modeling dna structure, elasticity, and deformations at...
TRANSCRIPT
PHYSICAL REVIEW E 68, 021911 ~2003!
Modeling DNA structure, elasticity, and deformations at the base-pair level
Boris Mergell,* Mohammad R. Ejtehadi, and Ralf Everaers†
Max-Planck-Institut fu¨r Polymerforschung, Postfach 3148, D-55021 Mainz, Germany~Received 24 January 2003; published 20 August 2003!
We present a generic model for DNA at the base-pair level. We use a variant of the Gay-Berne potential torepresent the stacking energy between the neighboring base pairs. The sugar-phosphate backbones are takeninto account by semirigid harmonic springs with a nonzero spring length. The competition between these twointeractions and the introduction of a simple geometrical constraint lead to a stacked right-handedB-DNA-likeconformation. The mapping of the presented model to the Marko-Siggia and the stack-of-plates model enablesus to optimize the free model parameters so as to reproduce the experimentally known observables such aspersistence lengths, mean and mean-squared base-pair step parameters. For the optimized model parameters,we measured the critical force where the transition fromB- to S-DNA occurs to be approximately 140 pN. Weobserve an overstretchedS-DNA conformation with highly inclined bases which partially preserves the stack-ing of successive base pairs.
DOI: 10.1103/PhysRevE.68.021911 PACS number~s!: 87.14.Gg, 87.15.Aa, 87.15.La, 61.41.1e
nn
ioa
ofe
e
thurtharrtiote
c-
g
m
or
in
d
th
bedis-
rlyble
inpla-
cro-veal
raleteden-
bleaints
of
st-s.ill
re-nce
cep-of
ultssd inf vanigh-gar-
lass
I. INTRODUCTION
Following the discovery of the double helix by Watsoand Crick@1#, the structure and elasticity of DNA has beeinvestigated on various length scales. The x-ray-diffractstudies of single crystals of DNA oligomers have led todetailed picture of the possible DNA conformations@2,3#with atomistic resolution. Information on the behaviorDNA on larger scales is accessible through nuclear magnresonance@4# and various optical methods@5,6#, such asvideo @7# and electron microscopy@8#. An interesting devel-opment of the last decade is the nanomechanical experimwith individual DNA molecules@9–13# which, for example,reveal the intricate interplay of supercoiling on large lengscales and local denaturation of the double-helical struct
The experimental results are usually rationalized inframework of two types of models: base-pair steps and vants of the continuum elastic wormlike chain. The first, molocal, approach describes the relative location and orientaof the neighboring base pairs in terms of intuitive paramesuch as twist, rise, slide, roll, etc.,@14–17#. In particular, itprovides a mechanical interpretation of the biological funtion of particular sequences@18#. The second approach models DNA on length scales beyond the helical pitch aswormlike chain~WLC! with empirical parameters describinthe resistance to bending, twisting, and stretching@19,20#.The results are in remarkable agreement with the nanochanical experiments mentioned above@21#. WLC modelsare commonly used in order to address biologically imptant phenomena such as supercoiling@22–24# or the wrap-ping of DNA around histones@25#. In principle, the two de-scriptions of DNA are linked by a systematic coarse-grainprocedure. From the given~average! values of rise, twist,slide, etc., one can reconstruct the shape of the corresponhelix on large scales@14,18,26#. Similarly, the elastic con-stant characterizing the continuum model is related to
*Electronic address: [email protected]†Electronic address: [email protected]
1063-651X/2003/68~2!/021911~15!/$20.00 68 0219
n
tic
nts
e.ei-en
rs
-
a
e-
-
g
ing
e
local elastic energies in a stack-of-plates model@27#.Difficulties are encountered in situations which cannot
described by a linear response analysis around the unturbed (B-DNA! ground state. This situation arises reguladuring cellular processes and is therefore of considerabiological interest@18#. A characteristic feature, observedmany nanomechanical experiments, is the occurrence ofteaus in force-elongation curves@10,11,13#. These plateausare interpreted as structural transitions between the miscopically distinct states. While atomistic simulations haplayed an important role in identifying the possible locstructures such asS- and P-DNA @11,13#, this approach islimited to relatively short DNA segments containing sevedozen base pairs. The behavior of longer chains is interpron the basis of stack-of-plates model with step-type depdent parameters and free energy penalties for non-B steps.Realistic force-elongation curves are obtained by a suitachoice of parameters and as the consequence of constrfor the total extension and twist~or their conjugate forces!@28#. Similar models, describing the nonlinear responseB-DNA to stretching@29# or untwisting@30,31#, predict sta-bility thresholds forB-DNA due to a combination of morerealistic, short-range interaction potentials for rise with twirise coupling enforced by the sugar-phosphate backbone
Clearly, the agreement with the experimental data wincrease with the amount of details which is properly repsented in a DNA model. However, there is a strong evideboth from atomistic simulations@32# as well as from theanalysis of oligomer crystal structures@33# that the base-pairlevel provides a sensible compromise between the contual simplicity, the computational cost, and the degreereality. While Lavery and co-workers@32# have shown thatthe base pairs effectively behave as rigid entities, the resof Hassan and Calladine@33# and of Hunter and co-worker@34,35# suggest that the dinucleotide parameters observeoligomer crystals can be understood as a consequence oder Waals and electrostatic interactions between the neboring base pairs and constraints imposed by the suphosphate backbone.
The purpose of the present paper is to propose of a c
©2003 The American Physical Society11-1
t t
acorloerd
nmims
cy
bs
enb
onc
caw
edo
-
y
-
re
twistius
ing
t
ack-Ri
the
he
p, abyorl:
s:
e,ainor
is-
ivehese-
or
MERGELL, EJTEHADI, AND EVERAERS PHYSICAL REVIEW E68, 021911 ~2003!
of ‘‘DNA-like’’ molecules with simplified interactions re-solved at the base or base-pair level. In order to represenstacking interactions between the neighboring bases~basepairs!, we use a variant@36# of the Gay-Berne~GB! potential@37# used in the studies of discotic liquid crystals. The sugphosphate backbones are reduced to semirigid springsnecting the edges of the disks/ellipsoids. Using Monte Ca~MC! simulations, we explore the local stacking and the gbal helical properties as functions of the model parametIn particular, we measure the effective parameters neededescribe our systems in terms of stack-of-plates~SOP! andwormlike chain models, respectively. This allows us to costruct DNA models which properly represent the equilibriustructure, fluctuations, and linear response. At the same twe preserve the possibility of local structural transitione.g., in response to the external forces.
The paper is organized as follows. In Sec. II, we introduthe base-pair parameters to discuss the helix geometrterms of these variables. Furthermore, we discuss howtranslate the base-pair parameters in macroscopic variasuch as bending and torsional rigidity. In Sec. III, we propoa model and discuss the methods~MC simulation, energyminimization! we use to explore its behavior. In Sec. IV, wpresent the resulting equilibrium structures, the persistelengths as a function of the model parameters, and thehavior under stretching.
II. THEORETICAL BACKGROUND
A. Helix geometry
To resolve and interpret the x-ray-diffraction studiesDNA oligomers, the relative position and orientation of sucessive base pairs are analyzed in terms of rise~Ri!, slide~Sl!, shift ~Sh!, twist ~Tw!, roll ~Ro!, and tilt ~Ti! @38# ~seeFig. 1!. In order to illustrate the relation between these loparameters and the overall shape of the resulting helix,discuss a simple geometrical model in which DNA is viewas a twisted ladder in which all bars lie in one plane. Fvanishing bending angles with Ro5Ti50, each step is characterized by four parameters: Ri, Sl, Sh, and Tw@18#. Withinthe given geometry, a base pair can be characterized b
FIG. 1. Illustration of all six base-pair parameters and the cresponding coordinate system.
