Yu-Jane Sheng, Jeff Z. Y. Chen, and Heng-Kwong Tsao- Open-to-Closed Transition of a Hard-Sphere Chain with Attractive Ends

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Open-to-Closed Transition of a Hard-Sphere Chain with Attractive EndsYu-Jane Sheng, Jeff Z. Y. Chen, and Heng-Kwong Tsao

Macromolecules , 2002, 35 (26), 9624-9627 • DOI: 10.1021/ma025604x

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O pe n-t o -C lo s e d T ra ns i t i o n o f a H a r d-Sphe r eC ha i n w i t h A t t r a c t i v e E nds

Yu - J a n e S h e n g ,† Je ff Z. Y. Che n, ‡ a n dH e n g -K w o n g T s a o *,§

Departm ent of Chemical Engineering, National TaiwanUniversity, Taipei, T aiwan 106, Republic of China; Departm ent of Physics, Un iversity of Wa terloo, Waterloo,Ontario, Canada N2L 3G1; and Department of Chemical

and Materials E ngineering, N ational Central University,Chung-li, Taiw an 320, Republic of Chin a

Received July 24, 2002

Revised Manuscript Received November 5, 2002

The h airpin str ucture is frequent ly observed in bothsingle-stra nd DNA (ssDNA) and RNA and part icipatesin many biological functions.1-4 A typical hairpin loopconsists of a l inear polymer segment connected t obinding monomers. I t fluctuat es thermodynamicallybetween different conformat ions, wh ich, in a sim plifieddescription, ar e divided into two main sta tes: the openstate where the binding monomers are separated andthe closed one where the binding monomers form abond. Characterization of the nature of the open-to-

closed tr ansit ion is essential to our understanding of biopolymer dynamics.

Recently, pa rticularly simple DNA ha irpin-loop str uc-tures were purposefully designed to investigate theirconformational fluctuations.5,6 The stem is formed by afew bases, complementar y to each oth er, at t he two endsof ssDNA. The loop is m ade of a homogeneous sequ encesuch as poly(T) (polydeoxythymidines). By att achingdonor and acceptor fluorophores to both ends of a ssDNAhairpin, the open-to-closed conforma tional dynamicshave been investigated.5,6 The fraction of the open stateis displayed in ter ms of the melting curve, which depictsthe variation of the static fluorescence intensity withtemperature. The m elting temperature T m of the struc-

t ur e i s def i ned as t he t em per at ur e w her e t he pr ob-abilities of closed a nd open stat es a re equa l.The all-or-none t wo-sta te model is comm only adopted

to describe th e t herm al equilibrium between closed (c)and open (o) conformations,1-3,5 -7

where k if j is the rate coefficient jumping from i t o jstate. The fraction of open state is f ) 1 / (1 + K ).However, other physical mechanisms may cause thedeviation from the two-state model. For example, theopen state may actually be further classified into an

open, random-coil and a mismatched-loop state. On theother hand, it is also difficult to tell whether the closedstate corresponds to a ha irpin structure with some basepairs int act or wheth er it is merely a compa ct, collapsedstru cture, where th e donor a nd a cceptor are qu ite close.7

Moreover, factors such as transit ion state7 and bas estacking5,7 also complicate the thermodynamics of the

open-to-closed transition. In this Communication, werule out those complications a nd concentra te on inves-tigating the open-to-closed tra nsition by considering ahard-sphere chain with attra ctive ends.

