between the base planes! and the length,0.h ofthe backbone segment~described as an elastic rod of re
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length ,0) that connects the attachment points of the baalong each strand. The ratio,0 /h fixes the strand slant antherefore the resulting helicity of the structure. This helicis accounted for by the angle of rotation of a base-pair wrespect to the previous one, namely theequilibrium twistangleu, equal to 2p/10.4 in B-DNA at room temperature.
In the model, bases are described as pointlike particleequal massesm joined by elastic rods along each stranBases lying on the same plane are coupled by hydrobonds leading to an attractive force that tends to maintheir equilibrium distance equal to the DNA diameter 2R0.We assume that the two bases in each pair move symmcally. To describe base-pair opening and helical torsion unthose constraints, it suffices to introduce two degrees of frdom per base-pair: these arer n andfn , i.e., the radial andangular positions and of thenth base. The total number odegrees of freedom is thus 2N, N being the total number obase pairs. The restriction imposed by the assumptionsymmetric motion preserves the essential feature of the hcal structure, the coupling between torsion and openwhile it keeps the model sufficiently simple to allow anexacttreatment of its thermodynamics.
We consider the Lagrangian@26#
L5m(n
~ r n21r n
2fn2!2D(
n~exp@2a~r n2R0!#21!2
2K(n
~ l n,n212,0!22S(n
~r n2r n21!2
3exp@2b~r n1r n2122R0!#, ~1!
where ,05Ah214R02sin2(u/2) and l n,n21 are respectively
the equilibrium and the actual distance between thebasesn andn21 along a strand,
l n,n215Ah21r n212 1r n
222r n21r ncos~fn2fn21!. ~2!
For later purposes, it is convenient to introduce the lotwist angle defined asun5fn2fn21.
The first term in the Lagrangian is the kinetic energy. Tsecond term is intended to describe the hydrogen bond inaction between the two bases in a pair. Following Refs.@13#
FIG. 1. Schematic representation of the fixed-planes DNA hcoidal model.
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THERMAL DENATURATION OF A HELICOIDAL DNA MODEL PHYSICAL REVIEW E 68, 061909 ~2003!
and@18#, a simple Morse potential form is chosen. The qudratic term in (l n,n212,0)2 represents the elastic energythe backbone rods between neighboring base-pairs onstrand. Finally, the last term models a stacking interactbetween neighboring base pairs. Its effect is to decreasestiffness of the open parts of the chain relatively to the cloones and to stabilize the latter with respect to the denaation of a single base-pair. Terms of this type increasecooperative effects close to the melting transition@15,27#.
In the present paper we restrict our attention to a chwith free boundary conditions, which corresponds to theperimental situation when DNA denaturation is studiedsolution. However, the model described above can be eamodified to account for an external torqueG applied at thebase pairs at the two ends@20#, as it is done in some singlemolecule experiments.
The geometrical parameters of the model can be straiforwardly fixed according to the available structural da@18#. Much more delicate is instead the choice of the paraetersb,D,S andK gauging the effective forces. We selectvalues similar to those previously considered for the fixplanes case, which have been discussed elsewherechoice of theK parameter can be independently derived@27#from the twist persistence length@28#, while the choice of theother two parameters was based@29# on a comparison withrecent mechanical denaturation experiments@2#. In particu-lar, the important parameterD which sets the main energscale, has been tuned to reproduce as closely as possibexperimental value of the denaturation temperatureTD5350 K. The full set of parameters used in the following asummarized in Table I.
For the numerics it is convenient to work in dimensioless units. A suitable choice is to measure lengths and egies in the natural units of the Morse potential,a21 andD,respectively, whereby time is expressed in units ofAm/Da2.With the parameters of Table I, one time unit~t.u.! .2.3 ps.
III. THE DENATURATION TRANSITION: AQUALITATIVE DISCUSSION
Before going on, it is instructive to briefly discuss ththermodynamic state variables of the chain as well as~possible! analogies between denaturation and the moremiliar liquid-gas transition.
TABLE I. The parameter set used throughout the paper.
Parameter Symbol Value
Morse potential range a 6.3 Å21
Stacking interaction range b 0.5 Å21
Morse potential depth D 0.15 eVStacking interaction coupling S 0.65 eV Å22
Interplane distance h 3.4 ÅElastic coupling K 0.04 eV Å22
Equilibrium distance R0 10 ÅTwist angle u 0.60707 radBase masses m 300 amu
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As already pointed out@20#, the applied torqueG playsthe role of the pressureP for the liquid-gas system. Its conjugate variable is thedegree of supercoilings. Since we donot consider the curvature of the axis of the helix, it reducsimply to theaverage twist:
s5(n
N~^un&2u!
Nu. ~3!
This variable thus plays the role of the volumeV. Followingthis analogy we can therefore establish the following corspondences:
DNA model liquid-gas
T ↔ T
G ↔ P
2s ↔ V
The sign ofs is chosen for convenience as the degreesupercoiling vanishes for B-DNA and is negative for a ptially denaturated chain@see also Eq.~4! below#.
Two natural scenarios may thus be expected: an isotmal, torque-induced denaturation atG5GD or a fixed-torque,thermally-induced one atT5TD . For the liquid-gas casethese two situations correspond to crossing of the coexence curve in the (P,T) plane with an horizontal or a verticaline, leading to the transitions classically described by itherms in the (P,V) plane or by isobars in the (V,T) one,respectively. In both cases, the presence of a consttemperature or a constant-pressure/torque domain is asated to the phase coexistence.
