Pulling hairpinned polynucleotide chains: Does base-pair stacking interaction matter?

8 Pages·0.24 MB·English

by  Haijun Zhou

Pulling hairpinned polynucleotide chains: Does base-pair stacking interaction matter?
Haijun Zhou
∗
Max-Planck-Institute of Colloids and Interfaces, D-14424 Potsdam, Germany
Yang Zhang
Institute of Theoretical Physics, The Chinese Academy of Sciences, P.O.Box 2735, Beijing 100080, China
(February 1, 2008)
1
0
Force-induced structural transitions both in relatively random and in designed single-stranded
0
DNA (ssDNA) chains are studied theoretically. At high salt conditions, ssDNA forms compacted
2
hairpin patterns stabilized by base-pairing and base-pair stacking interactions, and a threshold

n external force is needed to pull the hairpinned structure into a random coiled one. The base-pair
a stacking interaction in the ssDNA chain makes this hairpin-coil conversion a discontinuous (first-
J
order) phase transition process characterized by a force plateau in the force-extension curve, while

8 lowering thispotentialbelowsomecriticallevelturnsthistransition intocontinuous(second-order)
1
type,nomatterhowstrongthebase-pairinginteractionis. Thephasediagram (includinghairpin-I,

-II,andrandomcoil)isdiscussedasafunctionofstackingpotentialandexternalforce. Theseresults
]
t are in quantitative agreement with recent experimental observations of different ssDNA sequences,
f
o and they reveal thenecessity to consider thebase-pairstacking interactions in orderto understand
s thestructuralformation ofRNA,apolymerdesignedbynatureitself. Thetheoreticalmethodused
. may be extended to study the long-range interaction along double-stranded DNA caused by the
t
a topological constraint of fixed linking number.
m
PACS: 87.15.By, 05.50.+q, 64.60.Cn, 87.10.+e
-
d
n I. INTRODUCTION is therefore of both biological and theoretical interest.
o Recent measurements demonstrated that the struc-
c
turesofpolynucleotidesaresensitivetotheionicstrength
[ Single-macromolecular manipulation techniques, such oftheaqueoussolution.1,10–12 Inahighsaltenvironment
as atomic force microscopy and optical tweezer meth-
1 (for example, 150 mM Na+ or 5 mM Mg2+), electro-
ods, were recently used by many authors to investigate
v staticrepulsiveforcesbetweenthenegatively-chargednu-
6 the mechanical properties of double-stranded DNA (ds-
cleotide phosphate groups are largely shielded and base-
8 DNA) and to understand how the tension in dsDNA in-
pairing is favored; furthermore, experimental observa-
2 fluencestheinteractionsbetweenDNAandproteinssuch
tions suggested the formation of hairpinned configura-
1 as DNA/RNA polymerases, type II topoisomerases, and
0 tions in ssDNA.1 To understand the hairpin formation
RecAproteins(forabriefreviewseeRef.[1]). Manyuse-
1 in ssDNA and RNA, Montanari and M´ezard suggested
ful insights have been obtained through a detailed anal-
0 very recently a model for these polynucleotides.14 Base-
ysis of the experimental data generated by these precise
/ pairinginteractionisconsideredintheirmodel,andtheir
t measurements (see, for example, Refs. [2-9]). These new
a theoretical calculation can reproduce the experimental
m experimentaltechniqueswererecentlyusedalsoonRNA
force-extensioncurvemeasuredonassDNAchainwhose
and single-stranded DNA (ssDNA) to explore the struc-
d- tureformationofthesepolynucleotides.1,10–12 ARNAor nucleotidebasesarearrangedinrelativelyrandomorder.
It can be shown (see below and also Ref. [14]) that the
n ssDNAisalinearchainofnucleotidebases. Atphysiolog-
o icalconditions(concentration 0.1MNa+ andtempera- force-inducedstructuraltransitioninssDNAisasecond-
c ∼ order continuous phase transition process, characterized
ture 300K),RNA moleculesin biologicalcells canfold
: ∼ by a gradual decrease in the number of base-pairs as
v into stable native configurations (as in the case of pro-
Xi teins) to fulfill their various biological functions.13 The theexternalforceisincreasedbeyondacertainthreshold
value.
