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Physics Letters A
www.elsevier.com/locate/plaCurrent–voltage characteristics of double-strand DNA sequences
L.M. Bezerril
a, D.A. Moreira
a, E.L. Albuquerque
a,
∗
, U.L. Fulco
b, E.L. de Oliveira
c, J.S. de Sousa
c aDepartamento de Física, Universidade Federal do Rio Grande do Norte, 59072-970, Natal-RN, BrazilbDepartamento de Biofísica e Farmacologia, Universidade Federal do Rio Grande do Norte, 59072-970, Natal-RN, Brazil cDepartamento de Física, Universidade Federal do Ceará, 60455-760, Fortaleza-CE, Brazil
a r t i c l e
i n f o
a b s t r a c t
Article history:
Received 12 May 2009
Received in revised form 7 July 2009 Accepted 13 July 2009
Available online 15 July 2009 Communicated by R. Wu PACS: 72.15.Rn 72.20.Ee 87.14.Gg 87.15.-v Keywords: Tight-binding Hamiltonian DNA molecule Hopping mechanism Electronic properties Quasiperiodic structures
We use a tight-binding formulation to investigate the transmissivity and the current–voltage (I–V) characteristics of sequences of double-strand DNA molecules. In order to reveal the relevance of the underlying correlations in the nucleotides distribution, we compare the results for the genomic DNA sequence with those of artificial sequences (the long-range correlated Fibonacci and Rudin–Shapiro one) and a random sequence, which is a kind of prototype of a short-range correlated system. The random sequence is presented here with the same first neighbors pair correlations of the human DNA sequence. We found that the long-range character of the correlations is important to the transmissivity spectra, although the I–V curves seem to be mostly influenced by the short-range correlations.
©2009 Elsevier B.V. All rights reserved.
Due to their potential applications in nanoelectronics, there has been a growing interest in the synthesis, characterization, and the electronic properties of DNA-based molecules with periodic nucleotide sequences [1–4]. Using a full range of physical and biochemical methods, studies have now established that double helical DNA is a medium for the efficient transport of electrons, triggering a series of experimental and theoretical investigations [5,6]. The earliest studies involved physical measurements of cur-rent flow in DNA fibers, and led to a mixture of conclusions, some suggesting high electron mobility through DNA, others indicating no conductivity [7–10]. In fact, one experiment even pointed to DNA as a superconductor[11]. These physical studies have not yet been reconciled with one another. The variations probably depend heavily upon the connections between the DNA and the electrodes used, as well as upon the integrity of the DNA itself in the ab-sence of water and exposed to very high voltages. Besides, these experiments have underscored not only that the DNA base pair stack can mediate hole and electron transport chemistry, but also
*
Corresponding author. Tel.: +55 84 32153793; fax: +55 84 32153791.E-mail address:[email protected](E.L. Albuquerque).
the exquisite sensitivity of the charge transport through the DNA structure.
In addition to its own structural complexity, measuring charge transport in a DNA chain is strongly biased by the invasive role of contacts, interaction with some inorganic substrate, and tem-perature/atmosphere experimental conditions. As a result, a focus has been now shifted from asking whether DNA can mediate long-range charge transport to questioning how it works. Specifically, we are seeking answers to how do DNA structure and sequence affect this reaction, and how important is DNA-mediated charge transport.
To this end, several experimental groups have reported mea-surements of the current–voltage (I–V) characteristics obtained from electrical transport measurements throughout individual or small numbers of DNA-like molecules captured between two metal nanoelectrodes. The traditional molecular view of electron trans-fer between donor and acceptor species gives rise to a novel view of the molecule as a current-carrying conductor, and observables such as electron-transfer rates are replaced by the conductivities, or more generally by current–voltage relationships, in molecular junctions. Of primary importance is the need to understand the interrelationship between the molecular structure of such junc-tions and their function, i.e. their transmission and conduction 0375-9601/$ – see front matter ©2009 Elsevier B.V. All rights reserved.
properties. Such investigations, in which single molecules or small molecular assemblies operate as conductors connecting ‘tradition-al’ electrical components such as metal or semiconductor contacts, constitute a major part of what has become the active field of molecular electronics. Their diversity, versatility, and amenability to control and manipulation make molecules and molecular assem-blies potentially important components in nanoelectronic devices. A standard electron transfer system thus containing a donor, an ac-ceptor, and a molecular bridge connecting them. Our focus is the electron transfer between the two conducting electrodes (donor and acceptor) through a molecular medium, which bears strong similarity to the more conventional systems that involve at least one molecular species in the donor/acceptor pair. These develop-ments have attracted much attention from the semiconductor in-dustry and there is a great interest from an applied point of view to model and understand the capabilities of molecular conductors. At the same time, this is also a topic of great interest from the point of view of basic physics. Three possible mechanisms might be considered:
(a) Superexchange: the charge tunnels from the donor (DN) elec-trode to the acceptor (AC) elecelec-trode through the DNA segment in a nonadiabatic process. An exponential decrease in the rate of charge transport with increasing length of the DNA segment is predicted.