Within this analogy, the transition isotherms correspondthe curves in Fig. 3 of Ref.@20#, and reproduced schematcally, on the (G,2s) plane, in Fig. 2~c!. Conversely, thethermal denaturation at constant torqueG50 are correctlydescribed in the (2s,T) plane@Fig. 2~a!#. Notice that, withthe convention that a negativeG corresponds to an untwisting torque, bothGD and TD must increase upon increasintemperature and torque respectively@see Fig. 2~b!#. Howeverthe negative torque cannot exceed some critical value wout leading to an instability of the helix associated tochange of the sign of the helicity. In the following, we wifocus on the thermal denaturation transition atG50.
FIG. 2. A sketch of the behavior of first-order transition curvin the three planes defined by the DNA model parameters2s, G,andT. GD andTD represent the transition torque and temperaturespectively;~a! thermally-induced denaturation ‘‘isobar’’ at zertorque;~b! coexistence curve;~c! torque-induced denaturation isotherm atT5TD , corresponding to a zero critical torque.
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It is important to remark that the helical constraintscluded in the model roughly impose, at vanishing extertorque,un'u for a closed chain segment, andun'0 for thedenatured one. This follows from the geometry of the heand the stiffness of the strands: since the distance betwconsecutive bases is constrained by the elastic rods toapproximately equal to,0 , un is of order,0 /r n and hencevery small forr n@R0. Let us denote bynd the average number of open bases in a chain of lengthN. Provided that openand closed regions coexist along the helix, and that theyspatially well separated~this is well confirmed by simula-tions as we show below!, then, to a good approximation, whave
2s'2nd~2u!
Nu5
nd
N. ~4!
The latter quantity in nothing but the average fractionopen base pairsr5nd /N and the ‘‘isobar’’2s(T) can thusbe identified with the familiar denaturation curver(T). Inother words, the supercoiling and the fraction of open bpairs are equivalent order parameters.
Before going further there is one crucial issue that shobe addressed. One may argue that for a one-dimensimodel like the one at hand no phase transition of any tshould be observed. However, all usual arguments agathe existence of singularities in thermodynamic potenthave been showednot to hold for the PB model@9#. Sincethe latter is in many respects similar to the helicoidal modthe same arguments apply and a genuine phase transitinot forbiddena priori. This is confirmed by the transfer integral approach which can be carried exactly~although partlynumerically! for this simple model.
IV. TRANSFER INTEGRAL APPROACH
A. ‘‘Apparent’’ thermodynamics
Because the model is one dimensional, a direct calction of the partition function can be performed by the tranfer integral~TI! method, as it was done for the simpler Pmodel @15#. The calculation proceeds along the same linbut it is more involved because the model has two degreefreedom (r n andfn) per unit cell. It nevertheless reducesa one-dimensional TI equation because the contributiontroduced by the angular part can be diagonalized by a Frier transform@20,29#. As the calculation has already beepresented in these earlier studies we do not discussmethod here and we confine our attention on its resuwhich point out new aspects of the transition that had boverlooked.
In this subsection we restrict ourselves to results obtaivia numerical solution of the TI equation. The accuracythis approach is limited first by discretization errors in tintegrations and by the need to numerically evaluate ingrals over an infinite domain. As discussed in the next ssection, this second restriction can be partially lifted byfinite size scaling analysis, which involves a properly cotrolled approach to infinity. Integration methods, such asGauss-Legendre quadrature which select appropriate ab
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for the evaluation of the function according to the numberpoints involved in the calculation, are also useful to integrover a large domain with a reasonable number of points.this first study, in order to ensure that all integrals are evaated with the same discretization error, we have computhem with a 10th order Bode’s method@30# with a fixedspatial step dr 50.032 Å and a minimum valuer min59.7 Å ~due to the strong repulsion between bases descrby the Morse potential,r cannot take values significantlbelow the equilibrium length of 10 Å!. The maximumr maxof the integration range depends on the number of integrapoints, which has been varied from 631 to 3601 leading29.9<r max<124.9 Å. The eigenvalues of the transfer intgral operator have been obtained either by diagonalizatiothe equivalent matrix problem or by the Kellog’s metho@31# to get the two lowest eigenvalues.
The eigenvaluesL of the TI operator will henceforth bewritten asL5exp(2e/kBT) wherekB is the Boltzmann con-stant. With this notation the free energy per particle is demined by the smalleste eigenvaluee0 and is given by
f 52kBT ln~4pmkBT!1e0 . ~5!
Relevant thermodynamic quantities like the entropy and scific heat are then evaluated from the standard relations
s52] f
]T, cv52T
]2f
]T2. ~6!
The mean base-pair stretching is given by
^r &5E0
1`
r uf0~r !u2dr, ~7!
wheref0 is the eigenfunction associated withe0.The free energyf and the mean value of the base pa
stretching^r & for our model are shown in Fig. 3, for temperatures going from 349 to 352 K with a step of 0.02Within the accuracy of the calculation, a cusp inf at TD5350.74 K is distinctly seen. It is associated with a shajump of the entropy at the transition. A jump in the speciheat is also observed. Evaluating numerically the first asecond left and right derivatives of the free energy, onetains the jump in entropyDs54.40kB , or 8.75 cal/K/mol,and the jump in specific heatDcv50.64kB . The specificheat drops from 2.14kB below TD to cv51.5 kB for T.TD as expected from equipartition because after denaation only the harmonic contributions of the hamiltonian stsignificant.
Figure 3~b! shows that, within numerical accuracy,^r &exhibits a discontinuous transition from a finite constavalue ~very close to the equilibrium valueR0) to a value ofthe order of the system size; in other words, the eigenfution f0 appears to become suddenly delocalized. The picof a sharp transition persists down to a temperature sampof 0.01 K.