02191
he
r-n-
lo-s.to
-
e,,
eintolese
cee-
-
le
r
its
positionr and the angle of its main axis with then/b axis (npoints to the direction of the large axis,b points to the di-rection of the small axis, andt, representing the tangent vector of the resulting helix, is perpendicular to then-b plane asit is illustrated in Fig. 1!. At each step the center points adisplaced by a distanceASl21Sh2 in the n-b plane. Theangle between the successive steps is equal to theangle, and the center points are located on a helix with radr 5ASl21Sh2/@2 sin(Tw/2)#.
In the following, we study the consequences of imposa simple constraint on the bond lengthsl 1 andl 2 representingthe two sugar-phosphate backbones~the rigid bonds connecthe right and left edges of the bars along then axis, respec-tively!. Ri is the typical height of a step which we will try toimpose on the grounds that it represents the preferred sting distance of the neighboring base pairs. We choose53.3 Å corresponding to theB-DNA value. One possibilityto fulfill the constraintl 15 l 25 l 56 Å is a pure twist. In thiscase, a relationship between the twist angle, the width ofbase pairsd, the backbone lengthl, and the imposed rise isobtained:
Tw5arccosS d222l 212Ri2
d2 D . ~1!
Another possibility is to keep the rotational orientation of tbase pair (Tw50), but to displace its center in then-b plane,in which case Ri21Sl21Sh2[ l 2. With Sh50, it results in askewed ladder with skew angle arcsin(Sl/l )/p @18#.
The general case can be solved as well. In the first stegeneral condition is obtained which needs to be fulfilledany combination of Sh, Sl, and Tw independently of Ri. Fnonvanishing Tw, this yields a relation between Sh and S
tan~Tw!5Sh
Sl. ~2!
Using Eq.~2!, the general equation can finally be solved a
Sl51
A2FcosS Tw
2 D 2AsecS Tw
2 D 2
~2l 22d22Ri2!G . ~3!
Equation~3! is the result of the mechanical coupling of slidshift, and twist due to the backbones. Treating the rise agas a constraint, the twist is reduced for increasing slideshift motion. The center-center distancec between the twoneighboring base pairs is given by
c5ARi21Sl2@11tan~Tw!2#. ~4!
For Tw50 and a given value of Ri, the center-center dtance is equal to the backbone lengthl, and for Tw5arccos@(d222l 212Ri2)/d2# one obtainsc5Ri.
B. Thermal fluctuations
In this section, we discuss how to calculate the effectcoupling constants of a harmonic system, valid within tlinear response theory, describing the couplings of the ba
-
1-2
hoeopngns
tslal
ivun
s
a-
tioheotrixla(T
ofiit
-in
w
h
hef
t
e-e
lcu-or-
-enbe
-e-
nted
nalay
t ad-
ghto
an-n.
air
sible
MODELING DNA STRUCTURE, ELASTICITY, AND . . . PHYSICAL REVIEW E68, 021911 ~2003!
pair parameters along the chain. Furthermore, we showto translate the measured mean and mean-squared valuthe six microscopic base-pair parameters into macroscobservables such as bending and torsional persistence leThis provides the linkage between the two descriptioWLC versus SOP model.
Within the linear response theory, it should be possiblemap our model onto a Gaussian system where all trantional and rotational degrees of freedom are harmoniccoupled. We refer to this model as the SOP model@27#. Theeffective coupling constants are given by the second dertives of the free energy in terms of base-pair variables arothe equilibrium configuration. This yields 636 matricesK nm, describing the couplings of the base-pair parameterthe neighboring base pairs along the chain:
K nm5]2F
]xin]xj
m. ~5!
Therefore one can calculate the (N21)3(N21) correlationmatrixC in terms of base-pair parameters, whereN is therebythe number of base pairs:
^C&5S K 11 K 12 K 13 K 14•••
K 12 K 22 K 23 K 24•••
�
D 21
. ~6!
The inversion ofC results in a generalized connectivity mtrix with effective coupling constants as entries.
The following considerations are based on the assumpthat one only deals with nearest-neighbor interactions. Tthe successive base-pair steps are independent of eachand the calculation of the orientational correlation matbecomes feasible. In the absence of spontaneous dispments (Sl5Sh50) and spontaneous bending angles5Ro50), as it is in the case forB-DNA, going from onebase pair to the neighboring implies three operations. Inder to be independent of the reference base pair, onerotates the respective base pair into the midframe wR(Twsp/2) (R is a rotation matrix,Twsp denotes the spontaneous twist!, followed by a subsequent overall rotationthe midframe,
A5S ti•ti¿1 ti•bi¿1 ti•ni¿1
bi•ti¿1 bi•bi¿1 bi•ni¿1
ni•ti¿1 ni•bi¿1 ni•ni¿1
D , ~7!
taking into account the thermal motions of Ro, Ti, and Tand a final rotation due to the spontaneous twistR(Twsp/2).The orientational correlation matrix between the two neigboring base pairs can be written as^Oi i 11&5R(Twsp/2)^A& R(Twsp/2). A describes the fluctuations around tmean values. As a consequence of the independence osuccessive base-pair parameters, one finds^Oi j &5„R(Twsp/2)^A&R(Twsp/2)…j 2 i , where the matrix producis carried out in the eigenvector basis ofR(Twsp/2)^A&R(Twsp/2). In the end, one finds a relationship btween the mean and mean-squared local base-pair param
02191
ws oficth.:
oa-ly
a-d
of
nn
her,
ce-i
r-rsth
,
-
the
ters
and the bending and torsional persistence length. The calation yields an exponentially decaying tangent-tangent crelation function ^t(0)•t(s)&5exp(2s/lp) with a bendingpersistence length
l p52^Ri&
~^Ti2&1^Ro2&!. ~8!
In the following, we will calculate the torsional persistence length. Making use of a simple relationship betwethe local twist and the base-pair orientations turns out tomore convenient than the transfer matrix approach.
The ~bi!normal-~bi!normal correlation function is an exponentially decaying function with an oscillating term dpending on the helical repeat lengthh5p^Ri& and the helicalpitch p52p/^Tw&, respectively, namely ^n(0)•n(s)&5exp(2s/ln)cos(2p s/h). The torsional persistence lengthl n5 l b can be calculated in the following way. It can be showthat the twist angle Tw of two successive base pairs is relato the orientations$t,b,n% and$t8,b8,n8% through
cos~Tw!5n•n81b•b8
11t•t8. ~9!
Taking the mean and using the fact that the orientatiocorrelation functions and the twist correlation function decexponentially,
exp~21/l Tw!52 exp~21/l n!
11exp~21/l p!, ~10!
yields in the case of stiff filaments a simple expression ofl ndepending onl p and l Tw :
l n
25
l b
25S 2
l Tw1
1
l pD 21
, ~11!
where the twist persistence length is defined as
l Tw5^Ri&
^Tw2&. ~12!
III. MODEL AND METHODS
Qualitatively, the geometrical considerations suggesB-DNA-like ground state and the transition to a skewed lader conformation under the influence of a sufficiently histretching force, because this provides the possibilitylengthen the chain and to partially conserve stacking. Qutitative modeling requires the specification of a Hamiltonia
A. Introduction of the Hamiltonian
The observed conformation of a dinucleotide base-pstep represents a compromise between~i! the base stackinginteractions~bases are hydrophobic and the base pairs~bp!can exclude water by closing the gap in between them! and~ii ! the preferred backbone conformation~the equilibriumbackbone length restricts the conformational space acces
1-3
teuclleantro
a
Ngi
tu
a
in
a
o
teti
e
s
lly
ack-
igh-ck-
he
areuse
er-nsthetheithwemeovehy
ni-ids
in-from
MERGELL, EJTEHADI, AND EVERAERS PHYSICAL REVIEW E68, 021911 ~2003!
to the base pairs! @39#. Packer and Hunter@39# have shownthat roll, tilt, and rise are backbone-independent parameThey depend mainly on the stacking interaction of the scessive base pairs. In contrast, the twist is solely controby the constraints imposed by a rigid backbone. Slideshift are sequence dependent. While it is possible to induce sequence-dependant effects into our model, theyignored in the present paper.
