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Elastic Correlations in Nucleosomal DNA Structure Farshid Mohammad-Rafiee1,2 and Ramin Golestanian1,2 1Isaac Newton Institute for Mathematical Sciences, Cambridge, CB3 0EH, UK 2Institute for Advanced Studies in Basic Sciences, Zanjan 45195-159, Iran (Dated: February 9, 2008) The structure of DNA in the nucleosome core particle is studied using an elastic model that incorporatesanisotropyinthebendingenergeticsandtwist-bendcoupling. Usingtheexperimentally determinedstructureofnucleosomalDNA[T.J.RichmondandC.A.Davey,Nature423,145(2003)], it is shown that elastic correlations exist between twist, roll, tilt, and stretching of DNA, as well 7 as the distance between phosphate groups. The twist-bend coupling term is shown to be able to 0 capturethesecorrelationstoalargeextent,andafittotheexperimentaldatayieldsanewestimate 0 of G=25 nm for thevalueof thetwist-bend coupling constant. 2 PACSnumbers: 87.15.-v,87.15.La,87.14.Gg,82.39.Pj n a J 9 The DNA in eukaryotes is tightly bound to an equal 4.5◦ 2.7◦ 13 eˆ3 eˆ1 ] mDNasAs-opfrohtiesitnocnoempprloetxeeisnsc,alfloerdmniuncgleaosorempeesat[1in].gAarsrtareytcohf 44..55◦◦ 64..12◦◦ 14 eˆ2 M of147basepair(bp)DNAiswrappedin1.84left-handed 4.5◦ 9.5◦ 15 16 B superhelicalturnsaroundthehistoneoctamerthatforms 4.5◦ 2.9◦ Roll 17 bio. tlcihonerkeenriusDcalNeotAsinoytmoestizhceoedrneseppxoatorctliocwrleeit,phwaarhtiirccahled.iiusEscaoocnfhn5encuntcmeldeoavsnoidamaae 444...555◦◦◦ 1524..97.1◦◦◦ 1198 eˆ2 eˆ3 eˆ1 - height of 6 nm [2]. The wrapped DNA-histone octamer 20 q 4.5◦ 16.5◦ complexisessentiallyubiquitousinnatureandhasama- 21 [ 4.5◦ 5.2◦ jorroleinmanycelllifeprocessessuchasgeneexpression Uniformbending Kinkedbending Tilt 1 and transcription [3]. v Inarecenthighprecisionmeasurement,Richmondand FIG. 1: The conformation of a segment of the nucleosomal 4 1 Daveyhavedeterminedthestructureofthe147basepair DNA,betweenthe13th andthe21st basepairs, is compared 0 DNA in the nucleosome core particle with 1.9 ˚A resolu- withthatofasimplehomogenouselasticrodthatbendsuni- 1 tion [4]. They have observed that the structure of the formly. The bending angles between two consecutive base 0 pairs give a measure of the non-uniformity in the shape of bent DNA segment is modulated in the curvature, roll, 7 thehighly bentDNAsegment. Notethesharp bends(kinks) 0 andtilt, andthat the twist structureappearsto be most nearthe15thandthe20thbasepairs. Aschematicdefinition / affectedbythespecificinteractionswiththeproteinsub- of the bending components Roll and Tilt is presented in the o strate. This experiment provides a wealth of informa- right panel. i b tion about the conformational structure of such highly - bentandstronglyinteractingDNA,amongwhichwecan q : highlight a number of quantitative observations: (1) the DNA structure, a corresponding theoretical analysis is v period of modulation in curvature is set by half of the calledfor,and one naturally wonderswhichof the above i X DNA pitch ∼5 bp, where either the major or the minor approaches could more easily accommodate the addi- r groovesfacethehistoneoctamer,(2)rollappearstohave tional complications due to the high degree of bending a the main contribution to the curvature, as it is favored and the specific DNA-protein interactions. overtiltby1.9:1,and(3)theDNAsegmentisstretched Here,we attempt to use anaugmentedelastic descrip- by about1-2bpascomparedto its unbent conformation tion to account for a number