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Force and kinetic barriers to unzipping of the DNA double helix

Communicated by Curtis G. Callan, Jr., Princeton University, Princeton, NJ (received for review January 17, 2001)
Abstract
A theory of the unzipping of doublestranded DNA is presented and is compared to recent micromanipulation experiments. It is shown that the interactions that stabilize the double helix and the elastic rigidity of single strands simply determine the sequencedependent ≈12pN force threshold for DNA strand separation. Using a semimicroscopic model of the binding between nucleotide strands, we show that the greater rigidity of the strands when formed into doublestranded DNA, 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.
In cells, proteins apply forces to unzip and stretch DNA. These forces can be studied in singlemolecule experiments (Fig. 1) (1–8) 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, 5, 9–11). 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 sequencedependent force associated with unzipping of large DNAs, f_{u} ≃ 12 pN, can be simply deduced from the known free energy of DNA denaturation and the elasticity of singlestranded (ss)DNA. The unzipping experiments of EssevazRoulet et al. (Fig. 1A) (1, 2) and Rief et al. (Fig. 1D) (5) are accurately described at this macroscopic level.
Other experimentally observable aspects of unzipping can be investigated only by using a more detailed description of basepairing interactions. We therefore present a semimicroscopic model that accounts for hydrogen bonds and stacking interactions (12–16). We show that a freeenergy potential barrier originates from the greater range of conformational fluctuation of DNA strands when isolated, relative to when they are bound together to form doublestranded (ds)DNA (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 2r as well as the shape of the opening fork (Fig. 3). Because of the potential barrier, this force is much larger at small r (and can reach some hundreds of piconewtons) than the asymptotic value f_{u} at large r. Analysis of unzipping in thermal equilibrium at the high level of precision possible in atomic force microscopy (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 by using nucleation theory (17). 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. (8) and of Pörschke (18).
Extending Evans and Ritchie's theory for the breaking of single bonds (19) to the case of a onedimensional polymer (20), we then calculate the most probable rupture force when the DNA molecule is subjected to a force that increases at a constant “loading rate” (Fig. 6). The dependence of the rupture force on loading rate and molecular length could be quantitatively tested by AFM unzipping experiments; these results also shed light on the AFM DNAstretching experiments of Strunz et al. (6) and of Rief et al. (5).
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 g_{0}. Although it is dependent on sequence, we may consider g_{0} = −1.4 k_{B}T, the value determined from singlemolecule experiments on an ATrich sequence in λ phage (7), as a reference for the freeenergy difference between dsDNA and separated ssDNAs. Our emphasis is on an understanding of the freeenergy balance in unzipping rather than on studying inhomogeneous sequence effects (9, 10).
In the presence of applied torque Γ and unzipping force f (Fig. 1B), the freeenergy difference per base pair between unzipped and basepaired DNA strands is 1 When Δg < 0, opening is thermodynamically favorable. The last two terms represent mechanical work done per base pair unzipping the double helix. In order, they are the work done by the torque (θ_{0} = 2π/10.5 is simply the change in strand winding angle during conversion of dsDNA to separated strands) and the stretching free energy of the unzipped single strands.
The function g_{s}(f) in Eq. 1 is the ssDNAstretching free energy per base at fixed force. The leading factor of 2 simply reflects the fact that two bases of ssDNA are created for each base pair of dsDNA that is unzipped. The ssDNA elastic behavior is complicated by nucleotideinteraction effects (21), but experimental forceextension curves for λ phage ssDNA in 150 mM Na^{+} are well described by a freely jointed chainlike (FJCL) elastic response for forces >1 pN, with Kuhn length d = 15 Å (5, 22, 23). The corresponding free energy for forces up to ≈20 pN is 2 where the contour length per base pair is l_{ss} = 27 μm/48.5 kb ≃ 5.6 Å.
At zero applied force, g_{s}(0) = 0, thus ssDNA is stable when Γ < Γ_{u} = g_{0}/θ_{0} = −2.4 k_{B}T, in good agreement with an experimental estimate of the twisting torque needed to denature an ATrich sequence in a λ phage DNA (7) (the sign indicates a lefthanded dsDNAunwinding torque). In this case, the work done by the torque during opening is simply −g_{0} = 1.4 k_{B}T per base pair.