In the present paper, we propose a generic model for Dwhere the molecule is described as a stack of thin, riellipsoids representing the base pairs~Fig. 2!. The shape ofthe ellipsoids is given by three radiia, b, c of the main axesin the body frames, which can be used to define a strucmatrix
S5S a 0 0
0 b 0
0 0 cD , ~13!
where 2a corresponds to the thickness, 2b to the depthwhich is a free parameter in the model, and 2c518 Å to thewidth of the ellipsoid which is fixed to the diameter ofB-DNA helix. The thickness 2a will be chosen in such a waythat the minimum center-center distance for perfect stackreproduces the experimentally known value of 3.3 Å.
The attraction and the excluded volume between the bpairs are modeled by a variant of the GB potential@36,37#for ellipsoids of arbitrary shapeSi , relative positionrW12, andorientationA i . The potential can be written as a productthree terms:
U~A1 ,A2 ,rW12!
5U r~A1 ,A2 ,rW12!h12~A1 ,A2 , r 12!x12~A1 ,A2 , r 12!.
~14!
The first term controls the distance dependence of the inaction and has the form of a simple Lennard-Jones poten
U r54eGBF S s
h1gs D 12
2S s
h1gs D 6G , ~15!
where the interparticle distancer is replaced by the distanch of closest approach between the two bodies:
h[min~ urW i2rW j u! ;~ i , j ! ~16!
with i Pbody 1 andj Pbody 2. The range of interaction icontrolled by an atomistic length scales53.3 Å, represent-ing the effective diameter of a base pair.
In general, the calculation ofh is nontrivial. We use thefollowing approximative calculation scheme which is usuaemployed in connection with the GB potential:
h~A1 ,A2 ,rW12!5r 122s12~A1 ,A2 , r 12!, ~17!
s12~A1 ,A2 , r 12!5F1
2r 12
T G1221~A1 ,A2! r 12G21/2
, ~18!
02191
rs.-dd-re
Ad
re
g
se
f
r-al,
G12~A1 ,A2!5A1TS1
2A11A2TS2
2A2 . ~19!
In the present case of oblate objects with rather perfect sting behavior, Eq.~17! produces only small deviations fromthe exact solution of Eq.~16!.
The other two terms in Eq.~14! control the interactionstrength as a function of the relative orientationA1
TA2 and
position rW12 of interacting ellipsoids:
h12~A1 ,A2 , r 12!5det@S1#/s1
21det@S2#/s22
@det@H12#/~s11s2!#1/2, ~20!
H12~A1 ,A2 , r 12!51
s1A1
TS12A11
1
s2A2
TS22A2 , ~21!
s i~A i , r 12
f
g
isrin
gan
o-p
de
thnbr ohet
ert
e
lsas
th
retons-
n-s to
aintheord-
m-or-oftion
ain
cep-00Wend
aller
der
h
ra-ruc-al-
d to
gar-g thet-
MODELING DNA STRUCTURE, ELASTICITY, AND . . . PHYSICAL REVIEW E68, 021911 ~2003!
5.5 Å to 6.5 Å @18#. This is taken into account by two stifsprings with lengthl 15 l 256.0 Å, connecting neighboringellipsoids~see Fig. 2!. The anchor points are situated alonthe centerline in thenW direction~compare Figs. 1 and 2! witha distance of68 Å from the center of mass. The backbonethus represented by an elastic spring with nonzero splength l 056 Å,
Hel5k
2@~ ur1,i 112r1,i u2 l 0!21~ ur2,i 112r2,i u2 l 0!2#.
~26!
Certainly a situation where the backbones are broucloser to one side of the ellipsoid so as to create a minora major groove would be a better description of theB-DNAstructure. But it turns out that due to the ellipsoidal shapethe base pairs and due to the fact that the internal basedegrees of freedom~propeller twist, etc.! cannot relax, a non-B-DNA-like ground state is obtained where roll and slimotions are involved.
The competition between the GB potential that forcesellipsoids to maximize the contact area and the harmosprings with nonzero spring length which does not like tocompressed leads to a twist in either direction of the orde6p/5. The right handedness of the DNA helix is due to texcluded volume interactions between the bases andbackbone@18#, which we do not represent explicitly. Rathwe break the symmetry by rejecting the moves which leadlocal twist smaller than2p/18.
Thus we are left with three free parameters in our modthe GB energy depthe5min(U) which controls the stackinginteraction, the spring constantk which controls the torsionarigidity, and the depthb of the ellipsoids which influencemainly the fluctuations of the bending angles. All other prameters such as the width and the height of the ellipsoidthe range of interactions53.3 Å, which determines thewidth of the GB potential, are fixed so as to reproduceexperimental values forB-DNA.
FIG. 2. ~Color online! Illustration of DNA geometry for a diam-eter d516 Å: ~1! Twisted ladder with Sl5Sh50, Ri53.3 Å, Tw'2p/10. ~2! Skewed ladder with Tw5Sh50, Ri53.4 Å, Sl'5.0 Å. ~3! Helix with Tw52p/12, Ri53.4 Å, Sl'2.7 Å, Sh'1.6 Å.
02191
g
htd
fair
eicef
he
o
l:
-or
e
B. MC simulation
In our model, all interactions are local and it can therefoconveniently be studied using a MC scheme. In additiontrial moves, consisting of local displacements and rotatioof one ellipsoid by a small amplitude, it is possible to employ global moves which modify the position and the orietation of large parts of the chain. The moves are analogou~i! the well-known pivot move@40# and ~ii ! a crankshaftmove where two randomly chosen points along the chdefine the axis of rotation around which the inner part ofchain is rotated. The moves are accepted or rejected accing to the Metropolis scheme@41#.
Figure 3 shows that these global moves significantly iprove the efficiency of the simulation. We measured the crelation timet of the scalar product of the tangent vectorsthe first and the last monomer of 200 independent simularuns with N510, 20, 50 monomers using~i! only localmoves and~ii ! local and global moves~ratio 1:1!. The cor-relation time of the global moves is independent of the chlength withtglobal'78 sweeps, whereast local scales asN3.
Each simulation run comprises 106 MC sweeps where oneMC sweep corresponds to 2N trials ~one rotational and onetranslational move per base pair! with N denoting the numberof monomers. The amplitude is chosen such that the actance rate equals approximately 50%. After every 10sweeps, we store a snapshot of the DNA conformation.measured the ‘‘time’’ correlation functions of the end-to-edistance, the rise of one base pair inside the chain, andthree orientational angles of the first and the last monomand of two neighboring monomers inside the chain in orto extract the longest relaxation timetmax. We observetmax,1000 for all simulation runs.
An estimate for the central processing unit~CPU! timerequired for one sweep for chains of lengthN5100 on aAMD Athlon MP 20001 processor results in 0.026 s, whicis equivalent to 1.3331024 s per move.
C. Energy minimization
We complemented the simulation study by zero tempeture considerations that help to discuss the geometric stture, obtained by the introduced interactions, and to rationize the MC simulation data. Furthermore, they can be useobtain an estimate of the critical forcef crit that must beapplied to enable the structural transition fromB-DNA to theoverstretchedS-DNA configuration as a function of themodel parameters$e,k,b%.
