Denaturation transition of stretched DNA

0.2 MB·English

by  Andreas Hanke

Denaturation Transition of Stretched DNA
Andreas Hanke,1 Martha G. Ochoa,1 and Ralf Metzler2,3
1Department of Physics and Astronomy, University of Texas at Brownsville, 80 Fort Brown, Brownsville, USA
2Physics Department, Technical University of Munich, D-85747 Garching, Germany
3Center for NanoScience, Ludwig Maximilians University, D-80539 Munich, Germany
We generalize the Poland-Scheraga model to consider DNA denaturation in the presence of an
external stretching force. We demonstrate the existence of a force-induced DNA denaturation
transitionandobtainthetemperature-forcephasediagram. Thetransitionisdeterminedbytheloop
exponentcforwhichwefindthenewvaluec=4ν−1/2suchthatthetransitionissecondorderwith
8 c=1.85<2in d=3. Weshowthat afinitestretchingforce F destabilizes DNA,corresponding to
0 alowermeltingtemperatureT(F),inagreementwithsingle-moleculeDNAstretchingexperiments.
0
2 PACSnumbers: 87.14.G−,05.70.Fh,82.37.Rs,64.10.+h

n
a
Underphysiologicalconditionsthethermodynamically
J
stable configuration of DNA is the Watson-Crick double
0
1 helix. The constituent monomers of each helix, the nu-
cleotides A, T, G, C, pair with those of the complemen-

] tary helix according to the key-lock principle, such that
t
f onlythebase-pairs(bps)ATandGCcanform[1]. Upon
o
heatingortitrationwithacidoralkaliofdouble-stranded
s
. DNA, regionsofunboundbpsproliferatealongthe DNA
t
a untilfullseparationofthe twoDNAstrandsatthemelt-
m
ing temperature T ; depending on the relative content
m
- of AT bps, T ranges between some 60 to 110◦C [2].
d m
n The classical Poland-Scheraga (PS) model views DNA
o as an alternating sequence of intact double-helical and
c denatured, single-stranded domains (bubbles or loops).
[
Double-helical regions are dominated by the hydrogen

2 bonding of bps as well as base stacking, bubbles by the
v entropy gain on disruption of bps [3]. The PS model is
8
fundamental in biological physics and has been progres- FIG.1: StretchedDNAinthePSmodelwithboundsegments
5
sively refined to obtain a quantitative understanding of B and denatured loops Ω. The DNA is attached between O
29 the DNA melting process [4, 5]. DNA denaturation can and L and subject to the stretching force F in x-direction.
PerfectmatchinginheterogeneousDNArequiresbotharches
. also be induced mechanically, by longitudinal stretching
9 of a loop to haveequallength ℓ.
ofsingle DNA moleculesby opticalor magnetic tweezers
0
7 oratomicforcemicroscopes[6,7,8,9]. Atthetransition,
0 theplotofstretchingforceF versusmeanDNAextension
DNA melting experiments.
