On scaling laws at the phase transition of systems with divergent order parameter and/or internal length : the example of DNA denaturation

Transcription

1 On saling laws at the phase transition of systems with divergent order parameter and/or internal length : the example of DNA denaturation Sahin BUYUKDAGLI and Mar JOYEUX (a) Laboratoire de Spetrométrie Physique (CNRS UMR 5588), Université Joseph Fourier, BP 87, 3840 St Martin d'hères, Frane PACS numbers : Gg, Fr, h, Aa Abstrat : We used the Transfer-Integral method to ompute, with an unertainty smaller than 5%, the six fundamental harateristi exponents of two dynamial models for DNA thermal denaturation and investigate the validity of the saling laws. Doubts onerning this point arise beause the investigated systems (i) have a divergent internal length, (ii) are desribed by a divergent order parameter, (iii) are of dimension 1. We found that the assumption that the free energy an be desribed by a single homogeneous funtion is robust, despite the divergene of the order parameter, so that Rushbrooe s and Widom s identities are valid relations. Josephson s identity is instead not satisfied. This is probably due to the divergene of the internal length, whih invalidates the assumption that the orrelation length is solely responsible for singular ontributions to thermodynami quantities. Fisher s identity is even wronger. We showed that this is due to the d = 1 dimensionality and obtained an alternative law, whih is well satisfied at DNA thermal denaturation. (a) Mar.JOYEUX@ujf-grenoble.fr 1

2 I - Introdution It has long been reognized that there are mared similarities between the phase transitions of very different systems : antiferromagnets, liquids, superondutors and ferroeletris, to quote some of them, indeed all display a rather simple behavior in the region lose to the ritial point. A partial explanation omes from Landau's theory [1] and equivalent ones, lie Van der Waals' equation for liquids, Weiss' moleular field theory for ferromagnets, Ornstein-Zernie equations, random phase approximations [], and Ginzburg- Landau's equations for superondutors [3]. By supposing that the transition an be desribed by a so-alled order parameter [] and that the free energy an be expanded in power series in this parameter and the temperature gap T T (where T is the ritial temperature), these theories predit that most quantities (lie the speifi heat, the order parameter, the isothermal suseptibility, the orrelation length and the orrelation funtions) display power laws in the neighborhood of the phase transition. Experiments done on many systems onfirm the power laws predited by Landau, but show that real ritial exponents differ maredly from those predited by the theory [4]. These experiments furthermore suggest that the various ritial exponents are not independent but obey instead ertain onstraints. Phenomenologial senarii, whih explain these observations, were proposed by Widom [5,6], Fisher [7-9], Kadanoff [4,10] and Domb and Hunter [11]. Based on the assumption that the free energy and/or the orrelation length are homogeneous funtions, these theories lead to the onlusion that all ritial exponents an be expressed in terms of only two of them, thans to so-alled saling laws. Later, a method nown as the Renormalization Group theory, whih is based on Wilson's idea that the ritial point an be mapped onto a fixed point of a suitably hosen transformation of the system's Hamiltonian [1,13], has provided a oneptual framewor for understanding saling.

3 Yet, as far as we now, all the systems for whih the validity of the saling laws has been heed have two properties in ommon : (i) their phase transition is desribable by a finite order parameter, and (ii) these systems do not dissoiate at the ritial temperature. The fat that the order parameter remains finite is essential for most theories, whih assume that the free energy an be expanded in power series with respet to the order parameter and the temperature gap T T. Obviously, this assumption no longer holds when the order parameter diverges at the ritial point. Another entral assumption of saling theories is that the orrelation length is solely responsible for singular ontributions to extensive thermodynami quantities. While this is ertainly a reasonable assumption for bound systems, this might not be the ase for dissoiating ones. Indeed, a system that dissoiates at the ritial temperature possesses at least one physial internal length whih inreases infinitely at the ritial point and might therefore ontribute signifiantly to extensive thermodynami quantities. Whether the saling laws are valid or not for systems with divergent order parameter and/or internal length is therefore an open question. The purpose of this paper is to address this question through the alulation of the harateristi exponents of two realisti dynamial models for DNA denaturation. This phase transition, whih taes plae when DNA solutions are heated, orresponds to the separation of the two DNA strands, that is, to the dissoiation of the entangled polymers. Moreover, if the external stress depends expliitly on the distane between paired bases, then the orresponding order parameter diverges at the ritial point. DNA denaturation models are therefore partiularly well suited to investigate the appliability of the saling laws to suh unusual systems. The remainder of this paper is organized as follows. The two dynamial models for DNA denaturation are briefly desribed in setion II. The tehnique we used to ompute the harateristi exponents is the Transfer-Integral (TI) method. The details of the alulations 3