(b) Hopping: charge occupies the DNA segment in travelling from DN electrode to AC electrode by hopping between the DNA’s discrete molecular orbitals. If the rate of charge migration is faster than trapping, the charge should be able to migrate over long distances before getting trapped.
(c) Domain hopping: charge occupies the DNA segment by de-localizing over several bases, or a domain. This domain hops along the DNA segment to travel from DN electrode to AC elec-trode. As in a pure hopping mechanism, the charge should be able to travel long distances before getting trapped.
We are all aware that the DNA found in cells is a double helix consisting of two antiparallel strands held together by spe-cific hydrogen-bonded base pairs: adenine ( A) always pairs with thymine (T ), and guanine (G) always pairs with cytosine (C ). The specificity of this base pairing and the ability to ensure that it oc-curs in this fashion (and not some other) is key to the use of DNA in materials applications. The double helical arrangement of the two molecules leads to a linear helix axis, linear not in the geo-metrical sense of being a straight line, but in the topological sense of being unbranched. Such model (the two-leg ladder model) was also used by other groups to discuss the transport properties of DNA sequences [12–14]. Unlike proteins, a stacked array of DNA base pairs derived from these nucleotides can provide the way to promote long-range charge migration, which in turn gives impor-tant clues to mechanisms and biological functions of transport. So far, numerous algorithms have been introduced to characterize and graphically represent the genetic information stored in the DNA nucleotide sequence. The goal of these methods is to generate pat-terns for certain sequences or groups of sequences[15].
With this aim in mind, we report in this work an analyti-cal as well as numerianalyti-cal investigation of the one-electron states in double-strand binary DNA segments. Our theoretical model is based on a tight-binding Hamiltonian, together with a transfer matrix technique employed to simplify the algebra which can be otherwise quite involved. We consider a model in which the DNA molecule is sandwiched by two electrodes (donnor-DN and acceptor-AC, respectively), following a Fibonacci (FB) and a Rudin– Shapiro (RS) quasiperiodic structures[16–18], and compare them to the DNA sequence of the first sequenced human chromosome 22 (Ch22), whose arrangement was retrieved from the internet page
of the National Center of Biotechnology Information. We investi-gate the conductivity of the DNA molecule models through their electron transmittance coefficient. Furthermore, by solving numer-ically a time-independent Schrödinger equation, we compute also some basic properties of the I–V characteristics, for all DNA models considered here.
The Hamiltonian for a double-strand DNA chain is an effective tight-binding model describing one electron moving in a geometry, composed by two interconnected chains of sites, side by side, with a single orbital per site and nearest-neighbor interactions[19], i.e.:
H
=
nnα
ψ
nαψ
nα+
nβ
ψ
nβψ
nβ+
n wψ
nαψ
nβ+
ψ
nβψ
nα+
n,δ=±1 tψ
nαψ
nα+δ+
ψ
nβψ
nβ+δ,
(1)where
nα is the single energy at the orbital
ψ
nα (the upperin-dex refers to the chain, while the lower inin-dex refers to the site position in each chain). Also t and w are the intra-chain and the inter-chain first-neighbor electronic overlaps (hopping amplitude), respectively (for simplicity, we are considering that the charge car-riers are travelling from DN to AC electrodes by hopping between the DNA’s discrete molecular orbitals). The corresponding time in-dependent Schrödinger equation is given by:
t
ψ
nα+1+ ψ
nα−1+
wψ
nβ=
E−
nα
ψ
nα,
(2) tψ
nβ+1+ ψ
nβ−1+
wψ
nα=
E−
nβ
ψ
nβ.