In the opposite case, where torque is zero (i.e., for dsDNA with no constraint of its strandlinking number), the critical unzipping force [at which Δg(f) = 0] is f = 11 pN. These results are in good agreement with the midrange of unzipping forces encountered with experiments on inhomogeneoussequence DNAs by EssevazRoulet et al. (Fig. 1A): 12pN threshold to start, then 10–15 pN during unzipping λ DNA (1, 2). Data of Rief et al. [Fig. 1D; 20 ± 3 pN for poly(dGdC), 9 ± 3 pN for poly(dAdT) (5)] gives, via Eq. 1, denaturation free energies of 1.1 and 3.5 k_{B}T per AT and GC base pair, respectively, in good agreement with thermodynamical data (25). Finally, the projected length of one ssDNA nucleotide along the force direction at the unzipping transition is given as d ≃ 4 Å. Bockelman et al. used a similar theory to analyze unzipping force dependence on sequence (1, 2).
Fig. 3 Inset shows the curve in the torqueforce plane on which Δg = 0, which is the “phase boundary” separating dsDNA and unzipped ssDNAs. This boundary is predicted to have the shape f_{u} ∝ (Γ − Γ_{u})^{1/2} for small f_{u}.
For forces up to 15 pN, Eq. 2 is approximated to 0.15 k_{B}T accuracy by the simple quadratic form 3 where the ssDNA effective elastic constant is C = 0.12 k_{B}T/Å^{2}. Using this form allows analytical solution for the unzipping force, f = (Cg_{0}/2)^{1/2} = 12 pN (10, 11, 20); at this force, the projection of ssDNA monomer length along the force direction is d = (2g_{0}/C)^{1/2} ≃ 5 Å. This quadratic approximation is quantitatively nearly equivalent to the nonlinear model (Eq. 2) at forces up to ≈15 pN (e.g., note the accord between the torqueforce “phase boundaries” in Fig. 3 Inset); this will be key to the continuum theory below.
Semimicroscopic Model of Strand Binding
Features at the nucleotide scale relevant to the onset of unzipping are ignored in models like Eq. 1. We therefore move to a model that uses the distances 2r(n) between corresponding nth base pairs of the two strands as degrees of freedom. The energy of the DNA strands is: 4 This model is similar to models previously used to describe thermal denaturation (13, 14). The precise form of Eq. 4 follows from our previously developed model for denaturation by torque (15,‖ 16) by integration over angular degrees of freedom, followed by continuum limit for the base index n.
The first term in Eq. 4 describes interactions between neighboring bases along each strand and must depend strongly on the interbase halfdistance r(n), because conformational fluctuations of the strands are highly quenched inside the double helix, relative to those along ssDNA. The strand rigidity is m(r) = Ee^{−b(r−R0}^{)} + C, where R_{0} = 10 Å is the double helix radius, and where 1/b = 0.6 Å is the separation at which the strand rigidity changes from its doublehelix value E + C to the much smaller ssDNA value C. We use the value C = 0.12 k_{B}T/Å^{2} from the previous section. The rigidity of the strands inside the double helix has been determined from Raman measurements of internal vibrations of dsDNA to be E = 58 k_{B}T/Å^{2} (16, 26).
The second term in Eq. 4 is a potential, acting between the two strands, made up of the hydrogenbonding energy between corresponding bases plus a torque energy: U(r) = U_{H}(r) − ΓR_{1}/r, where R_{1} = 6 Å (15). We use the Morse potential form (27) for the hydrogenbonding interaction (12, 13): U_{H}(r) = D[(e^{−a(r−R)} − 1)^{2} − 1], with D = 5.84 k_{B}T and a = 6.3 Å^{−1} (Fig. 2 Inset).
In thermal equilibrium, strand unpairing is described by the partition function Z = ∫ 𝒟r(n) e^{−H/kB}^{T}. Z can be computed by use of a continuum transfer matrix technique along the n coordinate, leading to a Schrödingerlike equation: 5 The freeenergy potential V(r) = U(r) + (k_{B}T/2)ln[m(r)/m(∞)] includes an entropic contribution caused by the decrease in the rigidity m(r) with strand unbinding. This entropic potential (Fig. 2) arises when going from the path integral to the Schrödinger equation (24) with nonconstant mass (14), and here it generates a large force barrier to the initiation of unzipping, which strongly affects the kinetics of strand separation.