4 are sethed in setion III. Finally, the appliability of the saling laws to systems with divergent order parameter and/or internal length is disussed in setion IV on the basis of the ritial exponents that were obtained for the two models. II - The two dynamial models for DNA denaturation The potential energy E pot of the two dynamial models for DNA denaturation is of the general form ( y ) E pot = V ( y ) + W ( y, y + 1 ) + h f, (II.1) where y denotes the position of the partile at site, V y ) is the on-site potential, W ( y, y + 1) the nearest-neighbour oupling between two suessive partiles, and plays the role of an externally applied onstraint. The order parameter m is obtained as the first derivative of the free energy with respet to the external field, that is, here E pot f ( y)exp dy E pot BT m = B T ln exp dy = = f ( y) h. (II.) BT E pot exp dy BT In this wor, we used f ( y ) = y and ( y ) y ( h f ( ) y f =, whih lead to order parameters m = y and m = y, respetively. The first model for DNA denaturation was proposed by Dauxois, Peyrard and Bishop (DPB) [14-17]. Expressions for the on-site potential and nearest-neighbor oupling are V ( y W ( y ) = D, y ( 1 exp[ a y ]) K ) = ( y y ) 1+ ρexp[ α( y y )] ( ), (II.3) 4

5 where y represents the transverse strething of the hydrogen bond onneting the th pair of bases. Numerial values of the oeffiients are taen from Ref. [16], that is, D=0.03 ev, a=4.5 Å -1, K=0.06 ev Å -, α=0.35 Å -1 and ρ=1. Thans to the non-linear staing interation ( ρ > 0), this model displays a muh sharper transition at denaturation and is thus in better agreement with experiment than the older models on whih it is based [18-0]. The seond model for DNA denaturation was proposed by ourselves (JB) [1,] to tae into aount the fat that staing interations are neessarily finite. For homogeneous sequenes, it is of the form V ( y W ( y ) = D, y ( 1 exp[ a y ]) ΔH ) = 1 exp[ b( y y ) ] + 1 ( ) + K ( y y ), + 1 b + 1 (II.4) where D=0.04 ev, a=4.45 Å -1, Δ H =0.44 ev, b=0.10 Å - and K b =10-5 ev Å -. The first term of W ( y, y + 1) desribes the finite staing interation, while the seond one models the stiffness of the sugar/phosphate babone. Most interestingly, we were able, by introduing in this model the site-speifi staing enthalpies Δ H dedued from thermodynami alulations [3], to reprodue the multi-step denaturation proess that is experimentally observed for inhomogeneous DNA sequenes. III - Tranfer-integral (TI) alulations The transfer-integral (TI) method (see for example Ref. [4] for a general desription and Ref. [5] for a disussion regarding the appliability of the method to systems with unbound on-site potentials) onsists in finding the eigenvalues λ and eigenvetors of the φ symmetri TI operator, whih satisfy 5