FIG. 3. ~Color online! ~left! Illustration of the underlying idea.The base pairs are represented as rigid ellipsoids. The suphosphate backbone is treated as semirigid springs connectinedges of the ellipsoid.~right! Introduced interactions lead to a righhanded twisted structure.
1-5
ei-ren
ix
e
u-
urm
seseur
ithf
refo
uaunsn he
r-il-la-ichn
rr,
e
MERGELL, EJTEHADI, AND EVERAERS PHYSICAL REVIEW E68, 021911 ~2003!
IV. RESULTS
In the following, we will try to motivate an appropriatparameter set$e, k, b% that can be used for further investgations within the framework of the presented model. Thefore we explore the parameter dependence of experimeobservables such as the bending persistence lengthB-DNA, l p'150 bp, the torsional persistence lengthl t'260 bp @42#, the mean values and correlations of all sbase-pair parameters and the critical pulling forcef crit'65 pN @11,43–45# which must be applied to enable thstructural transition fromB-DNA to the overstretchedS-DNA configuration. In fact, static and dynamic contribtions to the bending persistence lengthl p of DNA are stillunder discussion. It is known thatl p depends on both theintrinsic curvature of the double helix due to spontaneobending of particular base-pair sequences and the thefluctuations of the bending angles. Bensimonet al. @46# in-troduced disorder into the WLC model by an additionalof preferred random orientation between successivements, and found the following relationship between the ppersistence lengthl pure , i.e. without disorder, the effectivepersistence lengthl e f f , and the persistence lengthl disordercaused by disorder:
l e f f
l pure55 12
A l pure
l disorder
2,
l pure
l disorder!1
2
l pure
l disorder
, l pure
l disorder@1.
~27!
Since we are dealing with intrinsically straight filaments w1/l disorder50, we measurel pure . The recent estimates ol disorder range between 430@47# and 4800@48# bp usingcryoelectron microscopy and cyclization experiments,spectively, implicating values between 105 and 140 bpl pure .
A. Equilibrium structure
As a first step, we study the equilibrium structure of ochains as a function of the model parameters. To investigthe ground state conformation, we rationalize the MC simlation results with the help of the geometrical consideratioand minimum energy calculations. In the end, we will chooparameters for which our model reproduces the experimevalues ofB-DNA @18#:
^Ri&53.323.4 Å,
^Sl&50 Å,
^Sh&50 Å,
^Tw&52p/10.522p/10,
^Ti&50,
02191
-talof
sal
tg-e
-r
rte-setal
^Ro&50.
We use the following reduced units in our calculations. Tenergy is measured in the units ofkBT, lengths in the unitsof Å, forces in the units ofkBTÅ21'40 pN.
We start by minimizing the energy for the various confomations shown in Fig. 4 to verify that our model Hamtonian indeed prefers theB form. Since we have only loca~nearest-neighbor! interactions, we can restrict the calcultions to two base pairs. There are three local minima whhave to be considered:~i! a stacked-twisted conformatiowith Ri53.3, Sl, Sh, Ti, Ro50, Tw5p/10, ~ii ! a skewedladder with Ri53.3, Sl55.0, Sh, Tw, Ti, Ro50, and~iii ! anunwound helix with Ri56.0, Sl, Sh, Ti, Ro50, Tw50.
FIG. 4. ~Color online! Time correlation functions of the scalaproduct of the tangent vectors of the first and the last monomet
5 tW(0,1)• tW(t,N) with N510 ~red, plus!, N520 ~green, crosses!,N550 ~blue, stars! for ~a! global and~b! local moves. It is observedthat tglobal is independent of the chain lengthN, whereast local
scales asN3. The ‘‘time’’ is measured in units of sweeps wherone MC sweep corresponds toN trials. The CPU time forone sweep scales asN2 in case of global moves and asN in case oflocal moves. Thus the simulation timet scales ast local}N4 andtglobal}N2.
1-6
is
Gnt
yed
h
Rng
eseon-
ccuraseter-ionsller
. Inller
areaseh,I A.
ingheionwist
lingto
pairble
wistTheuc-
hat
the
fand
iand bys
ters
an
MODELING DNA STRUCTURE, ELASTICITY, AND . . . PHYSICAL REVIEW E68, 021911 ~2003!
Without an external pulling force the global minimumfound to be the stacked-twisted conformation.
We investigated the dependence of Ri and Tw on theenergy depthe that controls the stacking energy for differespring constantsk. Ri depends neither one, nor onk, and noron b. It shows a constant value of Ri'3.3 Å for all param-eter sets$e,k,b%. The resulting Tw of the minimum energcalculation coincides with the geometrically determinvalue under the assumption of fixed Ri up to a criticale. Upto that value the springs behave effectively as rigid rods. Tcritical e is determined by the torquet(k,e) that has to beapplied to open the twisted structure for a given value of
Using MC simulations, we can study the effects arisifrom the thermal fluctuations. PlottingRi& and ^Tw& as a
FIG. 5. ~Color online! ~a! Rise~Å! and~b! twist as a function ofe @kBT# for 2b58, 9, 10, 11 Å~red, green, blue, purple!. For eachb, there are two data sets fork532 ~plus!,64 ~circles! @kBT/Å2#.The dotted line corresponds to the minimum energy value.^Ri&depends only one. In the limit of e→`, the minimum energyvalue is reached.~b! In addition to the MC data and the minimumenergy calculation, we calculated the twist with Eq.~1! using themeasured mean rise values of~a! ~red solid lines!. One can observethat^Tw& changes with all three model parameters. Increasingy andk decreases especially the fluctuations of Tw and Sh so that^Tw&increases as a result of the mechanical coupling of the shifttwist motions. In the limit ofe,k→`, the minimum energy value isreached.
02191
B
e
i.
function of the GB energy depthe, one recognizes that ingeneral^Ri& is larger than Ri(T50). It converges only forlarge values ofe to the minimum energy values. This can bunderstood as follows. Without fluctuations the two bapairs are perfectly stacked taking the minimum energy cfiguration Ri53.3 Å, Sl, Sh, Ti, Ro50, and Tw5p/10. Asthe temperature is increased the fluctuations can only oto larger Ri values due to the repulsion of neighboring bpairs. A decrease of Ri would cause the base pairs to insect. Increasing the stacking energy reduces the fluctuatin the direction of the tangent vector and leads to sma^Ri& value. In the limite→`, it should reach the minimumenergy value which is observed from the simulation dataturn the increase of the mean value of rise results in a smatwist angle^Tw&. We can calculate with the help of Eq.~1!the expected twist using the measured mean values of^Ri&.Figure 5 shows that there is no agreement. The deviationsdue to the fluctuations in Sl and Sh which cause the bpairs to untwist. This is the mechanical coupling of Sl, Sand Tw due to the backbones already mentioned in Sec. IIt is observed that a stiffer springk and a larger depthb ofthe ellipsoids result in larger mean twist values. Increasthe spring constantk means decreasing the fluctuations of ttwist and, due to the mechanical coupling, of the shift motaround the mean values which explains the larger mean tvalues. An increase of the ellipsoidal depthb in turn de-creases the fluctuations of the bending angles. The coupof the tilt fluctuations with the shift fluctuations leadslarger values for^Tw&. The corresponding limit where^Tw&→Tw(T50) is given byk,e→`.
The measurement of the mean values of all six base-step parameters for different temperatures is shown in TaI. One can see that with increasing the temperature, the tangles decrease while the mean value of rise increases.increase of the center-center distance is not only due to fltuations in Ri but also due to fluctuations in Sl and Sh. Tis why there are strong deviations of^c& from ^Ri& eventhough the mean values of Sl and Sh vanish. Note thatmean backbone lengthl & always amounts to about 6 Å.
