Poland-Scheraga models and the DNA denaturation transition

Transcription

1 arxiv:cond-mat/ v2 [cond-mat.stat-mech] 17 Apr 2003 Poland-Scheraga models and the DNA denaturation transition C. Richard Institut für Mathematik und Informatik Universität Greifswald, Jahnstr. 15a Greifswald, Germany A. J. Guttmann Department of Mathematics and Statistics University of Melbourne, Victoria 3010, Australia February 29, 2008 Abstract Poland-Scheraga models were introduced to describe the DNA denaturation transition. We give a rigorous discussion of a family of these models. We derive possible scaling functions in the neighbourhood of the phase transition point and review common examples. We introduce exactly solvable directed examples, which fully account for excluded volume effects. In particular, we present a two-dimensional Poland- Scheraga model with a first order phase transition, as observed experimentally. This illuminates recent suggestions as to how the Poland-Scheraga class might be modified in order to display a first order transition. 1 Introduction When a solution of DNA is heated, the double stranded molecules denature into single strands. In this process, looping out of AT rich regions of the DNA segments first occurs, followed eventually by separation of the two strands as the paired segments denature. This denaturation process corresponds to a phase transition [28]. A simple model of the DNA denaturation transition was introduced in 1966 by Poland and Scheraga [25, 26] (hereinafter referred to as PS) and refined by Fisher [11, 10]. The model consists of an alternating sequence (chain) of straight paths and loops, which idealise denaturing DNA, consisting of a sequence of double stranded and single stranded molecules. An attractive energy is associated with paths. Interactions between different parts of a 1

2 chain and, more generally, all details regarding real DNA such as chemical composition, stiffness or torsion, are ignored. It was found that the phase transition is determined by the critical exponent c of the underlying loop class. Due to the tractability of the problem of random loops, that version of the problem was initially studied by PS [26]. The model displays a continuous phase transition in both two and three dimensions. It was argued [11] that replacing random loops by self-avoiding loops, suggested as a more realistic representation accounting for excluded volume effects within each loop, sharpens the transition, but does not change its order. However, the sharp jumps observed in the UV absorption rate in DNA melting experiments 1 [28], which correspond to a sudden breaking of large numbers of base pairs, indicated that a first order phase transition would be the appropriate description. (The question whether such an asymptotic description, which implies very long chains, is valid for relatively short DNA sequences, has been discussed recently [17, 20]). Nevertheless, a directed extension of the PS model, being essentially a one-dimensional Ising model with statistical weighting factors for internal loops, is widely used today and yields good coincidence of simulated melting curves with experimental curves for known DNA sequences [28, 1, 5]. Another recent application of PS models analyses the role of mismatches in DNA denaturation [13]. With the advent of efficient computers, it has more recently been possible to simulate analytically intractable models. One of these is a model of two self-avoiding and mutually avoiding walks, with an attractive interaction between different walks at corresponding positions in each walk [9, 8, 4, 2]. The model exhibits a first order phase transition in d = 2 and d = 3. The critical properties of the model are described by an exponent related to the loop length distribution [22, 21, 19, 8, 4, 2]. This exponent is called c again and can, for PS models, be shown to coincide with the loop class exponent, see Section 2.4 and Fisher s review article [10]. Within a refined model, where different binding energies for base pairs and stiffness are taken into account, the exponent c seems to be largely independent of the specific DNA sequence and of the stiffness of paired walk segments corresponding to double stranded DNA parts [8]. There are, however, no simulations of melting curves for known DNA sequences which are compared to experimental curves for this model. An approximate analytic derivation of the exponent related to this new model was given by Kafri et al. [22, 21, 19] using the theory of polymer networks. They estimated the excluded volume effect arising from the interaction between a single loop and two attached walks. This approach (refined recently [4]) yields an approximation to the loop length distribution exponent c, which agrees well with simulation results of interacting self-avoiding walk pairs [8, 4, 2]. There is a recent debate about the relevance of this approximation w.r.t. real DNA [17, 20]. The polymer network approach, as initiated previously [22], led to a number of related applications [21, 19, 4, 3, 16]. 1 In recent years, a number of other properties of DNA molecules have been studied by refined techniques such as optical tweezers and atomic force microscopy, and theoretical descriptions of underlying effects such as unzipping [21, 19, 4] have been proposed. These will not be discussed here. 2

3 In this article, we reconsider PS models for three principal reasons. Firstly, the older articles are short in motivating the use of particular loop classes, which may be misleading in drawing conclusions about the thermodynamic effects of different loop classes. In fact, the loop classes discussed in the early approaches [26, 11] are classes of rooted loops. We will argue that the resulting models lead to chains which are not self-avoiding. This seems unsatisfactory from a biological point of view, since real DNA is self-avoiding. Secondly, the common view holds that self-avoiding PS models cannot display a first order transition in two or three dimensions. This is incorrect, as we demonstrate by introducing a selfavoiding PS model with self-avoiding loops and a two-dimensional directed, exactly solvable example. All specific PS models to be introduced here are self-avoiding and hence by definition take into account excluded volume effects between all segments. Thirdly, the two exponents c, extracted from different expressions as described above, are used in the literature without distinction, although there are subtle differences, which we will point out. This paper is organised as follows. In the next section, we give a rigorous discussion of PS models and their phase transitions. We derive the scaling functions which describe the behaviour below the critical temperature, as the critical point is approached. We then derive the loop statistics, thereby analysing the occurrence of the loop class exponent in the loop length distribution. The third section reviews the prevailing PS models, with emphasis on motivation for the underlying loop classes. We introduce a directed PS model with self-avoiding loops which shows a first order phase transition in d = 2 and d = 3. Then, two exactly solvable directed PS models are introduced. One of them is derived from fully directed walks and is solvable in arbitrary dimension. The second, two-dimensional model is derived from partially directed walks, and one of its variants displays a first order phase transition. We finally summarise and interpret our results, concluding with a discussion of some open questions. 2 Poland-Scheraga models: general formalism We define PS models and consider analytical properties of their free energies, thereby extending previous expositions [25, 26, 11, 10] in a rigorous manner. This leads to a classification of possible phase transitions. We extract the scaling functions which describe the asymptotic behaviour of chains at temperatures near the phase transition. Finally, we consider the loop statistics of chains. 2.1 Definition of the model Within the model as described by PS [26], the DNA structure is replaced by a chain structure as shown in Fig. 1, see below for a definition on a lattice. It consists of an alternating sequence of double stranded segments, which are separated by single stranded segments. We assume that the leftmost and rightmost segment are double stranded, and that different segments do not interact. (The case of an open end affects the behaviour 3

