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Phase diagrams for DNA denaturation under stretching forces

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J. Stat. Mech. (2009) L04001

(http://iopscience.iop.org/1742-5468/2009/04/L04001)

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ournal of Statistical Mechanics:An IOP and SISSA journalJ Theory and Experiment

LETTER

Phase diagrams for DNA denaturationunder stretching forces

D Marenduzzo1, A Maritan2, E Orlandini2, F Seno2 andA Trovato2

1 SUPA, School of Physics, University of Edinburgh, Mayfield Road,Edinburgh EH9 3JZ, UK2 Dipartimento di Fisica, CNISM and Sezione INFN, Universita’ di Padova,Via Marzolo 8, 35131 Padova, ItalyE-mail: dmarendu@ph.ed.ac.uk, amos.maritan@pd.infn.it,enzo.orlandini@pd.infn.it, flavio.seno@pd.infn.it and antonio.trovato@pd.infn.it

Received 2 February 2009Accepted 21 March 2009Published 21 April 2009

Online at stacks.iop.org/JSTAT/2009/L04001doi:10.1088/1742-5468/2009/04/L04001

Abstract. Double-stranded DNA denaturation induced by an external forceis modeled as a pair of complementary self-avoiding polymers with inter-strand binding potential. The resulting force–temperature phase diagram isdetermined using heuristic arguments, exactly solvable models and 3D MonteCarlo simulations. The results are qualitatively different for the cases wherethe force is applied to only one strand or to both strands. Crucial ingredientsfor getting a qualitative agreement with experimental results are the persistencelengths and the monomer sizes of single-stranded and double-stranded DNA. Thesubtle dependence of the small force behavior on the latter parameters, on thestretching mode and on the presence of denaturation bubbles is analyzed in detail.

Keywords: mechanical properties (DNA, RNA, membranes, bio-polymers)(theory)

c©2009 IOP Publishing Ltd and SISSA 1742-5468/09/L04001+10$30.00

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Phase diagrams for DNA denaturation under stretching forces

Single-molecule experiments have enriched our knowledge of biopolymer physics at thenanoscale [1]–[3]. DNA micromanipulation allows one to, for example, accurately measurephysical properties associated with its structure, such as end-to-end distance and fractionof unbound base pairs as a function of applied external forces.

Arguably the best known examples are the stretching of double-stranded DNA(dsDNA) and single-stranded (ssDNA) in the quasi-static limit, i.e. when dynamical effectssuch as hysteresis and trapping into meta-stable free energy minima are negligible. Fromthe theoretical point of view it has been recognized that simple statistical mechanicsmodels, e.g. the worm-like chain or the freely jointed chain, are able to provide asatisfactory description of the biopolymer elasticity [1], for chain lengths larger than thepersistence length, ∼50 nm (150 bases) for dsDNA and ∼2 nm (4 bases) for ssDNA.At quite large forces (∼65 pN) DNA elongates to about 1.7 times the B-DNA contourlength: this regime is usually referred to as overstretching. Overstretched DNA hasbeen interpreted as either a ds distorted form of B-DNA, named S-DNA, or ssDNA [3]–[7].

On a different level, theoretical work focused on simpler models for finding the(T–f) phase diagram of biopolymers perturbed by external forces [8]–[15]. In suchmodels one studies how the melting transition from dsDNA to ssDNA is affected bythe perturbing force. On the basis of a continuum model [13], it was proposed thatwhen one strand is being pulled, there should be a ‘re-entrance’ at low T , similarto that found for DNA unzipping [9]. Another question is whether the ds phase isdestabilized or not by a stretching force [3, 13]. A further issue is the nature of themelting transition, which is first order at f = 0, and was predicted to become secondorder at f �= 0 [13, 14].