(3)Within this framework, the Schrödinger equation can be writ-ten as:
⎛
⎜
⎜
⎜
⎝
ψ
nα+1ψ
nβ+1ψ
nαψ
nβ⎞
⎟
⎟
⎟
⎠
=
M(
n)
⎛
⎜
⎜
⎜
⎝
ψ
nαψ
nβψ
nα−1ψ
nβ−1⎞
⎟
⎟
⎟
⎠
,
(4)where M
(
n)
is the transfer matrixM
(
n)
=
⎛
⎜
⎜
⎝
(
E−
α n
)/
t−
w/
t−
1 0−
w/
t(
E−
nβ
)/
t 0−
1 1 0 0 0 0 1 0 0⎞
⎟
⎟
⎠ .
(5)After successive applications of the transfer matrix M
(
n)
, we have⎛
⎜
⎜
⎜
⎝
ψ
nα+1ψ
nβ+1ψ
nαψ
nβ⎞
⎟
⎟
⎟
⎠
=
M(
n)
M(
n−
1)
· · ·
M(
2)
M(
1)
⎛
⎜
⎜
⎜
⎝
ψ
1αψ
1βψ
0αψ
0β⎞
⎟
⎟
⎟
⎠
.
(6)Calculation of the products of these transfer matrices is com-pletely equivalent to solving the Schrödinger equation for the system. The criterion for allowed energy is when
(
1/
2)
Tr[
P] <
1, with Tr[
P]
meaning the trace of the matrixP
, andP =
M
(
N)
M(
N−
1)
· · ·
M(
2)
M(
1)
[20].The first sequenced human chromosome 22 (Ch22) contains about 3
.
49×
107nucleotides, its largest segment (NT011520) about 2.
33×
107 nucleotides, and the second largest one about 4.
25×
106 nucleotides. Here, we consider a very short sequence (the maximum number of nucleotides is 512) selected from the largest segment NT011520 (for a statistical study of this sequence see Ref.[21]), starting from the G (guanine) nucleotide. The ener-giesn are chosen from the ionization potential of the respective
bases [22], i.e.,
A
=
8.
24,T
=
9.
14,C
=
8.
87, andG
=
7.
75,all units in eV, representing the adenine, thymine, cytosine, and guanine bases, respectively. We have considered the hopping term t describing the aromatic base stacking between adjacent nucleotides equal to 1 eV, its single-strand value[20,21]. We em-phasize that our results also hold for the more realistic coupling term, e.g. t
=
0.
4 and 0.1 eV [23,24]. Observe that, depending on the DNA sequence composition, its length, and the effective tem-perature, the value of the hopping integral t between stacked bases can vary over a relatively broad interval, ranging from 0.01 to 0.4 eV [25,26]. The hopping potential w, describing the hy-drogen bonding between complementary bases, was taking equal to 0.1 eV.To setup a quasiperiodic chain of Rudin–Shapiro type, we con-sider that the energies
n take four different values
G,
A,
C,
and
T as in the DNA genomic sequence. With the intention of
comparing this sequence with the genomic one, we assume that their numerical values are the same. Starting from a G (gua-nine) nucleotide as seed, the quasiperiodic RS sequence can be built through the inflation rules G
→
GC , C→
G A, A→
T C , and T→
T A. The RS sequence belongs to the family of the so-calledsubstitutional sequences, which are characterized by the nature of their Fourier spectrum. It exhibits an absolutely continuous Fourier measure, a property which it shares with random sequences[27]. It should be contrasted with the Fibonacci sequence (another sub-stitutional sequence) which displays a dense pure point Fourier measure, characteristic of a true quasicrystal-like structure (for a review of the physical properties of these and others quasiperiodic structures see Ref.[16,17]). This important difference has been dis-cussed in the literature in connection with the localization prop-erties of both elementary excitations[28]and classical waves[29] in the RS sequence, as compared to other substitutional sequence. The quasiperiodic Fibonacci sequence is constructed starting from a G (Guanine) nucleotide as seed and following the inflation rule
G
→
GC , C→
G.We also constructed a random sequence with the same pair correlation of the Ch22 chromosome sequence[20]. To this end we firstly measured the fraction of pairs p
(
I,
J) (
I,
J=
G,
A,
C and T)
in one of the strands of the Ch22 sequence considered. After that, we generate an artificial sequence starting with a cytosine (C ) site. During the sequence construction, the nucleotide following a type
I nucleotide is chosen to be of type J with probability P
(
J,
I)
=
p
(
I,
J)/
kp(
I,
K)
. The resulting pair correlated random sequencehas only short-range correlations which, by construction, are the same as the original Ch22 sequence.