: L exhibits a plateau at 60-90pN [10, 11]. Force-induced
v We treatthe chain in the grand canonicalensemble in
i destabilization of DNA has become a valuable tool, e.g.,
X whichthetotalnumberN ofbpsandtheend-to-endvec-
toprobetheinteractionofproteinsthatspecificallybind
torL fluctuate. The partitionfunction ind=3 becomes
r tosingle-strandedDNA,atphysiologicalmeltingtemper-
a
atures T (F) well below T (0) of free DNA [12]. ∞
m m
Z(z,F)= d3LZ (N,L)zN exp(βFL ) (1)
can x
In this Letter we consider the force-assisted denatura- N=1Z
X
tiontransitionofdouble-strandedDNAintheframework
with β =1/(k T). Z (N,L) is the canonicalpartition
of the PS model (Fig. 1). The thermodynamic state of B can
functionofachainofN bpswithfixedend-to-endvector
the DNA molecule now depends on both temperature T
Landz is the fugacity. We assumethat the forceF acts
and stretching force F. We find a bounded region of
inthe positivex-direction,andL isthe x-componentof
bound states in the (T,F) plane (Fig. 2). The shape x
L (Fig. 1). If bound segments and bubbles are indepen-
of the transition line implies that finite stretching forces
dent, Z factorizes:
F indeed lower the melting temperature T (F), and
m
the calculated force-extension relations F(L) exhibit a
∞
plateauoveracertainrangeofDNAextensionL(Fig.3). Z(z,F)=Ω +Ω [BΩ]n BΩ , (2)
e e e
Both observations are in agreement with force-induced ( )
n=0
X
2
the last term equaling Ω2B/(1−BΩ). The alternating Li (u) = ∞ uℓℓ−c is the polylog function [14], con-
e c ℓ=1
sequenceofboundsegmentsandbubbles withweightsB vergingfor |u|<1 for anyc. For u=1 three casesexist:
P
andΩinEq.(2)iscomplementedbytheweightΩ ofan (i) c ≤ 1: Li (1) diverges; (ii) 1 < c ≤ 2: Li (1) con-
e c c
′
open end unit at both ends of the chain. Note that only verges but Li (u) diverges; (iii) c > 2: Both Li (1)
c u=1 c
onestrandoftheendunitis boundtothe, say,magnetic and Li′(u) converge. The limit u = 1 corresponds
c u=1 (cid:12)
bead, while the other strand is moving freely. to the value z (F(cid:12))=exp(−αy2)/s of the fugacity; thus,
(cid:12) m
We model a bound segment with k = 1,2,... bps as a Ω(z,F) is o(cid:12)nly well-defined for z ≤ z (F) and diverges
m
rigidrodoflengthakwherea=0.34nmisthelengthofa for z > z (F). The statistical weight Ω of an end unit
m e
bound bp in B-DNA [10]. For simplicity we assume that modeledasidealrandomwalkmaybederivedinasimilar
the bindingenergyE0 <0perbpis thesameforallbps. way and one obtains Ωe(z,F)=AeLi0(u).
The statistical weight of a segment with fixed number k ForfreeDNAitwasfoundthatthenatureofthedenat-
and fixed orientation is then ωk with ω = exp(βε) and urationtransitionisdeterminedbytheanalyticbehavior
ε = −E0 > 0. Assuming that k fluctuates with fixed of Lic(u) at u=1: for c≤1 there is no phase transition
fugacity z, and rotates around one end while subject to inthe thermodynamic sense;for1<c≤2 the transition
theforceF (Fig.1),thestatisticalweightofthesegment issecondorder,andforc>2itisfirstorder[3,13]. One
for fixed z and F becomes finds c = 3/2 < 2 if the loops are ideal random walks.
∞ Self-avoidinginteractionswithinaloopmodifythisvalue
(ωz)k
B(z,ω,F) = dΩ exp(βFx) (3a) to c = 3ν = 1.76 with ν = 0.588 in d = 3 [15]. In both
4π
Xk=1 ZΩ cases the transition is second order. Self-avoiding inter-
1 1−ωze−y actionsbetweendenaturedloopsandtherestofthechain
= ln , y ≡βFa. (3b) were found to produce c = 2.12 > 2, driving the tran-
2y 1−ωzey
(cid:18) (cid:19) sition to first order [13, 16]. These results suggest that
Integration in Eq. (3a) is over the unit sphere with area the inclusionofself-avoidinginteractionsgenerallyshifts
4π, and x = akcosθ where θ is the polar angle between the loop exponent c to larger values, possibly effecting a
segment and x-axis. At F =0, B(z,ω,0)=ωz/(1−ωz) change of the transition from second to first order.
asfoundpreviouslyforthedenaturationtransitionoffree To see how c changes when self-avoiding interactions
DNA[13]. NotethatB isonlywell-definedforωzey <1; within a loop are included for the case F >0, we obtain
in what follows we assume z <e−y/ω. the weightsΩ andΩe foraself-avoidingwalkforℓ→∞.