Although the numerical results strongly suggest the occrence of a first order transition, caution is necessary: previstudies of the related PB model have shown that the non
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THERMAL DENATURATION OF A HELICOIDAL DNA MODEL PHYSICAL REVIEW E 68, 061909 ~2003!
ear stacking produces an extremely sharp, first-order-likehavior which masks the underlying second-order transitas long as one stays out of a very narrow domain inimmediate vicinity of the critical temperatureTD ~exponen-tial crossover@32,33#!. A more complete picture of the properties of the transition will therefore be given in what folows.
B. The ‘‘underlying’’ transition
We first address the question of what happens in thesence of the nonlinear stacking, i.e., atS50. Preliminarynumerical investigations suggest a smooth behavior, botthe lowest eigenvaluee0 and of the next-to-lowest,e1, asfunctions of temperature; an ‘‘avoided crossing’’ betwethem appears, with a small, but finite gap which has a m
FIG. 3. The free energy per particlef ~a!, and the mean value othe base pair stretchingr & ~b! of the model evaluated by the tranfer integral method in the temperature range 349<T<352 K with astepDT50.02 K. The model parameters are those listed in Tab~calculation with the Bode’s method,dr 50.032 Å, r min59.7 Å,and r max5124.9 Å).
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mum at a certain temperature. Before we proceed to anathe data obtained in detail, it is necessary to provide sobackground and notation.
The order of the phase transition of the ideal systemunconstrained transverse spatial extent is determined bycritical exponentn which characterizes the gapDe[e12e0}(TD2T)n at temperatures belowTD ; a valuen51implies a cusp in the free energy and a discontinuoustropy; a value equal to 2 implies a discontinuity in the spcific heat, i.e., a usual 2nd order transition, etc.
The ‘‘raw’’ data provided by numerical solution of the Tequation refer to a particular transverse system sizeL5r max determined by the imposition of an upper cutoffthe integration. On the other hand, near a critical point ofinfinite system, the transverse fluctuations of the orderrameter also diverge. The quantityj'5A^r 2&2^r &2 providesa measure of the distance from the critical point with dimesions of length. According to the finite-size scaling hypoesis@34#, size-dependent properties of a system in the vicity of the transition should depend solely on the ratioL/j' ,i.e.,
DeL~T!5L2s f gS L
j'D , ~8!
where the exponents characterizes the rounding of the ga~cf. below!, f g(0) is a nonzero constant, andf g(x)}xs asx@1 guarantees size independence in the limitL→`. In thesimplest cases, the positions of the minima of the gaprelated to the type of divergence ofj' ; according to theabove scaling scenario, the temperatureTm(L) where the gapminimum occurs, is such that
j'~TD2Tm~L !!'L. ~9!
We have numerically solved the TI equation for a wide ranof system sizes, using Gauss-Legendre quadratures, wheLis defined as the largest grid point value provided byGauss-Legendre algorithm for the numberN of grid pointschosen for the calculation. Again, in order to ensure unifoaccuracy,N grows proportionately to the system size, wir max5350 Å corresponding toN52048 points. Figure 4shows the dependence of the minimal gapDeL(Tm), and thecorresponding temperatureTm on L. The gap appears to depend quadratically on 1/L @i.e., s52 in Eq. ~8!#, to verygood accuracy, with an estimated limit of 1.431025 ~with astandard deviation 0.831025) asL→`. We note in passingthat the disappearance of the gap between the boundeigenvalue and the one belonging to the bottom of the ctinuum band provides numerical evidence for the true occrence of an exact, thermodynamic phase transition@35#. Thetemperatures which correspond to the gap minima canwell fitted to the function
Tm~L !5TDF 12a2
lnS L
R0cD G ~10!
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with TD5577.8 K, a250.170 andc50.94. This type of de-pendence ofTm on L immediately suggests@cf. Eq. ~9!# that
j'5cR0ea2 /utu, ~11!
wheret5T/TD21.Figure 5 shows that data taken from a wide range of s
tem sizes scale well if plotted according to Eqs.~8! and~11!.The numerical evidence thus strongly suggests that thederlying transition manifests itself as an essential singulaof the gap, of the Kosterlitz-Thouless~KT! type. Equations~11! and ~8! then imply that, in the limitL→`,
De}e22a2 /utu. ~12!
In the Appendix, it will be possible to identify the origin othis particular behavior as an inverse-square attractive inaction between the stretching coordinates of successivepairs.
C. Finite stacking revisited
It is now reasonable to conjecture, by analogy with whhappens in the PB model, that the effects of the nonlinstacking interaction will depend on its range. For the stdard parameter set of Table I, the ratiob/a50.079 is verysmall indeed. What happens at a less extreme regime,b/a50.190, is shown in Fig. 6. Scaling according to the ans~11! holds within a fairly narrow rangeutu,0.05 around thedenaturation point; note that it is the smaller magnitudethe nonuniversal parametera2 which is responsible for thenarrowing of the asymptotic critical region. Figure 7 summrizes what happens atb50.9 Å21, i.e., b/a50.143, onlyslightly above the value of Table I. The gap exhibits an aparent critical exponent very close to unity down tot
FIG. 4. Dependence of the minimal difference between the loest eigenvalues of the transfer integral operatorDeL(Tm) ~opensquares, lefty axis! and temperaturesTm ~closed circles, rightyaxis! versus 1/L2.