The lowest eigenvalue g_{0} is the equilibrium free energy, and the corresponding ψ_{0}(r) describes thermal fluctuations of distance between the two strands. For g ≥ 0, the eigenvalue spectrum becomes continuous, corresponding to the appearance of completely separated ssDNAs at g = 0. When DNA is the thermodynamical favorable state, g_{0} < 0 represents the free energy per base pair of a long dsDNA, relative to separated ssDNAs. The fluctuations of r are confined to the Morse well, so g_{0} is well approximated if we take V(r) = V(R_{0}) + U_{H}(r) − U_{H}(R_{0}) and m(r) = m(R_{0}), for which Eq. 5 is exactly soluble (27). Inside the well, ψ_{0}(r) is the Morse ground state (27); outside the well, it can be computed by using the Wentzel, Kramers, Brillouin approximation (28). By using the parameters listed above, and at zero torque, the lowest eigenvalue is g_{0} = −1.4 k_{B}T, in accord with the corresponding number assumed in the previous section. Application of an unwinding torque gradually increases g_{0} until at Γ_{u} = −2.4 k_{B}T, it becomes zero and the DNA unwinds, exactly as occurs in the thermodynamic model of the previous section (Fig. 3 Inset).
Force Required to Hold ssDNA Ends at a Given Distance.
This situation can be analyzed simply in terms of ψ_{0}(r), without further computation. This is because, for a semiinfinite dsDNA, the function ψ_{0}(r) corresponding to g_{0} is the probability distribution for the two ssDNA ends to fluctuate a distance 2r apart. Therefore, the free energy associated with a fluctuation that separates the two ssDNA ends by a distance 2r (or equivalently the total work done separating the two ends to a distance 2r) is 6 up to an additive constant that is unimportant for our analysis. Note that, neglecting surface interactions, this is half the free energy associated to separate a distance 2r in the middle of a long dsDNA, because an interior “bubble” is made of two “forks.”
The unzipping force that must be supplied to hold the two ssDNA ends a distance 2r (Fig. 1B) apart is thus just the derivative of Eq. 6, f(r) = dW(r)/d(2r). This is in Fig. 3, which displays a large force barrier of ≈270 pN as the strands are forced apart. The barrier peak occurs for a halfseparation r − R_{0} = 0.5 Å and then decays to the longmolecule unzipping force f_{u} ≃ 12 pN by r − R_{0} ≈ 4 Å. At large distances, the work done per base pair by unzipping is 2f_{u}d_{u} ≃ 3 k_{B}T, twice as much as the denaturation −g_{0} free energy; this is because the force must unzip the DNA and extend the highly flexible ssDNAs. The peak force is large compared to the fluctuations in force associated with sequence (1, 2). The force barrier will not be observable in largescale unzipping experiments (1, 2) but should be observable in AFM studies. A stiff cantilever with roughly 0.1 Å thermal noise should be used to measure the force barrier as a function of essentially fixed opening distance.
In this fixeddistance experiment, one might also measure the shape of the opening “fork” by determining the relation between opening distance and base position, n(r) (Fig. 1B). The most probable configuration n(r) satisfies the equation of motion associated with Eq. 5, which expresses force balance along the chain, 7 We integrate Eq. 7 to obtain r(n), the shape of the opening “fork.” Starting from the opening point where r = R_{0} and dr/dn = 0, n(r) = 2[E(r − R_{0})/V′(R_{0})]^{1/2}. Far from the opening point, n(r) = r/d ≃ r/(5 Å) (11).
Unzipping Kinetics at Fixed Force.
Many experiments on short (10 to 100bp) dsDNAs (refs. 4, 6, 8, 18 and caption of Fig. 1) probe the kinetics of strand separation. The equilibrium results discussed above are a starting point for a kinetic theory of unzipping based on nucleation theory (17). The Schrödinger equation (Eq. 5) with the fluctuationcorrected potential V(r) describes dsDNA and ssDNA as locally stable molecular states. The general problem faced in unzipping kinetics is the transition from an initially metastable state (dsDNA or ssDNA, depending on the force; see below) to a final stable (lowerfreeenergy) state. Strand dissociation requires the whole polymer chain to cross the freeenergy potential barrier of V(r) (Fig. 2), which makes the transition rate strongly length and force dependent. Using the effective potential V(r) (including the entropic barrier) corresponds to averaging over microscopic fluctuations of individual bases, restricting us to consider unzipping rates slow compared to those of these microscopic fluctuations (the experimentally relevant regime).
