We address the relation between long range correlations and charge transfer
efficiency in aperiodic artificial or genomic DNA sequences. Coherent charge
transfer through the HOMO states of the guanine nucleotide is studied using the
transmission approach, and focus is made on how the sequence-dependent
backscattering profile can be inferred from correlations between base pairs.Comment: Submitted to Phys. Rev. Let

Dominique Bicout

Efim Kats

Enrique Maciá

H. E. Hurst

M. Di Ventra

P. Carpena

Stephan Roche

W. Feller

W. Li

English

arXiv

Crossref

Long Range Correlations in DNA: Scaling Properties and Charge Transfer Efficiency

We address the relation between long range correlations and charge transfer efficiency in aperiodic artificial or genomic DNA sequences. Coherent charge transfer through the HOMO states of the guanine nucleotide is studied using the transmission approach, and focus is made on how the sequence-dependent backscattering profile can be inferred from correlations between base pairs

Roche, Stephan

Bicout, Dominique

Macia, Enrique

Kats, Efim

HAL-CEA

Long range correlations in DNA : scaling properties and charge transfer efficiency

©2003 The American Physical Society.We address the relation between long-range correlations and charge transfer efficiency in aperiodic artificial or genomic DNA sequences. Coherent charge transfer through the highest occupied molecular orbital states of the guanine nucleotide is studied using the transmission approach, and the focus is on how the sequence-dependent backscattering profile can be inferred from correlations between base pairs.Depto. de Física de MaterialesFac. de Ciencias FísicasTRUEpu

Roche, S.

Bicout, D.

Maciá Barber, Enrique Alfonso

Kats, E.