4 Figure 1: Schematic representation of the chain structure above the critical temperature [20] and will not be considered here [4]). In the case of double stranded segments, an attractive energy proportional to their length is added. We consider a discrete model, defined on the hypercubic lattice Z d, where double stranded segments are paths, and single stranded segments are loops on the edges of the lattice. Each loop is assumed to have two marked vertices to indicate where paths are attached. (Like PS, we do not a priori assume the DNA condition that the marked vertices divide the loop into parts of equal lengths. There is a recent discussion about the effect of mismatches [13].) Any alternating sequence of paths and loops is called a chain. A chain is called self-avoiding if there are no overlaps, i.e., every two non-neighbouring segments have no vertex in common, and every two neighbouring segments have exactly the marked vertex in common. A PS model consists of all chains obtained by concatenation of paths and loops from a given path class and a given loop class, where the initial segment and the final segment of a chain are both paths. Note that, in general, such chains are not self-avoiding, in contrast to real DNA. We call a PS model self-avoiding if all chains of the model are self-avoiding. The requirement of self-avoidance restricts the admissible path classes and loop classes. A simple subclass of self-avoiding PS models are directed PS models: We call a PS model directed if there is a preferred direction such that the order of the chain segments induces the same order on the vertex coordinates (w.r.t. the preferred direction), for each pair of vertices taken from two different chain segments. For example, a directed model may be obtained if the double stranded segments are bridges, which are paths whose starting and ending points have, respectively, minimal and maximal x-coordinates [24]. These result in a directed model if paths are attached to loop vertices with extremal x-coordinate. We will discuss below several examples of directed PS models, which arise from classes of directed walks and loops derived therefrom. Let z m,n denote the number of such chain configurations with m contacts and length n. The generating function is defined by Z(x, w) = m,n z m,n w m x n, (2.1) where the activity x is conjugate to the chain length n. The Boltzmann factor w = e E/kT takes into account the attractive interaction (achieved by setting the energy E < 0) between 4

5 bonds. T is the temperature, and k is Boltzmann s constant. Over the relevant temperature range 0 < T <, we have > w > 1. The model is completely determined by properties of the paths and of the loops. As we show below, the generating function Z(x, w) can be expressed in terms of the generating functions for paths and loops. In the case of a chain consisting solely of paths, i.e., m = n in Eqn. (2.1), its generating function is given by V (wx) = b n (wx) n, (2.2) n=0 where b n is the number of paths of length n. The path generating function V (x) generalises that of the original PS model [25, 26], where b n = b, independent of n. We denote the loop generating function by U(x) = p 2n x n, (2.3) n=1 where p 2n is the number of loops of length 2n. The generating function of the model is a geometric series in V (wx)u(x), i.e., it is given by Z(x, w) = V (wx) 1 U(x)V (wx) = Z n (w)x n. (2.4) Since we want to analyse phase transitions of the model, which can only occur in the infinite system, we define the free energy of the model as n=1 1 f(w) = lim n n log Z n(w) = log x c (w) (2.5) where, for fixed w, x c (w) denotes the radius of convergence of Z(x, w). The existence of the free energy follows from the supermultiplicative inequality m m 1 =1 z m1,n 1 z m m1,n 2 z m,n1 +n 2, (2.6) which expresses the fact that two chains may be concatenated to obtain another admissible chain, but that not all chains can be obtained in this way. This can be used to show that the free energy exists [18]. Moreover, the free energy is a convex function of log w. If it is finite, its left- and right derivatives with respect to w exist everywhere, and the derivatives exist at almost every point where f(w) is finite [18]. We investigate properties of the free energy in terms of properties of the loop generating function U(x) and the path generating function V (x). This has been done before for specific choices of U(x) and V (x) by use of graphical arguments [25, 26, 21, 10]. However here we give a general derivation, starting with assumptions on U(x) and V (x). (The reader may find it illuminating in following this very general discussion that now follows to refer to a specific model, discussed in Section 4.1) 5