6 V ( x) + V ( y) + W ( x, y) + h f ( x) + h f ( y) φ ( x)exp dx = λ φ ( y). (III.1) BT For this purpose, we used the proedure desribed in Appendix B of Ref. [4], whih is based on the diagonalization of a symmetri matrix with elements M ij = δ 1/ i δ 1/ j V ( ui ) + V ( u j ) + W ( ui, u j ) + h f ( ui ) + h exp BT f ( u j ), (III.) where the u i define a grid of non-neessarily equally-spaed values of the position oordinate and the δ stand for the intervals δ = ( u u 1 ) /. The eigenvalues of the symmetri i i i+ 1 i TI operator oinide with the eigenvalues of the λ { M ij } matrix, while the eigenvetors φ of the symmetri TI operator are onneted to the normalized eigenvetors { V }, of the { M } 1/ matrix through the relation ( i ) i i the form λ = exp[ ε ( B T )] / eigenvalue (e.g. λ0, ε 0, φ u = δ V,. It is onvenient to rewrite the eigenvalues in i, and to label with a zero the quantities related to the largest φ and { V 0,i }) and with a 1 those related to the seond largest 0 eigenvalue (e.g. λ1, ε 1, ). In the thermodynami limit of an infinite number of sites, the ij singular part of the speifi heat V, the longitudinal orrelation length ξ, the average g(y) of any funtion g ( y), and the stati struture fator S( q, T ) are obtained aording to ε0 V = T T l ξ = BT ε1 ε0 ( y) = g( y) φ0 ( y) dy = g( ui ) g V0, i i λ0 λ S( q, T ) = R os( ), 0 λ0 + λ λ0λ q l (III.3) 6

7 where l denotes a harateristi length of the system (we assumed without loss of generality that l=1 Å), and R stands for the integral f ( y) φ y) φ0 ( y) dy = f ( ui ) R = ( V, iv0, i (III.4) i Note, that the derivative in the expression for, as well as the derivative dm / dh (see setion IV), were omputed from finite differenes rather than from the omplex expressions in Appendix B of Ref. [4]. V The harateristi exponents were estimated by drawing log-log plots of the various quantities in Eq. (III.3) and measuring the slopes in the regions where power laws are satisfied. For obvious physial reasons, these regions do not extend far from the ritial point. Unfortunately, numerial onsiderations also forbid the observation of these regions too lose from the ritial point. Indeed, an infinite range of y values would be needed to numerially onverge the quantities in Eq. (III.3) at the ritial point. Sine the dimension of the { M ij } matrix is neessarily finite, numerial results an be aurate only up to a ertain distane from it. Consequently, large grids of points extending to large values of y are mandatory for the interval on whih power laws are observed to be broad enough to allow a preise estimation of the harateristi exponents. This point is absolutely ruial. For example, some of the harateristi exponents for the DPB model have already been reported [16]. However, the authors note that several quantities diverge smoothly at the transition, beause of transients whih mas the leading-order asymptotis. As a onsequene, they only provide rough estimates for the exponents, whih sometimes differ by a fator from exat values. In the light of our alulations, it appears that the so-alled transients atually result from the numerial limitation mentioned above. In order to ahieve better preision, we used grids of 400 ui values regularly spaed between y = 00 / a and y = 4000 / a or, alternately, grids of the same length but with spaings whih inrease exponentially from δi = 0. / a at y 0 7

8 to δ = 4 a at y = 5067 / a (both grids lead essentially to the same result). We estimate on i / the basis of all our trials, that we were able to ompute the exponents (see Eq. (IV.1) below) with an unertainty smaller than 5 %. IV - Results and disussion A Charateristi exponents The six fundamental harateristi exponents α, β, γ, δ, η and ν (we omit the prime symbols although ) are traditionally defined aording to T < T V ~ ( T m ~ ( T dm dh m ~ h ~ ( T ξ ~ ( T S( q, T 1/ δ T ) T ) ) ~ T ) β T ) q ν α γ η. (IV.1) α, β, γ and ν are omputed at zero field (h=0), while δ and η are omputed at ritial temperature T. From the numerial point of view, T was obtained as the temperature where the longitudinal orrelation length ξ is maximum (at h = 0). With the exponentially spaed grid of length 400, we alulated T = K for the DPB model and = K for the JB model. As indiated in Set. III, the harateristi exponents were estimated by drawing log-log plots of the quantities in Eq. (IV.1) and measuring the slopes in the regions where power laws hold. For the sae of illustration, some plots for α, β, γ and ν are shown in T Figs. 1 and. Fig. 1 deals with the DPB model with external onstraint f ( y ) = y, while 8