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FIG. 5. Scaling of the difference between the two lowest eigvalues of the transfer integral operator~open symbols, lefty scale!,and the order parameter~solid symbols, righty scale!. The differentpoints have been obtained by transfer integral calculationsformed with 5 values ofL/R0516, 35, 70, 100, and 140. Thdotted lines have slopes 2 and21, respectively, in accordance witthe finite-size scaling hypothesis.
FIG. 6. Scaling of gap~open symbols, lefty scale!, and orderparameter~solid symbols, righty scale!; model parameters are thosof Table I, with the exception ofb51.2 Å21. The different pointshave been obtained by transfer integral calculations performeL/R0525, 35, and 50. The dashed lines have slopes 2 and21,respectively, in accordance with the finite-size scaling hypothes
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THERMAL DENATURATION OF A HELICOIDAL DNA MODEL PHYSICAL REVIEW E 68, 061909 ~2003!
50.01; closer to the denaturation point, the effective sloincreases significantly; it is reasonable to conjecture thatemperatures even closer toTD , the asymptotic behavior wilbe dominated by the underlying essential singularity. Atphysically relevant value ofb50.5 Å21, crossover to theKT regime has moved belowt51025 and is practically un-observable.
The analysis of this section demonstrates that, in spitethe very different mathematical properties of their ‘‘barversions, both the ‘‘straight’’~PB! and the helicoidal DNAmodels, are effectively dominated by the stacking interactwhen the latter is of sufficiently long range; because of it,all practical purposes, the transition has all the characterisof a first order transition, including a practically infinite dicontinuity of the mean base pair stretching, and a latent hSimilarly, at the transition temperature, a very small tempeture gradient~of the order of the width of the transition region, i.e.,DT,0.001 K) leads to an apparent phase coexence and hence to many features that one would be temto qualify as ‘‘typical of a first-order transition’’ as shown ithe next sections. These properties are very reminiscensome results found on models of martensitic phase trations @36–38#, but because we are dealing with a ondimensional model that has a genuine phase transition,phenomenon is more remarkable here.
Our results provide an excellent example of the distition between the ‘‘experimental’’ perspective and the ‘‘theretical’’ one, regarding the definition of the order of a phatransition: although, in theory, the transition of the helicoid
FIG. 7. Dependence of the gap on the reduced temperaT/Tm21 for b50.9 Å21 and system sizesL/R0512,49, 17.5, 25,35, 50, and 70. The dashed and dotted lines have slopes 1 arespectively. Apparent first-order behavior prevails forutu.0.01;closer to the critical point there is a clear increase in the slobefore the onset of finite size rounding.
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DNA model is of infinite order~essential singularity!, theactual temperature range over which it manifests its contious character is far beyond the limits of either experimenor numerical observation.
V. MOLECULAR DYNAMICS
In this section we report the results of direct simulatioof the model. They bring complementary information on tnature of the transition and allow us to study its dynamicsdiscussed in Sec. VI. As said above, we consider the casthermal denaturation forG50 and free boundary conditionsMicrocanonical and canonical simulations were performbecause they allow the observation of the phase spacedifferent viewpoints.
In the microcanonical ensemble, the Euler-Lagranequations derived from Eq.~1! were integrated directly withthe standard fourth-order Runge-Kutta method with a smenough time step~typically 0.02 t.u.! in order to insure thatthe relative energy drift is negligible~usually better than1025) on the time scales of each run, i.e., 105 to 106 t.u.@39#. Initially, all particles are set in their equilibrium postions (r n5R0 , fn5nu) with random Gaussian distributevelocities ~with zero average! in the radial direction. Thevariance of the distribution serves to fix the energy pergree of freedome. The averaging of the quantities of intereis only started after a long enough transient to let the sysequilibrate. After equilibration, the thermal energykBT iscomputed in the usual way as twice the average kineticergy per degree of freedom.
Constant-temperature~canonical! results were obtainedthrough an extended Nose´-Hoover method using a thermostat chain@40# which is specifically designed to constrain thtotal kinetic energy to fluctuate aroundNkBT, insuring at thesame time the correct~canonical! distribution of its fluctua-tions. A chain of 3 thermostats was employed wthe first thermostat typical frequency equal to the highphonon frequency of the lattice,vM5$a2D12K„R0(12cosu)/,0…
2#/m%1/2. The integration of the correspondinequations of motion was again performed with a fourth-orRunge-Kutta scheme with typical time step of 0.01 t.u., athermalization is achieved by a long-enough transieChanges in temperature were performed in a sequentialupon heating the chain with a temperature ramp and relaxafterward.
As discussed in Sec. IV, strictly speaking the transitionsmooth. However the temperature range of the crossovegion to a smooth behavior is so small~less than DT50.001 K for a stacking parameterb50.5 Å21) that thenumerical experiments, as well as actual denaturation expments on DNA, show all the character of a first order trasition. Therefore in this section we shall use the languagefirst order transitions which is the appropriate languagediscuss the results.
Measured caloric curves showing the temperature~in en-ergy units! as a function of the energy per degree of freedokBT(e), for a chain ofN5128 base pairs are reported in Fi8. They distinctly show a flat part atkBT50.2012D corre-
re
2,
e,
9-7
dcae-
a
th
n
o
he
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BARBI et al. PHYSICAL REVIEW E 68, 061909 ~2003!
sponding to the temperatureTD5350 K that has been founto be the denaturation transition by the transfer integralculation. In analogy with a liquid-gas transition, the flat rgion occurring betweeneB50.20D and eD50.64D is thusidentified as the curve of coexistence between the closedthe denatured phases. The slope of the two branches,kBT/eshould be equal to 2/cv ~the factor 2 appears becausee, en-ergy per degree of freedom is12 of the energy per unit cell!.The figure shows that this is in good agreement withvalues given by the TI calculation, i.e.,cv.2.2 belowTDand 1.5 aboveTD . We have also compared the melting etropy per particle,Ds52D(] f /]T), as obtained from thetransfer integral calculationDsth , with that obtained fromthe microcanonical simulationsDsnum for the finite chain,i.e., the ratio 2(eD2eB)/TD :
Dsnum53.7031024 eV/K, Dsth53.8031024 eV/K.~13!
