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An intermediate phase in DNA melting

We predict a novel temperature-driven phase transition of DNA below the melting transition. The additional, intermediate phase exists for repetitive sequences, when the two strands have different lengths. In this phase, the excess bases of the longer strand are completely absorbed as bulge loops inside the helical region. When the temperature is lowered, the excess bases desorb into overhanging ends, resulting in a contour length change. This continuous transition is in many aspects analogous to Bose Einstein condensation. Weak sequence disorder renders the transition discontinuous.

The base-pairing interaction between the two strands of DNA is not only pivotal to its biological function [1], but also leads to intriguing applications in nanotechnology [2]. One approach to probe this interaction is to monitor the DNA conformation as a function of temperature. Experimentally, one can observe the number of basepairs formed (using UV absorption [3,4]), as well as changes of intra-molecular distances on the nanometer scale (using modern single-molecule techniques [5]). On the theoretical side, the temperature dependence of DNA conformations has been studied for almost fifty years, using models of various degrees of complexity [6,7,8,9,10,11,12,13]. Particular attention has been paid to the characteristics of the melting transition, where the two strands separate completely. Whereas early models yielded only a crossover [12], the Poland-Scheraga (PS) model [7] was the first to display a phase transition, albeit a continuous one, which appeared to be at variance with the experimentally observed sharp jump in the fraction of bound basepairs [4]. Only recently have mechanisms been proposed [10,11] which yield an abrupt, first order transition. So far, however, most analyses of DNA melting have incorporated only native interactions, i.e. base pairs that occur in the ground state of the molecule (see [6,13] for notable exceptions). It is our aim here to show that such non-native interactions can introduce an intermediate phase in the melting behavior of DNA, associated with an additional conformational transition before strand separation.
Non-native interactions are particularly relevant for repetitive DNA sequences, which are common in genomes [14]. Periodic DNA, with e.g. a single base repeat such as TTT... or a higher order repeat such as CAGCAG..., can take on basepairing patterns with asymmetric loops and the two complementary strands can be shifted relative to each other. Here, we consider the general situation where the two strands can have arbitrary lengths N , M . We describe the DNA using a generalized PS model [13] and calculate its equilibrium behavior analytically. We find that for N = M , the bound phase splits into two separate phases. The low temperature phase is characterized by an extensive length of the unbound end on the longer strand, whereas in the new intermediate phase these overhanging bases are absorbed into the helical region. Mathematically, and also conceptually, many aspects of this transition are analogous to Bose-Einstein condensation (BEC), as "particles" (bases) condense into a single "state" (the overhanging end), which thereby acquires macroscopic "occupation" (length). Obviously, the analogy extends only to the behavior of the partition function, as there is no quantum coherence in the DNA problem. Effectively, the transition amounts to a temperature sensitive change in the contour length of the DNA molecule, which should be observable with optical or single molecule methods. While the transition is continuous for perfectly periodic sequences, we find that it becomes a first order transition once (weak) sequence disorder is introduced. We also show that the non-native interactions can change the order of the melting transition, as has been conjectured previously [15].
DNA model.-We consider two DNA strands with lengths N and M ≥ N , respectively, and describe their interaction with a generalized PS model [13,16]. Specifically, a base i ≤ N of the lower strand can form a base pair (i, j) with every complementary base j ≤ M of the upper strand, whereas the formation of base pairs within a strand can be neglected (since we are interested only in sequences with a high degree of complementarity and a low degree of self-complementarity). Due to geometrical constraints, we may neglect the 'crossing' of base pairs, e.g. two base pairs (i 1 , j 1 ) and (i 2 , j 2 ) with i 1 < i 2 but j 1 > j 2 . The basepairing pattern S, i.e. the set of all formed base pairs, then creates a DNA conformation consisting of bound segments alternating with (possibly asymmetric) loops, see Fig. 1. To simplify the discussion, we enforce the base pair (N, M ) at the right end, so that we need to consider only one overhanging end. Experimentally, this boundary condition would be realized e.g. by a few particularly strong basepairs at one end.