9 Fig. deals with the JB model with the same external onstraint. Measurement of the last two exponents δ and η were performed on similar plots, but with field (h) or wave-vetor (q) absissa. Note that we used two different external onstraints for eah model, namely ( y ) y ( ) f = and f y = y, whih orrespond to order parameters m = y and m = y, respetively. Exponents β, γ and δ depend on the hoie of the external onstraint, while α, ν and η do not. The two sets of harateristis exponents that were obtained for eah model are summarized in Table I. At that point, two omments are in order. First, the harateristi exponent for speifi heat, α, is signifiantly larger than 1 for both the DPB and the JB models. This onfirms that both models predit a first-order phase transition at DNA denaturation temperature [16,1,6]. Moreover, the signs in Eq. (IV.1) were hosen suh that that exponents are usually positive (although α and η are sometimes slightly negative). For the DPB and JB models, the order parameter m however diverges at the ritial point, so that β and δ are strongly negative. B Rushbrooe s and Widom s identities The first two saling laws, nown as Rushbrooe s and Widom s identities, an be written in the form α + β + γ = γ β( δ 1) = 0, (IV.) respetively. To obtain these relations, one just needs to assume that the singular part of the free energy, f, an be desribed by a single homogenous funtion in T T and h, that is, sing α ( ) = ( ) h fsing T, h T T G. (IV.3) ( ) Δ T T 9

10 Eq. (IV.), as well as the additional relation Δ = βδ, then arise naturally from the interonnetions between f,, m and dm / dh via thermodynami derivatives. Eq. sing (IV.3) is atually a generalization of what is observed within the saddle-node approximation of the Ginzburg-Landau model, whih leads to V ( 3 / ) ( T, h) ( T T ) G h ( T T ) fsing / =. The models investigated in this paper differ maredly from the Ginzburg-Landau one, but we heed that the homogeneity assumption of Eq. (IV.3) is nevertheless well satisfied. This is α illustrated in Fig. 3, whih shows the plots of ( ) ( ) βδ f sing / T T versus the logarithm of h / T T for the JB model with external onstraint f ( y ) = y and three values of h sing ranging from 10 D to D. Note that in the TI formalism, f is obtained from ( T h) = ε ( T, h) ε ( T = T, h 0) f. (IV.4) sing, 0 0 = The fat that the points orresponding to different values of h all lie on the same line indiates that the homogeneity assumption is orretly satisfied. It therefore omes as no surprise that Rushbrooe s and Widom s identities are also satisfied by the measured exponents. This is learly seen in Table, whih displays, for eah polynome α + β + γ and γ β( δ 1), the value predited by the orresponding saling law (olumn ) and those obtained from the measured values of the harateristi exponents (olumns 3-6). Table also provides qualitative unertainties obtained by assuming that all exponents have additive 5% errors. It is seen that in all ases the values predited by Rushbrooe s and Widom s identities lie well inside the unertainty range. C Josephson s identity Josephson inequality [7,8] states that 10

11 α + νd, (IV.5) where d is the dimensionality of the system (here d = 1). This inequality onverts to the equality nown as Josephson s identity α + νd =, (IV.6) if the generalized homogeneity assumption holds, that is, if (i) the only important length near the ritial point is the orrelation length ξ, and (ii) ξ is solely responsible for all singular ontributions to thermodynami quantities. Note that if the generalized homogeneity assumption is satisfied, then the homogeneity assumption of Eq. (IV.3) is also satisfied, so that Rushbrooe s and Widom s identities are true. Quite interestingly, examination of Table shows that the omputed exponents satisfy the inequality of Eq. (IV.5) but not Josephson s identity. Indeed, the differene between the omputed values of α + νd and that predited by the saling law (i.e. ) is larger than three times the 5% unertainty for both models. This indiates that, in ontrast with many systems, the generalized homogeneity assumption does not hold for DNA denaturation. As we antiipated in the Introdution, this is not unexpeted for systems whih dissoiate at the ritial point. Indeed, these systems possess at least one physial internal length whih inreases infinitely at the ritial temperature, so that it is no longer justified to assume that everything is a funtion only of the ratio of a typial finite mirosopi length to the orrelation length ξ. We unsuessfully tried to figure out, on the basis of the numerial values reported in Tables 1, what quantity ould replae the orrelation length ξ in the generalized homogeneity assumption (this quantity should obviously have length dimension and a harateristi exponent equal to α ). D Fisher s identity 11