We perform off-latt ice Mont e Ca rlo (MC) simula tionsto obtain th e melting curves and transit ion rates. Thebiopolymer is modeled as a freely jointed, hard-spherechain with N beads of diameter σ . Bonded beads i a n di + 1 intera ct via an infinitely deep squa re-well poten-

tial8

The terminal beads 1 and N interact via a square-wellpotential

At each MC step, a ra ndomly selected bead on th e chain

was allowed to move around its previous position witha r estriction of the bond fluctua tion between σ and 1.2σ .The number of MC steps per bead is more than 107. Theconformation is clearly identified as the closed statewhen U 1 N < 0, i.e., σ e |r 1 - r N | e 1.2σ , and as t he openstate otherwise. Without the loss of generality, weassume the binding energy ) 15. Figure 1 depicts t hemelting curve ( f -T ) for different chain lengths. Theresult is qualitatively consistent with experimentalobservations. As the temperature r ises, the probablestructure shifts from a stable closed state to a stableopen one. Moreover, the melting temperature of thestructure decreases with increasing chain length. Thedegree of th e melting temperature decline becomesmuch less substantial when the chain length is long

enough.Since the two-stat e model is perfectly fulfilled, a

thermodynamic theory can be adopted to obtain themelting curve, which is examined by simu lation results.At equilibrium, th e principle of detailed balan ce gives

where Pi i s t he pr obabi l i t y of t he s t at e i. T h e r a t ecoefficient is assumed to follow the Arrhenius kinetics,k if j ) k *if j exp(- βF if j), where β is the inverse temper-a t u r e β ) 1/ k T an d F if j denotes the free energy barrierassociated with jumping from the i t o j states. Whenthe conformation changes from the open state to theclosed one, th e free ener gy barr ier corresponds only to

the entr opy loss from a ran dom coil to a r ing polymer, βF ofc ) -(S c - S o)/ k . Hence, k ofc is temperature-independent. Moreover, the entropy cost varies withchain length and is logarithmic.9,10 In a Gau ssian chain,the closed-conformation probability is proportional to

N -3/2 and the open-conformation probability is ap-proximately 1 - O( N -3/2).10,11 As a result, ∆S / k ≈ R ln

N with R ) 3 / 2. However, for a ha rd-sphere chain, R ≈

2, as will be shown later. On the other hand, when thestructure fluctuates from the closed state to the openone, t h e fr ee ener gy bar r ier i s s im ply t he bindingenergy, βF cfo ) β.

* To whom correspondence should be addr essed. E-mail hkt [email protected].

† National Taiwan University.‡ University of Waterloo.§ National Centra l Un iversity.

open 798

k ofc

k cfo

closed, with K (T ) )k ofc

k cfo

U i,i+1 ) {∞, r < σ

0, σ er e1.2σ

∞, r > 1.2σ (1)

U 1, N ) {∞, r < σ

-, σ er e1.2σ

0, r > 1.2σ

(2)

k ofcPo ) k cfoPc (3)

9624 Macrom olecules 2002, 35 , 9624-9627

10.1021/ma025604x CCC: $22.00 © 2002 American Ch emical SocietyPublished on Web 11/21/2002



Since f ) Po a n d Po + Pc ) 1, using eq 3, one obtains

∆S is th e ent ropy chan ge from open to closed sta te. AtT ) T m, f ) 1 / 2. Consequently, the preexponential factorin eq 4 is related to the melting temperatu re,

Combing eqs 4 and 5 yields the melting curve,

Equation 5 also shows that the melting temperaturedecreases with increasing chain length logarithmically,

Thermodynamic quantities commonly analyzed at thetra nsition of polymer conformations ar e th e energy permonomer and the specific heat. Because the meltingcurve represents t he probability of the open stat e, heatcapacity curve can be evaluat ed directly from eq 6,

The rate constants are evaluated from MC simula-tions and compared to our thermodynamic theory. Forthe two-state model, the rate constant k if j is inverselyrelated to the mean l ifetime of the state i, i.e., k if j )

τ i-1. As shown in Figure 2a, when ln k cfo-1 is plotted

against β, all the ra te consta nts evaluated from differentchain lengths collapse into a single line with a slope of the binding energy . This consequence indicates thattherm al fluctuations pr ovide a probability of exp(- β)to u nbind the closed conformation regardless of th echain length. On t he other ha nd, the ra te consta nt fromt he open t o cl os ed s t at e k ofc is independent of the

temperature but decreases with increasing chain length.This result clearly shows tha t thermal fluctuat ionsfurnish a constant probability of exp[-∆S ( N ) / k ] to formthe closed state. When ln(k ofc

