Chaos, Solitons and Fractals 34 (2007) 104–111 www.elsevier.com/locate/chaos

Metal–insulator transition in DNA molecules induced by long-range correlations in the sequence of nucleotides V.M.K. Bagci *, A.A. Krokhin Center for Nonlinear Science, Department of Physics, University of North Texas, P.O. Box 311427, Denton, TX 76203, United States

Abstract We propose an analytical approach for calculation of the electron localization length in the DNA molecules. In the fishbone model the localization length is directly related to the binary correlation function of the sequence of nucleotides. Application of the proposed method to some known DNA sequences shows sharp maxima in the energy dependence of the localization length. These maxima can be considered as mobility edges. Existence of the mobility edge is rather rare event and it requires that the correlation function apart from an inverse power law decay, also oscillates as a function of the distance between the nucleotides. Although many DNA sequences exhibit slow decay of correlations, only very few of them possess the mobility edge. It is not clear yet what is the biological role of the mobility edge. Ó 2007 Elsevier Ltd. All rights reserved.

1. Introduction Electrical conductivity of the DNA molecules remains an intriguing physical property in spite of numerous experimental and theoretical studies, see recent review [1]. Although, it is well-understood now that the current in DNA is due to light particles, i.e. electrons or holes, [2] the physical mechanism that provides a wide variety of conducting (or non-conducting) properties for different DNA is still unclear. The electrical conductivity of the DNA molecules exhibits strong environmental dependence that includes variation with temperature, chemical composition of the solution, humidity, quality of the metallic contacts, etc. It is not clear whether these properties play a role in living nature or they are artifacts of the experiments. At the same time it is well-known that the DNA plays a crucial role in the process of coding of the genetic information. In the past few years, there have been many attempts to directly measure the conductivity of dry DNA by applying a voltage between two gold electrodes. There are reports on DNA as being an insulator, ohmic, and non-ohmic conductor [2,3] and a semiconductor [4]. There are several theoretical models proposed to explain the direct conductivity measurement experiments. DNA has been viewed as a 1D disordered system where a variable range hopping mechanism and a dependence of localization lengths upon temperature is proposed [5]. A fishbone model with backbone disorder has been considered in Ref. [6], which is in agreement with the experimental results of Ref. [4]. The role of correlations in the fishbone model was numerically studied in Ref. [7].

*

Corresponding author. E-mail address: [email protected] (V.M.K. Bagci).

0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.01.053

V.M.K. Bagci, A.A. Krokhin / Chaos, Solitons and Fractals 34 (2007) 104–111

105

The information in a DNA sequence is written by four-letters alphabet. The letters A, C, G, and T are associated with basis nucleotides – Adenine, Cytosine, Guanine, and Thymine. These four bases form a long chemical chain (a few billions of elements in the case of human DNA), where the exact sequence of the letters is very important. Even a weak damage of the ‘‘right’’ natural sequence may lead to the irreversible changes for a healthy living organism. What physical quantity may by sufficiently sensitive in order to detect a damage of a segment of the length of a few units in a chain of the length of a few billions? At this point one should take into account that a chain of nucleotide bases can be considered as a 1D disordered sequence of potential sites with four different on-site energies, A, C, G, and T. A famous feature of any disordered 1D system is Anderson localization [23] of a quantum particle at a finite length l(E), where E is the energy of the particle [8]. If the length of the system, L, exceeds the localization length l(E), the transmission coefficient of this system is exponentially suppressed, T(E) ’ exp(2L/l(E)). Besides, the transmission coefficient, T(E) is not a self-average quantity. The latter means that the transmission coefficient, electrical resistance, and the conductance strongly fluctuate, i.e. even a slight variation in the sequence of the sequence of nucleotides gives rise to a considerable change of the transport properties. In other words, the contribution of a short segment to the electrical resistance of a long chain is not washed out. This property makes the role of electron transport through the DNA molecules extremely important. We can speculate that the resistance of a nucleotide sequence is just the appropriate quantity, which controls the state of the DNA molecule and enables detection of a damage region even if it contains only a few sites. This conjecture is supported by the experimental results of Ref. [9] where it was observed that damaged DNA strands show significantly different electronic behavior to healthy DNA. Experimental measurements of the DNA resistance always include some effects of the environment or of the experimental set up. In what follows we are interested in the intrinsic resistance, i.e. in the ability of the electron to travel through the DNA molecule. This quantity depends on the internal molecular structure only, i.e. on the particular sequence of the nucleotides. Thus, an electron traveling through the DNA molecule may ‘‘read out’’ the genetic information. Of course, at the present stage we do not know how to extract the genetic information by analyzing the electrical resistance but it is important that this possibility does exist in principle. In this paper we demonstrate that statistical correlations in the sequence of nucleotides are directly related to the localization length l(E) and propose to classify the DNA molecules by the presence or absence of the metal–insulator transition in the energy spectrum. The metal–insulator transition is the critical point E = Ec (also called the mobility edge), which separates region of electron energies corresponding to the localized states, l(E) < L, from that, corresponding to the extended states, l(E) > L. The role of correlations on the electron transport through DNA molecules has been intensively studied in Refs. [7,10–18].