Docta Complutense

P H Y S I C A L R E V I E W L E T T E R S week ending28 NOVEMBER 2003VOLUME 91, NUMBER 22Long Range Correlations in DNA: Scaling Properties and Charge Transfer EfficiencyStephan Roche,1 Dominique Bicout,2 Enrique Maciá,3 and Efim Kats2,41Commissariat à l’Énergie Atomique, DSM/DRFMC/SPSMS, 17 avenue des Martyrs, 38054 Grenoble, France2Laue-Langevin Institute, 6 rue Jules Horowitz, BP 156, F-38042, Grenoble, France3Departamento de Fisica de Materiales, Facultad de Fisicas, Universidad Complutense, E-28040 Madrid, Spain4L. D. Landau Institute for Theoretical Physics, RAS, Moscow, Russia(Received 12 June 2003; published 26 November 2003)228101-1We address the relation between long-range correlations and charge transfer efficiency in aperiodicartificial or genomic DNA sequences. Coherent charge transfer through the highest occupied molecu-lar orbital states of the guanine nucleotide is studied using the transmission approach, and the focusis on how the sequence-dependent backscattering profile can be inferred from correlations betweenbase pairs.DOI: 10.1103/PhysRevLett.91.228101 PACS numbers: 87.14.Gg, 72.20.Ee, 72.80.Lethe evaluation of the second moment of the fluctuations of Hn  H1 for 1  n < nc and Hn  H2 for n  nc.During the past few years, the nature of long rangecorrelations in DNA sequences has been the subject ofintense debate [1–3]. Scale invariant properties in com-plex genomic sequences with thousands of nucleotideshave been investigated, in particular, with wavelet analy-sis [2], and have been argued to play a crucial role in generegulation and cell division. Besides, among the manyphysical, chemical, or biological phenomena that mightbe inferred from sequence correlations, charge transferproperties deserve particular concern. Indeed, a preciseunderstanding of DNA-mediated charge migration wouldhave a strong impact on the description of damage rec-ognition process and protein binding, or on engineeringbiological processes [4,5]. The -stacked array of DNAbase pairs (bp) (made up from nucleotides: guanine G,adenine A, cytosine C, thymine T) provides an ex-tended path to convey long-range charge transport al-though dynamical motions of base pairs, or energeticsequence-dependent heterogeneities, are expected to re-duce long-range efficiency. Photoexcitation experimentshave unveiled that charge excitations can be transmittedbetween metallointercalators, preferentially through theguanine highest occupied molecular orbitals (G-HOMO)of the DNA bridge [5,6]. Such experiments and meso-scopic transport measurements on single artificial orgenomic DNA sequences contacted between metallicelectrodes have also been the subject of controversialdebate [7]. While accurate determination of absolute val-ues of conductivity is important, characteristic sequencedependences of charge transport could provide valuableclues to mechanisms and biological functions of trans-port. Such an issue has up to now been poorly addressedexperimentally and theoretically. In that perspective, thepossible role of long-range correlations on electronicdelocalization has been recently anticipated [8]. In thisLetter, the electronic transport properties are proven to becritically related to the nature and range of correlations.Rescaling coefficients have been introduced as a usefulmeasure of correlations in DNA sequences [1]. It relies on0031-9007=03=91(22)=228101(4)$20.00 sequence composition. The statistical method consists ofconstructing a mapping of the nucleotide sequence onto awalk. A DNA walk is initiated from the first to the lastnucleotide of the sequence with the rule that the walkersteps down vi  1 if a purine (A, G) occurs atposition i along the sequence, whereas the walker stepsup vi  1 if a pyrimidine (T, C) occurs at position i.Given a nucleotide sequence of size N, the net displace-ment xn of the nucleotide walker after n steps is xn Pni1 vi, 1  n  N. Recently, Hurst’s analysis [9] wasargued to be more reliable for determining the preciserescaling coefficients [10]. We thus follow the prescriptionof Hurst’s analysis to construct adjusted variables asXm; k  xm; k  kn xm; n , 1  k  n, and definethe range Sm; n for random walks of lengths n asSm; n  max1knXm; k  min1knXm; k .Now, the rescaled range function Rn is defined as [9]Rn hSnin/ nH; (1)where hSni PNnm1 Sm; n=N  n and 2n is thestandard deviation of vi over walks of lengths n, andaveraged over the entire sequence. The Hurst exponent Hof the process is then defined through the scaling inEq. (1). Interestingly, for short-ranged correlated randomwalk the exact result for the rescaled range function readsRn n=2p 1 [9,11]. Thus, H  1=2 for the ordi-nary Brownian motion. The existence of power-law be-haviors suggests that there is no characteristic lengthscale associated with properties under consideration. Itis clear at the first glance that DNA sequences are unlikelyfully characterized by a single scaling exponent. Oneexpects that the scaling behavior be different for differentlength scales of the sequence; i.e., the rescaling exponentis itself a function of the length scale n. In the case wherea characteristic size nc can be defined, one may postulatethat Rn is still described by the power law in Eq. (1), butwith a scale dependent rescaling exponentsHn such that2003 The American Physical Society 228101-1101102103104n100101102R(n)RandomCh22FibonacciFIG. 