higher-order structures in RNA and ssDNA are caused
ar by the possibilities to form base-pairs between the com- Ontheotherhand,Riefetal.15 observedaquitediffer-
plementary nucleotide bases, A and U (T, in the case of entphenomenon: whentheypulledassDNAchainmade
ssDNA), and G and C. These structures can be further of poly(dA-dT) or poly(dG-dC) sequence apart with an
stabilized by the vertical stacking interactions between atomic force microscope, they found that the distance
nucleotide base-pairs. In a long polynucleotide chain, between its two ends suddenly changes from nearly zero
because there are a great many different ways to form to a value comparable to its total contour length during
base-pairs, the accessible configurational space is very a very narrow force range.
large,and the equilibrium configurationsare determined A question thus arises: Why is the force-induced
by the competition between entropy and the interaction structural transition of ssDNA gradual and continuous
energy. To investigate how base-pairing and base-pair in one case and highly cooperative in the other case,
stackinginfluence structure formationsofpolynucleotide even though the solution conditions in these two kinds
1
of experiments are comparable? We noticed that there on RNA secondary structure formation. We suggest the
may be strong base-pair stacking interactions in the de- possibility to extend the theoretical method in this pa-
signedssDNA chains,while stacking interactionsare ab- per toaddressthe long-rangeinteractionsalongtopolog-
sent or negligible in the relatively random sequences. ically constrained double-stranded DNA molecules, i.e.,
We have previously shown that the short-ranged base- dsDNA with fixed linking number.
pair stacking potential in dsDNA dominates the energy
contribution16anditisthereasonbothforthestabilityof
dsDNAanditshighextensibilityatlargeforces.7,8 Isthe II. MODEL OF SSDNA WITH BASE-PAIR
dramaticallydifferentexperimentalobservationofRiefet STACKING INTERACTION
al.15 (compared with those of [1,11,12]) caused by base-
pair stacking interaction? Given the fact that the stack- The model used in the present study is similar with
ingpotentialisakindofshort-ranged(nearest-neighbor) thatdevelopedearlierbyMontanariandM´ezard,14albeit
interaction, does the cooperative transition reported in with base-pair stacking potential included. We discuss a
[15]forsuchan1Dsystemreallydifferqualitativelyfrom fictitious polymer chain made of N tiny beads (bases)
that observed in [1,11]? with index i from 1 to N (each base in the model may
To address these questions, we extend the work of correspondtoseveralbasesinarealisticpolynucleotide).
Montanari and M´ezard14 to include the effect of stack- Betweenany two adjacent bases (i and i+1) there is an
ing interactions, and thus obtain a more realistic model elasticbondofequilibriumlengthbandlengthvarianceℓ
for polynucleotides. We find that when the base-pair (ℓ b, i.e., the bond is considerably rigid). For any two
≪
stacking potential is small, the ssDNA chain is in a bases i and j, if their mutual distance r is less than
ij
| |
hairpinned macro-state with the base-pairs only weakly aninteractionradiusatherecouldbe apairingpotential
stacked (hairpin-I), while when the stacking potential is Vi,j (r ) (and we say that an i-j base-pair is formed);
pair ij
large, the ssDNA is in a hairpinned macro-state with and if any two base-pairs are nearest neighbors to each
almost complete pairing and stacking (hairpin-II). The other (that is, i-j and (i+1)-(j 1)), there is an ad-
−
force-induced transition between hairpin-II and random ditionalstacking interactionJij,,ij+−11(rij,ri+1,j 1). In this
coil is first-order, while the transition between hairpin-I work, as what is assumed in other previous−theoretical
andrandomcoiliscontinuous. Thereisalsoacontinuous treatments,18–20 only the secondarypairing patterns are
phase transition between hairpin-I and hairpin-II, which considered, which leads to the following constraints on
may be inducedby changingthe nucleotide arrangement thepairingbetweenbases: (1)eachbasecanbeunpaired
or by environmental variations in temperature or ionic or be paired to at most one another base; (2) if base i is
strength. A simplified treatment of the same question paired to base j (suppose i<j) and k to l (k <l), then
was reported earlier.17 either i < j < k < l (“independent”) or i < k < l < j
The paper is organized as following: Section II in- (“nested”).