of observations made by [4]. Richmond and Davey. We consider an elastic energy ex- The conformational properties of relatively long DNA pression that includes anisotropic bending rigidities and segments, as well as their elastic response to mechan- twist-bendcoupling. Weshowthattheanisotropicbend- ical stresses such as pulling forces and torques, have ingelasticity isresponsiblefor the modulationsincurva- been successfully studied using a coarse-grained elastic turewiththeperiodof5bp[17],andcalculatetheshape description [5, 6, 7, 8, 9] that could take into account of DNA, an example of which is shown in Fig. 1. Using thermal fluctuations [10, 11, 12]. Other approaches for ananalysisofthe experimentaldataofRef. [4], weshow studying DNA structure include first-principle computer thatthespecificfeaturesintwistandbendarecorrelated simulations [13, 14] andphenomenologicalmodelings us- toalargeextentthroughthetwist-bendcoupling,andex- ing base-stacking interactions [15, 16]. In light of the tract an estimate of G = 25 nm for the twist-bend cou- recent experimental determination of the nucleosomal pling constant that best describes this correlation. We 2 the arclength s and at each point, an orthonormal basis 30 experiment a calculation(G=0) is defined with the unit vectors eˆ (s), eˆ (s), and eˆ (s), calculation(G=25nm) 1 2 3 20 where eˆ1 shows the direction from minor groove to major groove, and eˆ (s) is the unit tangent to the axis (see ees) 10 Fig. 1). Note tha3t due to the helical structure of DNA, gr (de 0 eˆ1(s)andeˆ2(s)rotatewiththehelix. Thedeformationof oll the double helix is characterized by the angular strains R -10 Ω (s) corresponding to bending in the plain perpen- 1,2 dicular to eˆ (s) and Ω (s) corresponding to twist and -20 1,2 3 torsion. We should mention that each slice of the rod is -30 10 20 30 40 50 60 70 labeled by s, which is corresponding to arc length along Base-pairstep the unstressed helix axis and so always changes from 0 20 to L. The actual arc length along the deformed axis is experiment b 15 ccaallccuullaattiioonn((GG==02)5nm) givenbyds′,whichintermsofaxialstrainα(s)onehave ds =[1+α(s)]ds [18]. The elastic energy for the defor- 10 ′ mation of DNA in units of thermal energy is written as s) 5 ee [7, 8] egr 0 d Tilt( -5 EDNA = 1 Lds A Ω2+A Ω2+C(Ω −ω )2 k T 2Z 1 1 2 2 3 0 -10 B 0 (cid:2) +Bω2α2+2Dω (Ω −ω )α -15 0 0 3 0 -20 10 20 30 40 50 60 70 +2Gω0(Ω3−ω0)Ω2i, (1) Base-pairstep e whereA andA arethebendingrigiditiesforthe“hard” 1 2 experiment c and “easy” axes of DNA cross section, C is the twist 6 calculation(G=0) calculation(G=25nm) rigidity, B is the stretch modulus, D is the twist-stretch 4 coupling, and G ≡ Gω is the twist-bend coupling. The 0 ˚A) 2 spontaneous twist oef the helix is defined via its pitch P ( as ω =2π/P, andfor B-DNA we have ω =1.85nm 1. Pa 0 0 0 − P Note that the twist-bend coupling G can be ruled out ∆ -2 by symmetry(ψ to −ψ)for a non-chiralrodω =0,and 0 -4 thusitspresenceisadirectconsequenceofaspontaneous twist structure, as manifested by the form G≡Gω . -6 0 ThelocalstrainsΩ canbewrittenintermsofthecur- 10 20 30 40 50 60 70 i e Base-pairstep vatureκ(s), the torsionτ(s), andthe twistangleψ(s) as Ω =κsinψ, Ω =κcosψ, and Ω =τ +∂ ψ [7]. It can 1 2 3 s be shown that the mean curvature and torsion imposed FIG.2: (a)Roll,(b)tilt,and(c)thesuper-helix-componentof by the nucleosomal structure are κav = R/ R2+ν2 thedifferenceofthelengthsofthetwophosphodiesterchains. and