6 Assume that U(x) and V (x) are generating functions with non-negative coefficients and with radius of convergence 0 < x U < x V 1. At the critical point x = x V, assume that V (x V ) := lim x x V (x) =. At the origin x = 0, assume that U(0) = V V (0) = 0. We will relax the assumption V (0) = 0 later on. Since U(x) is a generating function, this implies that U(x) > 0 and that the derivative satisfies U (x) > 0 for 0 < x < x U. A corresponding statement holds for V (x). The assumption that 0 < x U < 1 reflects the requirement that the number of configurations p 2n grows exponentially with length n. The models commonly discussed [25, 26, 11, 21] have x V = 1. For loops and paths of the same type, we have x U = x 2 V < x V < 1, see also the examples discussed below. The assumption V (x V ) = is satisfied for typical classes of paths, (see also below). Note that typically U(x U ) := lim x x U(x) <, but there are loop classes where U(x U U ) = such as convex polygons [6] in two dimensions. Let us now fix 0 < x < x U and determine the radius of convergence w c (x) of Z(x, w). The function x c (w), which determines the free energy Eqn. (2.5), is then given by the inverse function of w c (x), whose existence we show below. The curve w c (x) is given by either the location of the singularities of U(x), V (wx), or by the smallest value of x where the denominator 1 U(x)V (wx) vanishes. A curve w 1 (x) of singularities deriving from V (wx) is given by w 1 (x) = x V /x. The dominant curve of singularities w 2 (x) < w 1 (x) derives however from the vanishing denominator of Z(x, w), U(x)V (w 2 (x) x) = 1. (2.7) The above equation has a unique solution w 2 (x) > 0, since V (x V ) = and V (0) = 0, and V (x) is a strictly increasing function, hence invertible. We have w 2 (x) x < x V = w 1 (x) x, so that w 2 (x) < w 1 (x). Consider now the associated inverse functions x 1 (w) and x 2 (w). We have x 1 (w) = x V /w, and the inverse of w 2 (x) exists, since w 2 = 1 ( ) U V x U V + w 2 < 0. (2.8) This inequality is valid for 0 < x < x U, the functions V, V are evaluated at argument xw 2 (x), and U, U are evaluated at argument x, as is readily inferred from Eqn. (2.7). Note that, as x 0, w 2 (x). Define w c := lim x x w 2 (x). If U(x U U ) =, we have w c = 0. This follows from the criticality condition Eqn. (2.11), which gives V (w c x U ) = 0. Since V (0) = 0 and V (x) is a monotone increasing function of x for x > 0, we conclude w c = 0. Otherwise, w c is implicitly given by U(x U )V (w c x U ) = 1. (2.9) Thus x 1 (w) > x 2 (w) for w c < w <. Eqn. (2.7) has no solution with w 2 (x) < w c. Thus, for w < w c, the singularity of smallest modulus of Z(x, w) is given by x c (w) = min{x U, x V /w}. Noting that w c x U < x V, we obtain x c (w) = x U. To summarise: 6

7 Under the above assumptions, Z(x, w) has free energy f(w) = { log xc (w) (w c < w < ) log x U (0 < w w c ), (2.10) where x c (w) is the unique positive solution of U(x c (w))v (w x c (w)) = 1. (2.11) The critical point w c 0 is given by w c = 0 if U(x U ) =, and otherwise implicitly given by U(x U )V (w c x U ) = 1. Note that we may relax the assumption V (0) = 0. If V (0) = a, the above reasoning can be applied as long as U(x) 1/a for x < x U. (If U(x) > 1/a, the free energy is determined by the singularity of V (x)). We will discuss examples with a = 1 in Section Phase transitions Let us focus on the critical point w c. For w > w c, there is no phase transition, since the critical point x c (w) > 0 is analytic for w c < w <, as can be inferred from Eqn. (2.11). Hence, the free energy f(w) is analytic for w c < w <. Recall that U(x U ) = implies w c = 0. Since the relevant parameter range is w > 1, we conclude that in this case there is no phase transition. A necessary condition for a phase transition within the relevant temperature range is that w c > 1, which is equivalent to U(x U )V (x U) < 1. The nature of the transition at w = w c is exhibited by the fraction of shared bonds θ(w), for w w c defined by θ(w) = df(w) { dw = d log x dw c(w) (w c < w < ) (2.12) 0 (1 < w < w c ). We compare the left- and right derivatives of the free energy f(w) at w = w c. A short calculation shows that for w > w c θ(w) = x c (w) ( ) U 1 x c (w) = V U V + w 2 > 0. (2.13) Consider the limit w w c +. We have 0 < V <, since UV = 1 and U is finite. Since V (x) is a generating function with non-negative coefficients, this implies 0 < V <. We distinguish the cases of a finite or an infinite derivative U (x U ). If U =, it follows that θ(w) 0, such that the phase transition is continuous. If U <, θ(w) θ c > 0 as w w c +, and the phase transition is of first order since θ(w) = 0 for w < w c. If U(x U ) = or U(x U )V (x U) 1, no phase transition will occur at a strictly positive temperature. Otherwise, if U(x U ) < and U (x U ) <, a continuous phase transition will occur. If U(x U ) < and U (x U ) =, a first order phase transition will occur. 7

8 Thus, the nature of the transition is determined by the singularity of the loop generating function U(x). It can be more directly related to the asymptotic properties of loops, which are typically of the form 2 p 2n Ax n U n c (n ), (2.14) for some constants A > 0 and c R. The exponent c determines the singularity at x = x U, which is, to leading order and for c Q \ N, algebraic. Assume that, for c R \ N, the singular part of U(x) is, as x x U, asymptotically given by U (sing) (x) U 0 (x U x) c 1 (x x U ) (2.15) for some constant U 0 0. It follows from Eqn. (2.15) that for c < 1 we have U =. For 1 < c < 2 we find U finite and U infinite, whereas for c > 2 we get both U and U finite. We thus have the following result. Under the above assumptions, no phase transition occurs for 0 < c < 1, a continuous phase transition occurs for 1 < c < 2, and a first order transition occurs if c > 2. If c = 1 in (2.14), we get a logarithmic singularity in U(x), hence no phase transition. If c = 2 in (2.14), we get a logarithmic singularity in the derivative U (x), resulting in a continuous phase transition. 2.3 Scaling functions In the vicinity of a phase transition point, critical behaviour of the form Z(x, w) (x c x) θ F((w w c )/(x c x) φ ) (x, w) (x c, w+ c ), (2.16) uniformly in w, is expected with critical exponents θ and φ. Here F(s) is a scaling function which only depends on the combined argument s = (w w c )/(x c x) φ. Let us assume that the scaling function F(s) can be expanded in a Taylor series F(s) = k=0 f ks k. It can be shown [18] that under mild assumptions the scaling function implies a certain asymptotic behaviour of the function Z n (w) for large n. That is to say, ( ( ) φ n Z n (w) xc n θ n θ 1 h (w w c )) (n, w w c + ), (2.17) uniformly in w, where h(x) is the finite size scaling function defined by x c h(x) = f k Γ(kφ + θ) xk, (2.18) k=0 2 A n B n for n means that lim n A n /B n = 1. Similarly, f(x) g(x) for x x c means that lim x xc f(x)/g(x) = 1. 8

