Bubble dynamics in DNA

The formation of local denaturation zones (bubbles) in double-stranded DNA is an important example for conformational changes of biological macromolecules. We study the dynamics of bubble formation in terms of a Fokker-Planck equation for the probability density to find a bubble of size n base pairs at time t, on the basis of the free energy in the Poland-Scheraga model. Characteristic bubble closing and opening times can be determined from the corresponding first passage time problem, and are sensitive to the specific parameters entering the model. A multistate unzipping model with constant rates recently applied to DNA breathing dynamics [G. Altan-Bonnet et al , Phys. Rev. Lett. 90, 138101 (2003)] emerges as a limiting case. Submitted to: J. Phys. A: Math. Gen. PACS numbers: 87.15.-v, 82.37.-j, 87.14.Gg Introduction. Under physiological conditions, the equilibrium structure of a DNA molecule is the double-stranded Watson-Crick helix. At the same time, in essentially all physiological processes involving DNA, for docking to the DNA, DNA binding proteins require access to the “inside” of the double-helix, and therefore the unzipping (denaturation) of a specific region of base pairs [1, 2]. Examples include the replication of DNA via DNA helicase and polymerase, and transcription to single-stranded DNA via RNA polymerase. Thus, double-stranded DNA has to open up locally to expose the otherwise satisfied bonds between complementary bases. There are several mechanisms how such unzipping of double-stranded DNA (dsDNA) can be accomplished. Under physiological conditions, local unzipping occurs spontaneously due to fluctuations, the breathing of dsDNA, which opens up bubbles of a few tens of base pairs [3]. These breathing fluctuations may be supported by accessory proteins which bind to transient single-stranded regions, thereby lowering the DNA base pair stability [2]. Single molecule force spectroscopy opens the possibility to induce denaturation regions of controllable size, by pulling the DNA with optical tweezers [4]. In this way, the destabilising activity of the ssDNA-binding T4 gene 32 Letter to the Editor 2 protein has been probed, and a kinetic barrier for the single-strand binders identified [5]. Denaturation bubbles can also be induced by under-winding the DNA double helix [6]. A recent study of the dynamics of these twist-induced bubbles in a random DNA sequence shows that small bubbles (less than several tens of base pairs) are delocalised along the DNA, whereas larger bubbles become localised in AT-rich regions [7]. Finally, upon heating , dsDNA exhibits denaturation bubbles of increasing size and number, and eventually the two strands separate altogether in a process called denaturation transition or melting [8, 9]. Depending on the specific sequence of the DNA molecule and the solvent conditions, the temperature Tm at which one-half of the DNA has denatured typically ranges between 50C and 100C. Due to the thermal lability of typical natural proteins, thermal melting of DNA is less suited for the study of protein-DNA interactions than force-induced melting [4]. On the other hand, the controlled melting of DNA by heating is an important step of the PCR method for amplifying DNA samples [10], with numerous applications in biotechnology [11]. The study of the bubble dynamics in the above processes is of interest in view of a better understanding of the interaction with single-stranded DNA binding proteins. This interaction involves an interplay between different time scales, e.g., the relaxation time of the bubbles and the time needed for the proteins to rearrange sterically in order to bind [12]. Dynamic probes such as single-molecule force spectroscopy [4] and molecular beacon assays [13] may therefore shed light on the underlying biochemistry of such processes. In a recent experiment by Altan-Bonnet et al [14], the dynamics of a single bubble in dsDNA was measured by fluorescence correlation spectroscopy. It was found that in the breathing domain of the DNA construct (a row of 18 AT base pairs sealed by more stable GC base pairs) fluctuation bubbles of size 2 to 10 base pairs are formed below the melting temperature Tm of the AT breathing domain. The relaxation dynamics follows a multistate relaxation kinetics involving a distribution of bubble sizes and successive opening and closing of base pairs. The characteristic relaxation time scales were estimated from the experiment to within the range of 20 to 100μs. Also in reference [14], a simple master equation of stepwise zipping-unzipping with constant rate coefficients was proposed to successfully describe the data for the autocorrelation function, showing that indeed the bubble dynamics is a multistate process. The latter was confirmed in a recent UV light absorption study of the denaturation of DNA oligomers [15]. In this work, we establish a general framework to study the bubble dynamics of dsDNA by means of a Fokker-Planck equation, based on the bubble free energy function. The latter allows one to include microscopic interactions in a straightforward fashion, such that our approach may serve as a testing ground for different models, as we show below. In particular, it turns out that the phenomenological rate equation approach, corresponding to a diffusion with constant drift in the space of bubble size n used by Altan-Bonnet et al [14] to fit their experimental data corresponds to a limiting case of our Fokker-Planck equation. However, the inclusion of additional microscopic interactions in such a rate equation approach is not straightforward [16]. In what follows, Letter to the Editor 3 we first establish the bubble free energy within the Poland-Scheraga (PS) model