The two quantities are in very good agreement.In addition, the transition markedly displays a signature
metastability and hysteretic effects. Indeed, the B-DNAbranch extends well aboveTD ~up to about 500 K!. Markedhysteretic effect upon heating are also observed for the tmostated chain~crosses in Fig. 8!. Either in microcanonicaland canonical simulations, the system appears to be sponeously ‘‘trapped’’ into this metastable state for low enouenergies~or temperatures! over the transition one. Direct inspection of the system configuration reveals that the chacompletely closed and we can refer to it as an overheastate.
An undercooled branch exists as well below the denaation temperature. To detect it in the microcanonical schewe employed the following procedure. The initial conditioin the B phase, is evolved for a certain time after whichchain is ‘‘annealed’’ by multiplying all velocities by an as
FIG. 8. Result of microcanonical simulations~open symbols!:kinetic temperature as a function of the energy per degree of fdom for a molecule ofN5128 base pairs. To show the convergenof averages, data for two transient durations are reported. Pland crosses respectively refer to canonical simulations with cloinitial conditions and with the initial insertion of an artificial bubbof length 120 base pairs~transient 105 time units!. Solid line is thetransfer integral result~see text for details!.
06190
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nd
e
-
f
r-
ta-
isd
r-e,
e
signed factor smaller than 1~we set it equal to 0.8!. Theaverages are thus computed after a further transient~seeagain Fig. 8!. Similar results can be obtained for the thermstated chain by artificially imposing on standard initial coditions the presence of a denaturated bubble of given len, in the middle of the chain at temperatureT.TD . We willgive more details on this procedure at the end of this sect
Canonical and microcanonical results are therefore csistent, apart from some deviations at high temperaturesT.TD , which can be expected because after denaturathe model becomes almost purely harmonic as the Mopotential linking the bases plays no role for base-pair dtancer corresponding to the plateau of the potential, andstacking contribution also tends to vanish. Therefore forT.TD a microcanonical equilibrium cannot be achieved, uless we force it by averaging over thermalized initial contions ~for instance obtained by a Monte Carlo procedure! orby a temporary switch to a canonical simulation during a r
The results were checked to be robust with respect totransient duration as well as to the rate at which temperais changed through the ramp. Alternative thermalizatschemes do not change the outcomes as well. For examsimulations where microcanonical runs are alternated tononical ones, yield the same results~except atT.TD asmentioned above!. In such a case the computed averagesmicrocanonical as the thermostated dynamics only servea way to change the system energy.
Obviously, a crucial issue is the dependence of the reson the chain size. We observed that upon increasing the clength up toN5256 or 512, the only difference with respeto theN5128 case is a slower convergence of the averain the high and intermediate energy region. Nonetheless,coexistence line is practically reached within comparasimulation times. This is presumably influenced by the initconditions and could be improved by a more sensible cho
In order to precise the nature of the transition from a vipoint close to experiments, we measured the average fracof open base pairsr. This is indeed a quantity which ismeasurable by UV absorption. As in previous stud@15,18,20#, we consider a base pair to be open wheneverradial displacementr n is larger than the inflection point othe Morse potential, i.e., whenr n /R0.8 ln 2, and we aver-age the counting during the run. A simple reasoning shothat the order parameterr should obey a ‘‘lever rule’’@41#
e5~12r!eB1reD , eB,e,eD , ~14!
thus implying thatr(e) increases linearly between 0 andalong the coexistence line. In a similar way, we expect tthe average twist per base pair^un&5^fN2f1&/N5u(11s) should decrease linearly from a value close tou to 0.This is illustrated in Fig. 9. Notice once again the hystereeffects.
From Fig. 8 it is clear that the microcanonical ensemhas the merit of allowing to investigate the dynamics of tchain in the coexistence region. For illustration, Fig.shows a snapshot of the state of the chain fore/D50.5where, from formula~14!, we expect around 40% of the baspairs to be denaturated. The fact that the chain opens a
e-
esed
9-8
n
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oit
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of
THERMAL DENATURATION OF A HELICOIDAL DNA MODEL PHYSICAL REVIEW E 68, 061909 ~2003!
sides is clearly caused by the free boundary conditioMoreover, this figure can give a hint on why we observetransition that has all the features of a true first order tration ~coexistence of phases, metastability! although the tran-sition is actually second order. From a theoretical pointview, the order is determined in the thermodynamic limi.e., for an infinite system. This correspond to the identifition, in the transfer integral method, of the free energy wthe lowest eigenvaluee0, Eq. ~5!. In numerical simulationsas well as in experiments, one is dealing with a finite systWhen the thermodynamic transition is extremely sharp~as itis the case for the stacking parameterb50.5 Å21), the in-homogeneity caused by the free ends is sufficient to leaan apparent coexistence of phases, i.e., a first-order-likesition, presumably because the boundary effects inducperturbation~in particular on the average local torque! whichis sufficient to change the local transition temperature byvery small amount which separates the domain of cloDNA from the domain where the molecule denaturates. Tis why the molecular dynamics simulations are useful
FIG. 9. Fraction of open base pairsr and average twistun&from microcanonical simulations. The simulation parameters aresame as in Fig. 8. The circles were obtained with an initially clochain while the squares refer to annealed initial conditions.