troduces the theoretical model of ssDNA; and Sec. III Becauseoftheserestrictions,thepartitionfunctioncan
presents some qualitative arguments concerned with the becalculatediteratively.14,18,19Inthecasewhenthepair-
structural transitions in the model of ssDNA; in Sec. IV ing interactionradius a is far less than bond equilibrium
weperformadirectcomparisonwithexperimentalobser- length b (a b), the total statistical weight Z (r) for
j,i
≪
vations. The main points of the paper are summarized a polynucleotide segment (from base i to j) whose ends
in Sec. V, accompanied by some qualitative discussions being separated by a distance r is expressed as
Z (r)= duµ(u)Z (r u)+f (r) du du µ(u )µ(u )Z (r u u )
j,i j 1,i ji 1 2 1 2 j 1,i+1 1 2
− − − − −
Z Z
j 2
−
+ du du du dvµ(u )µ(u )µ(u )f (v)Z (r u v)Z (v u u )
1 2 3 1 2 3 jk k 1,i 1 j 1,k+1 2 3
− − − − − −
k=i+1Z
X
+s(a−|r|)exp(−βVpja,iir(r)) du1du2µ(u1)µ(u2)gij,,ij+−11(r,r−u1−u2)Zj(−p)1,i+1(r−u1−u2)
Z
j 2
−
+ du1du2du3dvs(a−|v|)exp(−βVpja,kir(v))gkj,,jk−+11(v,v−u2−u3)×
k=i+1Z
X
Z (r u v)Z(p) (v u u ). (1)
k−1,i − 1− j−1,k+1 − 2− 3
Here, µ(r) exp( (r b)2/2ℓ2) is the prob- tion between bases i and j (this viral expansion form f
∝ − | |−
ability density for the bond vector r; f (r) = is introduced because that, in the first integral on the
ji
exp[ βVj,i (r )] 1 characterizes the pairing interac- right-hand side of the equality in Eq. (1), an vanishing
− pair ij −
2
bpaasireinigs pasostuenmteiadlbanetdwtehene tchoenternibdubtaiosnesjoafndalalntyheasneopthoesr- pexapir[−stβaJckij,,iijn+−g11(inrtije,rraic+t1io,jn−1(t)h]e−re1ascohnarfaocrttehriezeinstrtohdeubctaisoen-
sible base-paired configurations are included. Such spu- of the viral coefficient g can be similarly understood as
rious contributions should be removed since the pairing (p)
that for the introduction of the viral coefficient f). Z
j,i
potential is actually nonzero); s(x) is a signal function, is the statistical weight for a ssDNA segment whose two
s(x)=1ifx≥0and0otherwise;gij,,ij+−11(ri,j,ri+1,j−1)= end bases (i and j) forming a base pair:
Z(p)(r)=s(a r)exp( βVj,i (r)) du du µ(u )µ(u )Z (r u u )
j,i −| | − pair 1 2 1 2 j−1,i+1 − 1− 2
Z
+s(a−|r|)exp(−βVpja,iir(r)) du1du2µ(u1)µ(u2)gij,,ij+−11(r,r−u1−u2)×
Z
Z(p) (r u u ). (2)
j−1,i+1 − 1− 2
The statistical property of this model system is deter- the stacking potential to be a constant J and denote
mined by the coupled equations (1) and (2). In the gen- g =gij,,ij+−11(ri,j,ri+1,j 1)=const.
eral case where the pairing and stacking potentials are The Fourier trans−forms of the generating functions
dependent onbase index,it is certainly ofno hope to in- (Laplacetransforms)ofthestatisticalweightsaredefined
vestigatethepropertyofthesystemanalytically. Herewe as
simplifyourtaskbyassumingthatthepairingandstack-
ingpotentialsbetweenanybaseshavethesameform,i.e., ∞
Ξ(ζ,p)= dr Z (r)ζn exp(ip r), (3)
assuming the polymer chain to be homogeneous.21 Then n ! ·
Z n=0
in Eqs. (1) and (2) all the subscripts specifying specific X
bases can be dropped, especially we can write fji(r) as and
f(r), Z (r) as Z (r) and Z(p)(r) as Z(p)(r). In the
j,i j i j,i j i
model, the stackin−g potential is a function−of the inter- Ξ(p)(ζ,p)= dr ∞ Z(p)(r)ζn exp(ip r). (4)
base distances for the two base-pairs concerned. Since n ! ·
Z n=0
the base-pair’s inter-base distance is usually very small X
(less than a), we can approximate the stacking potential Considering the iterative expressions for Z and Z(p) in
n n
to be constant in this range, and hence we just denote
Eqs. (1) and (2), we can show that
Ξ(ζ,p)=[1+ζσ(p)Ξ(ζ,p)]
×
ζ2b3
1+ dqσ2(q)γ(p q)Ξ(ζ,q)+g dqσ2(q)γ′(p q)Ξ(p)(ζ,q) , (5)
(2π)3 − −
(cid:18) (cid:20)Z Z (cid:21)(cid:19)
and
ζ2b3
Ξ(p)(ζ,p)= dqσ2(q)γ′(p q) Ξ(ζ,p)+gΞ(p)(ζ,q) , (6)
(2π)3 −
Z h i
where σ(p) = drµ(r)exp(ip r) = η =gγb3(2π) 3 dqσ2(q)=gγ(4π) 3/2(b/ℓ), (9)
(sinpb/pb)exp( p2l2/2), with σ(0) = 1;· γ(p) = 1 − −
rSbe−ing3acRer|dra|e≤d≪aadsrbfi,n(wdr−ee)pehexanpvd(eiepnγt′·or=f)Rmγanoamdnedγn′bt(upomt)h=pγ.γaT(nphde)nγ+′E(cqaas/n.b()b53e). η2(D)=Dη1(cid:20)1+Z(b2ℓ/gπ3/2)Z dq1−σ2σ((qq))D(cid:21). (10)
and (6) lead to
Equations (7) and (8) are the central equations of this
article.