To each basepairing pattern S, we assign a statistical weight Q(S), which takes the form of a product with factors of four different types: (i) a Boltzmann factor q = e ε b /kB T for every basepair with binding energy −ε b < 0, (ii) a Boltzmann factor g 2 = e −ε ℓ /kB T for every loop with loop initiation cost ε ℓ > 0, (iii) an entropic factor B ℓ (m) = s m m −c for each loop, which is the increase in the number of polymer configurations when m bases form a (floppy) loop instead of being in a (rigid) double helical conformation, (iv) and a similar entropic factor A(n) = s n n −c for a single-stranded end of n bases. Here, the exponents c,c in the entropic factors are universal in that they are independent of the detailed polymer properties, but are sensitive to excluded volume interactions. For interacting self-avoiding loops one has c ≈ 2.15, whilec ≈ 0.1 [15]. Whereas the value of c determines the critical behavior at the melting transition [10], the non-universal constant s has no qualitative effect on the melting behavior (we use s = 10 in all numerical examples). In the following, we apply the DNA model to perfectly periodic sequences, where each repeat unit can be treated as an effective base with renormalized parameters (we use ε b = 6 and ε ℓ = 3 in temperature units, k B = 1). We emphasize that our simplistic model for the involved energies and entropies is meant to illustrate the physical phenomena in a transparent way, but leads to an unrealistic temperature scale. With a more detailed description [3], we find that all of the interesting behavior happens at accessible temperatures [17].
Free energy of periodic DNA.-To obtain the equilibrium properties of the DNA model, we calculate the partition sum over all basepairing patterns, Z M N = S Q(S). By separating the single stranded ends from the double stranded part, see Fig. 1, we write Z M N as Here, W t r is the partition function of two complementary and periodic strands of length r and t with the first and last base pair formed. W t r obeys the recursion relation with the initial conditions W 1 1 = q and W i 1 = W 1 i = 0 for i > 1 [13,16]. Eqs. (1) and (2) can be used to calculate Z M N for finite lengths N , M . To extract the thermodynamic behavior in the limit of long strands, we take the The contour integration required for the inverse transformation ofẐ(x, y0) is given by the sum of the integral along the branchcut [s −1 , ∞] ⊂ R and the integral encircling the singularity ofŴ (x, y0) at x = x * (y0). This isolated singularity only exists below the melting temperature.
Compared to the related case of a single self-complementary RNA strand folding back onto itself [18,19], we need two instead of one transformation variables here, due to the second strand of DNA. One obtainŝ where the transforms of the entropic factors are given bŷ A(z) = φc(sz) + 1 andB(z) = φ c (sz), with the polylogarithm φ c (z) = ∞ n=1 z n n −c . The z-transformation carried out above amounts to a change from the canonical to the grand canonical ensemble. The transformation variables x, y play the role of fugacities for bases in the lower and upper strand, respectively. However, for the ensuing discussion, it is advantageous to keep the length N of the shorter strand fixed as a reference. Hence, we perform the inverse transformation for the lower strand by contour integration in x, see Fig. 2, to obtain the partition sum Z N (y 0 ) for N bases on the lower strand and the upper strand coupled to a "nucleotide reservoir" with fixed fugacity y 0 . Whenever both strands are bound,Ẑ(x, y) has a singularity at x * (y 0 ) < s −1 , see Fig. 2. For large N , the contour integration is dominated by the residue at x * (y 0 ), leading to Z N (y 0 ) =Â(y 0 )x * (y 0 ) −N . Hence, the free energy of the bound phase is given by N f b (y 0 ) − T lnÂ(y 0 ), where the first term is the contribution of the helical region with a free energy per length f b (y 0 ) = T ln x * (y 0 ), and the second term is the contribution from the unbound end of the longer strand. The free energy for given N and M is then obtained by saddle point integration, where the fugacity y 0 is determined by Phase diagram.-To extract the physical behavior of the DNA model from Eqs. (4) and (5), we focus on two observables, the total number of base pairs, N θ, and the length of the single-stranded overhang. The fraction θ of bound base pairs is calculated from the free energy per length of the helical region as To obtain the overhang length, we note that the right hand side of (5) decomposes the total length M of the upper strand into two contributions, where the first term is the expected overhang length and the second term corresponds to the number of bases in the helical region. The dashed line in Fig. 3 (top) shows the overhang length as a function of temperature, for N = 1000 and M = 1150. At low temperatures, the two DNA strands are completely aligned, so that all M − N excess bases of the longer strand form an overhanging end. However, we observe that the overhang length decreases with increasing temperature, dropping almost to zero before it rises again sharply at even higher temperature. We see in Fig. 3 (bottom) that this drop occurs in a temperature range where almost all possible base pairs are formed, and the rise occurs when the two strands separate. These observations suggest that a temperature-driven conformational transition occurs before the melting transition. This transition is in fact completely analogous to BEC, as Eq. (5) parallels the behavior of the equation of state for an ideal Bose gas: If we divide Eq. (5) by our system size N and introduce the "particle density" α = M/N , we obtain whereᾱ ≥ 1 is the density inside the helical region. In Eq. (7), the first term on the right hand side corresponds to the occupation of the ground state of an ideal Bose gas, whereasᾱ(y 0 ) is analogous to the occupation of the excited states. The fugacities of a Bose gas and our DNA are bounded: for the former, by the energy of the ground state, and for the DNA by the weight of an unbound monomer, i.e. y 0 ≤ s −1 . The population of the excited states increases monotonically with the fugacity, and attains a finite maximal value, in our caseᾱ max =ᾱ(s −1 ) (provided the loop exponent c > 2). If the density α exceeds this maximal value the length of the unbound end has to diverge in order to accommodate the remaining bases. In other words, the length of the unbound end becomes extensive. In an analogous way, the ground state of a Bose gas is macroscopically populated at low temperatures. In this "condensate" phase, the fugacity is locked to the value s −1 in the thermodynamic limit (N, M → ∞, α = const.). The deviation for finite systems scales as s −1 − y 0 ∼ 1/N , see Fig. 4 (left). In the opposite case, where α <ᾱ max , there is a solution to Eq. (7) with y 0 < s −1 and the unbound end remains finite for all system sizes.