1. Rescaled range function Rn versus n. Dashed linecorresponds ton=2p 1.TABLE I. Hurst exponents calculated from data in Fig. 1.Hnc  300Sequence N Purines H1 H2Ch22 182 617 91 029 0:60 0:75Random 182 617 91 118 0:50 0:50Fibonacci 46 368 28 657 0:085 0:011P H Y S I C A L R E V I E W L E T T E R S week ending28 NOVEMBER 2003VOLUME 91, NUMBER 22In our study, we consider three sequences: a DNAsequence of the first completely sequenced human chro-mosome 22 (Ch22) containing about 33:4  106 nucleo-tides entitled NT011520 retrieved from the National Centerfor Biotechnology Information, a random DNA se-quence (where A, T, C, G are evenly chosen probability1=4) and a Fibonacci polyGC quasiperiodic sequenceconstructed starting from a G nucleotide as seed andfollowing the inflation rule G ! GC and C ! G. Thisgives successively G, GC, GCG, GCGGC, GCGGCGCG,GCGGCGCGGCGGC, . . ., for sequences of length 1; 2; 3;5; 8; 13; . . . , respectively, such that its characteristic self-similar order introduces correlations on a broad scalerange. The ratio number of G=number of C approachesthe golden mean value 1 5p=2 ’ 1:618 in the limit ofan infinite sequence. The random and Fibonacci sequen-ces are used as prototypes of short-range (or uncorre-lated) and strongly correlated systems, respectively.The computed functions Rn for the three sequencesdescribed above are reported in Fig. 1 and values ofH aresummarized in Table I. It clearly appears from thesecalculations that the random sequence is indeed uncorre-lated following then=2plaw, whereas the Fibonaccisequence is strongly correlated with a ‘‘ballistic behav-ior’’ and correlations in the Ch22 sequence exhibit apower-law behavior with a scaling exponent dependingon the length scale. The Ch22 sequence has long-rangecorrelations characterized by Hurst exponents greaterthan 1=2 (see Table I). Given the huge amount of nucleo-tides of the Ch22 sequence, the physically relevant ques-tion seems rather to address to which extent charge228101-2transport can be efficient through the G-HOMO, in com-parison with uncorrelated random or quasiperiodic se-quences. To have some elements of response, we now turnto the examination of charge transfer properties in thesesequences. To this end, we consider an effective tight-binding Hamiltonian describing the energetics of a holelocated at nucleotide site n [12,13],H Xn"ncyncn Xnt0cyncn1  H:c:; (2)where cyn (cn) is the creation (annihilation) operator of ahole at site n. The hole site energies "n are chosen accord-ing to the ionization potentials of respective bases [13],"A  8:24 eV, "T  9:14 eV, "C  8:87 eV, and "G 7:75 eV, while the hopping integral, simulating the stacking between adjacent nucleotides, is taken ast0  1 eV. The DNA sequences are further assumed to beconnected to two semi-infinite electrodes whose energies"m are adjusted to simulate a resonance with the G-HOMO energy level, "m  "G, and with hopping inte-grals such that tm  t0. Note that ab initio studies suggestthat t0  0:1–0:4 eV [13], but the choice tm=t0  1 re-duces backscattering of holes at the contact electrodesand allows for a larger accessible transmission spectrumand a better characterization of DNA’s intrinsic conduc-tion [12]. Sites comprised between 1; 0 [ N 1;1 belong to the leads, whereas sites i  1; N areassociated with the sequence of size N under study. Thetransmission coefficients are computed using the transfermatrix formalism in which the time independentSchrödinger equation is projected into a localized basisby properly accounting for the boundary conditions [14].Let  n denote the wave function with energy E at site n,we obtain from Eq. (2) the recurrent equation,  N2 N1 MN N1 N  MN   M1 1 0, where Mn is a 2  2 matrixwith elements Mn1; 1  E "n=tn1, Mn1; 2 tn=tn1, Mn2; 1  1, and Mn2; 2  0. The transmis-sion coefficient TNE that gives the fraction of tunnelingelectrons transmitted through the N-site DNA is relatedto the Landauer resistance as h=2e21  TNE=TNE,where h=2e2 is the quantum resistance and [14]TNE 4 E "m2t2mE "m2t2mP 12P 21  1 E "mtmP 11  P 22P 12  P 21 Xi;j1;2P 2ij  2	(3)with P  MNMN1 . . .M1. For a given energy, TNE reflects the level of backscattering events in the hole transportthrough the sequence. As metallic leads are adjusted to