Denaturedloops areconsideredasclosedrandomwalks Then, Eq. (4) holds with C0(2ℓ) ∼ µ2ℓℓ−dν being the
with 2ℓ monomers, corresponding to ℓ broken bps. This number of self-avoiding loops with 2ℓ monomers. The
loop starts at O and visits the point r after ℓ monomers probability density pℓ(r) scales as pℓ(r) = R−dg(r/R)
(Fig. 1). The number of configurations of a loop is where R = bℓν is the scaling length of a self-avoiding
walk and g(x) a scaling function. The function g(x) is
Ω(ℓ,r)=C0(2ℓ)pℓ(r) (4) not known for a self-avoiding loop. In what follows we
assume g(x) ∼ xφexp[−λxδ] for x → ∞ where λ > 0, φ
under this constraint, where C (2ℓ) counts the con-
0 is an exponent, and δ = 1/(1−ν) is determined by an
figurations of a loop of length 2ℓ starting at O and
argument by Fisher [15]. This form of g(x) is consistent
p (r) is the probability that the loop visits r after ℓ
ℓ withp (r)foraGaussianloop(ν =1/2,φ=0)obtained
monomers. For anidealrandomwalk ind=3,C (2ℓ)∼ ℓ
0 above. For the related linear self-avoiding walk starting
µ2ℓℓ−3/2 (µ is the connectivity constant) and p (r) ∼
ℓ atOandendingatrafterℓmonomers,theaboveformof
R−3exp[−λ(r/R)2] where λ>0, r =|r|, and R=bℓ1/2
g(x)alsoholdsandφcanbeexpressedintermsofknown
is the scaling length of the walk. The amplitude b is
exponents[15,17]. Forthepresentcaseofaself-avoiding
proportionaltothe persistencelengthofthe walk. Thus,
loopδ =1/(1−ν)stillholdsbutφisunknown. However,
Ω(ℓ,r)∼sℓℓ−3exp[−λ(r/R)2]wheres=µ2. Weassume
wewillseethatφdropsoutfromtheresultforΩ(z,F)in
that r moves freely and is subject to the force F in the
the limit ℓ →∞ at F > 0. The integral in Eq. (5) is no
positive x-direction. The weightofanidealrandomloop
longerGaussian,butcanbe evaluatedusingthe steepest
for fixed ℓ and F is given by the Gaussian integral
descentmethod atκ=βFbℓν →∞. It turns outthat in
this limit the integralis dominated by values r/ℓν →∞.
Ω(ℓ,F)= d3rΩ(ℓ,r)eβFx =Asℓℓ−cexp(αy2ℓ) (5)
With the above behavior of g(x) at x→∞ we find for a
Z self-avoiding loop [cf. Eq. (5)]
where A is an amplitude, c = 3/2, and α = b2/(4λa2).
Finally, we sum Ω(ℓ,F) over ℓ with weight zℓ to obtain Ω(ℓ,F)=Asℓℓ−cy1/(2ν)−1exp αy1/νℓ (7)
the statistical weight for an ideal random loop
(cid:16) (cid:17)
∞ for κ→∞ with the new loop exponent in d=3,
Ω(z,F)=A uℓℓ−c =ALi (u), u=szexp(αy2). (6)
c
c=4ν−1/2=1.85 . (8)
ℓ=1
X
3
FIG. 3: Force-extension curves f = Fa/ε as function of l =
hLxi/(ahNi) at fixed t1 < t2 < t3 < t4 (cf. Fig. 2b). Open
circles mark second-order transitions (c = 1.85 < 2). Inset:
f(l) for c=2.5>2 where the transition is first order.
fugacity is now given by z (F) = exp(−αy1/ν)/s. The
m
weight Ω (z,F) for an end unit obtains similarly, the
e
resultbeingEq.(9)withcreplacedbyζ =3/2+ν−2γ =
−0.232,using γ =1.16.