12 Fisher s identity onnets γ, η and ν aording to γ ν( η) = 0 (IV.7) Examination of Table shows that this equality is very far from being satisfied by the models for DNA denaturation. The reason for these disrepanies is that Fisher s identity is based on the assumption that the orrelation funtion ( y ) f ( y ) f ( ) G( x) = f + y (IV.8) j j x falls off, lose to the ritial temperature, as 1 G ( x) ~. (IV.9) d +η x While orret for the Ginzburg-Landau Hamiltonian with d, this assumption is just wrong for a system with d = 1 and η 0 beause, for these values of d and η, Eq. (IV.9) diverges with inreasing values of x. In the TI formalism, G(x) may be obtained from [4] f ( ) ( ) λ y j f y j + x = R λ0 x. (IV.10) Numerially, we found that Eq. (IV.10) atually leads to onstant values of G(x) for the two investigated models lose to the ritial temperature. Evaluating these onstants at x = 0, one gets ( y) f ( y) ~ f ( ) G( x) = f y. (IV.11) Writing, as usual, that dm dh ~ ξ G( x) dx 0, (IV.1) one thus obtains, instead of Fisher identity, the relation γ = μ + ν, (IV.13) where μ is the harateristi exponent for f ( y) 1

13 μ ( y) ~ ( T T ) f. (IV.14) The measured values of μ are reported in Table 1 (note that the exponent μ for f ( y ) = y is the opposite of β for f ( y ) = y ). The validity of the saling rule in Eq. (IV.13) is heed in Table. The agreement is exellent. V - Conlusion We investigated the validity of the saling rules for two dynamial models of DNA thermal denaturation. These models indeed display several harateristis, whih shed doubts on this question : (i) the distane between paired bases, that is, the physial length in terms of whih the Hamiltonian is expressed, diverges at the melting temperature, (ii) the expressions we assumed for the external onstraint lead to order parameters, whih also diverge at the ritial temperature, (iii) the dimensionality is d = 1. Conlusions are : - the assumption that the free energy an be desribed by a single homogeneous funtion seems to be rather robust, despite the divergene of the order parameter. Consequently, Rushbrooe s and Widom s identities are valid relations. - Josephson s identity is instead not satisfied. We argued that this is probably due to the divergent internal length, whih invalidates the assumption that the orrelation length is solely responsible for singular ontributions to thermodynami quantities. - Fisher s identity is still farther from being satisfied. We showed that this is due to the d = 1 dimensionality and obtained an alternative law, whih is well satisfied at DNA thermal denaturation. Of ourse, one annot derive general onlusions from a single study, and additional wor is ertainly needed to asertain the robustness of the homogeneity assumption for free 13

14 energy and/or improve Josephson s identity. This wor still indiates that saling laws must be handle with are when dealing with systems with unusual harateristis. 14

15 REFERENCES [1] L.D. Landau and E.M. Lifshitz, Statistial Physis (Pergamon Press, London, 1958) [] R. Brout, Phase Transitions (W.A. Benjamin, New-Yor, 1965) [3] E.A. Lynton, Superondutivity (Wiley, New-Yor, 1964) [4] L.P. Kadanoff, W. Götze, D. Hamble,, R. Heht, E.A.S. Lewis, V.V. Paliausas, M. Rayl and J. Swift, Rev. Mod. Phys. 39, 395 (1967) [5] B. Widom, J. Chem. Phys. 43, 389 (1965) [6] B. Widom, J. Chem. Phys. 43, 3898 (1965) [7] J.W. Essam and M.E. Fisher, J. Chem. Phys. 39, 84 (1963) [8] M.E. Fisher, Rep. Prog. Phys. 30, 615 (1967) [9] M.E. Fisher, J. Appl. Phys. 38, 981 (1967) [10] L.P. Kadanoff, Physis, 63 (1966) [11] C. Domb and D.L. Hunter, Pro. Phys. So. (London) 86, 1147 (1965) [1] K.G. Wilson, Rev. Mod. Physis 55, 583 (1983) [13] A. Lesne, Renormalization Methods : Critial Phenomena, Chaos, Fratal Struture (Wiley, New-Yor, 1998) [14] T. Dauxois, M. Peyrard and A.R. Bishop, Phys. Rev. E 47, R44 (1993) [15] T. Dauxois and M. Peyrard, Phys. Rev. E 51, 407 (1995) [16] N. Theodoraopoulos, T. Dauxois and M. Peyrard, Phys. Rev. Lett. 85, 6 (000) [17] M. Peyrard, Nonlinearity 17, R1 (004) [18] M. Tehera, L.L. Daemen and E.W. Prohofsy, Phys. Rev. A 40, 6636 (1989) [19] M. Peyrard and A.R. Bishop, Phys. Rev. Lett. 6, 755 (1989) [0] T. Dauxois, M. Peyrard and A.R. Bishop, Phys. Rev. E 47, 684 (1993) [1] M. Joyeux and S. Buyudagli, Phys. Rev. E 7, (005) 15