Our aim in this work is to study the DNA melting transition induced by stretchingforces by using coarse grained models, some of which are exactly solvable. Our approachallows us to introduce phenomenological parameters such as persistence lengths andmonomer sizes of dsDNA and ssDNA, and determine the qualitative changes inducedin the phase diagram in the (T, f) plane. In our models, dsDNA molecules are anchoredon one side and subject to stretching forces, f1 and f2, on the two strands on the other side(figure 1). Accurate analysis in [3] showed that only one strand was pulled, though bothstretching modes could occur in general. Therefore, we use two pulling modes: (A) f1 = fand f2 = 0, i.e. one of the two strands is stretched by a force (of modulus f) and theother is free (this mode was theoretically studied in [12, 13]); (B) f1 = f2 = f/2, i.e. bothstrands are stretched by equal forces (studied in [14, 15]). As torsion is not present in ourmodel, mode B would describe an experiment in which both strands are grabbed but onestrand is allowed to rotate freely around the other. Each strand will be considered at somecoarse grained level and modeled as a self-avoiding polymer. Strand complementarity isconsidered by allowing a pair of monomers in different strands to be bound with an energy−ε < 0 only if their monomer indexes along the two strands are equal [16]. The monomersizes (length of 1 bp) and persistence lengths are here the only parameters controlling thebiophysics of dsDNA and ssDNA. We choose monomer sizes in two fundamentally distinctways. In models A0, B0 we assume that the monomer sizes are equal for both ssDNA anddsDNA, this is roughly consistent with dsDNA being in the S form [4]. In models A1, B1,we assign to dsDNA the monomer size of the B form, thereby assuming that melting tossDNA occurs from the B form. The values of monomer sizes and persistence lengths

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Figure 1. Schematics of our model. A dsDNA molecule is anchored at one of itsends. At the other end, either one strand is stretched with a force f1 (stretchingmode A, top), or both strands are forced by two equal forces f1 = f2 (stretchingmode B, bottom) At a finite T bubbles may appear in typical DNA configurationsas in the figure.

mostly impact on the large and small f behavior respectively. T will be measured in unitsof ε/kB.3

Unlike all previous workers, we study models without any a priori ansatz for thepartition function, and this allows us to clearly associate each model (together with itsset of assumptions) with a phase diagram valid over the whole T range. We can thusaccurately find the behavior near T = 0 and T = Tm. One of our main results is that thetwo different pulling modes considered in this work yield amazingly different behaviors,both qualitatively and quantitatively.

We begin with simple heuristic arguments to determine the critical force, fc(T ),separating the zipped phase from the unzipped/denaturated one. At f = 0 we havethe standard thermal denaturation phase transition occurring [16, 17] at T = Tm,i.e. fc(Tm) = 0. For case A0, at large forces we have that the (almost) completely stretchedstrand acts as a 1D substrate for the adsorption of the free (unstretched) strand. In thiscase the only allowed conformation is a Y-like configuration with no bubbles. A simpleenergy–entropy argument leads to a vertical asymptote for fc(T ) at Ta = ε/ss, where ss isthe entropy per monomer of the single DNA strand. In case B0, both strands are pulledand bubbles (see figure 1) are stretched out, so the two strands are on average closerto each other with respect to the f = 0 case and one may argue that dsDNA shouldbe stabilized by f . Another expectation is that at very large f , both strands become

3 These scales may be related to physical units by assuming, e.g., that Tm = 70 ◦C. In this way, for instance, infigure 4, ε would be set to 2.7 kcal mol−1, giving a unit of temperature of 1345 K. Using σ ∼ 1 nm (the size ofssDNA) leads to a force unit of 18.4 pN.