The calculation of the probability distribution functions oall six base-pair parameters shows that especially the risetwist motions do not follow a Gaussian behavior~Fig. 7!.The deviation of the distribution functions from the Gaussshape depends mainly on the stacking energy determinee. For smaller values ofe, one observes larger deviationthan for largee values.
TABLE I. Dependence of mean values of all six step parameand of the mean center-center distance^c& on the temperature for2b511 Å, e520kBT, k564kBT/Å2. ^Ri&, ^Sh&, ^Sl&, and^c& aremeasured in Å,l p in base pairs.
T ^Ri& ^Sh& ^Sl& ^Tw& ^Ti& ^Ro& ^c& l p
0 3.26 0.0 0.0 0.64 0.0 0.0 3.26 `
1 3.37 0.01 20.01 0.62 0.0 0.0 3.47 172.82 3.76 20.01 20.03 0.47 0.0 0.0 4.41 25.33 4.10 20.01 0.01 0.34 0.0 20.01 5.07 14.45 4.30 0.03 20.02 0.27 0.0 0.01 5.39 13.6
d
1-7
r-m
ioerlie-
inseee
n-
eir-ed
thesivto
o
ain
a-
it
nc
atei
si
anne
uec
oru-
MERGELL, EJTEHADI, AND EVERAERS PHYSICAL REVIEW E68, 021911 ~2003!
It is worthwhile to mention that there are mainly two corelations between the base-pair parameters. The first is acroscopic twist-stretch coupling determined by a correlatof Ri and Tw, i.e., an untwisting of the helix implicates largrise values. A twist-stretch coupling was introduced in earrod models@49–51#, motivated by experiments with torsionally constrained DNA@52# which allow for the determinationof this constant. Here it is the result of the preferred stackof neighboring base pairs and the rigid backbones. Theond correlation is due to the constrained tilt motion. If wreturn to our geometrical ladder model, we recognize immdiately that a tilt motion alone will always violate the costraint of fixed backbone lengthl. Even though we allow forbackbone fluctuations in the simulation, the bonds are vrigid which makes tilting energetically unfavorable. To ccumvent this constraint, tilting always involves a directshift motion.
Figure 6 shows that we recover the anisotropy ofbending angles Ro and Ti as a result of the spatial dimsions of the ellipsoids. Since the overlap of the succesellipsoids is larger in case of rolling, it is more favorableroll than to tilt.
The correlations can be quantified by calculating the crelation matrixC of Eq. ~6!. Inverting C yields the effectivecoupling constants of the SOP model,K5C 21. Due to thelocal interactions, it suffices to calculate the mean and mesquared values of Ri, Sl, Sh, Tw, Ro, and Ti, characterizthe ‘‘internal’’ couplings of the base pairs:
C5~s! i j ; i , j P$1, . . . ,6%, ~28!
with sx,y5^xy&2^x&^y&.
B. Bending and torsional rigidity
The correlation matrix of Eq.~28! can also be used tocheck Eqs.~8! and~11!. Therefore we measured the orienttional correlation functions t i•t j&, ^ni•nj&, ^bi•bj& andcompared the results to the analytical expressions asillustrated in Fig. 8. The agreement is excellent.
The simulation data show that the bending persistelength does not depend on the spring constantk. But itstrongly depends one being responsible for the energy thmust be paid to tilt or roll two respective base pairs. Sincchange of twist for constant Ri is proportional to a changebond length, the bond energy contributes to the twist pertence length explaining the dependence ofl Tw on k ~compareFig. 9!.
We also measured the mean-square end-to-end dist^RE
2& and find that RE2& deviates from the usual WLC chai
result due to the compressibility of the chain. So as to invtigate the origin of the compressibility, we calculate^RE
2& forthe following geometry. We consider two base pairs withospontaneous bending angles such that the end-to-end vRW E can be expressed as
RW E5(i
cW i5(i
~Ri t i1Shbi1Slni !. ~29!
02191
i-n
r
gc-
-
ry
en-e
r-
n-g
is
e
ans-
ce
s-
ttor
FIG. 6. ~Color online! Contour plots of the measured clouds frise-twist, shift-tilt, and roll-tilt to demonstrate the internal coplings and the anisotropy of the bending angles (2b511 Å,e520kBT,k564kBT/Å2).
1-8
reement
MODELING DNA STRUCTURE, ELASTICITY, AND . . . PHYSICAL REVIEW E68, 021911 ~2003!
FIG. 7. Comparison of the probability distribution functions of all base-pair parameters fore520kBT, k564kBT/Å2, 2b58 Å. TheGaussians~black solid line! are plotted with the measured mean and mean-squared values of the MC simulation, and are in good agwith the simulation data~circles! except for Ri and Tw.
ain,
-
The coordinate system$t i ,bi ,ni% is illustrated in Fig. 1.cW idenotes the center-center distance of two neighboring bpairs. Since successive base-pair step parameters arependent of each other and Ri, Sh, and Sl are uncorrelatedmean-square end-to-end distance^RE
2& is given by
^RE2&5(
i~^ci
2&2^Ri&2!1(i
(j
^Ri&2^t i•t j&
5N^Ri&
g12N^Ri& l p22l p
2F12expS 2N^Ri&
l pD G ,
~30!
whereN denotes the number of base pairs. Note that^Sl& and
02191
sede-the
^Sh& vanish. Using ci2&5^Ri2&1^Sh2&1^Sl2&, the stretch-
ing modulusg is simply given by
g5^Ri&
~^Ri2&2^Ri&2!1^Sh2&1^Sl2&. ~31!
We compared the data for different temperaturesT to Eq.~30! using the measured bending persistence lengthsl p andstretching modulig ~see Fig. 10!. The agreement is excellent. This indicates thattransverseslide and shift fluctuationscontribute to thelongitudinal stretching modulus of thechain.
1-9
deurpthis
e-.o
aiblidamalveo
an
l
ofhe
ter-eniningm
ns
e in
MERGELL, EJTEHADI, AND EVERAERS PHYSICAL REVIEW E68, 021911 ~2003!
C. Stretching
The extension experiments on the double-stranB-DNA have shown that the overstretching transition occwhen the molecule is subjected to stretching forces of 65or more@45#. The DNA molecule thereby increases in lengby a factor of 1.8 times the normal contour length. Thoverstretched DNA conformation is calledS-DNA. Thestructure ofS-DNA is still under discussion. First evidencof the possibleS-DNA conformations were provided by Lavery et al. @11,43,44# using atomistic computer simulations
In principle, one can imagine two possible scenarios hthe transition fromB-DNA to S-DNA occurs within ourmodel. Either the chain untwists and unstacks resulting inuntwisted ladder with approximately 1.8 times the equilrium length, or the chain untwists and the base pairs sagainst each other resulting in a skewed ladder with the sS-DNA length. The second scenario should be energeticfavorable since it provides a possibility to partially conserthe stacking of successive base pairs. In fact, molecular meling of the DNA stretching process@11,43,44# yielded both
FIG. 8. ~Color online! Comparison of the analytical expressio@Eqs. ~8! and ~11!# for l p and l n ~solid lines! with numericallycalculated orientational correlation functions~data points! for 2b58 Å, k564kBT/Å2, and e520, . . . ,60@kBT# ~red, plus; . . . ;black, diamonds!.
02191
dsN
w
n-ee
ly
d-
a conformation with strong inclination of base pairs andunwound ribbon depending on which strand one pulls.