16 [] S. Buyudagli, M. Sanrey and M. Joyeux, Chem. Phys. Lett. 419, 434 (006) [3] R.D. Blae, J.W. Bizzaro, J.D. Blae, G.R. Day, S.G. Delourt, J. Knowles, K.A. Marx and J. SantaLuia, Bioinformatis 15, 370 (1999) [4] T. Shneider and E. Stoll, Phys. Rev. B, 5317 (1980) [5] Y.-L. Zhang, W.-M. Zheng, J.-X. Liu and Y.Z. Chen, Phys. Rev. E 56, 7100 (1997) [6] D. Cule and T. Hwa, Phys. Rev. Lett. 79, 375 (1997) [7] B.D. Josephson, Pro. Phys. So. 9, 69 (1967) [8] A.D. Soal, J. Stat. Phys. 5, 5 (1981) 16

17 TABLE CAPTIONS Table 1 : Values of the six fundamental harateristi exponents α, β, γ, δ, η and ν for the DPB and JB models with external onstraints f ( y ) = y and ( y ) y f =. The seventh exponent μ haraterizes the behavior of f ( y) lose to the ritial temperature (see Set. IV-D). Table : Values of α + β + γ, γ β( δ 1), α + νd, γ ν( η) and γ ( μ + ν) predited by saling laws (olumn ) and obtained from the measured harateristi exponents reported in Table 1 for the DPB and JB models with external onstraints f ( y ) = y and f ( y ) = y (olumns 3-6). The unertainties orrespond to additive 5% errors for all the exponents. The last saling law, γ ( μ + ν) = 0, is introdued in Set. IV-D. 17

18 FIGURE CAPTIONS Figure 1 (olor online) : Log-log plots used to determine the ritial exponents α, β, γ and ν for the DPB model with external onstraint f ( y ) = y. Figure (olor online) : Log-log plots used to determine the ritial exponents α, β, γ and ν for the JB model with external onstraint f ( y ) = y. α Figure 3 (olor online) : Plots of ( ) f sing / T T versus the logarithm of h / ( T T ) βδ for the JB model with external onstraint f ( y ) = y and three values of h ranging from 4 D 10 6 sing 10 to D. f and h are expressed in units of D. The fat that the points orresponding to different values of h all lie on the same line indiates that the homogeneity assumption of Eq. (IV.3) is orretly satisfied by the model. 18

19 TABLE 1 DPB model JB model f ( y) = y f ( y) = y f ( y) = y f ( y) = y α β γ δ η ν μ

20 TABLE saling law DPB model JB model f ( y) = y f ( y) = y f ( y) = y f ( y) = y Rushbrooe : α + β + γ.17±0.3.01± ± ±0.51 Widom : γ β( δ 1) ± ± ± ±0.63 Josephson : α + νd.57± ± ±0.1.36±0.1 Fisher : γ ν( η) ± ± ±0.9.38±0.36 γ ( μ + ν) 0 0.0± ± ± ±0.48 0

21 FIGURE 1 1

22 FIGURE

23 FIGURE 3 3