D(ζ)
Ξ(ζ,p)= , (7) When a external force F is applied at the end of
1 σ(p)D(ζ) the ssDNA chain, the total partition function is ZF =
− N
drZ (r)exp( βF r). The Laplace transform of this
N
where − ·
partition function is calculated to be
R
D(ζ)=ζ+ζΞ(p)(ζ)= η ζ3+η ζ2+ζ, (8)
1 2
− ∞
ZFζN =Ξ(ζ, iβF), (11)
with coefficient η a constant and η related to D: N −
1 2 N=0
X
3
where Ξ(ζ, iβF) is determined by Eq. (7). in the hairpinned state and the free energy density is
−
For a linear polymer system, the free energy is an ex- φ(F) = (1/2β)lnη (1/2β)ln(gγ) (again indepen-
1
− ∝ −
tensivequantityproportionaltothenumberofmonomers dentofF). ForF >F the systemisinthe randomcoil
cr
N in the thermodynamic limit. This indicates that state, and φ(F) = (1/β)ln(ζ ). Here ζ is deter-
pole pole
the smallest positive singularity point of the function mined by Eq. (12) (with D =ζ ). At F =F there is
pole cr
Ξ(ζ, iβF) in the variable ζ corresponds to the free en- a first-order hairpin-coil phase-transition, resulted from
−
ergy density of the ssDNA chain.22,23 This point is used the fact that dζ/dD =1.
|D=1/√η1
in the next section.
Comparing case A and case B, we have the impres-
sion that the inclusion of base-pair stacking interaction
may dramatically change the statistical property of the
III. QUALITATIVE ANALYSIS OF THE
ssDNA system, even the order of the hairpin-coil phase
HAIRPIN-COIL TRANSITION
transition. This is understandable qualitatively. The
stacking interaction has two effects: (1) it makes base-
It is evident that Ξ(ζ, iβF) has a pole ζ deter-
pole pairingevenmorefavorable;and(2)itcausestheformed
−
mined by
base-pairs to aggregate into large stacked blocks. Since
βFb theorderofthehairpin-coiltransitionisrelatedtothein-
D(ζ )=1/σ( iβF)= exp( β2F2ℓ2/2),
pole tensity ofthe stackingpotential,the hairpinmacro-state
− sinh(βFb) −
at low stacking intensity is anticipated to be different to
(12) the hairpin macro-state at high stacking intensity, and
a continuous phase transition between these two hairpin
where F = F. The function at the right side of the
| | macro-states can be predicted as the average base-pair
equalitymonotonouslydecreaseswithF from1atF =0
stacking intensity changes. This insight is confirmed by
to 0 asF . Onthe other hand,the functionD(ζ) is
→∞ observinghowtheorderparameterofthissystemchanges
related to the Laplace transform of the hairpinned con-
withstackingpotential,asshowninFig.1. Laterwewill
figurations as indicated by Eq. (8), therefore D(ζ) has
refer to the hairpin macro-state at low stacking inten-
a finite convergence radius, and it is an increasing func-
sity as hairpin-I and that at high stacking intensity as
tionof ζ before this radius is reached. The singularityof
hairpin-II.
D(ζ) is related to the roots of Eq. (8). Although analyt-
icalexpressionsfor the roots ofthis third-orderequation In the general case, the average number of base-pairs
are available, they are lengthy and here we discuss the (in units of N/2) is calculated according to N =
bp
statistical property of the system through an alternative 2∂φ/∂lnγ, and the average number of stacked base-
−
routine. First we consider two extreme cases: pairs (also in units of N/2) is N = 2∂φ/∂lng,
sbp
−
Case A: the base-pair stacking potential J = 0. and the relative extension of each ssDNA bond along
In this case g = 0 and η = 0 and Eq. (8) reduces to the direction of the external force is obtained by Ex =
1
second-order. This situation has been studied by Mon- ∂φ/∂F.