the G-HOMO energy level, the hole transport will experience asequence-dependent contribution of backscattering according to the distribution of C, T, and A potential barriers over228101-2P H Y S I C A L R E V I E W L E T T E R S week ending28 NOVEMBER 2003VOLUME 91, NUMBER 22the length scale of the sequence. To compare transmis-sion properties of different chains, the behavior ofthe Lyapunov coefficient, NE 12N lnTNE, is alsocalculated. NE has been extensively investigatedto sort out the main features of complex localizationpatterns [15,16]. For systems with uncorrelated dis-order, NE provides the localization length E 1=limN!1NE. In the presence of scale invarianceproperties, the underlying structure of NE reflects theself-similarity of the spectrum [16].Following our analysis on correlations, the TNE forthe three sequences of Table I have been computed, vary-ing the sequence length. The random and Fibonacci qua-siperiodic based sequences are generated starting fromthe first nucleotide of the sequence up to N bp, while theCh22-based sequences are constructed by starting fromthe bp  15 000 of the full Ch22 sequence and thenextracting the first N bp, namely AGGGCATCGCT-AACGAGGTCGCCGTCCACA GCATCGCTATCGAG-GACACCACACCGTCCA for N  60 bp. Figures 2 and3 present the comparison of TNE between the quasi-periodic and Ch22 sequences and between uncorrelatedrandom DNA and Ch22 sequences, respectively, withthe same number of bp. Lyapunov coefficients for quasi-periodic and Ch22-based sequences are also displayedin Fig. 4.General trends of Figs. 2 and 3 are that TNE ischaracterized by an energy spectrum of resonant peakswith high transmission. As the sequence length increases,fewer states will present good transmittivity, due to theprogressive fragmentation of the spectrum, although sev-eral peaks with high transmission remain at certain en-ergy values, and new ones may appear. For Fibonacci andCh22-based sequences, these resonant energies are robustenough to persist against backscattering effects due to6 7 8 9 1000.20.40.60.816 7 8 9 1000.20.40.60.816 7 8 9 1000.20.40.60.816 7 8 9 1000.20.40.60.815.75 6.75 7.75 8.75 9.7500.20.40.60.81TN(E) N=60 bp N=60 bpN=180 bp N=180 bpEnergy(eV)Energy(eV)TN(E) (Ch22)(quasiperiodic)(Ch22)FIG. 2. Transmission coefficient for Fibonacci polyGC quasi-periodic (left frames) and Ch22-based sequences (rightframes). Inset: TNE in Eq. (4) for a periodic approximantof length N  50 bp.228101-3interspersed bases along the sequence. This point is illus-trated in Figs. 2 and 3 where one observes that Fibonacci(Ch22-based sequences) of 180 bp (360 bp) exhibit stateswith better transmission properties than those present in a60 bp (300 bp) long sequence. In addition, NE shown inFig. 4 illustrates intrinsic properties of the two correlatedsequences albeit of different nature. Indeed, the series ofmain elliptic bumps found in the Fibonacci sequence with60 bp are reproduced in the 480 bp sequence, whichpresent additional features associated with the partition-ing of spectrum. While self-similarity fully characterizesthe quasiperiodic sequence, the scaling properties inCh22 rely on a totally different kind of long-range cor-relations, with no hints of self-similar patterns.In contrast, the fragmentation of the spectrum stronglyaffects the transmittivity of the uncorrelated randomsequence. All resonant states (when any) are evenly af-fected and the corresponding transmission decreases asthe sequence length gets longer. From a statistical analy-sis over many random sequences, it clearly appears thatCh22-based sequences exhibit a much higher chargetransfer efficiency over much longer distances in com-parison with uncorrelated random sequences.Nevertheless, to improve our understanding and gainsome physical insights about characteristic features ex-hibited by these sequences, we now focus on quasi-periodic sequences since it has been shown that theglobal structure of the electronic spectrum of such chainscan be obtained in practice by considering very shortperiodic approximants to infinite quasiperiodic chains[16]. These sequences are characterized by long-rangecorrelations that manifest themselves on electronicproperties in terms of power-law localization of eigen-states in the thermodynamic limit or power-law increaseof Landauer resistance in finite samples [16]. For this6 7 8 900.20.40.60.816 7 8 900.20.40.60.816 7 8 900.10.20.30.45 6 7 8 900.050.10.150.26 7 8 905e061e051.5e056 7 8 90e+002e044e046e048e046 7 8 900.0050.010.0150.026 7 8 9 1000.10.20.3TN(E) N=300 bp N=360 bpN=420 bp N=480 bp Energy(eV)Energy(eV)TN(E)  (160nm)FIG. 3. Transmission coefficient for Ch22-based sequences(main frames) and typical results (over about 50 sequences)for uncorrelated DNA random chains (insets) with the samenumber of nucleotides.228101-36.5 7.5 8.5 9.500.050.10.156.5 7.5 8.5 9.500.050.10.15Ch22based chainsEnergy(eV)γΝ(E) N=480bpN=60 bpQuasiperiodic chainsFIG. 