We expect that the critical forcef crit where the structuratransition fromB-DNA to overstretchedS-DNA occurs de-pends only on the GB energy depthe, controlling the stack-ing energy. So as a first step to find an appropriate valueeas input parameter for the MC simulation, we minimize tHamiltonian with an additional stretching energyEpull5 f ci ,i 11, where the stretching force acts along the cenof-mass axis, with respect to Ri, Sl, and Tw for a givpulling force f. Figure 11 shows the resulting stress-stracurve. First, the pulling force acts solely against the stackenergy up to the critical force where a jump froL( f crit 2)/L0'1.05 toL( f crit 1)/L05ARi21Sl2/Ri'1.8 oc-
FIG. 9. ~Color online! Dependency of~a! bending persistencelength l p and ~b! torsional persistence lengthl n on the spring con-stantk, the width of the ellipsoidsb, and the energy depthe. Wemeasured the persistence lengths for varying width sizes 2b58, 9,10, 11 Å ~red, green, blue, purple; from bottom to top! and for twodifferent spring constantsk532 ~plus!, 64 ~circles! @kBT/Å2#. Thebending persistence length depends solely onb ande. It gets largerfor largere andb values. But it does not depend onk ~the curves fordifferent k values corresponding to the same widthb lie one uponthe other!. The torsional persistence length in turn depends onk,since a change of twist for constant Ri is proportional to a changbond length.
1-10
se
e
nelobfo
ene
in
ta
e
aiotc
osbe
mk-gthatapnsha
et to
fol-
eeter-
sis-
aen-
re-es,a-
irs,
Our
s.
unc-tam-heye
y
ia-
h
gre
MODELING DNA STRUCTURE, ELASTICITY, AND . . . PHYSICAL REVIEW E68, 021911 ~2003!
curs, followed by another slow increase of the length cauby the overstretching the bonds.L05L(F50)5Ri denotesthe stress-free center-of-mass distance. As already mtioned, three local minima are obtained:~i! a stacked-twistedconformation,~ii ! a skewed ladder, and~iii ! an unwoundhelix. The strength of the applied stretching force determiwhich of the local minima becomes the global one. The gbal minimum for small stretching forces is determined tothe stacked-twisted conformation and the global minimastretching forces larger thanf crit is found to be the skewedladder. Therefore the broadness of the force plateau depsolely on the ratio ofl /Ri determined by the geometry of thbase pairsSand the bond lengthl 56.0 Å. A linear relation-ship is obtained between the critical force and the stackenergye so that one can extrapolate to smallere values inorder to extract thee value that reproduces the experimenvalue of f crit'65 pN. This suggests a value ofe'7.
The simulation results of the previous sections show seral problems when this value ofe is chosen. First of all, itcannot produce the correct persistence lengths as the chfar too flexible. Second, the undistorted ground state is nB-DNA anymore. The thermal fluctuations suffice to unstaand untwist the chain locally. That is why one has to cholargere values even though the critical force is going tooverestimated.
Therefore we choose the following way to fix the paraeter set$b,e,k%. First of all, we choose a value for the stacing energy that reproduces correctly the persistence lenAfterwards the torsional persistence length is fixed toexperimentally known values by choosing an approprispring constantk. The depth of the base pairs has alsoinfluence on the persistence lengths of the chain. If the deb is decreased, larger fluctuations for all the three rotatioparameters are gained such that the persistence lengthsmaller. Furthermore, the geometric structure and the be
FIG. 10. ~Color online! Comparison of the simulation data wite520kBT, k564kBT/Å2, 2b511 Å, and T51 ~purple, dia-monds!, 2 ~blue, triangles!, 3 ~green, circles!, 5 ~red, squares! toEqs. ~8!, ~30!, and ~31! ~solid lines!. Using the measured bendinpersistence lengths and the stretching moduli, we find a good agment with the predicted behavior. ForT51, we obtain g56.02 Å21.
02191
d
n-
s-er
ds
g
l
v-
n isa
ke
-
th.eenthalgetv-
ior under pulling is very sensitive tob. Very small valuesprovoke non-B-DNA conformations or unphysicalS-DNAconformations. We choose forb a value of 11 Å for thosereasons. Fore520 and k564, a bending stiffness ofl p5170 bp and a torsional stiffness ofl n5270 bp are obtainedclose to the experimental values. We use this parameter ssimulate the corresponding stress-strain relation.
The simulated stress-strain curves for 50 bp show thelowing three different regimes~see Fig. 11!.
~a! For small stretching forces, the WLC behavior of thDNA, in addition with the linear stretching elasticity of thbackbones is recovered. This regime is completely demined by the chain lengthN. Due to the coarse-grainingprocedure that provides analytic expressions of the pertence lengths depending on the base-pair parameters@seeEqs.~8! and~11!#, it is not necessary to simulate a chain offew thousand base pairs. The stress-strain relation of thetropic and WLC stretching regime~small relative extensionsL/L0 and small forces! is known analytically@20,53#. Sincewe have parametrized the model in such a way that wecover the elastic properties of DNA on large length scalthe simulation data for very long chains will follow the anlytical result for small stretching forces.
~b! Around the critical forcef crit'140 pN, which ismainly determined by the stacking energy of the base pathe structural transition fromB-DNA to S-DNA occurs.
~c! For larger forces the bonds become overstretched.MC simulations suggest a critical forcef crit'140 pN whichis slightly smaller than the valuef crit'180 pN calculated byminimizing the energy. This is due to entropic contribution
In order to further characterize theB-to-S transition, wemeasured the mean values of rise, slide, shift, etc., as a ftion of the applied forces. The evaluation of the MC dashows that the mean values of shift, roll, and tilt are copletely independent of the applied stretching force and tvanish for allf. Rise increases at the critical force from th
FIG. 11. ~Color online! Force-extension relation calculated bthe minimum energy calculation~black! and obtained by MC simu-lation for 50 ~red, plus! and 500~blue, crosses! bp. The red solidline represents the analytical result of the WLC. Inset: The devtion between energy minimization~black dotted line! and MC in thecritical force is due to the entropic contributions.e-
1-11
eeed
ee
.ese
ingthesi-toex-
-ons
ingysisAs-ion,er-
Eqs.om a
eucethis-
ll sixruc-
MERGELL, EJTEHADI, AND EVERAERS PHYSICAL REVIEW E68, 021911 ~2003!
undisturbed value of 3.3 Å to approximately 4.0 Å and dcays subsequently to the undisturbed value. Quite interingly, the mean value of slide jumps from its undisturbvalue of 0 to65 Å ~no direction is favored! and the twistchanges at the critical force fromp/10 to 0. The calculationof the distribution function of the center-center distancec oftwo neighboring base pairs forf 5140 pN yields a double-peaked distribution~see Fig. 12!, indicating that a part of thechain is in theB form and a part of the chain in theS form.The contribution of the three translational degrees of frdom to the center-center distancec is shown in Fig. 12. TheS-DNA conformation is characterized by Ri53.3 Å, Sl565 Å, and Tw50 ~Fig. 13!. In agreement with Refs@11,43#, we obtain a conformation with highly inclined baspairs still allowing for partial stacking of successive bapairs.
FIG. 12. ~Color online! ~a! Probability distribution function ofthe center-center distance of successive base pairs forf 50 ~red,squares!, 140 ~green, spheres!, 200 pN~blue, triangles!. ~b! Mean-squared values of rise~red, plus!, shift ~green, crosses!, slide ~blue,stars!, and center-of-mass distance~purple, squares! for neighboringbase pairs as a function of the stretching forcef. The dashed linecorresponds to theS-DNA center-of-mass distance.^Tw& of theresultingS-DNA conformation vanishes as predicted by Eq.~3!.
02191
-st-
-
V. DISCUSSION
We have proposed a simple model Hamiltonian describthe double-stranded DNA on the base-pair level. Due tosimplification of the force field and, in particular, the posbility of nonlocal MC moves, our model provides accessmuch larger length scales than atomistic simulations. Forample, 4h on a AMD Athlon MP 20001 processor are sufficient in order to generate 1000 independent conformatifor chains consisting ofN5100 bp.