FIG. 10. Snapshot of the chain of 128 bps in the coexisteregione50.5D.
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.
ton-a
ed
iso
complete the transfer integral study, and to provide resthat can be compared with experiments.
Another interesting aspect which can be studied throusimulation is the dynamics of opening events. This allowslook for analogies with the classical nucleation mechanisthat drives relaxation from metastable states at ordinary fiorder transitions@41#. Figure 11 reports the distribution othe length of denaturated bubbles for subsequent timesing the same run of Fig. 10. There is a clear tendencysmaller bubbles to close~or merge! until only a few largeones remain. A similar measure in the overheated metastphase shows instead that the size of bubbles is pretty sand decrease systematically in time.
To further investigate this aspect, we performed simutions in the canonical ensemble, starting from a thermalistate at temperatureT and artificially seeding a denaturatebubble of given length, in the middle of the chain. Toaccomplish this, given the geometry of the model, weun50 in the central region and impose a triangular profifor the r ns designed in such a way that the resulting stressthe backbone springs is approximatively zero. The flankregions are initially at equilibrium and free from any addtional supercoiling.
In a first series of simulations we checked that forT,TD the B-DNA form is very robust with respect to suclocal perturbation: for instance, atT5200 or 300 K bubblesof length,<12 base pairs on a chain 128 base-pair long teto shrink and the system rapidly returns to its completclosed state. Very large bubbles (,;110 base pairs! maytend to close as well, although on longer times. Howevseveral cases are found where perturbations as large,5120 base pairs are able to drive the system into the mstable, undercooled, denaturated state. These experimelow the observation of the second metastable state incanonical scheme: thus completing the correspondencethe microcanonical results~see again data shown in Fig. 8!.
The situation is, as expected, the opposite in the ov
ed
e
FIG. 11. Distributions of the lengths of the denaturated regioat different times in the coexistence regione50.5D. To improvethe statistics, histograms were cumulated over a time windowaround 103 time units.
9-9
rtheel
ghm,tice
do
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ic
. Ateh
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al
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BARBI et al. PHYSICAL REVIEW E 68, 061909 ~2003!
heated regionTD,T&600 K. Here, the insertion of a shobubble suffices to destabilize the B-DNA form and let tsystem switch to its equilibrium state, i.e., the completopen chain. For instance, atT5400 or 500 K a bubble oflength,.8 on a chain of 128 base pairs is generally enouApproximated estimates seem to indicate that the minilength of the bubble tend to decrease with temperatureintuitively expected, but this behavior is not very systemaStatistics over a very large number of events would be nessary to conclude quantitatively.
VI. DYNAMICAL STRUCTURE FACTORS
One of the motivations for considering mechanical moels is the possibility to probe microscopic and collective mtion in different phases. In this section, we focus on dynacal correlation functions that usually reflect different typesexcitations. More precisely, we computed the radial structfactor
Sr~q,v!5K U E (n
r nei (qn2vt)dtU2L ~15!
and the angular oneSc(q,v) wherecn5fn2nu is the an-gular displacement from the equilibrium position. Brackedenote an average over an ensemble of independent molar dynamics trajectories~typically hundreds!. All the resultsreported in this section are obtained in the microcanonensemble.
Let us first consider the low temperature native phaseshown in Fig. 12, the spectral analysis display, as expecsharp lines at the frequencies of the two phonon brancv6(q) that can be computed atT50 in the harmonic ap-proximation@27# ~see the vertical lines in Fig. 12!. Acousticvibrations in the angular variables are only weakly coupto the radial~optical! ones. Interesting enough, the radspectra also displays a large peak at a frequency lying inphonon gap and independent on the wave [email protected]., thelarge peak atv.0.7 in Fig. 12~a!#. Its origin can be tracedback to the excitation of a localized surface mode. Thisconfirmed by direct inspection of the chain configuratioActually, the mode is found to slowly decay in time duenonlinear interaction leading to a systematic decrease ospectral component.
Upon increasing the energy, the optical branch gradushifts towards lower values of the frequency~softening! andhigher-harmonics appear. Furthermore, the resonanceboth the radial and angular peaks are substantially broaddue to increasing anharmonicity that enhances the effecdamping. More importantly, a large low-frequency compnent, acentral peak, arises in the radial structure functioThe temperature dependence ofSr(q,v) across the denaturation transition is illustrated in Fig. 13. The three differeenergies correspond toT5300, 357, and 535 K. The lattevalue is well into the metastable overheated region. For fiq, the position of the central peak is unchanged upon increing temperature but its width broadens. Furthermore,v22 behavior at low frequencies~see the inset of Fig. 13!suggests a Lorentzian line shape. The origin of this cen
06190
y
.alas.c-
--i-fe
scu-
al
sd,es
d
e
s.
its
ly
ined
ve-
t
ds-e
al
peak, also found in the simpler PB model, and its properare still unclear although it is tempting to assign it to tslow dynamics of the bubble boundaries.
An even more sizable central component appears wclosed and open form coexist~see Fig. 14!. This is accom-
FIG. 12. Structure factorsSr(q,v) ~a! and Sc(q,v) ~b! for N5256 at very low energy,e50.01D ~corresponding toT518 K).The different curves correspond to wave numbersq52.5676,1.66896, 0.88356, and 0.09812~right to left!.