2. Correlated disorder and localization length Any infinitely long disordered 1D system with white-noise spectrum of fluctuations of the potential does not conduct since all electron states are localized, independently on the degree of disorder [8]. Correlations in the spectrum of fluctuations usually increase the localization length and may give rise to the extended states. In the latter case the system may conduct in the interval of energies corresponding to the extended states. The interval may be just a single point (or a few discrete points) if the correlations are of the short-range. This was demonstrated in the random dimer model [19].A continuum of the extended states may appear in the energy spectrum only if the correlations are of the long-range [20,21], i.e. the binary correlation function decays as inverse power law. The relation between the correlation function and the inverse localization length (Lyapunov exponent) obtained in the tight-binding model, has the following form [21], l1 ðEÞ ¼ l1 0 ðEÞuðlÞ:

ð1Þ

Here l(E) is the decay length of the envelope for the eigenfunction wi which satisfies Shro¨dinger equation wiþ1 þ wi1 ¼ ði  EÞwi :

ð2Þ

All the energies here are normalized to the hopping parameter t, which is assumed to be coordinate-independent (diagonal disorder). If the on-site potential is uncorrelated, i.e. hi k i ¼ 20 dik all the states are localized on the length l0(E). For weak disorder, 0  1, the localization length l0(E) was calculated by Thouless [22] l1 0 ðEÞ ¼

20 : 8 sin2 l

ð3Þ

106

V.M.K. Bagci, A.A. Krokhin / Chaos, Solitons and Fractals 34 (2007) 104–111

The parameter l plays the role of the Bloch wave number for the unperturbed (periodic) system and it is related to the energy, E ¼ 2 cos l; 0 < l < p:

ð4Þ

The allowed energies form a band, jEj < 2. The modulation function u(l) in Eq. (1) takes into account the correlations in the site potential n. It is expressed through the binary correlator hi k i ¼ 20 nði  kÞ as follows [21]: 1 X nðkÞ cosð2klÞ: ð5Þ uðlÞ ¼ 1 þ 2 k¼1

A continuum of extended states (where l(E) = 1) appears if the function u(l) vanishes within a finite interval of l (energies). Then, it follows from the properties of the Fourier series, that the Fourier coefficients n(k) of the function u(l) decay as inverse power law. This means that the correlations extend for long distances. In particular, it was shown theoretically [21] and experimentally [24], that a sharp mobility edge [25] in the spectrum requires the correlation function of the following form, nðkÞ ¼

sinðakÞ ak

ð6Þ

The constant a determines the position of the mobility edge within the interval of the allowed energies, jEj < 2. If the mobility edges are at the points E = ±1, the value of a = 2p/3. Eq. (6) emphasizes two principal features which are necessary in order the mobility edge exists in the spectrum of 1D disordered system: slowly-decaying correlations and oscillations. The latter means periodic alternation of correlations and anti-correlations. It is clear that these two requirements are not easy to fulfill in a disordered system. It is well-known that as a rule the DNA sequences are correlated with a power-law decay of the correlation function [7,10–18]. However, only very few of them possess a mobility edge in the energy spectrum. In what follows we demonstrate that the correlation functions of those with mobility edges indeed exhibit almost regular oscillatory behavior.

3. Fishbone model of DNA Electron transport in a DNA molecule can be considered as discrete jumps between the neighboring base nucleotides. There are two coupled strands (double-helix structure of DNA) that form two propagating channels for the electrons. Due to strong A–T and C–G coupling, it is usually considered that there is a single electron channel with a binary sequence of the on-site potentials with energies eA–T = 8.69 eV and eG–C = 8.31 eV. Each site in this sequence has a link to the sugar molecule, see Fig. 1. The sugar molecules maintain the structure stability, and they are called backbones. From the base site an electron may jump either to the neighboring sites or to one of the backbone sugars. The jumps occur due to overlapping between electron p orbitals. The backbone sugars are not coupled. Fig. 1 represents the main features of this so-called fishbone model. The Hamiltonian of the fishbone model is written as follows [6,7]:

Fig. 1. The Fishbone model for DNA. The base pairs and backbones have disordered on-site energies while the hopping parameters t and tb are constant. The base pairs are either A–T or G–C, with energies eA–T = 8.69 eV and eGC = 8.31 eV respectively.