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Phase diagrams for DNA denaturation under stretching forces

Figure 2. Geometry for the exact calculations. Notice that we pull along the [1 1]direction just because of the directed nature of the lattice. Pulling along the [0 1]or [1 0] directions would give qualitatively the same phase diagrams. (b) Phasediagram for model A0 (solid line) and B0 (dashed line). (c) Fraction of zippedbases as a function of T for f = 0 (solid line), for model A0 and f = 1 (dashedline), for model B0 and f = 2 (dot–dashed line). (d) End-to-end distance of thepulled strand per monomer as a function of f for model A0 and T = 2 (solidline), T = 3 (dashed line); for model B0 and T = 4 (dot–dashed line; the arrowdenotes the transition).

straight in the same direction and the zipped phase is stabilized at all finite T , since basepairing does not lead to further entropy cost. All this holds, we stress, in the case inwhich the monomer sizes of the zipped and unzipped phases are assumed to be exactlythe same.

In order to confirm these heuristic arguments and obtain a complete phase diagramfor our models A0 and B0 we introduce a simplified version which can be exactlysolved.

In these models the two strands are mutually avoiding directed walks along the (1, 1)diagonal of a 2D lattice (figure 2(a)). When two beads are in contact they gain a pairingpotential ε, and one of the ends of the strands is fixed at the origin, while the other twostrands are stretched by two forces (f1 and f2; see figure 2(a)). If f2 = 0 we get thecase corresponding to our model A0. Similar exactly solvable models were very useful formapping out the phase diagrams for DNA unzipping [9].

One can write down a recursion for the partition function of a system with only onedegree of freedom, the open end distance x. If the strands are subjected to two forces f1

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Phase diagrams for DNA denaturation under stretching forces

and f2 (e.g. identified by their projection along the positive (−1, 1) vector; figure 2(a)),then the recursion is

ZN+1(x) = ZN(x + 1)y2/y1 + ZN(x − 1)y1/y2

+ ZN(x) (y1y2 + 1/(y1y2)) for x > 0 (1)

ZN+1(0)e−1/T = ZN(1)y2/y1 + pZN(0) (y1y2 + 1/(y1y2)) . (2)

In equations (1) and (2), ZN(x) indicates the partition function of two N -step walks withopen end distance x, y1 = eβf1 , y2 = eβf2 , and p is a parameter which controls the weightof dsDNA segments and which we temporarily fix equal to 1.

If we define a grand canonical partition function Z(x, z) ≡∑

N zNZN(x), then itis known that the thermodynamic behavior is identified by the smallest singularity inZ(x, z) [9, 14, 16]. The grand canonical partition function may be found, for instance, byusing the ansatz Z(x, z) = A(z)λ(z)x, with the functions A(z) and λ(z) to be determined,as in [9]. The solution shows that there are two competing singularities, z1 and z2, whichare associated with the zipped and denatured phase respectively, and which are given bythe following formulae:

z1 =

√w2 + 4 exp(−1/T ) (1 − exp(−1/T )) − w

2(3)

z2 =1

y1/y2 + y2/y1 + y1y2 + 1/ (y1y2)(4)

w = [1 − exp(−1/T )]

(

y1y2 +1

y1y2

)

. (5)

The phase diagrams for models A0 and B0 are plotted in figure 2(b). The criticalforce can be found explicitly for both of these 1 +1-dimensional models by equating z1 toz2 and we obtain

fA,c (T ) =T

2log

[2(1 − e−1/T

)

2e−1/T − 1

]

(6)

fB,c (T ) = T arccosh

[2e−1/T − 1

2 (1 − e−1/T )

]

(7)

for stretching modes A and B respectively (we note that the phase diagram for stretchingmode A has been found, in another context, in [12]).

Broadly, the phase diagrams confirm our conjectures. As expected, the phase diagramfor stretching mode A displays a vertical asymptote, at T = Ta = (1/log 2), thetransition temperature for polymer adsorption on a wall. Figure 2(b) shows that thephase diagram for stretching mode B is completely different from the mode A one, sinceany force (whether small or large) stabilizes dsDNA (see figure 2, dashed line): the regionof stability of the zipped phase, in the (T, f) plane, increases, and the denaturationtemperature appears to be a monotonically increasing function of f , as expected fromheuristic arguments.