The transition rate (equivalently the inverse lifetime) has the form familiar from transitionstate theory (17): 8 It requires the activation free energy G* of a transition state, relative to the initial metastable state. The transition state is the saddlepoint configuration of the free energy, with one unstable direction leading monotonically down to the initial and final states, and is the dominant transition pathway (17).
Our transition states are just the partially unzipped configurations r*(n) determined from Eq. 7 for forcedependent boundary conditions consistent with the initial (r_{i}) and final (r_{f}) states. The activation free energy is 9 i.e., the free energy associated with Eq. 7 minus the free energy of the metastable state (G_{m} = Ng_{m}) from which the transitions occur. G* is in practice the free energy of the fewbasepair “bubble” portion of the transition state, which separates the unzipped and doublestranded regions.
Finally, the rate prefactor ν_{0} is the linear growth rate of unstable perturbations around the saddlepoint configuration (17). We assume viscous dynamics with a friction coefficient per base ζ = 6πηR_{0} [water viscosity η = 1 × 10^{−3} kg/(m⋅sec)]. From a detailed calculation, we find ν_{0} ≈ Da^{2}/(4ζ) = 1 × 10^{12} sec^{−1}, essentially the ratio of the negative curvature of the Morse potential near the top of the well, to the friction coefficient. We now describe how to compute the dissociation rate and the nucleation bubble shape (i.e., boundary condition r_{i}, r_{f}), depending on the unzipping force.
Kinetics of Unzipping Where ssDNA Is Stable (f > f_{u}).
If a steady force is applied that is slightly bigger than the equilibrium unzipping threshold f_{u}, then the initial dsDNA is metastable relative to separated strands: Δg(f) = −g_{0} − 2f^{2}/C < 0 (Eq. 1), and formula (Eq. 8) is directly applicable (Fig. 4A) to calculation of the dissociation rate ν_. The free energy of the initial metastable state is g_{m} = g_{0}. The transition state in this case is a short ssDNA “nucleus” of n* bases at the open end (n = 0) of the dsDNA, with boundary conditions at the ends of fixed force f [dr/dn = 2f/m[r(n)]] and free energy equal to the metastable dsDNA value [V(r(n)) − 2f^{2}/m[r(n)] = g_{0}]. This determines r_{i} = r(n*) and r_{f} = r(0).
The size of the nucleation “bubble” depends weakly on the force and is n*(f) ≃ 4 bases, independent of the overall DNA molecule length. Therefore, the dissociation time t_ = 1/ν_ (Fig. 5) is lengthindependent for forces above f_{u} = 12 pN. The decrease in dissociation time with forces > f_{u} is because of reduction of G* by the applied force. Beyond f_{b} ≃ 230 pN, the barrier in V(r) is completely overcome, and unzipping is immediate (t_ = 1/ν_{0}). Our computation addresses only the initial unzipping barriercrossing event and does not include the time necessary to push the fork down the dsDNA, which would introduce a weak molecularlength dependence for forces > f_{u} in Fig. 5. Fork motion near f_{u} may also be retarded by additional barriers introduced by inhomogeneous sequence.
Kinetics of Unzipping Where ssDNA Is Metastable (f < f_{u}).
Below the equilibrium unzipping force threshold, an infinitely long molecule is stable as a dsDNA. However, relatively short (<20 bp) oligomers have a finite stranddissociation time that can be accelerated by applied force. Here the final ssDNA state is metastable (although with a possibly long lifetime) relative to the initial dsDNA state. Therefore our calculation scheme applies not to unzipping (offrate ν_) but instead to annealing of ssDNAs (onrate ν_{+}, Fig. 4B). We compute the offrate from the onrate, using the equilibrium condition ν_/ν_{+} = e^{−NΔg(f)/KBT}, where N is the number of base pairs in the molecule, and where Δg(f) is the forcedependent freeenergy difference per base pair between the paired and unpaired states.
The calculation of the onrate requires computation of a saddlepoint configuration with a nucleation bubble of dsDNA at the end of ssDNAs to which the force is applied (Fig. 4B). The relevant boundary conditions are analogous to those above, apart from the requirement that the ssDNAs be in close proximity but not basepaired (i.e., slightly outside the potential barrier of Fig. 2), and the free energy of the metastable initial state equals g_{m} = 2g_{s}(f) = −2f^{2}/C. The dsDNA nucleation “bubble” remains near 4 bp as the force f decreases from f_{u} to zero, leading to an essentially Nindependent onrate ν_{+} (again the time necessary for the actual “zipping up” is not included). Then, the N dependence from the energy difference of paired and unpaired DNA results in a strong molecularlength dependence of ν_, as shown in Fig. 5 for forces less than f_{u}.