V.M.K. Bagci, A.A. Krokhin / Chaos, Solitons and Fractals 34 (2007) 104–111

107

10.4

Energy , eV

10

9.6

9.2

8.8

0

30

μ

60

90

Fig. 2. The dispersion curves for the fishbone model.

HF ¼ 

L X

ti jiihi þ 1j 

i¼1

L X

tqi jqi ihij þ

X

ei jiihij þ eqi jqi ihqi j þ h:c:

ð7Þ

i;qi

i¼1;q¼u;d

Here jii and jqii are the wave functions at the base and backbone site respectively, ti is the hopping parameter between base sites i and i + 1, tqi is the hopping between the upper, q = u (lower, q = l)) backbone site and the base site, and ei and eqi are the on-site energies of the bases and the backbones, respectively. In what follows we will consider the case of the diagonal disorder when the hopping parameters ti = t and tui ¼ tdi ¼ tb are coordinate- independent constants. The role of the off-diagonal disorder was considered in Ref. [16]. It is convenient to chose t as a unit of energy, so as all the energies become dimensionless. The randomness in the structure of the DNA molecules is due to the weak fluctuations 2 u;d 2 of the on-site energies, ei = e0 + dei and eiu;d ¼ eb þ deu;d i , where hdei i ; hdei i  1. After these simplifications the Schrodinger equation corresponding to Hamiltonian Eq. (7) can be written in the tight-binding model Eq. (2), where the onsite energy i is replaced by [7] ni ¼ dei þ

t2b ðE  eb Þ2

fdeui þ dedi g;

ð8Þ

and the eigenenergy E is replaced by 2 e ¼ E  e0  2tb : E E  eb

ð9Þ

e being the eigenenergy of the tight-binding model Eq. (2), lies within the interval 2 < E e < 2. The effective energy E, e has a standard form, E e ¼ 2 cos l. The dispersion relation for the electron energy E is obThe dispersion relation for E tained from Eq. (9), qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e0 þ eb 1 ðeb þ e0 þ 2 cos lÞ2  4½eb ðe0 þ 2 cos lÞ  2t2b : þ cos l  ð10Þ E ðlÞ ¼ 2 2 In the fishbone model the allowed energies form two equivalent conduction bands [6], thus modelling semiconductor behavior of DNA observed in the experiment [4]. Eq. (8) is a result of linear expansion over weak fluctuations, therefore both terms are supposed to be small. Because of the singularity in the second term, a narrow interval of energies close to E = eb is excluded from the consideration. This limitation of the fishbone model was already mentioned in Ref. [7]. The genetic information is coded in the sequence of potentials ei, therefore the fluctuations dei are correlated. Unlike this, the fluctuations of the potentials in the two backbone sequences do not carry genetic information, therefore they are considered to be uncorrelated and statistically independent. Thus, the correlations that exist in the sequence of the random numbers ni are due to the correlations between the base pairs only, hni nk i ¼ n20 /ði  kÞ þ r2

t4b ðE  eb Þ2

dik :

ð11Þ

108

V.M.K. Bagci, A.A. Krokhin / Chaos, Solitons and Fractals 34 (2007) 104–111

Here n20 ¼ hde2i i, r2 ¼ hðdeui Þ2 i þ hðdedi Þ2 i, and /(k) is the normalized (/(0) = 1) binary correlation function of the base sequence.