τ = −ν/ R2+ν2 , where R is the rad(cid:0)ius of th(cid:1)e av The filled circles are the experimental data taken from Ref. nucleosome and(cid:0)2πν den(cid:1)otes the pitch of the wrapped [4], the hollow triangles (squares) are calculated points with DNA around the histone octamer. For the nucleosome (without) thetwist-bend coupling. case, R ≃ 41.9 ˚A and 2πν ≃ 25.9 ˚A [4], which yields τ /κ =0.098. Since typical values of ∂ ψ is of the or- av av s der of ω , we can estimate the relevance of torsion with 0 calculate the components of curvature as well as the ax- respect to twist by the ratio τ /ω = 0.012. These es- av 0 ialstrainusingtheexperimentalvaluesfortwistasinput, timates justify neglecting torsion with respect to twist and find that roll is favored over tilt by 1.7:1, and that and curvature. After defining A ≡ 1(A + A ), and the overall stretching of DNA is about 1 bp, both in en- A ≡ 1(A −A ), we can write Eq. (1)2as 1 2 couraging quantitative agreement with the experiment. ′ 2 1 2 We have also studied another parameter called ∆PP , E 1 L a DNA = ds (A−A cos2ψ)κ2+Bω2α2 which shows the super-helix-componentof the difference k T 2Z ′ 0 B 0 (cid:2) of the lengths of the two phosphodiester chains [4], and +C(∂ ψ−ω )2+2Dω (∂ ψ−ω )α s 0 0 s 0 found good agreement with the experimental data. +2G(∂ ψ−ω )κcosψ]. (2) To study the structure of the nucleosomal DNA, we s 0 consider a simple model in which the molecule is repre- For the coupling parameters, we use A = 50 nm [10], sentedasanelasticrod[17]. Therodisparameterizedby A = 30 nm [19], B = 78 nm [20], C = 75 nm [21], and ′ 3 D = 15 nm [22]. Since the only estimate available for thevalueofthetwist-bendcouplingconstantGhasbeen ratherindirect[7],wetreatGasatuning parameterand find its value by fitting to the experimental data of Ref. [4]. To find the shape of the DNA, we need to incorpo- rate the interactions with the binding substrate. The wrapping of DNA by an overall angle of Θ = 2π×1.84 around the protein octamer can be imposed as a global L constraint, which reads ds(1+α)κ = Θ. To take ac- 0 count of the specific anRd local interactions, we assume that it is mostly the twist degree of freedom that is af- fectedbythoseinteractions[23];aviewthatissupported bythe experimentalobservationsofRef. [4]. This allows FIG.3: TheFouriertransformκq ofthecurvature. Thefilled ustowritetheinteractionenergyterm,whichisthefinal circles are the experimental data taken from Ref. [4], the additiontotheenergyexpressionfortheDNAconforma- hollowtriangles(squares)arecalculatedpointswith(without) tion Eq. (2), as VDNA Histone[ψ]. We can now minimize the twist-bend coupling. The peak at q = 15.6 corresponds − the sumofE +V [ψ]withrespecttoκ,α, to a periodicity of 5 bp. (A large peak at zero corresponding DNA DNA Histone and ψ, subject to the wr−apping constraint above. This to the average of the curvature is eliminated to enhance the resolution of thefigure.) yields expressions for the curvature and the axial strain as functions of the twist angle as κ = (∂sψ−ω0) µω0D+BGω02cosψ −µBω02, (3) of 10 bp, which is imposed by the near periodicity in ψ. µ2(cid:2)−Bω2(A−A cos2ψ(cid:3)) The best fit to the experimental data yields G = 25, 0 ′ (∂ ψ−ω )[Dω (A−A cosψ)+µGcosψ]−µ2 whichresultsinthe rollbeingfavoredovertiltby1.7:1, s 0 0 ′ α = µ2−Bω2(A−A cos2ψ) , to be compared with the experimental value of 1.9 : 1. 