Phase diagram. We now obtain the transition line be-
tweenboundanddenaturedstates in the (T,F)-plane in
the thermodynamic limit N → ∞. For given fugacity z
the averagenumber of bps (open and closed) becomes
hNi=∂lnZ(z,ω,F)/∂lnz, (10)
FIG.2: Transitionlinesfm =Fma/εasfunctionoft=kBT/ε where we explicitly include the argumentω from Eq.(3)
for α = 1, s = 5 for denatured loops modeled as (a) ideal inthepartitionfunction(2). IfN issetonehastochoose
random walks and (b) self-avoiding walks (cf. Fig. 3).
afugacityz suchthatN =hNi; inthis casez becomesa
functionofω,F,andN. We denotethevalue ofz inthe
∗
limit N →∞ by z (ω,F)≡ lim z(ω,F,N). Similar to
Thus, with self-avoiding interactions within a denatured
N→∞
∗
loop and F > 0 the transition remains second order, the case F = 0 [13], z (ω,F) is the lowest value of z for
but moves closer to first order compared to free DNA which expression (10) diverges. In the bound state the
(with c = 3ν = 1.76 obtained within the same ap- divergence turns out to occur when the denominator in
proach). The amplitude A in Eq. (7) is proportional Ω2B/(1−BΩ)vanishes[seetextbelowEq.(2)],implying
e
to the cooperativity parameter σ ≪ 1 quantifying the z∗(ω,F) to satisfy
0
initiation of a loop in a previously intact double strand
∗ ∗
B[z ,ω,F]Ω[z ,F]=1, bound state. (11)
in the PS model [4, 5], such that also A ≪ 1. More-
over, α = 0.6...1.7 ≈ 1 using α = b2/(4λa2) obtained
Conversely,in the denatured state the divergence occurs
for an ideal random walk where b2/λ = 2L x /3; here,
p ss because ∂ Ω (z,F) diverges, which implies
z e
x = 0.6nm is the length of a base in single-stranded
ss
DNA [10] and values for the persistence length Lp for z∗(ω,F)=zm(F), denatured state, (12)
single-strandedDNAwerefoundtorangebetween0.7nm
[6] and 2nm [18]. where zm(F) is the critical fugacity obtained above
Finally,wesumΩ(ℓ,F)overℓwithweightzℓ toobtain (which is independent of ω). Thus, starting in a bound
thestatisticalweightforaself-avoidingloop[cf.Eq.(6)], state in the (T,F)-plane and approaching the transition
∗
linebyvaryingT andF,thevaluez (ω,F)isdetermined
Ω(z,F)=Ay−θLic szexp(αy1/ν) , (9) byEq.(11)andincreasesuntilitreachesthevaluezm(F)
from Eq. (12). At this point the denaturation transi-
h i
∗
where θ = 1 − 1/(2ν) = 0.15 in d = 3. The critical tion occurs. In the denatured state z (ω,F) is given by
4
∗ ∗
Eq.(12). RightatthetransitionbothEqs.(11)and(12) N. ForN →∞,dlnz +Θdlnω+ldy=0,z (ω,F)being
hold simultaneously. Using Ω[z (F),F] = Ay−θLi (1) the fugacity for hNi → ∞ as discussed above; Θ(ω,F)
m c
by definition of z (F) this implies A(βFa)−θLi (1) = andl(ω,F)are the bound bp fractionand reducedDNA
m c
1/B[z (F),ω,F], relating F and ω, or,equivalently, the extension in the same limit. For constant y = βFa (or
m
∗
reduced force f =Fa/ε and temperature t=k T/ε, for y =0)onefindsΘ=−∂lnz (ω,F)/∂lnω(sothatΘ=0
B
the transition line in the (t,f)-plane. in the denatured state due to Eq. (12) as expected). For
The shape of the transition line fm(t) depends on A, constant ω, corresponding to constant t = kBT/ε, we
α,ands. Fig.2ashowsf (t)forA=1,α=1,ands=5 find l(ω,F) = −(aβ)−1∂lnz∗(ω,F)/∂F. Based on this
m
forthecasethatdenaturedloopsareidealrandomwalks resultFig. 3 shows force-extensionrelations f(l)at fixed