4. Numerical calculations Our numerical calculations are based on the information about DNA sequences available from the GenBank database [27]. We have analyzed dozens of DNA’s in order to find a few of them possessing a mobility edge. It was already well understood that the DNA sequences, being random, reveals long-range correlations. Different methods have been used to study these correlations: the Hurst’s analysis, [14], diffusion [11] and information [28] entropy, random-walk analysis, [10] hierarchial structure theory, [29] signal processing, [30] etc. The effect of the long-range correlations on the charge transfer has been numerically studied in Ref. [14], where the authors obtained enhancement of the electron transmission due to correlations. Our approach, based on the analytical result Eq. (1), establishes a direct relation between the strength of correlations in the nucleotide chain and the electron localization length. To the best of our knowledge this is the only approach which gives an explicit relation between these quantities. To calculate the correlation function /(k) a sequence of nucleotides was mapped into a binary one with energies ±(eA–T  eG–C)/2t. The value of t = 0.37 eV [7] and the average backbone energy is taken equal to 0 since the backbones and the nucleotides in the base sequence are chemically connected. The correlation functions for two different DNA are shown in Figs. 3 and 4. Both correlation function oscillates with small but slowly decaying amplitude. These oscillations turn out to be the source of the sharp metal–insulator transition predicted in Ref. [21] for the correlation function Eq. (6), which oscillates with power-decaying amplitude. In the numerical calculations the value of k in Eq. (5) does not run to infinity but is cut at the value of kmax = 400. This provides good convergence of the Fourier series and the accuracy <1% for the localization length. The localization length, corresponding to the correlation functions in Figs. 3 and 4 is plotted in Figs. 5 and 6. Only the region of high-energy band (see Fig. 2) is shown, since the dependence l(E) in the low-energy band is exactly the same. In the both cases the major part of the allowed energies is occupied by the states localized at the distances of 30–70 base pairs. At these energies DNA macromolecule of the length of 103 behaves like an insulator. However, there are sharp peaks near E = 10.2 eV. At the peak regions the localization length is orders of magnitude larger. Our estimates show that it can be as long as 104–105 base pairs. In the experiments usually much shorter segments are examined, which, thus, may exhibit metallic behavior within the peak regions of energies. In Figs. 5 and 6 the peak regions are shown in the insets. However, the localization length there remains finite, very sharp increase of the localization length should be considered as a mobility edge. Within the peak region the states with localization length l(E)  105 are practically delocalized. The region of the delocalized sates occupies a narrow interval of energies 20 meV. For the both DNA’s shown in Figs. 5 and 6 the presence of the sharp mobility edges is due to the oscillatory behavior of the correlation function. The long-range correlations by themselves do not necessary lead to the metal–insulator

1 BRCA

0.8 Correlation Function

1 0.04

0.6

0.02

0.5

0

0.4

-0.02 0 0

2

4

6

8

10

100

k

0.2

110

120

k

0 -0.2

0

200

400

600

800

1000

k

Fig. 3. Binary correlation function of the sequence of nucleotides of Human BCRA gene. The correlation function drops from the value of 1 at k = 0 to /  0.1 at k P 1, left inset. Correlations extend to distances of a few thousands of base pairs, decaying very slowly. An important feature of this correlation function is close to regular oscillations about zero, right inset.

V.M.K. Bagci, A.A. Krokhin / Chaos, Solitons and Fractals 34 (2007) 104–111

109

1 Ch22

0.8 Correlation Function

1

0.05

0.6

0.045 0.5 0.04

0.4

0.035

0 0

2

4

6

8

10

300

310

k

0.2

320

k

0 -0.2

0

100

200

300

400

500

k

Fig. 4. Binary correlation function of the sequence of nucleotides of 51 kbp of non-coding region of Ch22. The correlation function exhibits behavior similar to that shown in Fig. 3 but with oscillations about a small positive value.

BRCA

1000

Localization Length

1000

100 10

100

10.18

10.19

10.2

eV

10

1

10

10.1

10.2

10.3

10.4

Energy, eV

Fig. 5. Localization length vs Energy for Human BRCA gene.

Localization Length

Ch22 1000

1000

100 10 10.2

10.21

10.22

eV

100

10

1

10

10.1

10.2 Energy, eV

10.3

10.4

Fig. 6. Localization length vs Energy for Ch22 DNA.

110

V.M.K. Bagci, A.A. Krokhin / Chaos, Solitons and Fractals 34 (2007) 104–111

transition. The sharp, step-like transition requires the correlation function given by Eq. (6). Any deviation from this specific form leads to the ‘‘softening’’ of the mobility edge and to its disappearance.

5. Conclusion We proposed a novel method for the calculation of the localization length of electrons in the DNA macromolecules. The fishbone model of DNA is used that leads to the tight-binding approximation for the electron Hamiltonian. The method requires calculation of the binary correlation function for the sequence of the ionization potentials of the nucleotides. In agreement with previous studies, we obtained that most of the DNA molecules behave like insulators since the electron transfer is limited by a distance of 30–70 base pairs only. However, if the correlation function oscillates with slowly-decaying amplitude, the localization length may have narrow regions of energies with localization length 104– 105. Within these regions the electrons are practically delocalized, i.e. metallic behavior is also available. It is not clear yet what is the biological role of the extended states and how the metal–insulator transition is employed in genetics.

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