0 ′ Thisshowsthatinsuchahighlybentstructure,theDNA (4) prefersbyaratioofnearly2to1tousethebendingover where µ is the Lagrange multiplier for the constraint, the easy axis as opposed to the hard one. as well as an equation for the twist that contains un- To make a more refined quantitative comparison with known interaction terms arising from V [ψ]. the experiment, we consider the net curvature (as op- DNA Histone Instead of elaborating on the possible forms−of this in- posed to its components) and take its Fourier transform teraction, we simplify the procedure by directly read- defined as κq = √1n ns=1 κse2πi(s−1)(q−1)/n for a list κs ing off the twist angle from the experimental results of oflengthn(n=73hPere),tobetterresolveitsfeatures. A Richmond and Davey. This allows us to make a direct plotof the absolute value of the curvature Fouriertrans- comparison between the calculated values for the cur- form is shown in Fig. 3, comparing the experimental vature and stretching and the corresponding experimen- data with the calculated ones. Note that the absolute tallymeasuredvaluesforthem,andhenceputtheelastic value of the Fourier transformis symmetric with respect model to a stringent test. to the transformation q → n − q, hence only the first Thetwistangleisobtainedfromthetwiststrainbyin- half of the plot is shown. The Fourier transform of the tegration, and the corresponding integration constant is curvature shows a distinct peak at q = 73 +1 = 15.6 5 chosenbynotingthatatthedyadaxesofthenucleosome corresponding to a periodicity of 5 bp, which is a result the major groove-minor groove direction is perpendicu- of the anisotropic bending elasticity [17]. Moreover,Fig. lar to the surface of the nucleosome core particle. This 3 shows that while the simple elastic description fails to means that the first base pair in either the left or the accountforthedetailedfeaturesofthecurvaturewithout right half of the nucleosomalDNA should have an offset atwist-bendcoupling,onceequippedwithsuchatermit twist angle of ψ = 2π× 1bp = 36 . Finally, we note can give a considerably improved account, with the best 0 10bp ◦ that followingRef. [4]the calculationsareperformedfor fitcorrespondingtoG=25nm. We alsonote thatusing half of the DNA length corresponding to 73 base pairs, the values for the curvature the shape of the bent DNA to make a direct comparison possible. can be determined upon integration. We have provided In Figs. 2a and 2b, we have plotted the calculated such an example in Fig. 1, where the presence of two roll R = 180bΩ = 180κbcosψ and tilt T = 180bΩ = kinks at the distance of5 base pairs fromeachother can π 2 π π 1 180κbsinψ [24] as functions of the position of the base be visibly noted. π pairs, together with the experimental data of Ref. [4], Using Eq. (4), one can also determine the stretch- whereb=3.4˚Aisthebase-pairstepforB-DNA.Thetwo ing of the nucleosomal DNA. In Fig. 4, the axial strain quantitiesappeartobemodulatedwithanearperiodicity α is plotted as a function of the base-pair position. A 4 0.08 a full microscopic computer simulation of such a large 0.06 DNA-protein complex appears to be out of reach with 0.04 the computational power at hand, such simplified phe- (α) 0.02 nomenological approaches could be helpful in under- n strai 0 standing the structural properties of biomolecules. xial-0.02 We thank R. Bruinsma,H. Flyvbjerg, T.B. Liverpool, a P. D. Olmsted, Z.