9 where Γ(z) denotes the Gamma function. Note that from the finite size scaling function h(x) all asymptotic quantities about the phase transition point can be computed. For PS models whose loop generating function displays an algebraic leading singularity, the critical exponents and the scaling function can be explicitly computed. Let us assume as in (2.15) that U(x) behaves as U(x) = U(x U ) + U 0(x U x) c 1 + o ( (x U x) c 1) (x x U ), (2.19) and 1 < c R \ N. (If c < 1, no phase transition occurs at positive temperature according to the considerations above). In this case, the constant U 0 is negative. In order to extract the scaling function F(s), we analyse the leading singularity in Z(x, w). This is done by expanding the denominator 1 U(x)V (wx) about x = x U and w = w c to leading order in (x U x) and (w w c ). The contribution with smallest exponent in (x U x) determines the exponent θ. We then introduce the scaling variable s = (w w c )/(x U x) φ and obtain an expansion of the above form. This leads to the following result. Assume a leading singularity (2.15) of the loop generating function U(x) with an exponent c > 1. If 1 < c < 2, the critical exponents and the scaling function are θ = φ = c 1, F(s) = For c > 2, the critical exponents and the scaling function are θ = φ = 1, F(s) = 1 U 0 U 2 (x U )V (w c x U )x U s. (2.20) 1 U 2 (x U )V (w c x U )[w c x U s]. (2.21) Note that, for c > 2, the critical exponents are independent of c. In both cases, the scaling function is proportional to F a (s) = 1/(1 as) with a positive constant a. This simple structure leads to finite size scaling functions of the form h a (x) = k=0 1 Γ((k + 1)θ) (ax)k. (2.22) If θ = 1, we get h a (x) = e ax. For 0 < θ = c 1 < 1 and θ rational, implying a leading singularity of algebraic type, the finite size scaling function can be expressed in terms of hypergeometric functions. For the case c = 3/2 = θ + 1, which will be relevant below, we get h a (x) = 1 π + axe (ax)2 (1 + erf(ax)), (2.23) where erf(x) = 2/ π x dt denotes the error function. t=0 e t2 For an application of these results, we derive the critical behaviour of the fraction of shared bonds above the phase transition. We have, for w w c +, ) 1 θ(w) = w lim n n Z n (w) Z n (w) = w lim n 9 ( ( ( ) θ h n 1 n n x c ) θ x c (w wc ) ). (2.24) h ( ( n x c ) θ (w wc )

10 If w > w c, the existence of the limit (for the right derivative) implies a power-law behaviour h (x)/h(x) Cx α for x, since powers of n must cancel. Collecting exponents of n, Eqn. (2.24) yields 1 + θ + αθ = 0. For 1 < c < 2, this leads to θ(w) = A(w w c ) 2 c c 1 (w w c + ), (2.25) with some constant A. This implies that for 3/2 < c < 2, θ(w) approaches zero with an infinite slope, whereas for 1 < c < 3/2, θ(w) approaches zero smoothly with a gradient which vanishes for w w + c [11]. 2.4 Loop statistics We consider the loop length distribution, i.e., the probability of loops of length 2l within chains of length n. This distribution is used for the analysis of DNA models extending the PS class [22, 21, 19, 8, 4, 3]. We show that, for PS models, the loop class exponent c also appears in the loop length distribution P(l, n). For models extending the PS class, this justifies the use of the same name c for the loop length distribution exponent. Within a given PS model, consider the set of all chains of length n with m contacts. Denote the number of loops of length 2l in this set by g m,n,l. Define the generating function L(x, w, y) = m,n,l g m,n,l w m x n y l. (2.26) Again, due to the directedness of the model, this generating function can be expressed in terms of the loop generating function U = U(x), Ū = U(xy) and the path generating function V = V (xw). An argument as in Section 2.1 yields L(x, w, y) = V ŪV + (V ŪV UV + V UV ŪV ) + = V ŪV (1 + 2UV + 3(UV )2 + ) = V ŪV 1 (1 UV ) 2 = U(yx)Z2 (x, w). (2.27) The generating function for the total number of loops is then given by setting y = 1, T(x, w) = L(x, w, 1) = U(x)Z 2 (x, w), (2.28) and the generating function for the sum of all loop lengths is given by S(x, w) = y d dy L(x, w, y) y=1 = xu (x)z 2 (x, w). (2.29) The finite size behaviour of these quantities about the phase transition can be computed from the scaling function results of Section 2.3. For the total number of loops, we get lim [x n ]T(x, w) A x n 2θ w w c + U n 2θ 1 (n ), (2.30) 10