(θ =0, ν =1/2). The transition line for the more realis- t1 < t2 < t3 < t4 for the case that denatured loops are
tic value A≪1 is also shown (here A= 0.01). The line self-avoiding random walks (cf. Fig. 2b). The two sets
f (t) separates a finite region of bound states from an of curves correspond to expansions of f(l) for small f
m
infinite region of denatured states. The point (t ,f =0) and close to the transition, respectively. The curves f(l)
0
with t =t (f =0)correspondsto the traditionalmelt- display flattened regions close to the transition, in qual-
0 m
ing transition for free DNA (F = 0). The line f (t) for itative agreement with experimental force-extension re-
m
A = 1 contains a region in which f (t) decreases with lations for DNA. These regions become less pronounced
m
t, such that increasedstretching forces f lower the melt- as t increases and vanish for t → t0 = tm(F = 0). A
ing temperature t (f), corresponding to force-induced force-extension relation for the case c>2, for which the
m
destabilization of DNA [10]. Interestingly, for A = 0.01, transition is first order, is also shown (here c = 2.5). In
applicationofasmallstretchingforcef firstincreasestm this casel(f)jumps discontinuouslyfromavaluel− to a
[19, 20]. Moreover, fm(t) vanishes for both t → t0 (as larger value l+ at the transition.
|t−t |1/2) and t → 0 (as α−1/2t1/2). This means that We have shown that a longitudinal stretching force F
0
for given 0<f0 <fmax, where fmax is the maximum of results in a reduced denaturation temperature Tm(F),
f (t),thechaindoesnotonlydenatureatalarget+(f ) corresponding to force-induced destabilization of DNA.
m m 0
but also at a small t−(f ), as indicated in [21]. This For the loop exponent in the presence of a finite F > 0
m 0
behavior can be traced back to a balance of the terms we found c = 4ν−1/2 = 1.85, so that the denaturation
(βFa)2 andβFa inz (F)=exp(−αy2)/s andEq.(3b), transition remains second order, but with an increased
m
respectively [22]. For (βFa)2 ≪ βFa, i.e., k T ≫ Fa, exponent. Itwouldbeinterestingtostudyhowthevalue
B
the melting transition at t+(f ) is mainly driven by the of c is modified when self-avoiding interactions between
m 0
entropy gain on creation of fluctuating loops, similar as a loop and the rest of the chain are included [13, 16].
for free DNA. For k T ≪ Fa the transition at t−(f ) ThisworkwassupportedbytheNIHthroughSCORE
B m 0
is due to the fact that B[z (F),ω,F] decreases with grantGM068855-03S1andbytheAFOSRthroughgrant
m
y = βFa = f/t in the denatured state, due to the rapid FA9550-05-1-0472(AH and MGO).
decay of z (F) [cf. Eq.(3b)] [23]. Fig. 2b shows the line
m
f (t) for self-avoiding loops with A = 1 and c = 1.85.
m
Note that Eq. (9) reduces to the known result for a free
self-avoiding loop (y =0) only if θ =0.15 is replaced by [1] A. Kornberg and T. A. Baker, DNA Replication
θ = 0; this is not a contradiction since Eq. (9) is based (W. H. Freeman, New York,1992).
on the assumption that κ = βbFℓν is large. To include [2] S. G. Delcourt and R. D. Blake, J. Biol. Chem. 266,
15160 (1991).
in Fig. 2b the behavior of fm(t) for f = Fa/ε → 0 we [3] D.Poland and H.A.Scheraga, J. Chem. Phys. 45, 1456
use Eq. (9) with θ =0.15 for y >1 and θ =0 for y ≤1.