-1 / N 2) is plotted against β, as i l l us t r at ed i n Fi gur e 2b, al l t he r at e cons t ant scom put ed fr om differ ent chai n l engt hs fal l i nt o aconstant line with zero slope. In accordance with oursimulation results, the entropy loss from a coil to a ring

is ∆S(N)/ k ∼ N 2. In comparison with the Gaussianchain, the scaling exponent of N rises from 3 / 2 to 2 dueto excluded volume.

The entropy loss from the open to closed state can befurther confirmed by considering the distribution func-tion associated with a self-avoiding walk of N steps.11

The probabili ty for a terminal point adjacent to theorigin is given by p N ∼ N -(3+g)ν. For good solvents, oneha s ν ≈ 3 / 5 a n d g ≈ 0.28. Consequently, ∆S / k ≈ R ln N with an entropy reduction exponent R ) 1.968, wh ichis quite close to our numerical value 2. For the idealchain, ν ) 1 / 2 an d g ) 0, one recovers R ) 3 / 2. This resultis also closely related to the mean first passage t imefor th e close a pproach of the ends of a polymer.12

The inverse rate constant k of

c

-1 is the mea n lifetimeof the open state τ o, which is irrelevant to the bindingenergy and expected to be closely related to the meanfirst passage time of cyclization τ R of a diffus ion-limit edintrachain reaction. Previous studies point out that τ Ris proportional t o N R with 3 / 2 e R e 2 for a Rouse chainof bond length b.12,13 When the contact distance a issmall compared to b / N 1/2, τ R is much greater than thelongest relaxation t ime τ m and a local equilibriumassumption is valid. Thus, one has τ R ∝ N 3/2. O n t heot her hand, i n t he l i m i t of l ar ge N at fixed a, τ R isbelieved to be comparable to τ m.13,14 Since τ m ∝ N 1+2ν,10

τ R is found to be N 2 for a Gaussian chain (ν ) 1 / 2). Fora hard-sphere chain adopted in our simulation, oneant i ci pat es t hat , s i m i l ar t o t he G aus s i an chai n, t he

F i g u r e 1 . Melting curves for different chain lengths.

f )1

1 + K )

1

1 + [k *ofc

k *cfo

exp(- ∆Sk )] exp( β)

(4)

k *ofc

k *cfo

exp(-∆S

k ) ) exp(- βm) (5)

f (T ) )1

1 + e( β- βm )

(6)

k T m) R ln N + ln (

k *cfo

k *ofc) (7)

C

k )

f (1 - f )2

(k T )2 ) ( β)2 e( β- βm )

[1 + e( β- βm )

]2 (8)

F i g u r e 2 . Variation of the ra te constant s with th e tempera-t u re β for different chain lengths. (a) τ c ) k cfo

-1; (b) τ o ) k ofc-1.

Macromolecules, Vol. 35, No. 26, 2002 Commu nications to t he E ditor 9625



exponent is in the range 2.0 e R e 2.2 for a swollen

chain (ν ) 3 / 5).15 The current result implies that ourcontact distance in eq 2 leads to a local equilibriumcondition, and hence R ) 2. Following this l ine of reasoning, one a lso expects that the contact distancew oul d s hi ft t he m elt i ng cur ve and hence al t er t hemelting temperature, as will be discussed later.