11 where A > 0 is some amplitude 3, and for the number of loops of length 2l, we get lim [x n y l ]L(x, w, y) p 2l x l n 2θ w w c + U n 2θ 1 (n ) A l c x n 2θ U n 2θ 1 (1 l n, n ). (2.31) Note that we obtained (2.31) under the assumption that l is asymptotically large but that l n. This is due to the estimate [y l ]U(yx) = p 2l x l p 2l x l U for x x U and l finite, but p 2l Ax l U l c for l, where we assumed (2.14) to be valid. For the sum of the loop lengths, we distinguish between the cases 1 < c < 2 and c > 2, where the derivative of the loop generating function U (x U ) is finite or infinite at the critical point. We find lim [x n ]S(x, w) w w c + { A x n c U n c 1 (1 < c < 2) A x n 2 U n (c > 2) (n ). (2.32) Let us compute the probability of a loop of length 2l within chains of length n. We get, in the limit w w + c, for the loop length distribution P(l, n) = lim w w + c [x n y l ]L(x, w, y) [x n ]T(x, w) Al c (1 l n, n ). (2.33) Furthermore, for the mean loop length l n we get, in the limit w w + c, l n = n l=0 l P(l, n) = lim w w + c { [x n ]S(x, w) A n [x n ]T(x, w) 2 c (1 < c < 2) A n 0 (c > 2) (n ). (2.34) The behaviour of the mean loop length reflects the nature of the phase transition: If 1 < c < 2, the mean loop length diverges, such that the transition is continuous. If c > 2, the mean loop length stays finite, indicating a first order transition. We note that we obtained (2.33) only for 1 l n, whereas this behaviour is commonly assumed to hold for l n arbitrarily large [22, 21, 19, 3]. If c > 2, such an assumption is however not compatible with (2.34), as follows from performing the sum in (2.34) using (2.33). 3 Two prominent loop classes Since the critical behaviour of PS models is determined by the properties of loops alone, PS [25, 26], and later Fisher [11], were led to consider various loop classes (together with straight paths for the double stranded segments). Whereas PS analysed loop classes derived from random walks, Fisher considered loop classes derived from self-avoiding walks. The latter model and its implications for the order of the phase transition is most important in 3 In order to simplify notation, we will denote all following amplitudes by the letter A, with the convention that their values may be different. 11

12 the recent discussion about extending the PS class [9, 22, 21, 19, 8, 3, 4, 2, 17, 20]. On the other hand, as we will show below, there are serious problems with the interpretation of the above mentioned loop classes w.r.t. real DNA, since they lead to chains which are not self-avoiding. We will first discuss how walk classes can be used to define corresponding loop classes. We will then review the loop classes used by PS and by Fisher and derive the nature of the phase transition. This is followed by a an interpretation of their approach, which also introduces a directed PS model with self-avoiding loops. 3.1 Loops and walks For a given class of walks on Z d, a corresponding loop class is defined as follows [24]. A loop of length n is a walk of length n 1, whose starting point and end point are nearest neighbours. If we denote the number of n-step walks starting at 0 and terminating at x by b n (0, x), and the number of translationally inequivalent n-step loops by p n, we have 2n p n = 2d b n 1 (0, e), where e is one of the nearest neighbours of 0. This is due to the fact that a given loop has 2n possible realisations by walks (n possible starting positions and two orientations). Within a chain structure, two paths are attached to each loop. In general, there might be more than one possible way to do so. This freedom may be accounted for by marking the positions of possible path attachments on the loop, thereby increasing the number of loop configurations to p n p n. We emphasise that the loop classes used by both PS and Fisher are rooted loops p n = np n. This means that each path can connect to such loops in n ways, resulting in chains which are not self-avoiding, as the chains will usually intersect the loops. These chains cannot therefore represent real (self-avoiding) DNA. 3.2 Rooted random loops The first simple example of loops, which are discussed by PS [26], are rooted random loops derived from random walks. They can be analysed in arbitrary dimension d. Consider the hypercubic lattice Z d. The number of random walks of length n is (2d) n. The asymptotic behaviour of the number of rooted loops of lengths 2n is given [24] by p 2n (2d) 2n (2n) d/2 (n ). (3.1) PS conclude that the exponent c = d/2 implies a continuous phase transition in dimensions d = 2 and d = 3, in contrast to a first order transition as suggested by experiment. 3.3 Rooted self-avoiding loops The PS results led to the question [11] whether including excluded volume effects within a loop increases the loop class exponent c, which might change the order of the phase transition. This led to considering self-avoiding loops. A self-avoiding walk on Z d is a walk which never visits a vertex twice. Self-avoiding loops are related to self-avoiding walks by 12

13 the above description. By definition, the self-avoiding loop class fully accounts for excluded volume interactions within a loop. There are a number of rigorous results known about self-avoiding walks and loops. Walks and associated loops have the same exponential growth constant µ d < 2d, where the latter number is the growth constant for the corresponding random objects. It has been shown [24] that for d > 4 the critical exponents of self-avoiding walks coincide with the values for random walks. In particular, for the number of configurations b n Aµ n d nγ 1, we have for d > 4 γ = 1, and the mean square displacement of walks of length n grows as n 2ν, where ν = 1/2. For d = 4, renormalisation group techniques predict logarithmic corrections to the random walk behaviour. For d = 2, conformal field theory arguments [24] and independent recent results from the theory of stochastic processes [23] predict γ = 43/32 and ν = 3/4. There is, however, no rigorous proof that the above exponents exist in low dimension. The situation is worse in d = 3, where only numerical estimates for the exponents are known. These are [18] γ = (6) and ν = (6), where the number in brackets denotes the uncertainty in the last digit. The behaviour of rooted self-avoiding loops, p 2n Bµ n d n c, is related to the walk exponent ν by the hyperscaling relation [24] c = dν, which results in d/2 d 4 c = (18) d = 3 (3.2) 3/2 d = 2 Fisher concludes that the above values of the loop class exponent c imply a continuous phase transition in d = 2 and d = 3, in contrast to a first order transition as suggested by experiment. 3.4 Interpretation We comment on the motivation for considering classes of rooted loops instead of ordinary loops. To this end, ignore the question whether attachments of paths to a loop yield admissible chains, but assume the DNA condition that the two paths attached to a loop dissect it into pieces of equal length. Then the number of possible attachments to a loop of length n is given by n, thus leading to rooted loop classes p n = np n. However, as explained above, the assumption of rooted loop classes leads to chains which are not self-avoiding. Indeed, in a random loop or in a self-avoiding loop, not all vertices are accessible to paths, so that the actual number of possible attachments must be less than n. One possible way to construct a self-avoiding PS model would be to attach bridges [24] to the extremal vertices of loops. As these extremal vertices will not be unique, we can impose uniqueness by specifying the bottom-most of the left-most vertices, and the top-most of the right-most vertices as the attachment points. We lose the DNA condition however. This would imply p n = p n and hence increase the previous exponents by one. Hence, such a directed PS model with self-avoiding bridges and self-avoiding loops displays a first order transition with c = 5/2 in d = 2 and with c in d = 3. 13

