Force and kinetic barriers in unzipping of DNA

A theory of the unzipping of double-stranded (ds) DNA is presented, and is compared to recent micromanipulation experiments. It is shown that the interactions which stabilize the double helix and the elastic rigidity of single strands (ss) simply determine the sequence dependent ≈ 12 pN force threshold for DNA strand separation. Using a semi-microscopic model of the binding between nucleotide strands, we show that the greater rigidity of the strands when formed into dsDNA, relative to that of isolated strands, gives rise to a potential barrier to unzipping. The effects of this barrier are derived analytically. The force to keep the extremities of the molecule at a fixed distance, the kinetic rates for strand unpairing at fixed applied force, and the rupture force as a function of loading rate are calculated. The dependence of the kinetics and of the rupture force on molecule length is also analyzed. Introduction: In cells, proteins apply forces to unzip and stretch DNA. These forces can be studied in singlemolecule experiments (Fig. 1) [1–7], and are of biophysical as well as biological interest. Our focus here is primarily on unzipping experiments where forces are applied across the double helix to adjacent 5’ and 3’ strands (Fig. 1A) [1,4,8–10]. In experiments, the control parameters may be the force f itself, the distance between the last base pairs 2r, or the rate of force increase or ‘loading rate’ (Fig. 1B). We discuss the results expected in all these situations. We first use a thermodynamical equilibrium approach to show that the sequence-dependent force associated with unzipping of large DNAs, fu ≃ 12 piconewton (pN), can be simply deduced from the known free energy of DNA denaturation and the elasticity of single-stranded DNA. The unzipping experiments of Essevaz-Roulet et al (Fig. 1A) [1] and Rief et al (Fig. 1D) [4] are accurately described at this macroscopic level. Other experimentally observable aspects of unzipping can only be investigated using a more detailed description of base-pairing interactions. We therefore present a semi-microscopic model which accounts for hydrogen bonds and stacking interactions [11–15]. We show that a free-energy potential barrier originates from the greater range of conformational fluctuation of DNA strands when isolated, relative to when they are bound together to form dsDNA (Fig. 2). Our model can be investigated in detail and allows precise calculation of the effects of this barrier for the initiation of unzipping and the kinetics of strand dissociation. We compute the force necessary to keep apart the two extremities of the DNA molecules at some distance 2 r, as well as the shape of the opening fork (Fig. 3). Due to the potential barrier, this force is much larger at small r (and can reach some hundreds of pN) than the asymptotic value fu at large r. Analysis of unzipping in thermal equilibrium at the high level of precision possible in AFM experiments would allow unambiguous verification of this predicted force barrier. The barrier makes strand dissociation an activated process with dynamics that can be analyzed using nucleation theory [16]. Unzipping starts with a transition ‘bubble’ a few (≤ 4) bases long (Fig. 4). We calculate the free energy of this bubble, and determine how the dissociation rate depends on applied force and molecule length (Fig. 5). Results are compared to the experiments of Bonnet et al. [7] and of Pörschke [17]. Extending Evans’ theory for the breaking of single bonds [18] to the case of a one-dimensional polymer [19], we then calculate the most probable rupture force when the DNA molecule is subjected to a force which increases at a constant ‘loading rate’ (Fig. 6). The dependence of the rupture force upon loading rate and molecular length could be quantitatively tested by AFM unzipping experiments; these results also shed light on AFM DNA-stretching experiments of Struntz et al. [5] and of Rief et al. [4]. Thermodynamic Description of Unzipping: An unstressed double helix is stabilized against spontaneously dissociating into its two strands by the interaction free energy per base pair, which from a thermodynamic perspective we may take to be some average amount g0. Although dependent on sequence, we may consider g0 = −1.4kBT , the value determined from single molecule experiments on an AT rich sequence in λ phage [6], as a reference for the free energy difference between dsDNA and separated ssDNAs. Our emphasis is on an understanding of the free-energy balance in unzipping rather than to study inhomogenous sequence effects [8,9]. In the presence of applied torque Γ and unzipping force f (Fig. 1B) the free energy difference per base pair between unzipped and base-paired DNA strands is