4. Lyapunov coefficient for Ch22-based (main frame)and Fibonacci polyGC quasiperiodic sequences (inset).P H Y S I C A L R E V I E W L E T T E R S week ending28 NOVEMBER 2003VOLUME 91, NUMBER 22purpose, we consider a periodic approximant whose unitcell is GCGGC. The corresponding dispersion relation ofthis approximant is given by, 2t50 cos5q E"G3E  "C2  t20E  "GE  "C5E  4"G  "C t405E3"G2"C. The energy spectrum of the GCGGCapproximant is composed of three broad bands (of band-width ’ 0:5–0:6 eV) centered at the energies E2 6: 915 eV, E3  8:143 eV, and E4  9: 527 eV, plus twonarrower bands (of bandwidth ’ 0:25 eV) located at theedges of the spectrum at E1  6:191 eV and E5 10:213 eV. These analytical results allow us to properlyassign the different resonant peaks appearing in thespectrum of the transmission coefficient (shown in theinset of Fig. 2) with respect to the four main subbands ofthe spectral window 5:75; 9:75 eV. States belonging tothe broader central bands around E2  6:915 eV andE3  8:143 eV turn out to be very robust to the progres-sive fragmentation of the energy spectrum. Accordingly,one is tempted to conclude from the simple inspection ofFig. 2 (left frames) that these states should exhibit goodtransport properties even in the thermodynamic limit. Tofurther substantiate such an assertion, we consider inaddition the transmission coefficient corresponding tothe GCGGC approximant,TNE  1  qx; yU2N=51w1; (4)where x  E  "C=2t0, y  E  "G=2t0, w 16x2y3  16xy2  4yx2  3y  2x, the Un1w is aChebyshev polynomial of the second kind, and qx;yA2=1y2B21 with A24xy316x2y26xy2x232x2y44y4y2 and B32x2y38x2y24xy24y33y2x. The resonance condition then readsqx; yU2N=51w  0, while the condition qx; y  0yields El  4: 317 eV (which does not belong to the spec-228101-4trum) and Eu  10: 158 eV (located near the center ofthe uppermost band, which is not included in our spec-tral window). On the other hand, the roots of theChebyshev polynomial label a full transmission peakseries according to the relationship w  cos5k=Nwith k  0; . . . ; N. This is illustrated in the inset ofFig. 2 (top left) where one observes oscillations in theenergy dependence of the transmission curve for a se-quence GCGGC with 10 units. By a deeper analysis, wefind that Fibonacci quasiperiodic sequences as long as160 nm, i.e., 450 bp, will still allow for nearly resonanttransmission around two specific energies E2 ’ 6:9 eVand E3 ’ 8:1 eV.In summary, long-range correlations in aperiodic DNAsequences seem to induce coherent charge transfer overlonger length scales, when compared to uncorrelatedsequences. Such a feature has been illustrated, in particu-lar, in Chromosome 22-based sequences. Given that thenature of long-range correlations differs in coding versusnoncoding regions of a given genomic DNA [3], a moresystematic study of charge transport should, however, beundertaken.[1] C. K. Peng et al., Nature (London) 356, 168 (1992); W. Liand K. Kaneko, Europhys. Lett. 17, 655 (1992).[2] R. F. Voss, Phys. Rev. Lett. 68, 3805 (1992); A. Arneodoet al., Phys. Rev. Lett. 74, 3293 (1995); B. Audit et al.,Phys. Rev. Lett. 86, 2471 (2001).[3] S.V. Buldyrev et al., Phys. Rev. E 51, 5084 (1995);H. Herzel and I. Grosse, Phys. Rev. E 55, 800 (1997);D. Holste and I. Grosse, Phys. Rev. E 67, 061913 (2003).[4] E. Braun et al., Nature (London) 391, 775 (1999).[5] C. Treadway, M. G. Hill, and J. K. Barton, Chem. Phys.281, 409 (2002).[6] M. A. Ratner, Nature (London) 397, 480 (1999); D. J.Bicout and E. Kats, Phys. Lett. A 300, 479 (2002).[7] M. Di Ventra and M. Zwolak, in Encyclopedia ofNanoscience and Nanotechnology, edited by H. S.Nalwa (American Scientific Publishers, San Francisco,2003).[8] P. Carpena et al., Nature (London) 418, 955 (2002); 421,764 (2003).[9] H. E. Hurst, Proc. Am. Soc. Civ. Eng. 76, 1 (1950).[10] A. Montagnini et al., Phys. Lett. A 244, 237 (1998).[11] W. Feller, Ann. Math. Stat. 22, 427 (1951).[12] Y. A. Berlin, A. L. Burin, and M. A. Ratner, SuperlatticesMicrostruct. 28, 241 (2000); S. Roche, Phys. Rev. Lett.91, 108101 (2003).[13] H. Sugiyama and I. Saito, J. Am. Chem. Soc. 118, 7063(1996); A. Voityuk et al., J. Chem. Phys. 114, 5614 (2001).[14] A. Stone et al., Phys. Rev. B 24, 5583 (1981).[15] J. L. Pichard, J. Phys. C 19, 1519 (1986).[16] E. Maciá et al., Phys. Rev. B 49, 9503 (1994).228101-4

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Long range correlations in DNA: scaling properties and charge transfer efficiency

Long range correlations in DNA : scaling properties and charge transfer
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Long range correlations in DNA : scaling properties and charge transfer efficiency

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