14 Due to its directedness, this model overestimates the excluded volume effects between different parts of the chain structure, in comparison to the more realistic model of selfavoiding walk pairs with DNA condition [9, 2, 4, 8]. This model has a loop length distribution exponent c = 2.44(6) in d = 2 [4] and c = 2.14(4) in d = 3 [2]. One might thus be tempted to argue that relaxing excluded volume effects decreases the value of c, but does not change the order of the transition. We stress, however, that one of these models obeys the DNA condition, whereas the other does not. Therefore, an immediate comparison is misleading. Also, given a particular class of paths and loops within a directed model, there will in general be several accessible loop sites at which paths can be attached to loops. Accounting for this effect increases the number of loop configurations, which might decrease the exponent c (by not more than one). Imposing the DNA condition in addition will decrease the number of loop configurations. If this restriction does not change the loop class entropy (which we believe to be true), then this effect might increase the loop class exponent c. It seems plausible that the net effect of the two contributions will result in an increase of c, and then the conclusion proposed in the previous paragraph would be valid. It would thus be interesting to analyse in detail the critical behaviour of such loops, which incorporate the two opposite effects mentioned in this paragraph. 4 Directed exactly solvable Poland-Scheraga models We present two directed PS models which are exactly solvable and, by definition, take into account excluded volume interactions between all parts of the structure. Whereas the first model of fully directed walks and loops is solvable in arbitrary dimension, a solution for the second one is described only in two dimensions. The first model displays a first order phase transition only for d > 5, whereas the second model displays a first order phase transition in d = Fully directed walks and loops This directed PS model consists of fully directed walks for the paths in the chain. These only take steps in positive directions. The corresponding loops are staircase polygons, which consist of two fully directed walks, with common starting points and end points which do not intersect or touch. Note that this model satisfies all the conditions discussed in Section 2, and also satisfies the DNA condition that the two segments of the loops are the same length. In d = 2, explicit formulae can be obtained [18]. The generating function of fully directed walks is given by V (wx) = 2 n (wx) n = n= wx = 1 + 2wx + 4w2 x (4.1) 14

15 We have x V = 1/2. The generating function for the loops is (see, for example, [14]) U(x) = 1 ( ) 1 2x 1 4x = x 2 + 2x 3 + 5x (4.2) 2 This leads to the generating function Z(x, w) = V (wx) 1 U(x)V (wx) = x(1 2w) + 1 4x. (4.3) We have x U = 1/4 and w c = 3/2. Moreover, we have V (0) = 1 and U(x U ) = 1/4 < 1. The exponent is c = 3/2. The free energy f(w) is given by { ( log 2w + 1 f(w) = (w 1) 1) ( 3 < w < ) 2 2 log 4 (0 < w 3). (4.4) 2 The fraction of shared bonds follows as { w (w 1)(w θ(w) = 1 )(w 3) 2 (3 < w < ) (0 < w < 3), (4.5) 2 which approaches zero linearly in w w c. The asymptotic behaviour of Z n (w) about w = w c is given by Z n (w) 4n+1/2 n 1/2 h n(w w c ) (n, w w c + ), (4.6) uniformly in w, where h a (x) is given by Eqn. (2.23). It can be shown that this leads to Eqn. (4.5) in the limit w w c +. The PS model of fully directed walks and loops is exactly solvable in arbitrary dimension, as follows from previous results [14]. Consider fully directed walks on Z d. If there are k i steps in direction i, the number of distinct walks, starting from the origin, is given by the multinomial coefficient ( k 1 +k k d ) k 1,k 2,...,k d. The generating function of a pair of such walks is given by ( ) 2 k1 + k k d Z d (x) = x k k d. (4.7) k 1, k 2,...,k d k 1,k 2,...,k d =0 The generating function Z d (x) can be interpreted as a chain, see Fig. 2, where each link consists of a staircase polygon or a double bond. Thus, it is related to generating function for staircase polygons U d (x) by Z d (x) = 1 1 (dx + 2U d (x)). (4.8) It can be shown by applying standard techniques of series analysis [15] that the functions Z d (x) satisfy Fuchsian differential equations of order d 1, from which the singular behaviour of Z d (x), and hence of U d (x) can be derived. In dimensions 2 < d < 5, U d (x) can be expressed in terms of Heun functions. We get in d = 3 Z 2 3 (x) = ( 2 π) 2 (1 9x) 1 (1 x) 1 K(k + )K(k ), (4.9) 15

16 Figure 2: A chain of fully directed walks and staircase polygons where K(k) is the complete elliptic integral of the first kind, and k± 2 = 1 2 ± x 1 4 (4 x 1) (2 x 1)(1 x 1 ) x x 1 = 9x 2 10x + 1 (4.10) We have x U = 1/9. In arbitrary dimension d > 1, it has been shown that the models have an exponent c = (d 1)/2 for d > 2, with logarithmic corrections in odd dimensions. In d = 2, we have c = 3/2. Noting that U d (x U ) < 1/2 < 1 = 1/V d(0), we thus conclude for the corresponding PS model that the phase transition is first order for d > 5, and that we have a continuous phase transition in dimensions 1 < d 5. The value c = (d 1)/2 for fully directed loops might appear unexpected in comparison to the value c = (d + 2)/2 for (unrooted) random loops. One might argue that restricting the number of possible loops should increase the exponent c, since the number of configurations decreases. This argument is not applicable here because both loop classes are fundamentally different, namely directed and non-directed. These classes have a different loop entropy, which makes an immediate comparison of exponents impossible. 4.2 Partially directed walks and loops We consider a directed PS model, where paths are partially directed walks on the square lattice Z 2. These walks are self-avoiding and not allowed to take steps in the negative x direction. The number b n of paths of length n can readily be obtained by considering the ways a walk of length n can be obtained from a walk of length n 1 [18]. This yields the path generating function V (x) = b n x n = n=0 1 + x 1 2x x2, (4.11) whose dominant singularity is a simple pole at x V = 1 2. The associated loop class is column-convex polygons, which has been analysed in [7, 6]. A column-convex polygon is shown in Fig. 3. We assume that partially directed walks are attached to the extremal 16