Using ) 15, eq 6 gives t he m elt i ng cur ves i nexcellent a greement with th ose in Figure 1. The meltingtemperature obtained by the fi t t ing procedure is con-sistent with that determined directly from the simula-tion data at f ) 1 / 2. Using the pr edetermined T m, all themelting curves associated with different chain lengthscollapse into a stra ight line with a slope as shown inFigure 3, when we plot ln( f -1 - 1) against ( β - βm). The

dat a becom e s li ght l y s cat t er ed at l ow t em per at ur ebecau se of the meta stable stru cture of the closed sta te.The variation of the melting temperature βm with chainlength ln N is depicted in Figure 4. The da ta points canbe well represented by a straight l ine with a slope R

equal t o 2. This result is predicted in eq 7 an d consistentwith that obtained from the chain length dependenceof k ofc

-1 in Figure 2b.

It is natural to identify the peak temperature of heatcapacity curve T c as th e tran sition temperatu re of open-to-closed transition. When we plot (C / k )( β)-2 against( β - βm) in Figure 5, all data points of different chainlengths collapse into a single curve as indicated by eq8. Solving eq 8 yields T c ≈ T m /[1 + 4( βm)-2] a s ( βc -

βm

) < 1. Consequently, the peak temperatu re is a lwaysless than the melting temperature. Figure 1 and eq 6show that open-to-closed transition is continuous for achain of finite length. As the chain length N increases,th e en tr opy loss of forming a closed loop also increases,and the formation of a closed conforma tion becomesmore difficult . To ma intain a finite T m, t he bi ndi ngenergy must also increase as well, according t o eq 7.In reali ty, this is usually achieved by increasing thenumber of binding monomers at the stem. In the currentmodel, we consider two single attractive ends with aneffective binding energy which should increase loga-rithmically in N . In th e ther modynamic limit of N f ∞,the derivative of the melting curve yields (T d f /dT )T )T m

) (1/4)( / k T m) f ∞ based on eq 6. The smooth open-to-

cl os ed t r ans it i on of a fi nit e chai n becom es a s t epfunction. In the meantime, T c approaches T m, and t hepeak posit ion in h eat capacity curve represents thetran sit ion t emperature. One can furth er show that th efree en ergy un dergoes a finite jump across the closed-to-open transit ion temperature in t he thermodynamiclimit, indicating a first-order pha se tr ansition.

In this Communication, we have established a simpleth erm odynamic th eory for th e open-to-closed tr an sitionbased on a two-sta te picture. It describes excellently th emelting curves of a hard-sphere chains with attractiveends, which are obtained by MC simulations. For achain with N ) 15, we obtain k T m / ) 0.15 for thecontact distance a ) 1.2σ . In a fluorescence experimen t,Wallace et al.6 attached a single strand of poly-T tostems consisting of five-base sequence TTGGG a t oneend and i ts complementary AACCC at the other end.T hey eval uat ed t he appar ent ent hal py change ∆ H ≈-40 k T a t T ) 298 K. If we adopt t he binding ener gy

) -∆ H in our model, the estimation of the meltingtemperature is evidently too high. To obtain a reason-able m elt i ng t em per at ur e bet w een t he fr eezing and

F i g u re 3 . Melting curves for different chain lengths arereplotted with ln( f -1 - 1 ) as a fu n ct i on of ( β - βm) a n dcompared to eq 6.

Figure 4. Inverse melting temperature is plotted against thechain length. The straight line denotes eq 7 with R ) 2.

F i g u r e 5 . Heat capacity curves for different chain lengthsare p l o t t ed w i t h (C / k )( β)-2 as a fu n ct i on of ( β - βm) a n dcompared to eq 8.