It is easily shown thatᾱ max approaches 1 at low temperatures, and consequently all excess bases of the longer strand are condensed in the overhang, as illustrated in Fig. 4 (right). As T increases, more and more bases are absorbed in the helical region (ᾱ max increases), and the system enters the intermediate phase at T = T c , wherē α max = α. At T c the condensate fraction vanishes, as the solid line shows in Fig. 3 (top). If T is raised to the melting temperature T m , which is independent of α, the strands separate and θ vanishes (denatured phase). Note that the intermediate phase exists only when α is not too large.
It has been previously predicted [15] that the loop ex- ponent c is effectively reduced by one for periodic sequences compared to the standard PS-model with native base pairs only. This prediction is explicitly confirmed by our exact calculation of the free energy. We find [17], that there is no melting transition if c ≤ 2, that the transition is continuous if 2 < c ≤ 3 and of first order if c > 3. For 2 < c ≤ 3, we obtain θ ∼ |T − T m | 3−c c−2 , using the same method as [20] for the standard PS-model. To illustrate this, we plot θ for periodic sequences and for the standard PS-model in Fig. 3 (bottom). Whereas for the latter θ drops discontinuously to zero for c = 2.15, θ of periodic DNA vanishes with zero slope. Only after increasing c artificially to 3.15 does periodic DNA exhibits a similar first order transition.
Weak sequence disorder.-Is the intermediate phase identified above robust against sequence disorder? To address this question, we replace a small fraction of base pairs by bases that can pair with each other, but not with other bases in the sequence. Fig. 5 shows the average length of the overhanging end, obtained by Monte Carlo simulation, for evenly spaced mutations with densities 0.005 and 0.01 and mutation strengthε b = 2 . The plot suggests that in the presence of weak sequence disorder the transition described above remains, but is of first order instead of being continuous. The unbound end keeps its ground state length up to certain temperature, and then shortens rapidly. This is readily understood, when comparing the energy barriers for forming bulgeloops with and without mutations. The formation of a bulgeloop on the longer strand of a perfectly periodic molecule requires only the initiation energy ε ℓ . In the presence of mutation, however, shifting both strands breaks mutated basepairs. Hence, to form a bulge loop, all mutations to the left of the loop have to be broken and the energy barrier for loop formation grows with the distance from the end. Due to this extensive energy barrier for loop formation, mutated basepairs stay bound in a finite temperature range. For a sufficiently low density of mutations, there is a temperatureT c , at which the entropy gained by distributing excess bases in loops along the molecule outweighs the energetic costs [21]. BelowT Discussion.-We have identified a BEC-like conformational transition in periodic and nearly periodic DNA, which occurs below the melting transition. This transition leads to a change in the contour length of the DNA molecule, which is roughly proportional to M − N . We also expect an effect on the persistence length of the helical region due to the increased density of bulgeloops. The hallmark of the transition, i.e. the shortening of the unbound end, could be directly observed by resonant energy transfer between fluorescent dyes located at the ends of the two strands. We expect the existence of the intermediate phase to be independent of the details of our model. Furthermore, we have shown that the additional conformations possible for repetitive sequences change the critical behavior at the melting transition.
We are grateful for important comments by E. Frey and H. Wagner. We acknowledge financial support by the Deutsche Forschungsgemeinschaft through the Emmy Noether Program.