17 Figure 3: A column-convex polygon vertices of a polygon as explained in Section 3.4. The loop generating function U(x) (of unrooted loops) satisfies the algebraic equation of order four 0 = (x 1) 4 x 2 + (x 3 7x 2 + 3x 1)(x 1) 3 U(x) + 2(2x 3 11x x 4)(x 1) 2 U(x) 2 + (5x 3 35x x 21)(x 1) U(x) 3 + (2x 3 23x x 18) U(x) 4 (4.12) and has the series expansion U(x) = x 2 + 2x 3 + 7x x x x The dominant singularity at x U = x 2 V = is a square-root singularity with an exponent c = 3/2. Noting that U(x U ) < 1 = 1/V (0), this value implies a continuous phase transition for the PS model. However, the loop class of column-convex polygons does not obey the restriction that the upper and lower walks of a loop are of the same length (that is, the DNA condition). The question arises whether the introduction of this constraint increases the exponent c. Using standard techniques [7, 6], it is possible to derive a functional equation for the generating function of column-convex polygons counted by perimeter and, in addition, by their upper and lower walk length. The corresponding expression is, however, much more difficult to handle than Eqn. (4.12), and there seems to be no practical way to extract the coefficient c for this model. On the other hand, an additional restriction makes an analysis feasible. Let us further assume that the upper and lower walk are not allowed to touch the x-axis. This restriction will result in an exponent c = 5/2, and the associated PS model will thus display a first order phase transition. The loop class can be described in terms of bar-graph polygons [27]. These are column-convex polygons, whose lower walk is a straight line. Alternatively, loops may be thought of as consisting of partially directed walks on the half-plane y 0, which start and end on the x-axis. Loops are thus two bar-graph polygons, one of them in the half-plane y 0, the other in the half-plane y 0. Both polygons have the same starting points and end points and are constrained to have equal walk lengths, see Fig. 4. Paths of the model are partially directed walks, which are attached to the two loop vertices 17

18 with vertical coordinate zero Figure 4: A loop derived from two bar-graph polygons (see text) Let g m,n denote the number of bar-graph polygons with 2m horizontal and 2n vertical steps. The anisotropic perimeter generating function G(x, y) = g m,n x m y n satisfies an algebraic equation of degree two [27], G(x, y) = xy + (y + x(1 + y))g(x, y) + xg 2 (x, y). (4.13) A generalisation of the Lagrange inversion formula [12] can be used to obtain a closed formula for its series coefficients. It is n ( )( )( ) [x m y n 1 m + k m m ]G(x, y) = g m,n =. (4.14) m + k m n 1 k n k k=0 The number p 2n of all combinations of bar-graph polygons subject to equal walk lengths is then p 2n = (n 1)/2 k=1 g 2 n 2k,k. (4.15) The first few numbers p 2n are 1, 1, 2, 10, 38, 126, 483, 126, 483,... for n = 3, 4, 5,.... We used standard methods of numerical series analysis [15] to estimate the critical point and critical exponent of U(x). An analysis with first order differential approximants, using the coefficients p 2n for 70 n 80, yields the estimates x U = (1), c = (1) (4.16) This is, to numerical accuracy, indistinguishable from x U = = , as expected, and c = 5/2. For the unsolved model, corresponding to column-convex polygons chained together with partially directed walks so that the DNA condition is obeyed, we expect a value of 3/2 c 5/2, with a value of c = 2 seeming plausible. This PS model would then be at the border between a first order and a continuous phase transition. 18

19 5 Conclusion We discussed PS models with emphasis on the order of their phase transitions. We introduced several exactly solvable, directed examples, which fully account for excluded volume effects. One of these models displays a first order phase transition in d = 2. We also re-analysed the old model of rooted self-avoiding loops [11] and found that the resulting PS model is not self-avoiding. Hence the conclusions about the order of the phase transition, which are most relevant to the recent discussions [9, 22, 21, 19, 8, 3, 4, 2, 17, 20], rely on a model where most chain configurations are not self-avoiding and thus cannot represent real DNA. The directed PS model with self-avoiding bridges and (unrooted) self-avoiding loops as suggested in Section 3.4 yields a first order phase transition in both d = 2 and d = 3. It assumes that paths are attached to the extremal vertices of a loop. On the other hand, this violates the DNA condition, and there may be more than one possible way to attach paths to a loop, which are still consistent with a directed PS model. Presumably, accounting for these effects will not change the order of the phase transition, but a detailed (numerical) analysis is needed to answer this question, as in the discussion in Section 3.4. With respect to the more realistic model of pairs of interacting self-avoiding walks [9, 22, 21, 19, 8, 4, 2], our results suggest an interpretation of excluded volume effects, which complements the common one [22, 21, 19]. The self-avoiding PS model of Section 3.4 correctly accounts for excluded volume effects within a loop, but overestimates excluded volume effects between different segments of a chain, due to its directed nature. Whereas this leads to a first order phase transition in d = 2 and d = 3, the question arises whether the relaxation of excluded volume effects between different segments of the chain changes the nature of the transition. In contrast, the PS model with rooted self-avoiding loops [11] displays a continuous phase transition in d = 2 and d = 3, which leads to the opposite question whether introducing excluded volume effects between different segments of the chain can change the nature of the transition [9, 22, 21, 19, 8, 4, 2]. The above interpretation motivated simulations of self-avoiding walk pairs [9, 8, 4, 2] and perturbative analytic arguments using the theory of polymer networks [22, 21, 19], which revealed a first order phase transition in d = 2 and d = 3. The perturbative treatment yields a very good approximation to the value of the loop length distribution exponent found in simulations. This indicates that the main effect responsible for the change of critical behaviour w.r.t. the PS model of rooted self-avoiding loops is due to local excluded-volume effects arising from forbidden attachments of paths to loops. The number of such attachments is generously overestimated in the above model in low dimension, such that the majority of chains is not self-avoiding. For our directed PS model, the simulation result suggests that the relaxation of excluded volume effects between different segments of the chain does not change the nature of the phase transition. We emphasise that the self-avoiding walk pair models mentioned above are not PS models as defined in this paper, and that loop class exponents as defined in (2.14) do not make sense for these models. The appropriate generalisation is the exponent c of the loop length distribution P(l, n) of the chain, which is assumed to behave in the limit w w + c 19