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Phase diagrams for DNA denaturation under stretching forces

Interestingly, the phase transition between native and denatured phase is first orderfor stretching mode A, at f �= 0, and second order for stretching mode B. This can beseen from the plots of the average percentage of zipped bases (θ; figure 2(c)), and of theaverage open end distance per monomer (〈x〉/N ; figure 2(d)). (Note that this holds forthe specific exactly solvable model that we choose, and that the more general case of fullself-avoiding walks in 3D may differ.)

The phase diagrams for both models A0 and B0 predict that DNA can stay zipped evenat large forces. This would be compatible with early interpretations of DNA overstretchingexperiments, where it was hypothesized that the overstretched DNA was dsDNA in a formdifferent from B-DNA [4]. However, the reader should recall that up to now we have madethe strong assumption that the monomer sizes of dsDNA and ssDNA are the same.

We then turn to models A1 and B1 in which such an assumption is removed, anddifferent monomer sizes and persistence lengths are instead considered for dsDNA andssDNA.

The effective, or Kuhn, length of dsDNA segments4 is larger (�d ∼ 300 bp ∼ 100 nmfor B-DNA) than that of ssDNA ones (�s ∼ 8 bp ∼ 4 nm). The monomer sizes aread � 0.34 nm for dsDNA and as � 0.56 nm for ssDNA. The easiest way to incorporatethese elements into our models is to weight dsDNA segments in equation (2) with a force-dependent weight as follows:

p(β, f, ξ, a) = [2cosh (βfaξ)]1/ξ/2cosh (βf). (8)

In this way the entropy of a dsDNA portion is 1/ξ ≡ �s/�d � 0.04 times that of the samessDNA portion (as their persistence lengths, and thus Kuhn lengths, are different) and thefactor of a ≡ ad/as � 0.6 takes into account the different monomer sizes. The resultingphase diagrams are shown in figure 3(a). The most striking difference with respect to thephase diagrams in figure 2 is that there is now a bound region for the zipped phase, andat large force the DNA is always single stranded and denatured (in this phase diagramthe overstretching transition would coincide with the melting one in agreement with theinterpretation given in [3]). Instead, the order of the transitions remains the same as forthe case 0: first-order transition for mode A1 and second-order transition for mode B1.

At low T , in contrast to what occurs for DNA unzipping phase diagrams [9], thereis no re-entrance5. This can be understood as in the DNA denaturation models thatwe are considering here the entropy of the zipped phase is never larger than that of theunzipped one. The phase diagram found in [13] for DNA stretching models shows at lowtemperature a re-entrant transition line because of the approximations employed6. Onthe other hand, at low f near Tm, the pulling force stabilizes dsDNA for mode B also inthis case of different monomer sizes (this agrees with a thermodynamic analysis of DNAoverstretching experimental data [3, 14]). It is finally instructive to compare the phasediagrams in figure 3(a) with those in figure 3(b), where denaturation bubbles (figure 1) aresuppressed. This can be done in practice by tracing only over configurations where there

4 The effective segment length (or Kuhn length) is twice the persistence length for a freely jointed chain, hencethe factor of 2 difference from the numbers quoted in the introduction.5 No ‘re-entrance’ means here that dfc(T )/dT at T = 0 is ≤0. A small re-entrance is indeed seen at finite T inthe B1 case.6 As noted by the authors of [13], assuming a Gaussian probability distribution for the ends of the chains is notvalid at low T .

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(a) (b)

Figure 3. (a) (T, f) phase diagrams for models A1 (solid line) and B1 (dashedline), with ad = 0.34, ld = 50 nm, and as = 0.56, �s = 2 nm for dsDNAand ssDNA respectively. (b) Same phase diagrams, but now with denaturationbubbles suppressed throughout the (T, f) plane.

is only one double-stranded segment—i.e. after the two strands separate they may neverjoin again, yielding a Y-shaped configuration. This restriction renders the calculationmuch simpler (see [9] for the equivalent calculation in DNA unzipping models). Forreal DNA, bubbles at room temperature are rather infrequent, although not completelysuppressed. The phase diagrams in figure 3(b) coincide with those in figure 3(a) for smallT , as expected (bubbles are unlikely to occur in that region); however close to Tm thereare differences, and in the absence of denaturation bubbles also stretching one strand only(mode A) leads to dsDNA stabilization at small force.