The transmission coefficient TN
(
E)
, N being the generationnumber of the quasiperiodic sequence under consideration, that gives the transmission rate through the chain and is related with the Landauer resistance, is defined by[30]
TN
(
E)
=
|
Dα|
2+ |
Dβ|
2/
2,
(7) with Dα=
T
11Bα+
T
12Bβ+
T
13Aα+
T
14Aβ,
(8) Dβ=
T
21Bβ+
T
22Bβ+
T
23Aα+
T
24Aβ,
(9)where the matrix
T = Θ
−1S
−1PS
. We consider here Bα=
Bβ
=
1. Also, Aα= (−
T
41T
34−
T
42T
34+
T
44T
31+
T
44T
32)/(
T
43T
34−
T
33T
44),
(10) Aβ= (−
T
43T
31−
T
43T
32+
T
33T
41+
T
33T
42)/(
T
43T
34−
T
33T
44).
(11) Furthermore,S
=
⎛
⎜
⎜
⎝
e−ika 0 eika 0 0 e−ika 0 eika 1 0 1 0 0 1 0 1⎞
⎟
⎟
⎠ ,
(12)Θ
=
⎛
⎜
⎜
⎝
e−ikNa 0 0 0 0 e−ikNa 0 0 0 0 eikNa 0 0 0 0 eikdNa⎞
⎟
⎟
⎠ ,
(13) where k is given by k=
cos−1(
E−
S
)/(
2VS)
.
(14)We use the energy of the platinum electrode
S
=
5.
36 eV, whichis related to the work function of this metal[31]. Also the potential inside the electrode (VS) is considered to be 12 eV.
The transmission coefficient is a useful quantity to describe the transport efficiency in quantum systems. Nonetheless, TN
(
E)
isusually difficult to be directly measured experimentally. Access to transmission properties can be performed by measuring their I–V characteristics. However, applying a voltage bias in the conducting leads (DN and AC electrodes) contacting the DNA segment has also some influence on the scattering properties inside the molecule, and direct information on intrinsic effects of sequences on trans-mission should thus be considered with care. With the effective Hamiltonian given above, one can evaluate the I–V characteristics by applying the Landauer–Büttiker formulation[32]
I
(
V)
=
2e h +∞ −∞ TN(
E)
fDN(
E)
−
fAC(
E)
dE,
(15)where the Fermi–Dirac distribution fDN(AC)
= [
exp[(
E−
μ
DN(AC))/
kBT
] +
1]
−1. Here,μ
DN(AC) is the electrochemical potential of the two leads (donor-DN and acceptor-AC) fixed by the applied bias voltage V as|
μ
DN−
μ
AC| =
eV [33]. We are assuming the Fermi level energy equal to zero [34]. The current onset is crucially de-pendent on the electrochemical potentials of the leads that can be altered by the coupling to molecules, which is another important issue to be separately considered. For simplicity, before bias volt-age is applied, the electrochemical potential of the whole system is taken to be zero. It is important to emphasize that the trans-mittance TN(
E)
should be calculated also for negative values ofenergy.
We start by reporting the transmission coefficients TN
(
E)
, asgiven by Eq. (7), which are depicted in Fig. 1 as a function of the energy, in units of eV. We have considered the Fibonacci se-quence (Fig. 1a, with generation number NF
=
12, correspondingto nF
=
233 nucleotides), the Rudin–Shapiro one (Fig. 1b, withgen-eration number NRS
=
7, corresponding to nRS=
64 nucleotides), the random case (Fig. 1c, with nRD=
64 nucleotides), and the human chromosome Ch22 (Fig. 1d, with nCh22=
64 nucleotides),respectively. Observe that the transmission bands in all cases are fragmented, which is related to the localized nature of the elec-tron’s eigenstates in disordered chains, and reflects the number of passbands in each structure (when the localization factor is zero, the corresponding frequency intervals are known as pass-bands). It is relevant to stress that the presence of long-range correlations in the disorder distribution is a possible mechanism to induce delocalization in low dimensional systems [35]. How-ever, the actual correlations in our model (hopping mechanism) are not strong enough to produce this correlation-induced transition, and the stationary states remain all localized. Moreover, the pres-ence of long-range correlations enhances the localization length
Fig. 1. Transmittance coefficient TN(E)as a function of the energy E, in units of eV, for: (a) the Fibonacci sequence, with generation number NF=12, corresponding to
nF=233 nucleotides; (b) the Rudin–Shapiro structure, with generation number NRS=7, corresponding to nRS=64 nucleotides; the random case, with nRD=64 nucleotides; (d) the human chromosome Ch22, with nCh22=64 nucleotides. Notice that the presence of correlations contributes to the survival of resonant transmission peaks for
sequences up to hundreds of nucleotides.
and, therefore, transmission resonances survive in larger segments as compared with a non-correlated random sequence (see, for in-stance, the Fibonacci case). Observe also that the transmission coefficient for long-range correlated Rudin–Shapiro sequences, de-picts a trend similar to the one produced by the genomic Ch22 sequence.