20 like [22, 21, 19, 8, 4, 3] P(l, n) l c g(l/n) (1 l n, n ), (5.1) where g(x) is a scaling function, assumed to be constant in some analyses [22, 21, 19, 3]. Accepting this behaviour with a scaling function g(x) integrable over [0, 1], the same conclusions as in Section 2.2 about the nature of the phase transition determined by the loop length distribution exponent c hold. This follows from the behaviour of the mean loop length l n in the limit w w c +, which is obtained from (5.1) by integration, l n = n l P(l, n) A n 2 c (n ). (5.2) l=0 If 1 < c < 2, the mean loop length diverges, indicating a continuous transition. If c > 2, the mean loop length stays finite, indicating a first order transition [8, 3]. For PS models, the behaviour (2.33) is consistent with (5.1), which justifies the use of the same name for the two exponents. We found (5.2) to be satisfied for PS models if 1 < c < 2 in Section 2.4. If c > 2, however, the mean loop length exponent is independent of c for PS models see (2.34). This implies that a scaling form like (5.1), with a scaling function g(x) integrable over [0, 1], cannot hold for PS models with loop class exponent c > 2. The key question, as to which effects are responsible for the observed behaviour in real DNA, is only partially answered by these results. With respect to directed models, which also account for different bases pair sequences, it seems surprising that simulations of melting curves [28, 1] agree so well with experimental curves. In these simulations, a heuristic critical exponent accounting for the statistical weight of internal loops has to be inserted. It has recently been suggested [5] that a value more realistic than the commonly used Fisher loop class exponent [11] c might be the loop length distribution exponent of the polymer network approximation [22] c This suggestion implicitly assumes that the loop length distribution exponent of a non-directed model can be interpreted as an effective loop class exponent within a directed model. Although appearing plausible, it seems difficult to give this assumption a quantitative meaning. On the other hand, the simulation approach [1] is robust against a change of parameters. Indeed, melting curves can be simulated with satisfactory coincidence for both exponents, if the cooperativity parameter for the loops is adjusted accordingly [5]. It would certainly be illuminating to obtain such simulated melting curves from a self-avoiding walk pair modelling [9, 8, 4, 2]. We also mention the recent debate about the relevance of the self-avoiding walk pair model w.r.t. real DNA [17, 20]. The stiffness of the double stranded segments does not seem to qualitatively alter the critical properties of chains [8], as can be seen by comparing the PS model result of Section 2.1 showing that critical properties are largely independent of the path generating function. The corresponding question of introducing energy costs for bending and torsion to single stranded segments has, to our knowledge, not been discussed. Even within the PS class, it is not clear how this affects the loop class exponents, since the changes just discussed might 20

21 lead to loop classes with different exponential growth constants, but there is no argument as to how the exponents might change. In conclusion, the question of the mechanisms applying in real DNA which are responsible for the denaturation process and which explain multistep behaviour as observed in melting curves, are still far from being satisfactorily answered in our opinion. Acknowledgements We would like to acknowledge useful discussions with Stu Whittington. CR would like to acknowledge financial support by the German Research Council (DFG) and like to thank the Erwin Schrödinger International Institute for Mathematical Physics for support during a stay in December 2002, where part of this work was done. AJG would like to acknowledge financial support from the Australian Research Council, and helpful discussions with Marianne Frommer. We acknowledge clarifying comments on an earlier version of the manuscript by Marco Baiesi, Enrico Carlon, Andreas Hanke and Ralf Metzler. References [1] R. D. Blake et al., Statistical mechanical simulation of polymeric DNA melting with MELTSIM, Bioinformatics 15 (1999), [2] M. Baiesi, E. Carlon, Y. Kafri, D. Mukamel, E. Orlandini and A. L. Stella, Inter-strand distance distribution of DNA near melting, Phys. Rev. E 67 (2002), ; cond-mat/ [3] M. Baiesi, E. Carlon, E. Orlandini and A. L. Stella, A simple model of DNA denaturation and mutually avoiding walk statistics, Eur. Phys. J. B 29 (2002), ; cond-mat/ [4] M. Baiesi, E. Carlon and A. L. Stella, Scaling in DNA unzipping models: denaturated loops and end-segments as branches of a block copolymer network, Phys. Rev. E 66 (2002), ; cond-mat/ [5] R. Blossey and C. Carlon, Reparametrizing loop entropy weights: effect on DNA melting curves, preprint (2002); cond-mat/ [6] M. Bousquet-Mélou, A method for the enumeration of various classes of column-convex polygons, Discr. Math. 154 (1996), [7] R. Brak, A. L. Owczarek and T. Prellberg, Exact scaling behaviour of partially convex vesicles, J. Stat. Phys. 76 (1994), [8] E. Carlon, E. Orlandini and A. L. Stella, The roles of stiffness and excluded volume in DNA denaturation, Phys. Rev. Lett. 88 (2002), ; cond-mat/