At zero force, this becomes a calculation of dissociation time for free dsDNAs in solution with result t_ = 10^{0.6N−6.3} sec. Fig. 5 Inset shows this as a function of dsDNA length; a 10bp dsDNA has a lifetime of roughly 1 sec; the exponential length dependence results in a 30bp DNA being stable for ≈10^{12} sec ≈ 30,000 yr. Our estimate of t_ for N = 5, t_ ≈ 0.3 msec, is in agreement with the results of Bonnet et al. (8) (Fig. 1E). The prediction for the nucleation bubble size, n* = 4, is close to the value n* = 3 measured by Pörschke for a poly(A)poly(U) acid (18).
Rupture of dsDNA During Gradual Loading.
The previous two sections discuss fixedforce experiments; an alternate experiment is to steadily increase force (f = λt, where λ is the “loading rate” in piconewtons/sec), and then to measure the force at which rupture occurs. Generally, rapid loading rates result in rupture at large forces. We are able to predict the most probable rupture force versus loading rate and molecular length, and our results display a rich range of possibilities (Fig. 6).
Using the calculations of offrates presented above, the probability distribution for rupture with force is (19) 10 For a number of molecular lengths, Fig. 6 shows the location of the peak of this distribution, the most probable rupture force that would be measured experimentally. For sufficiently slow loading rate, rupture occurs at zero force simply by thermal dissociation; for molecules >20bp thermal dissociation is practically unobservable.
At some loading rate λ_{1}, the peak in P(f) jumps to finite force and then increases, with each length of molecule following a different curve. In this regime, the forces at rupture are typically below the equilibrium unzipping threshold f_{u}, because there is time for many thermal attempts at barrier crossing to the metastable ssDNA state during loading. The length dependence follows from the calculation of formation of metastable ssDNA discussed above and has been qualitatively observed for 5′–5′ pulling experiments (Fig. 1C) (6).
At a higher loading rate λ_{2}, the peak of the rupture force distribution hits f_{u}, which remains the most likely rupture force up to a loading rate λ_{3}. For λ > λ_{2}, dissociation is occurring at forces large enough that the ssDNA final state is stable, resulting in no Ndependence. Finally, beyond the very large loading rate λ_{3} ≈ 10^{5.5} pN/sec, the rupture force gradually increases simply because the molecule is unable to respond to the force before it becomes very large. λ_{3} separates equilibrium and nonequilibrium time scales for very long sequences (N → ∞): the rupture force is independent of the loading rate and equal to f_{u} when λ < λ_{3} and increases above.
Our framework treats nonequilibrium rupture of a onedimensional object, a development of previous theory (19) necessary for interpretation of unzipping experiments. In light of our detailed predictions, 3′–5′ AFM unzipping experiments (Fig. 1A) should yield interesting results and would be of help for an accurate determination of the freeenergy potential V(r).
Acknowledgments
We thank C. Bouchiat, G. Bonnet, V. Croquette, and F. Pincet for useful discussions. Work at the University of Illinois at Chicago was supported by National Science Foundation (NSF) Grant DMR9734178, by the Petroleum Research Foundation of the American Chemical Society, and by the Research Corporation. S.C. is partly funded by an A. della Riccia grant. R.M. is supported in part by the Materials Research Science and Engineering Center Program of the NSF under Award Number DMR9808595.
Footnotes

↵* To whom reprint requests should be addressed. Email: cocco{at}franklin.phy.uic.edu.

↵§ Permanent address: Centre National de la Recherche Scientifique—Laboratoire de Physique Theorique de l'Ecole Normale Superieure, 24 rue Lhomond, 75231 Paris cedex 05, France.

↵‖ Note that in this previous version of ref. 15, the basepair index is discrete. The passage to the continuum will be exposed in detail in a forthcoming work.
Abbreviations
 ds,
 doublestranded;
 ss,
 singlestranded;
 AFM,
 atomic force microscopy
 Received January 17, 2001.
 Accepted May 23, 2001.
 Copyright © 2001, The National Academy of Sciences
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