(1966).
Force-extension relations. In thermal denaturation of [4] D. Poland and H. A. Scheraga, Theory of Helix-Coil
DNA one measures the fractionΘ of bound bps as func- TransitionsinBiopolymers(Academic,NewYork,1970).
tion of T. From the partition function (1) the average [5] R. M. Wartell and A. S. Benight, Phys. Rep. 126, 67
(1985).
number hMi of bound bps is hMi = ∂lnZ/∂lnω and
[6] S.B.Smith,Y.J.Cui,andC.Bustamante,Science271,
Θ = hMi/hNi with hNi from Eq. (10). Conversely,
795 (1996).
stretchingexperiments onDNA revealits response to an
[7] J. R. Wenner, M. C. Williams, I. Rouzina, and V. A.
applied mechanical stress. The mean of the component Bloomfield, Biophys. J. 82, 3160 (2002).
of the DNA extension along F is hLxi = β−1∂lnZ/∂F. [8] H. Clausen-Schumann, M. Rief, C. Tolksdorf, and H. E.
The average extension per bp in units of the bp-bp dis- Gaub, Biophys. J. 78, 1997 (2000).
tance a is l =hL i/(ahNi). For comparison with exper- [9] T. R.Strick et al.,Rep.Prog. Phys. 66, 1 (2003).
x [10] I. Rouzina and V. A. Bloomfield, Biophys. J. 80, 882
iments and simulations we calculate l in the thermody-
(2001).
namic limit hNi→∞. Consider the Gibbs-Duhem rela-
[11] C. Limouse et al., E-print arXiv:0704.3753v1.
tion for the thermodynamic potential lnZ(z,ω,F) [24]:
[12] K.Pant,R.L.Karpel,andM.C.Williams, J.Mol.Biol.
Ndlnz+Mdlnω+βLxdF =0. IfN isfixedoneobtains 327,571(2003); I.M.Sokolov,R.Metzler,K.Pant,and
dlnz+Θdlnω+ldy =0wherezisafunctionofω,F,and M. C. Williams, Biophys. J. 89, 895 (2005).
5
[13] Y.Kafri,D.Mukamel,andL.Peliti,Phys.Rev.Lett.85, are always aligned along the x-direction. This produces
4988 (2000); Eur. Phys. J. B 27, 132 (2002). a quadratic equation for fm with t as a parameter, and
[14] Seehttp://functions.wolfram.com. fm(t) is a combination of both branchesof thesolution.
[15] M. E. Fisher, J. Chem. Phys. 44, 616 (1966). [23] This relies on the assumption that pℓ(r) is Gaussian for
[16] E. Carlon, E. Orlandini, and A. L. Stella, Phys. Rev. ideal random loops and as described above Eq. (7) for
Lett.88, 198101 (2002). self-avoiding loops. For very large βF denatured loops
[17] D. S. McKenzie and M. A. Moore, J. Phys. A 4, L82 are stretched out and aligned along F so that the parti-
(1971). tionfunctionisdominatedbyparametervaluesforwhich
[18] M. C. Murphy et al.,Biophys. J. 86, 2530 (2004). pℓ(r)deviatesfrom thisform.Asuitablepℓ(r)shouldbe
[19] J.-B. Lee et al., Nature(London) 439, 621 (2006). used to obtain thephase diagram in this regime.
[20] M. C. Williams, J. R. Wenner, I. Rouzina, and V. A. [24] AssuchthepotentiallnZ(z,ω,F)dependsonlyoninten-
Bloomfield, Biophys. J. 80, 1932 (2001). siveparametersandwouldvanishduetotheEulerequa-
[21] H.Mao et al.,Biophys. J. 89, 1308 (2005). tion. This is avoided by formally including the system
[22] One may study the interplay between (βFa)2 and βFa volume V in thepotential. Here V =∞ is understood.
explicitlyinasimplifiedmodelinwhichboundsegments