FIG. 13. Radial structure factorsSr(q,v) for N5256 and dif-ferent energiese50.17D, 0.20D, and 0.30D ~solid, dashed, anddot-dashed lines, respectively!. To reduce fluctuations, a smoothinof the data has been performed by averaging over 10 consecchannels.
9-10
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THERMAL DENATURATION OF A HELICOIDAL DNA MODEL PHYSICAL REVIEW E 68, 061909 ~2003!
panied by a stronger coupling between angular and radegrees of freedom, as manifested by the peaks at acofrequencies inSr . The birth of large low-frequency components bears strong resemblance with heterophase fluctuaobserved in other lattice models with~pseudo! first-ordertransition characterized by large entropy barriers@36–38#. Inother words, the motion of the interface between the tphases should be responsible for the slow dynamics.
To close this section, it is worth mentioning that a relatanalysis of collective modes for the helicoidal model hbeen recently reported@42#. The analytical calculations werperformed at room temperature and are based on the intaneous normal modes. At variance with our simulations,approach describe the short-time dynamics~on a time scaleof picoseconds! and a direct comparison is therefore nstraightforward.
VII. CONCLUSIONS AND DISCUSSION
The study of a simplified model of DNA has proved toextremely fruitful to unveil the basic features of the meltitransition at the single-molecule level. From the theoreti
FIG. 14. Structure factorsSr(q,v) andSc(q,v) for N5256 inthe coexistence regione50.50D. Vertical lines are the phonon frequencies~one from the acoustic and one from the optical brancheach q value! calculated atT50. Graphs have been arbitrarilshifted for clarity.
06190
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ns
o
ds
n-is
l
point of view, dealing with a one-dimensional model~withtwo degrees of freedom per base pair! turned out to be par-ticularly convenient as it allows an exact evaluation of tpartition function. Indeed, the angular variables can be elinated by Fourier transform, yielding a more tractable ovariable transfer integral problem. The latter cannotsolved analytically but the numerical and approximate respresented above provide a complete insight on the naturthe transition. In particular, the finite-size scaling analysisthe transfer integral turned out to be essential to takeaccount the finiteness of the integration range. Suchanalysis strongly suggests that the underlying transitioncontinuous, of the Kosterlitz-Thouless type. This behavcan be related to the existence of an effective attractive fowhich is directly connected to the helicoidal geometrybe-cause it appears when the angular degree of freedom isgrated out. Qualitatively it can be understood as comfrom the difficulty to disentangle the two helices. On thother hand, for physically relevant values of the parametthe temperature range over which the continuous aspecthe transition can be detected may become extremely na~less thanDT50.001 K). For all practical purposes the trasition appears to be perfectly sharp, and bears the hallmof a first order transition, in agreement with experiments.our view, this result is remarkable and attracts attentionhow numerical or experimental observations on finite stems and with a limited resolution may dramatically difffrom theoretical expectations.
Molecular dynamics simulations confirm this apparefirst-order character. They show hysteresis and metastabas well as a coexistence region between an open and a c‘‘phase’’ and, by varying the energy density in the criticregion, a gradual change of the volume fraction occupiedthe two which is reminiscent of, say, the liquid-gas trantions.
In addition to the very sharp transition found by the thoretical analysis, the finite-size effects certainly play a marole in the above phenomenology. The helicoidal modemore sensitive to these finite size effects than the ‘‘flat’’ Pmodel because the free ends allow a release of the torsienergy which appears when a segment of the chain opOne can understand the crucial role of boundary effectone considers that a finite closed loop of helicoidal DNcannot denaturate at all because the two strands aretangled.
The dynamics of the transition, as probed by the calcution of the radial and angular structure factors, shows soprominent features such as the existence of a central pthat is presumably due to the slow motion of the denaturabubbles. Moreover, the coupling between opening and twintroduces some additional spectral features that wouldserve further investigations.
Another point that should be reconsidered is the nucation of denaturation bubbles. The phenomenology descrat the end of Sec. V is, at least qualitatively, very mureminiscent of the nucleation mechanisms that drives reation from metastable states at ordinary first-order transiti@41#. For instance, simulation in the overheated state sgests the existence of a ‘‘critical size’’ of the denaturati
r
9-11
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el
dhieh
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isN
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BARBI et al. PHYSICAL REVIEW E 68, 061909 ~2003!
loops, above which they become unstable. Hence, metability stems from the fact that small enough bubbles clorelatively fast. Nevertheless, there are important differenthat one should keep in mind. Indeed, in classical nucleatheory the key role is played by the surface tension te~proportional to the square of the droplet’s radius!, wherebyin our one-dimensional case the bubble ‘‘surface’’ is indpendent of its length. The correspondence with the ustheory is probably due to the torsional energy, associatethe opening, which grows with the bubble size. This suggethat a ‘‘one-dimensional nucleation theory’’ could be devoped for helicoidal DNA.
The present study has focused on a DNA model thatscribes the molecule at the scale of the base pair. We tthat it is relevant because it is the scale of the genetic codwhich phenomena related to biological functions occur. Thelicoidal geometry itself is at this scale~or more precisely atthe scale of a few tens of base pairs!. On the other handthere are other phenomena that enter in the statistics of Dmelting, and they are related to the behavior of the molecat a much larger scale, on which the strands are regardeflexible strings. Recent studies have shown that the entrof the loops also contributes to lead to a first order transitiprovided that self-avoiding aspects between segments oloop and between open regions and closed domains are perly taken into account@11#. Our approach is complementarto these studies and shows that the observed sharp metransition of DNA may have multiple origins.