To gain more insight into what determines the small force behavior of the phasediagram, we present a Flory-like argument which is valid for both stretching modes in theabsence of denaturation bubbles (Y-shaped configurations only are considered), and in thelimit of small parallel pulling forces f1, f2. The energy of two pulled DNA strands, witha fraction θ of open bps, is E = −εN(1 − θ) − f1r1 − f2r2, where r1, r2 are the projectedpositions, along the force direction, of the unanchored ends of the stretched strands.We may expect the probability density, P (rd, r1, r2|N, θ), for the stretched strands to bedescribed at small forces by separate independent Gaussians for the initial ds portion andthe two ss tails,

P (rd, r1, r2|N, θ) ∝ exp

{

− r2d

2N(1 − θ)a2d�d

}

exp

{

−(rd − r1)2 + (rd − r2)

2

2Nθa2s�s

}

(9)

where rd is the projected position, along the force direction, of the end of the ds portion.

The total number of Y-like configurations is given, modulo power law corrections,by C(N |θ) ∝ exp{(N(1 − θ)/�d)s + (2Nθ/�s)s} where the entropies per monomer ofthe double-stranded and single-stranded DNA have been estimated to be s/�d and s/�s

respectively, s being the entropy per segment. The total entropy of the Y-like configurationwith fixed rd, r1, and r2 is estimated as S = log{C(N |θ)P (rd, r1, r2|N, θ)}. Minimizingthe free energy F = E − TS with respect to rd, r1, r2, and θ, we obtain the following

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form for the critical force, fc(T ):

f 2c (T ) =

T − Tm

Tm

2εT

�da2d − gf�sa2

s

, Tm =ε�d�s

s (2�d − �s)(10)

where gf = 1 for stretching mode A (f1 = f , f2 = 0), and gf = 1/2 for stretching modeB (f1 = f2 = f/2). Thus, if �da

2d > gf�sa

2s , as in reality, Tc increases with f and, as a

result, fc(T ) ∼ (T − Tm)1/2 at T ∼ T+m for both pulling modes. The behavior of fc given

by equation (10) is only valid for T ∼ Tm. Away from Tm the transition line has to turnleft and e.g. approach a vertical asymptote for as = ad.

Equation (10) can be obtained rigorously as the leading behavior, in the f → 0 limit,of the exact solution obtained for the directed walk models in D = 1 + 1 in the absenceof bubbles, provided that s = log 2. The Flory-like argument thus reproduces the smallforce shape of the phase diagrams shown in figure 3(b), with realistic values for a and ξ.

However, equation (10) is at odds with the phase diagrams for stretching mode Ain figures 2(b) and 3(a). In these cases we would respectively expect Tm not to dependon f close to the melting point (phase diagram of figure 2(b), mode A), or to increasewith f (phase diagram of 3(a), mode A). Why is the dsDNA phase destabilized by f inthe exact solution? The reason is that the argument that we have just proposed neglectsdenaturation bubbles. These are absent in figure 3(b), but are crucial in both of the othercases, as the melting transition at zero force is second order and there are ∼N1/2 bubblesat the transition. Bubbles increase Tm and the stability of the zipped phase at f = 0.Whenever f1 �= f2, we find from the exactly solved models fc(T ) ∼ (Tm−T ) at T ∼ T−

m asthe leading behavior in the presence of bubbles, so df/dT along the critical line changessign. In mode B (f2 = f1) dsDNA is stabilized in all cases (see figures 2 and 3) as thislinear contribution due to bubbles is suppressed and �da

2d − gf�sa

2s > 0 if a = ξ = 1 and

gf = 1/2.On the other hand, we recall that the melting transition for DNA is first order, due

to self-avoidance [17]; hence in this more realistic case bubbles should be unimportant atthe transition, and we would expect a small force to lead to stabilization for dsDNA inmode A as well.