In the data analysis, the main emphasis was on derivthe elastic constants on the elastic rod level from the analof the thermal fluctuations of base-pair step parameters.suming a twisted ladder as the ground state conformatone can provide an analytical relationship between the psistence lengths and the local elastic constants given by~8! and~11! @66#. Future work has to show, if it is possible tobtain suitable parameters for our mesoscopic model frocorresponding analysis of atomistic simulations@54# orquantum-chemical calculations@55#. In the present paper, whave chosen a top-down approach, i.e., we try to reprodthe experimentally measured behavior of DNA on lengscalesbeyondthe base diameter. The analysis of the perstence lengths, the mean and mean-squared values of abase-pair parameters, and the critical force, where the sttural transition fromB-DNA to S-DNA takes place, as afunction of the model parameters$b,k,e% and the appliedstretching forcef suggests the following parameter set:
2b511 Å, ~32!
e520kBT, ~33!
k564kBT/Å2. ~34!
FIG. 13. ~Color online! Contour plot of rise~Å! versus slide~Å!for the S-DNA conformation.
1-12
phethreevtuth
ith
r ore
h-
O
kitic
st-
din
thC-
rca
oThaex
ence
ilits
p-e-
vtokin-
d
e
r via
er-n-
reason
ead
heges
n,theible
e
ialsrvem,
on
tiontiveirs.
andlyz-y athenalre-by
alinga
t a
sedons
pu-ing, theer-heying
ismeatin
MODELING DNA STRUCTURE, ELASTICITY, AND . . . PHYSICAL REVIEW E68, 021911 ~2003!
It reproduces the correct persistence lengths forB-DNAand entails the correct mean values of the base-pair steprameters known by the x-ray-diffraction studies. While tpresent model does not include the distinction betweenminor and major grooves and suppresses all internal degof freedom of the base pairs such as propeller twist, it nertheless reproduces some experimentally observed feaon the base-pair level. For example, the anisotropy ofbending angles~rolling is easier than tilting! is just a conse-quence of the platelike shape of the base pairs and the twstretch coupling is the result of the preferred stacking ofneighboring base pairs and the rigid backbones.
The measured critical force is overestimated by a facto2 and cannot be improved further by fine tuning of the thfree model parameters$b,k,e%. f crit depends solely on thestacking energy valuee that cannot be reduced further. Oterwise neither the correct equilibrium structure ofB-DNAnor the correct persistence lengths would be reproduced.model suggests a structure forS-DNA with highly inclinedbase pairs so as to enable at least partial base-pair stacThis is in good agreement with the results of atomisB-DNA simulations by Lavery and co-workers@11,43#. Theyfound a force plateau of 140 pN for freely rotating ends@11#.The mapping to the SOP model yields the following twistretch ~Ri-Tw! coupling constant: kRi,Tw5(C 21)Ri,Tw5267/ Å. kRi,Tw is the microscopic coupling of rise antwist describing the untwisting of the chain due to ancrease of rise~compare also Fig. 6!.
The possible applications of the present model includeinvestigation of~i! the charge renormalization of the WLelastic constants@56#, ~ii ! the microscopic origins of the cooperativity of theB-to-S transition @57#, and ~iii ! the influ-ence of nicks in the sugar-phosphate backbone on foelongation curves. In particular, our model providesphysically sensible framework to study the intercalationcertain drugs or of ethidium bromide between base pairs.latter is a hydrophobic molecule of roughly the same sizethe base pairs that fluoresces green and likes to slip betwtwo base pairs forming an DNA-ethidium-bromide compleThe fluorescence properties allow to measure the persistlengths of DNA@6#. It was also used to argue that the forplateau is the result of a DNA conformational transition@11#.
In the future, we plan to generalize our approach todescription on the base level, which includes the possibof hydrogen-bond breaking between complementary baalong the lines of Refs.@30,31#. A suitably parametrizedmodel allows a more detailed investigation of DNA unziping experiments@58# as well as a direct comparison btween the two mechanism currently discussed for theB-to-Stransition: the formation of skewed ladder conformations~asin the present paper! versus local denaturation@59–61#.Clearly, it is possible to study the sequence effects and emore refined models of DNA. For example, it is possiblemimic the minor and major grooves by bringing the bacbones closer to one side of the ellipsoids without observthe non-B-DNA-like ground states. The relaxation of the internal degrees of freedom of the base pairs, characterizeanother set of parameters~propeller twist, stagger, etc.!,should help to reduce the artifacts which are due to the
02191
a-
ees-rese
st-e
fe
ur
ng.
-
e
e-
fesen.ce
ayes
en
-g
by
l-
lipsoidal shape of the base pairs. Sequence effects entethe strength of the hydrogen bonds (EGC52.9kBT versusEAT51.3kBT) as well as via base-dependent stacking intactions@35#. For example, one finds for guanine a concetration of negative charge on the major-groove edge, whefor cytosine one finds a concentration of positive chargethe major-groove edge. For adenine and thymine instthere is no strong joint concentration of partial charges@18#.It is known that in a solution of water and ethanol, where thydrophobic effect is less dominant, these partial charcause GG/CC steps to adoptA or C forms @62# by a negativeslide and positive roll motion and a positive slide motiorespectively. Thus by varying the ratio of the strengths ofstacking versus the electrostatic energy, it should be possto study the transition fromB-DNA to A-DNA and C-DNA,respectively.
VI. SUMMARY
Inspired by the results of Hassan and Calladine@33#and of Hunter and co-workers@34,35#, we have put forwardthe idea of constructing simplified DNA models on thbase~-pair! level where discotic ellipsoids~whose stackinginteractions are modeled via coarse-grained potent@36,37#! are linked to each other in such a way as to presethe DNA geometry, its major mechanical degrees of freedoand the physical driving forces for the structure formati@18#.
In the present paper, we have used energy minimizaand Monte Carlo simulations to study a simple representaof this class of DNA models with nonseparable base paFor a suitable choice of parameters, we obtained aB-DNA-like ground state as well as realistic values for the bendtwist persistence lengths. The latter were obtained by anaing the thermal fluctuations of long filaments as well as bsystematic coarse-graining from the stack-of-plates toelastic rod level. In studying the response of DNA to exterforces or torques, models of the present type are notstricted to the regime of small local deformations. Ratherspecifying a physically motivated Hamiltonian forarbitrarybase-~step! parameters, our ansatz allows for realistic locstructural transitions. For the simple case of a stretchforce, we observed a transition from a twisted helix toskewed ladder conformation. While our results suggessimilar structure forS-DNA as atomistic simulations@11#, theDNA model studied in this paper, of course, cannot be uto rule out the alternate possibility of local strand separati@59–61#.
In our opinion, the base~-pair! level provides a sensiblecompromise between the conceptual simplicity, the comtational cost, and the degree of reality. Besides providaccess to much larger scales than atomistic simulationsderivation of such models from more microscopic considations provides considerable insight. At the same time, tmay serve to validate and unify analytical approaches aimat ~averaged! properties on larger scales@28–31,57#. Finally,we note that the applicability of linked-ellipsoid modelsnot restricted to the base-pair level of DNA as the satechniques can, for example, also be used to study chrom@63–65#.
1-13
Kse
ate-n
MERGELL, EJTEHADI, AND EVERAERS PHYSICAL REVIEW E68, 021911 ~2003!
ACKNOWLEDGMENTS
We gratefully acknowledge extended discussions withKremer, R. Lavery, and A. C. Maggs. We thank H. Schies
a
oc
nc
ng
rt
Se
ys
e
02191
.l
for a careful reading of our paper. Furthermore, we are grful to the DFG for the financial support of this work withithe Emmy-Noether grant.