−
tanari and M´ezard14 and they found that D(ζ) has a
WedemonstrateinFig.1howthevaluesofN ,N ,
bp sbp
second-orderbranchingpointatζ equalingtothemax-
bp and the ratio N /N change with strength of base-
sbp bp
imum of the expression( 1+√1+4Dη )/2η , which is
2 2 pair stacking potential. In this figure γ is fixed to 1.9
−
reachedatD =D <1. When the externalforce is less
bp (just serves as an example) and the external force is
than the threshold value F determined by Eq. (12) at
cr set to zero. We see that as the stacking potential J
D = D , the polymer resides in the hairpinned phase
bp increases, all these three quantities increases and they
with zero extension, and the free energy density equals
approaches 1 as J reaches about J 6k T. The ra-
cr B
to φ(F) = (1/β)lnζ and is force-independent. At ≃
bp tio N /N could serve as an order parameter. Fig-
sbp bp
F = F there is a second-order continuous hairpin-coil
cr ure 1 shows that when stacking potential is high, almost
phase-transition (because dζ/dD = 0 at D ), and the
bp all the nucleotide bases are paired and stacked; lowering
free energy density is changed to φ(F)=(1/β)ln(ζ ).
pole stacking potential from J > J to J < J there is a
cr cr
Case B: the stacking potential J so large that g 1.
continuous phase transition where such a highly stacked
≫
Inthiscase,Eq.(10)indicatesthatη =Dη andEq.(8)
2 1 configurationisgraduallymeltedoutandagrowingfrac-
is equivalent to
tion of bases becomes unpaired and unstacked. This is
a structural transition between a highly stacked hairpin
(ζ+1/√η )(ζ D)(ζ 1/√η )=0. (13)
1 1
− − macro-state (hairpin-II) and a loosely stacked (or irreg-
We readily see that when D 1/√η1 the smallest ular) hairpin macro-state (hairpin-I). This hairpin I-II
≤
positive root of this equation is ζ = D; and when transition is induced by changing the average effective
D > 1/√η1, the smallest positive solution is a con- base-pair stacking interaction in the polymer. Giving a
stant ζ = 1/√η1 (gγ)−1/2. Most importantly, we polynucleotide chain, macroscopically the effective base-
∝
have dζ/dD = 1 as D approaches 1/√η1 from be- pairinginteractionisdeterminedmainlybyitsbasecom-
low. As a consequence, for F < F which is deter- position, while the effective stacking potential is deter-
cr
mined by Eq. (12) with D = 1/√η1, the polymer is minedmainlybytheparticulararrangementofthebases
4
alongthechain. Hence,fordifferentpolynucleotideswith thicksolidlineinFig.4). Suchacomparisonhasalready
the samebase composition,the folded hairpinconfigura- been performed by Montanari and M´ezardearlier.14 For
tions may be quite different and fall into two gross cat- random sequences the reason that the effective stacking
alogue, dependent on their particular base sequence ar- interaction is negligible may be explained as follows: In
rangement. This is consistent with the observation that the polynucleotide chain whose bases arrange randomly,
certain RNA chains have unique stable native configu- the formed base-pairs are usually seperated from each
rations. The transition between hairpins-I and -II can other and the possibility for two base-pairs to be adja-
also be induced by temperature changes or variations in cent to each other is quite small.
solution ionic concentrations. Even when the effective stacking potential J between
It should be mentioned that the critical stacking po- twoconsectivenode-pairsbecomescomparablewithther-
tential Jcr is not sensitive to the pairing potential. This mal energy kBT (the thin doshed line in Fig. 4), the
can be seen from the fact that the hairpin I-II transition gradual force-extension pattern is only slightly changed.
occurs when the second term in the square brackets of Itindicatesthatthe stackinginteractionstillplaysa less
Eq.(10)approacheszero,aconditionwhichissolelysat- important role than the pairing interaction. But the fit-
isfiedwhenthestackingpotential(andhenceg)becomes ting deteriarates much for J 2k T if the pairing po-
B
large. tential is kept constant (thin≥dotted line): for example,
The force-induced hairpin-coil transition is then at J =6K T (thin solid line) there is a wide plateau in
B
second-order or first-order depending on whether the the theoretical force-extension curve.