ACKNOWLEDGMENTS
We thank Franco Bagnoli, Simona Cocco, Thierry Dauois, and Stefano Ruffo for useful discussions. S.L. acknoedges partial financial support from the Region Rhoˆne-Alpesthrough the Bourse d’Accueil, No. 00815559. Part of thwork was supported by the EU through Contract No. HPRCT-1999-00163~LOCNET network!.
APPENDIX: RESULTS OBTAINED VIA GRADIENTEXPANSION
1. An approximate TI kernel
In the absence of an external torque, the nontrivial parthe partition function of the model is given by
ZP5E )n51
N
$drn@r nr n21#1/2dun%e2V($r i ,u i %)/kBT, ~A1!
where the potential energy consists of the three last termEq. ~1!, and the relative angle coordinate enters only viasecond term. Introducing sum and difference coordinater n5(r n1r n21)/2, dn5r n2r n21, it is possible to write the in-tegral overun as
E0
p
dune2K( l n,n212,0)2/kBT[F0~ r n!eV(dn2 , r n). ~A2!
06190
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-altots-
e-nkate
Aleas
py,
hep-
ing
-l-
-
f
ine
For the parameter values given in Table I, andT5480 K, thedimensionless ratiol5kBT/KR0
2 is equal to 0.01; this allowsus to use the leading-order low-temperature asymptoticpansion
F0~ r !;ApkBT
K
L0
k rS 12
k2
4r 2D 21/2
, ~A3!
wherek52R0 sin(u/2)55.98 Å, over the whole temperaturrange of interest. Note that due to the repulsive core ofMorse potential, the inequality
r @k/2 ~A4!
always holds. Furthermore, numerical evaluation revealsthe functionV is ~i! almost independent ofr , and~ii ! weaklydependent on temperature~cf. Fig. 15!. In the following wewill use the temperature-independent approximation
eV(d2)'e2(d/k)2, ~A5!
which misses the weak peak neard5k, but reproduces cor-rectly the second moment, which is central to what followWithin the approximations~A3!, ~A4!, the partition functionis dominated, in the thermodynamic limit, by the largestgenvalue of the one-dimensional TI equation
E0
`
dr8T~r ,r 8!cn~r 8!5Lncn~r ! ~A6!
with
T~r ,r 8!'ApkBT
K
L0
k S 11k2
8r 2D S 12d2
8r 2D3eV(d2)e2VM(r )/kBTe2VS( r ,d2)/kBT, ~A7!
FIG. 15. exp(V) as a function ofd/k for two different values ofthe dimensionless ratiol; the r dependence is not visible forl50.001.
9-12
c
endse
nds
es
th
o--o
o-
nolv
t
oat a
to
eratealpy
r-thanthatow-na-ireatectlyfne
areior,therbe-
oferent
THERMAL DENATURATION OF A HELICOIDAL DNA MODEL PHYSICAL REVIEW E 68, 061909 ~2003!
where VM(r )5D(12exp@2a(r2R0)#)2 and VS5Sd2
3exp@22b(r2R0)#.The form of V @cf. Fig. 15 and/or Eq.~A5!# establishes
that, in addition to the ranges 1/a, 1/2b of the Morse andstacking interactions, respectively, there is a third, mularger, characteristic length in the problem,k. Depending onthe strength of the various parameters, it may be possiblfurther simplify the general one-dimensional TI problem aelucidate the ensuing critical behavior. Two distinct cawill be considered below.
2. Strong stacking interaction: transformation to an ODE
A gradient expansion of Eq.~A6! involves~i! introducing
cn~r 1d!'cn~r !1cn8~r !d11
2cn9~r !d2, ~A8!
~ii ! changing the variable of integration fromr 8 (5r 1d) tod, and~iii ! performing the Gaussian integrals overd. Notingthat ~a! the combined effect of the stacking interaction athe Gaussian approximation~A5! can be described in termof the quantity
S
kBTm2~r 
or
thi
th
ultiardoac
ave
eesnottoy a
thee
sh-z-
BARBI et al. PHYSICAL REVIEW E 68, 061909 ~2003!
where
VL* ~r !52kBTk2
16r 2. ~A14!
In the absence of stacking, the system is subject to the Mpotential and the long-range attraction~A14!; the point tonote is that the attractive force is linearly dependent ontemperature, just as the coefficient of the 2nd derivativeEq. ~A13!; consequently, ifD50, the system will eitherhave a bound state or not, according to the value ofcoefficient in the denominator of Eq.~A14!. The value 16 ismarginal; if the interaction had been stronger, one wohave confinement at all temperatures; the Morse potenbeing of short range, could not change that; in other woone would obtain a near-transition at a temperature ctrolled by the Morse potential, but then the long-range attr
e
e
ta
s
06190
se
en
e
dl,
s,n--
tion would prevent dissociation at all temperatures. We hverified this by numerically solving Eq.~A13!. For weakerattractions~value of the coefficient 16 or higher in thesunits!, numerical work suggests that the transition becomhigher than second order; however, numerical accuracy issufficient to determine the detailed behavior. It is possibleguess what happens by substituting the Morse potential bnarrow well, i.e., the total potential in Eq.~A13! being equalto 2D for R0,r ,R011/a and equal to Eq.~A14! for largerr; this case is exactly solvable and shows that althoughshift in the value of the critical point is less than 1%, thnature of the transition is radically transformed: the vaniing of the lowest eigenvalue is now of the KosterlitThouless type
e0}2e2const/(TD2T). ~A15!
o
s on
and
l
s
ean
d
usebeuldhis
s.
o-
i-
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