. J.
on-
ol.
-e,
.
V.
rr.
rn,
V.
hys.
ev.
,
@1# J.D. Watson and F.H.C. Crick, Nature~London! 171, 737~1953!.
@2# R.E. Dickersonet al., Science216, 475 ~1982!.@3# R.E. Dickerson, Methods Enzymol.211, 67 ~1992!.@4# T.L. James, Methods Enzymol.261, 1 ~1995!.@5# D.P. Millar, R.J. Robbins, and A.H. Zewail, J. Chem. Phys.76,
2080 ~1982!.@6# J.M. Schurr and K.S. Schmitz, Annu. Rev. Phys. Chem.37,
271 ~1986!.@7# T.T. Perkins, S. Quake, D. Smith, and S. Chu, Science264,
822 ~1994!.@8# T.C. Boles, J.H. White, and N.R. Cozzarelli, J. Mol. Biol.213,
931 ~1990!.@9# S.B. Smith, L. Finzi, and C. Bustamante, Science258, 1122
~1992!.@10# S.B. Smith, Y. Cui, and C. Bustamante, Science271, 795
~1996!.@11# P. Cluzelet al., Science264, 792 ~1996!.@12# B. Essevaz-Roulet, U. Bockelmann, and F. Heslot, Proc. N
Acad. Sci. U.S.A.94, 11935~1997!.@13# J. Allemand, D. Bensimon, R. Lavery, and V. Croquette, Pr
Natl. Acad. Sci. U.S.A.95, 14152~1998!.@14# C.R. Calladine and H.R. Drew, J. Mol. Biol.178, 773 ~1984!.@15# R.E. Dickersonet al., EMBO J.8, 1 ~1989!.@16# X.J. Lu and W.K. Olson, J. Mol. Biol.285, 1563~1999!.@17# W.K. Olsonet al., J. Mol. Biol. 313, 229 ~2001!.@18# C.R. Calladine and H.R. Drew,Understanding DNA: The Mol-
ecule and How it Works~Academic Press, New York, 1999!.@19# J.F. Marko and E.D. Siggia, Macromolecules27, 981 ~1994!.@20# J.F. Marko and E.D. Siggia, Macromolecules28, 8759
~1995!.@21# T.T. Perkins, D.E. Smith, R.G. Larson, and S. Chu, Scie
268, 83 ~1995!.@22# N.R. Cozzarelli and J.C. Wang,DNA Topology and Its Biologi-
cal Effects~Cold Spring Harbor Laboratory Press, Cold SpriHarbor, NY, 1990!.
@23# T. Schlick and W.K. Olson, J. Mol. Biol.223, 1089~1992!.@24# G. Chirico and J. Langowski, Biopolymers34, 415 ~1994!.@25# H. Schiessel, J. Widom, R.F. Bruinsma, and W.M. Gelba
Phys. Rev. Lett.86, 4414~2001!.@26# M.A.E. Hassan and C.R. Calladine, Proc. R. Soc. London,
A 453, 365 ~1997!.@27# C. O’Hern, R. Kamien, T. Lubensky, and P. Nelson, Eur. Ph
J. B 1, 95 ~1998!.@28# A. Sarkar, J.F. Leger, D. Chatenay, and J.F. Marko, Phys. R
E 63, 051903~2001!.@29# Z. Haijun, Z. Yang, and O.-Y. Zhong-can, Phys. Rev. Lett.82,
4560 ~1999!.@30# M. Barbi, S. Cocco, and M. Peyrard, Phys. Lett. A253, 358
~1999!.@31# S. Cocco and R. Monasson, Phys. Rev. Lett.83, 5178~1999!.
tl.
.
e
,
r.
.
v.
@32# N. Bruant, D. Flatters, R. Lavery, and D. Genest, Biophys77, 2366~1999!.
@33# M.A.E. Hassan and C.R. Calladine, Philos. Trans. R. Soc. Ldon, Ser. A355, 43 ~1997!.
@34# C.A. Hunter and X.-J. Lu, J. Mol. Biol.265, 603 ~1997!.@35# C.A. Hunter, J. Mol. Biol.230, 1025~1993!.@36# R. Everaers and M.R. Ejtehadi, Phys. Rev. E67, 041710
~2003!.@37# J.G. Gay and B.J. Berne, J. Chem. Phys.74, 3316~1981!.@38# M.S. Babcock, E.P.D. Pednault, and W.K. Olson, J. Mol. Bi
237, 125 ~1994!.@39# M.J. Packer and C.A. Hunter, J. Mol. Biol.280, 407
~1998!.@40# D.P. Landau and K. Binder,Monte Carlo Simulations in Sta
tistical Physics ~Cambridge University Press, Cambridg2000!.
@41# N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.NTeller, and E. Teller, J. Chem. Phys.21, 1087~1953!.
@42# T.R. Strick, D. Bensimon, and V. Croquette, Genetica106, 57~1999!.
@43# R. Lavery and A. Lebrun, Genetica106, 75 ~1999!.@44# R. Lavery, A. Lebrun, J.-F. Allemand, D. Bensimon, and
Croquette, J. Phys.: Condens. Matter14, R383~2002!.@45# C. Bustamante, S.B. Smith, J. Liphardt, and D. Smith, Cu
Opin. Struct. Biol.10, 279 ~2000!.@46# D. Bensimon, D. Dohmi, and M. Mezard, Europhys. Lett.42,
97 ~1998!.@47# J. Bednaret al., J. Mol. Biol. 254, 579 ~1995!.@48# M. Vologodskaia and A. Vologodskii, J. Mol. Biol.317, 205
~2002!.@49# R.D. Kamien, T.C. Lubensky, P. Nelson, and C.S. O’He
Europhys. Lett.38, 237 ~1997!.@50# J.F. Marko, Europhys. Lett.38, 183 ~1997!.@51# P. Nelson, Biophys. J.74, 2501~1998!.@52# T.R. Strick, J.-F. Allemand, D. Bensimon, A. Bensimon, and
Croquette, Science271, 1835~1996!.@53# T. Odijk, Macromolecules28, 7016~1995!.@54# I. Lafontaine and R. Lavery, Biophys. J.79, 680 ~2000!.@55# C.F. Guerra and F.M. Bickelhaupt, Angew. Chem., Int. Ed.38,
2942 ~1999!.@56# R. Podgornik, P.L. Hansen, and V.A. Parsegian, J. Chem. P
113, 9343~2000!.@57# C. Storm and P. Nelson, e-print physics/0212032.@58# U. Bockelmann, B. Essevaz-Roulet, and F. Heslot, Phys. R
Lett. 79, 4489~1997!.@59# M.C. Williams, J.R. Wenner, I. Rouzina, and V.A. Bloomfield
Biophys. J.80, 874 ~2001!.@60# I. Rouzina and V.A. Bloomfield, Biophys. J.80, 882 ~2001!.@61# I. Rouzina and V.A. Bloomfield, Biophys. J.80, 894 ~2001!.@62# Y. Fang, T.S. Spisz, and J.H. Hoh, Nucleic Acids Res.27, 1943
~1999!.
1-14
d byts as-
MODELING DNA STRUCTURE, ELASTICITY, AND . . . PHYSICAL REVIEW E68, 021911 ~2003!
@63# G. Wedemann and J. Langowski, Biophys. J.82, 2847~2002!.@64# V. Katritch, C. Bustamante, and W.K. Olson, J. Mol. Biol.295,
29 ~2000!.@65# B. Mergell, H. Schiessel, and R. Everaers~unpublished!.
02191
@66# The general case where the ground state is characterizespontaneous rotations as well as spontaneous displacemenin theA-DNA conformation is more involved. This is the subject of ongoing work.
1-15