hairpinned configuration is type I or type II. In Fig. 2 Inanotherexperiment,Riefet al.15 synthesizedsingle-
theforcevsextensionrelationshipforssDNAatdifferent
strandedpoly(dG-dC)andpoly(dA-dT)DNAchainsand
pairingandstackinginteractionintensitiesareshown. As
investigated their elastic responses under external force
is expected, at large stacking intensity, there is a broad
field. Theyobservedahighlycooperativeelongationpro-
force-plateau and this force-plateau disappears as stack-
cess in both these two kinds of sequences, with the tran-
ing potential is lowered. It is striking to notice that only
sition force being 20 pN for poly(dG-dC) and 9 pN for
theinclusionofbase-pairstackinginteractioncanleadto
poly(dA-dT). Figure 5A and Fig. 5B shows the experi-
the appearanceof a highly cooperativehairpin-coiltran-
mentalrecordsaswellasthetheoreticalfittingsbasedon
sition. As shownalsoin Fig.2 (the dot-dashedcurve),if
the present model.
J =0theforce-extensioncurveisalwayscontinuousand
In the case of poly(dG-dC), the theoretical curve is
gradual no matter how strong the base-pairing interac-
obtained by calculating the total extension of a polymer
tion is.
connectedbyadsDNAsegmentof290.0nm(withpersis-
These theoretical predictions are summarized in the
tence length 53.0nm and stretch modulus S =1000.0pN
qualitative phase-diagramdepicted in Fig. 3.
as determined by previous experiments1) and a ssDNA
segment of 230.0nm (with Kuhn length 1.7nm and bond
length variance 1.105˚A as determined by the data of
IV. QUANTITATIVE COMPARISON WITH Fig. 4). (We have included a segment of dsDNA simply
EXPERIMENTAL OBSERVATIONS because the experiment in [15] was performed by insert-
ing a ssDNA segment between two dsDNA segments).
Inthehairpinnedstateatzeroforce,theaveragefreeen-
In the last section we have analyzed the main qual-
ergyperKuhnlengthisthusdeterminedbythetransition
itative predictions of the present polynucleotide elastic
force to be φ = (1/2β)ln(η ) = 5.59k T. Similarly,
model. In this section we apply these results to attain a 1 B
− −
in the case of poly(dA-dT), the fitting is obtained with
quantitative understanding of some reported experimen-
a dsDNA segment of 80.0nm and a ssDNA segment of
tal findings.
Bustamanteet al. andothers1,11,12 havepulledaplas- 445.0nm (b = 1.7nm, ℓ = 1.53˚A) with the average free
energy per Kuhn length being φ= 1.763k T.
mid ssDNA fragment of 10.4 kilobases under different B
−
ionic concentrations. They found that the elastic re- Assuming each Kuhn length contains three nucleotide
sponseofssDNAdeviatesfromthatofarandomcoilboth bases (i.e., each base in the model corresponds to three
athighandatlowionicconditions. Itwassuspectedthat realistic nucleotides), we can infer that the free energy
in high salt environments there is partial hairpin forma- perbase-pairinthepoly(dG-dC)chainis3.72kBT,while
tion in the ssDNA chain.1,11 As is consistent with this in the poly(dA-dT) chain is 1.18kBT. These values are
insight, Fig. 4 demonstrates that at highsalt conditions, close to what we estimated earlier17 and are close to
the ssDNA made of such a relatively random sequence the phenomenological parameters EG C 3.0kBT and
can be well modeled by the present model with an effec- EA T = 1.3kBT chosen by Bochelma−nn∼et al.25 to in-
tive base-pairinginteractioncharacterizedby the dimen- terp−ret their experimental results of separating the two
sionlessquantityγ =1.2andKuhnlengthb=1.7nmand strands of a dsDNA apart by mechanical force.
bondlengthvariationℓ=0.065b=1.105˚A. Theeffective
base-pairstackinginteractionforradomsequencesseems
to be quite small and is set to J = 0 in the fitting (the
5
V. CONCLUSIVE REMARKS have been observed.27
In this paper, we have presented an elastic model for VI. ACKNOWLEDGMENT
single-strandedDNAandRNApolymers,whereboththe
base-pairing and base-pair stacking interactions where
WearegratefultoProf. Z.-C.Ou-Yang. Oneoftheau-
considered. The theoretical calculations demonstrated
thors(H.Z.)appreciatesahelpfulcorrespondencewithA.
that,dependingontheintensityofbase-pairstackingpo-
Montanari, the helps of U. Bastolla and Jian-Jun Zhou,
tential,ssDNAorRNAcanformtwokindsofhairpinned
as well as the financial support of the Alexander von
structures, hairpin-I and hairpin-II, with the base-pairs
Humboldt Foundation. H.Z. is also grateful to Prof. R.