of DNA denaturation [8, 9], and then derive the Fokker-Planck equation to describe the bubble dynamics both below and at the melting temperature of dsDNA. Bubble free energy. In the PS model, energetic bonds in the double-stranded, helical regions of the DNA compete with the entropy gain from the far more flexible, singlestranded loops [8, 9]. The stability of the double helix originates mainly from stacking interactions between adjacent base pairs, aside from the Watson-Crick hydrogen bonds between bases. In addition, the positioning of bases for pairing out of a loop state gives rise to an entropic contribution. The Gibbs free energy Gij = Hij − TSij for the dissociation of two paired and stacked base pairs i and j has been measured, and is available in terms of the enthalpic and entropic contributions Hij and Sij [17]. In the following we consider a homopolymer for simplicity, as suitable for the AT breathing domain in reference [14]. For an AT-homopolymer (i = j = AT), the Gibbs free energy per base pair in units of kBT yields as γ ≡ βGii/2 = 0.6 at 37C for standard salt conditions (0.0745M-Na). Similarly, for a GC-homopolymer one finds the higher value of γ = 1.46 at 37C. The condition γ = 0 defines the melting temperature Tm [17, 18], thus Tm(AT) = 66.8 C and Tm(GC) = 102.5 C (we assume that Gij is linear in T , cf. reference [19]). Above Tm, γ becomes negative. For given γ = γ(T ), the statistical weight for the dissociation of n base pairs obtains as W (n) = exp(−nγ). (1) Additional contributions arise upon formation of a loop within dsDNA. Firstly, an initial energy barrier has to be overcome, which we denote as γ1 in units of kBT . From fitting melting curves to long DNA, γ1 ≈ 10 was obtained, so that the statistical weight for the initiation of a loop (cooperativity parameter), σ0 = exp(−γ1), is of order 10 [9, 17]. As the energy to extend an existing loop by one base pair is smaller than kBT , DNA melts as large cooperative domains. Below the melting temperature Tm, the bubbles become smaller, and long range interactions beyond nearest neighbours become more important. In this case, the probability of bubble formation is larger, and γ1 ranges between 3 and 5, thus σ0 . 0.05 [7] (cf. section 5 in [9]). According to reference [18], the smallness of σ0 inhibits the recombination of complementary DNA strands with mutations, making recognition more selective. Secondly, once a loop of n base pairs has formed, there is a weight f(n) of mainly entropic origin, to be detailed below. The additional weight of a loop of n open base pairs is thus Ω(n) = σ0 f(n) . (2) For large bubbles one usually assumes the form f(n) = (n+1) [9, 17]. Here, the value of the exponent c = 1.76 corresponds to a self-avoiding, flexible ring [8, 9, 20], which is classically used in denaturation modelling within the PS approach. Recently, the PS model has been considered in view of the order of the denaturation phase transition [21, 22, 23, 24, 25]. Reference [15] finds by finite size scaling analysis of measured melting curves of DNA oligomers that the transition is of second order. In reference [21], the value c = 2.12 was suggested, compare the discussion in references [7, 24, 18]. For smaller Letter to the Editor 4 bubbles (in the range of 1 to a few tens of base pairs), the appropriate form of f(n) is more involved. In particular, f(n) will depend on the finite persistence length of ssDNA (about eight bases), on the specific base sequence, and possibly on interactions between dissolved but close-by base pairs (cf. section 2.1.3 in [9]). Therefore, the knowledge of f(n) provides information on these microscopic interactions. For simplicity, we here adopt the simple form f(n) = (n + 1) for all n > 0, and consider the loop weight Ω(n) = σ0(n + 1) −c . (3) We show that at the melting temperature the results for the relaxation times for the bubbles are different for the available values c = 1.76 and c = 2.12 quoted above. This shows that the specific form of f(n) indeed modifies experimentally accessible features of the bubble dynamics [14]. In what follows, we focus on a single bubble in dsDNA, neglecting its interaction with other bubbles. Since due to σ0 ≪ 1 the mean distance between bubbles (∼ 1/σ0 [8]) is large, this approximation is justified as long as the bubbles are not too large [7]. It also corresponds to the situation studied in the recent experiment by Altan-Bonnet et al [14]. According to the above, the total free energy F(n) of a single bubble with n > 0 open base pairs follows in the form βF(n) = − ln [W (n)Ω(n)] = nγ(T ) + γ1 + c ln(n+ 1) , (4) where the dependence on the temperature T enters only via γ = γ(T ). We show the free energy (4) in figure 1 for c = 1.76 and for the parameters of an AT-homopolymer, for physiological temperature T = 37C, at the melting temperature Tm = 66 C, and at T = 100, compare the discussion below. Bubble dynamics. In the generally accepted multistate unzipping model, the double strand opens by successively breaking Watson-Crick bonds, like opening a zipper [26, 27]. As γ becomes small on increasing the temperature, thermal fluctuations become relevant and cause a random walk-like propagation of the zipper locations at both ends of the bubble-helix joints. The fluctuations of the bubble size can therefore be described in the continuum limit through a Fokker-Planck equation for the probability density function (PDF) P (n, t) to find at time t a bubble consisting of n denatured base pairs, following a similar reasoning as pursued in the modelling of the dynamics of biopolymer translocation through a narrow membrane pore [28]. To establish this Fokker-Planck equation, we combine the continuity equation (compare reference [28])