Current–voltage characteristics of double-strand DNA sequences are plotted inFig. 2for Fibonacci (Fig. 2a), Rudin–Shapiro (Fig. 2b), the random case (Fig. 2c) and the human chromosome Ch22 (Fig. 2d), respectively. We are assuming a linear voltage drop across the DNA molecules by means of the usual expression, numerically computed near zero temperature, as given by Eq.(15). To repro-duce the potential mismatch at zero bias, the energy difference between the guanine HOMO energy level and the metallic Fermi level of the electrode is set to 1.2 eV[36]. As the voltage drop is switched on, the transmission coefficient TN
(
E)
becomesvoltage-dependent, resulting in transmission band shifts (shown inFig. 2 for all cases studied here), which in turn lead to a voltage thresh-old modulation.
To extract the main features of tunneling currents in DNA chains, let us compare the behavior of the genomic Ch22 (Fig. 2d) with those characterizing the quasiperiodic and random structures (Figs. 2a, b, c) under the resonance condition given by the hopping term choice t
=
1 eV. In this case, if the potential barrier between the metallic contacts and the DNA is set to zero, a staircase in the plot I–V is found[34,37].As soon as a potential barrier between the DNA and the met-als is introduced (1.2 eV), the I–V characteristic curves show the profiles depicted in Fig. 2. The current threshold at a given voltage scale is not sensitive concerning the different structures considered here, mainly due to the electronic correlations pre-sented by the structures. However, such correlations shall depend strongly on the intra-chain coupling, and further studies consid-ering more realistic model parameters would be needed in or-der to infer about the actual relevance of this treshold
enhance-ment in DNA molecules. Observe the striking agreeenhance-ment between the I–V characteristic curves for the random and genomic case. Such agreement can be accounted by the short-range pair cor-relations shared by them, suggesting that the inclusion of just first-neighbors intra-strand pair correlations on the nucleotide dis-tribution provides an adequate description of the DNA electronic properties.
In summary, we numerically study the one-electron conduc-tivity dynamics in arrays of double-strand DNA segments. We have modelled the DNA sequences considering its four nucleotides following three artificial sequences, two of them characterizing the quasiperiodic structures of Fibonacci and Rudin–Shapiro types, both presenting a long-range pair-correlation. The other artifi-cial sequence is a pair-correlated random sequence (RD), with the same first neighbors pair correlations of the human Ch22 DNA sequence, which shows short-range pair correlation behav-ior. The I–V characteristic curve seems to be accounted by the short-range pair correlations, suggesting that the inclusion of just first-neighbors intra-strand pair correlations on the nucleotide dis-tribution provides an adequate description of the DNA’s elec-tronic properties. However, as the electron transmissivity depends strongly on the intra-chain coupling, further studies consider-ing more realistic model parameters would be needed in or-der to infer about the actual relevance of this behavior in DNA molecules.
Finally, we would like to stress that the double-strand model used here, does not account for variability of the hopping am-plitudes and their dependence on the electron energy. Such fea-tures may be included by explicitly taking into account the trans-port along the sugar phosphate side chains. Although, we do not expect any relevant change in the main features related to the one-electron dynamics, the energy band structure and the ac-tual localization length may be influenced, specially at the band edges. We expect to address this point in a future communica-tion.
Fig. 2. Current–voltage characteristics of double-strand DNA sequences for (a) Fibonacci; (b) Rudin–Shapiro; (c) the random case; (d) the human chromosome Ch22, respec-tively.
Acknowledgements
This work was partially financed by the Brazilian Research Agencies CAPES (PROCAD and Rede NanoBioTec), CNPq (Rede NanoBioestruturas Project No. 555183/2005-0 and INCT-Nano(Bio)-Simes, Project No. 573925/2008-9) and FAPERN/CNPq (Pronex).
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