To test this prediction, we performed 3D Monte Carlo simulation of model A, i.e. twopaired DNA strands, one of which is under a stretching force of modulus f . Each strandis modeled by a self-avoiding chain, made up by N beads of size σ connected by springs.To model homologous base pairing, we chose a truncated Lennard-Jones potential, withminimum value −ε, attained when the corresponding beads are at a mutual separationσ. The bonds between successive beads are harmonic springs with minimal and maximalelongations equal to 0.7σ and σ respectively (the maximum elongation is then (N − 1)σ).The simulations were performed by proposing a combination of crankshaft, pivot andreptation moves for each of the two strands. We used multiple Markov chains [18] toimprove sampling efficiency at low T or high f , and reweighted the simulation data priorto analysis according to the Ferrenberg–Swendsen algorithm [19]. The phase transitionpoints were estimated as the positions of the peaks of the specific heat.

We have considered the case in which the monomer sizes of the zipped and unzippedphases are equal. However due to the presence of the base pairing interaction the double-stranded phase is stiffer as in reality, even if to a lesser extent. The theoretical argumentfor the slope of the phase diagram at Tm then predicts that a small force should stabilize

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Phase diagrams for DNA denaturation under stretching forces

Figure 4. Solid line: f–T phase diagram in the continuum limit for a model withtwo self-avoiding chains kept together by a truncated Lennard-Jones potentialand a force stretching one of the strands. Temperature and force are measured inunits of ε/kB and ε/σ respectively. The pulling force was along the z direction (inthe continuum case there is isotropy at variance with the directed lattice case).

dsDNA, as a = 1, ξ > 1, and gf = 1. This is indeed what is observed in our numericalresults (figure 4). Self-avoidance would change the exponent governing the small forceleading behavior, fc(T ) ∼ (T − Tm)ν at T ∼ T+

m , from ν = 1/2 to ν � 0.588. This trendis not discernible due, possibly, to the distance from the thermodynamic limit.

It is not straightforward to quantitatively compare our results with typical pullingDNA experiments, as these are commonly performed with heterogeneous DNA (see e.g. [3]in which DNA from phage λ was used). In that case, it was argued that the DNA strandsare held together by GC regions for all probed values of f and T , so stretching modeB may effectively be the relevant one even when pulling on one strand only. However,in other cases or with homogeneous DNA [4], one of the strands may fall off the other,making mode A relevant as well. A general result is that the details of the attachment ofthe molecule to the AFM tip (the pulling end) and to the gold substrate (the anchoredend) are found to lead to very different force–extension curves in overall agreement withour predictions, as the melting transition occurs at a much larger force when stretchingmode B is considered at low pulling rates.

In conclusion, we have presented heuristic arguments, exact results, and Monte Carlosimulations for determining the phase diagram of model dsDNA denaturation in thepresence of an external force, stretching one or both of the strands, modes A or Brespectively. Our results show that these two cases lead to strikingly different results, bothqualitatively and quantitatively. We have also shown that the phase diagrams significantlychange when one considers the fact that dsDNA and ssDNA have different persistencelengths and monomer sizes. Finally, we have discussed extensively the role of the latterparameters, together with the pulling mode and the presence/absence of bubbles, in

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determining the small force dependence of the denaturation transition. Specifically, ourresults suggest that a model taking self-avoidance fully into account, as well as consideringrealistic values for the base pair size and persistence length of dsDNA and ssDNA, shoulddisplay dsDNA stabilization by force, close to the melting temperature.

We thank M Baiesi, S M Bhattacharjee, E Carlon, R Metzler, for stimulating discussions.We acknowledge support from University of Padova through Progetto di Ateneo 2008—CPDA083702.

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