9626 Commu nications to t he E ditor Macromolecules, Vol. 35, No. 26, 2002



boiling temperatures of water, one must have 0.023 <

k T m / < 0.031. Since th e melting tem perat ure declineswith decreasing the contact distance, we are able toobtain T m w i t hi n t hi s r ange as a f σ . For example,k T m / ≈ 0.10 for a ) 1.01σ an d k T m / ≈ 0.08 for a )

1.001σ . This small value of the conta ct distance (a - σ )seems unrealist ic. I t is because we have employed aspherical excluded volume associated with attractiveinteractions. In the ssDNA experiment, two stems cancome much closer to each other. We have performedsimulations for a h ard-sphere chain of N ) 40 with fiveattra ctive spheres at both ends. The melting tempera-ture for a ) 1.2σ is a pproximately k T m / ≈ 0.037, whichis smaller but still not within reasonable range. Noteth at th e effect of misfolded configura tion, th e AT bond-ing of the AA segment in the stem with any TT segmentin poly-T loop, is not considered in the present model.O ur M C s t udi es s how t hat s uch m i sm at ched l oopconformat ions can significantly reduce the meltingtemperat ure. The base-stacking interactions, which arealso not accoun ted for h ere, lower th e melting tem per-ature furthermore.5

Finally, it is worth mentioning that our work couldhave relevance beyond the world of DNA. For example,

i t m ay be helpful i n under st andi ng t he behavior of associating systems such as telechelic polymers.16 E a r -lier work focus mainly on interactions between manychains below the melting temperature.17 T hat i s , t hebinding energy is large compared to the thermal energy.Our present study provides th e effect of temperatur ek T / on conformat iona l kinet ics of a t elechelic polymer.

A c k n o w l e d g m e n t . Y.-J.S. a nd H.-K.T. th ank NSCof Taiwan for finan cial su pport, a nd J .Z.Y.C. gra tefullyacknowledges financial support from National Center

of Theoretical Science of Taiwan. We also thank B.-Y.Ha for bringing ref 13 to our attention.

R e f e re n c e s a n d N o t e s

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(3) Ansar i, A.; Kuznets ov, S. V.; Shen, Y. Proc. Natl. Acad. Sci.U.S.A . 2001, 98 , 7771.

(4) Zhang, W.; Chen, S.-J. Proc. Natl. Acad. S ci. U.S.A. 2002,99 , 1931.(5) Goddard, N. L.; Bonnet, G.; Krichevsky, O.; Libchaber, A.

Phys. Rev. Lett. 2000, 85 , 2400.(6) Wallace, M. I.; Ying, L.; Balasubra man ian, S.; Klenerman ,

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2001, 334 , 145.(8) Zhou, Y.; Hall, C. K.; Karplu s, M. Phys. Rev. Lett. 1996, 77 ,

2822.(9) Bundschuh, R.; Hwa, T. Phys. Rev. Lett. 1999, 83 , 1479.

(10) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics;Oxford University Pr ess: New York, 1988.

(11) De Gennes, P .-G. S caling Concepts in Polymer Physics;Cornell University P ress: Itha ca, NY, 1993. Des Cloizeaux,J . ; J a n n in k , G . Polymers in S olution: Their Modelling &S tru ctu re; Oxford University P ress: Oxford, 1990.

(12) Wilemski, G.; Fixman, M. J. C h em. Ph ys. 1974, 60 , 866.

Wilemski, G.; Fixman, M. J. C h em. Ph ys. 1974 , 60 , 878.Szabo, A.; Schulten, K.; Schulten, Z. J. C h em. Ph ys. 1980,72 , 4350.

(13) Pa stor, R. W.; Zwan zig, R.; Szabo, A. J. C h em. Ph ys. 1996,105 , 3878.

(14) Doi, M. Chem. Phys. 1975, 9, 455.(15) A. Dua; B. Cherayil, J. Chem. Phys. 2002, 116 , 399.(16) Winnik, M. A.; Yekta, A. Curr. Opin. Colloid Interface Sci.

1997, 2, 424.(17) Baljon-Haakman, A. R. C.; Witten, T. A. Macromolecules

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MA025604X

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Yu-Jane Sheng, Jeff Z. Y. Chen, and Heng-Kwong Tsao- Open-to-Closed Transition of a Hard-Sphere Chain with Attractive Ends

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