-C. Ou-Yang, W.C.K. Poon, M. Rao, -0.04 H. Schiessel, and A. Traversfor very helpful discussions. -0.06 -0.08 10 20 30 40 50 60 70 Base-pairstep FIG. 4: The calculated axial strain as a function base-pair [1] B. Albertset al.,Molecular Biology of the cell(Garland, position. While detailed experimental data is lacking for the New York,2002). stretching at each base pair, the overall stretching obtained [2] K. Luger, A.W.Mäder, R.K. Richmond, D.F. Sargent, by integration over the above data gives 1 bp in agreement T.J. Richmond,Nature 389, 251 (1997). with theobservations of Ref. [4]. [3] R.M.Saecker,M.T.Record,Curr.Opin.Struct.Biol.12, 311 (2002). [4] T.J. Richmond,C.A. Davey,Nature 423, 145 (2003). positive (negative) value of α shows a stretching (com- [5] C.J. Benham, Biopolymers 22, 2477 (1983). pression) for the corresponding base-pair. The overall [6] F. Tanaka, and H. Takahashi, J. Chem. Phys. 83, 6017 length of the DNA in the nucleosome can be found as (1985). 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Panyukov and Y. Rabin, Phys. Rev. Lett. 85, 2404 undepformed DNA. The calculated values of ∆PPa are (2000); Europhys. Lett. 57, 512 (2002); A. De Col and shown in Fig. 2c, which are in good agreement with the T.B. Liverpool, Phys. Rev.E 69, 061907 (2004). experimental values taken from Ref. [4]. [13] T. Schlick and W.K. Olson, Science 257, 1110 (1992). We have also examined the effect of other elastic [14] W.K. Olson, Curr. Opin.Struct.Biol. 6, 242 (1996). couplings—suchasthe bend-stretchcoupling—bytrying [15] C.S. O’Hern, R.D. Kamien, T.C. Lubensky, and P. Nel- son, Eur. J. Phys. B 195 (1998). tofittotheexperimentaldata,andhavefoundnosignif- [16] B.Mergell, M.R.Ejtehadi,andR.Everaers,PhysRevE icant effect. Higher order elastic terms such as the cubic 68 021911 (2003). terms in curvature etc. [7], are expected to add cor- [17] F. Mohammad-Rafieeand R.Golestanian, Eur.Phys.J. rections of the order of κd, which become (marginally) E 12, 599 (2003); J. Phys.: Condens. Matter, 17, S1165 important in the highly bent (kink) regions. While the (2005). addition of suchterms woulddefinitely help improvethe [18] The fact that the backbone is considered as inextensi- resultsquantitatively,thefactthatitwillintroducemore bledoesnotcontradictwiththeintroductionoftheaxial strain, which as a field defined on the inextensible coor- unknown coupling constants make such a direction not dinatesystemofthebackbonedescribestheextensionof particularly appealing. Moreover, there are also other the rod. structural properties of the nucleosomal DNA such as [19] W.K. Olson, N.L. Marky, R.L. Jernigan, and V.B. shift and slide [24], which appear to be beyond such a Zhurkin,J. Mol. Biol. 232, 530 (1993). simple elastic description. This suggests that a more [20] M.D. Wang, H. Yin, R. Landick, J. Gelles, S.M. Block, promisingdirectionforanimprovedtheorythatcanbet- Biophys. J. 72, 1335 (1997). ter describe the conformation of nucleosomal DNA is a [21] A.V. Vologodskii, S.D. Levene, K.V. Klenin, M. Frank- Kamenetskii, N.R. Cozzarelli, J. Mol. Biol. 227, 1224 generalization of the base-stacking model of O’Hern et (1992). al. [15, 25]. [22] J.F. Marko, Europhys.Lett. 38, 183 (1997). In conclusion, we have shown that an elastic theory [23] H.Schiessel,J.Phys.: Condens.Matter15,R699(2003). that takes into account anisotropic bending and twist- [24] R.E. Dickerson et al.,EMBO J. 8, 1 (1989). bend coupling can account to a considerable degree for [25] F. Mohammad-Rafiee and R. Golestanian, work in the observed structure of the nucleosomal DNA. Since progress.