in hairpin-I only slightly stacked and those in hairpin-II
Lipowsky for a critical reading of the manuscript and
almost completely stacked. The force-induced hairpin-
comments.
coil transition is a second-order process if the polymer
was originally in hairpin-I macro-state and a first-order
process if it was in hairpin-II macro-state. The phase
diagram and the force-extension relationship for this
polymer system were obtained and the theoretical re-
sults achieved good agreement with experimental obser-
vations. This work indicates the significance of base- ∗ Author for correspondences. Email address:
pair stacking interaction to the structural property of [email protected].
polynucleotide chains. The existence of such a hairpin- 1C. Bustamante, S. B. Smith, J. Liphardt, and D. Smith,
II macro-statein a designed polynucleotide also justified Curr. Opin.Struct.Biol. 10, 279 (2000).
ourpreviousphenomenologicalworkonthe samesystem 2C. Bustamante, J. F. Marko, E. D. Siggia, and S. Smith,
by considering only single-looped hairpin structures.17 Science 265, 1599 (1994).
3J. Marko and E. Siggia, Macromolecules 28, 8759 (1995).
Thepresentworkhasthefollowingimplications. Given
4J.D.MorozandP.Nelson,Proc.Natl.Acad.Sci.USA94,
a polynucleotide chain, if the nucleotide bases are ran-
14418 (1997).
domly arranged, then the possibility to forming stacked
5T. R. Strick, J. F. Allemand, D. Bensimon, and V. Cro-
patterns of base-pairs is very small, since even forming
quette,Biophys. J. 74, 2016 (1998).
the smallest stack of two base-pairs requires the correla-
6C. Bouchiat and M. M´ezard, Phys. Rev. Lett. 80, 1556
tionsamongfournucleotidebases. Therefore,inrandom
(1998).
polynucleotide chains, stacking potential is not impor- 7H. Zhou, Y. Zhang, and Z.-C. Ou-Yang, Phys. Rev. Lett.
tant. Thechainscanonlyformloosehairpins(hairpin-I),
82, 4560 (1999).
and structural fluctuations between these hairpins may 8H. Zhou, Y. Zhang, and Z.-C. Ou-Yang,Phys. Rev. E 62,
belargeandquick,makingitdifficultfortheexistenceof 1045 (2000).
any well-defined stable native configurations. However, 9Y. Zhang, H. Zhou, and Z.-C. Ou-Yang, Biophys. J. 78,
thingsmaybedramaticallychangedifthepolynucleotide 1979 (2000).
chain composed of the same bases as the random chain 10S.B.Smith,Y.Cui,andC.Bustamante,Scinece271,795
rearranges these nucleotide bases carefully. If the bases (1996).
in the chain is arranged in such a way that forming of 11G. L. Wuite, S. B. Smith, M. Young, D. Keller, and C.
base-pairsresultsinstackingofbase-pairs,thenitisvery Bustamante, Nature 404, 103 (2000).
likely that the polymer will fold into certain stable na- 12B. Maier, D. Bensimon, and V. Croquette, Proc. Natl.
tivestructureswhicharemuchlowerinstructuralenergy Acad.Sci. USA 97, 12002 (2000).
than other configurations. Is this part of the reasonwhy 13J. D. Watson, N. H. Hopkins, J. W. Roberts, J. A. Steitz,
some polynucleotide sequences are folded in a particular and A.M. Weiner, Molecular Biology of the Gene, 4th ed.
manner (such as transfer RNA or t-RNA13)? The nu- (Benjamin/Cummings, Menlo Park, CA, 1987).
cleotide sequences in RNA molecules of biological cells 14A.Montanari and M. M´ezard, e-print cond-mat/0006166.
15M. Rief, H. Clausen-Schaumann, and H. E. Gaub, Nature
are the results of millions of years of natural evolution
Struct.Biol. 6, 346 (1999).
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16P.J.Hagerman, Annu.Rev.Biophys. Biophys.Chem. 17,
all be well designed.
265 (1988).
Finally, we just mention that the theoretical method 17H. Zhou, Y. Zhang, and Z.-C. Ou-Yang, Phys. Rev. Lett.
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86, 356 (2001).
tionsinssDNAmaybeappliedtoinvestigatethetertiary 18P. Higgs, Phys. Rev.Lett. 76, 704 (1996).
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The topological constraint leads to a long-range inter- 20A. Pagnani, G. Parisi, and F. Ricci-Tersenghi, Phys. Rev.
action along the dsDNA chain and make the polymer Lett. 84, 2026 (2000).
to fold into compacted plectonemic structures.26 Similar 21Sincethisworkisconcernedwiththemacroscopicalelastic
force-extensionprofilesasthetheoreticalcurvesinFig.4 property of polynucleotides and each fictitious base here
6
FIG. 2. The relationship between the average end-to-end
represents an average of several real bases of the polynu-
extensionofapolynucleotideandtheappliedforceatdifferent
cleotide, this can be a well justified treatment.
22S. Lifson, J. Chem. Phys. 40, 3705 (1964). effectivebase-pairingandbase-pairstackingintensities. Here
23D. Poland and H. A. Scheraga, J. Chem. Phys. 45, 1456 b denotes the Kuhn length of a ssDNA segment, γ accounts
for the intensity of the pairing potential between nucleotide
(1966).
24Y.Zhang, H. Zhou, and Z.-C. Ou-Yang(in preparation). bases, and J is the effective stacking potential between nu-
25U. Bockelmann, B. Essevaz-Roulet, and F. Heslot, Phys. cleotide base-pairs.
Rev.Lett.79, 4489 (1997); Phys.Rev.E 58, 2386 (1998).
26J.F.MarkoandE.D.Siggia,Phys.Rev.E52,2912(1995).
27T. R. Strick, J. F. Allemand, D. Bensimon, A. Bensimon,
and V. Croquette, Science 271, 1835 (1996).
Random coil
F
1
discontinuous
continuous
0.8
Hairpin−II
0.6
No. of base−pairs (Nb p )
No. of stacked base−pairs (Ns b p ) Hairpin−I continuous
0.4
N s b p /N b p
0 Jcr J
0.2
FIG.3. Thequalitative phase-diagram ofa polynucleotide
at high-salt conditions. The phases hairpin-I and hairpin-II
are defined in the main text. F denotes the intensity of the
0
0 2 4 6 external force.
Stacking Potential (kB T)
FIG. 1. The relationship between the order parameter
Nsbp/Nbp and the effective stacking potential. Here Nbp is
thetotal numberof paired bases (in units of N/2), and Nsbp
isthetotalnumberofstackedbase-pairs(alsoinunitsofN/2). 100
In the numerical calculation of this figure, γ (it accounts for
thepairing intensity) is set to 1.9 and externalforce F =0.
10
7 N)
p
e ( J=0, γ=0
6 orc J=0, γ=1.2
F
1 J=1.0kB T, γ=1.2
5 J=2.0kB T, γ=1.2
J=6.0kB T, γ=1.2
T/b)4
k B
e (
orc3 00 0.5 1
F
J=6.0k T, γ=1.9 Relative Extension (b)
2 B FIG. 4. Force vs extension curves for a ssDNA polymer
J=5.0kBT, γ=1.9
withitsbasesrandomlyordered. Theexperimentaldatacome
J=3.0k T, γ=1.9
1 J=1.0kBBT, γ=1.9 fcreonmtraRtieofn. [i1s]5(mpMlusMes)g2a+n.dT[h1e2]th(seqoureatriecsa)l,caanlcdultahteioinonaircecpoenr--
J=0, γ=500.0 formed with b = 1.7nm and ℓ = 1.105˚A, and the other two
0
00 0.2 0.4 0.6 0.8 fittingparameters (γ and J) are listed in the figure.
Relative Extension (b)
7
50 30
A B
40
poly(dA−dT): experimental records
20
Theoretical fitting: ln(η )=3.52741
1
30
Force (pN) 20 Force (pN) 10
10
poly(dG−dC): experimental records
Theoretical fitting: ln(η )=11.18528 0
1
0
−10
200 250 300 350 400 450 500 550 −10
0 100 200 300 400 500
Total Extension (nm)
Total Extension (nm)
FIG. 5. Force vs extension curves for (A) poly(dG-dC)
and (B) poly(dA-dT) ssDNA chains. The experimental data
are from Ref. [15] and the theoretical fittings are done with
b=1.7nm,andℓ=1.105˚Ain(A)andℓ=1.530˚Ain(B).The
parameter lnη1 (see Eq. 9) is determined by the transition
force,andissetto11.18528 and3.52741, respectively. Toac-
countforthelowextensiondataasegmentofdouble-stranded
DNAisincludedinthefitting(sincetheexperimentwasper-
formedbyinsertingassDNAsegmentinbetweentwodsDNA
segments).
8