Revista de Ciencias Tecnológicas (RECIT). Volumen 3 (1): 10-22.
Revista de Ciencias Tecnológicas (RECIT). Universidad Autónoma de Baja California ISSN 2594-1925
Volumen 2 (3): 118-123 Julio-Septiembre 2019 https://doi.org/10.37636/recit.v23118123
118
ISSN: 2594-1925
Electronic transport through renormalized DNA chains
Transporte electrónico a través de cadenas de ADN renormalizadas
García Flores Daniel
1
, Iglesias Vázquez Priscilla Elizabeth
1
, Villarreal Sánchez Rubén César
2
1
Facultad de Ciencias, Universidad Autónoma de Baja California. Carretera Tijuana-Ensenada Km 106,
22800 Ensenada, Baja California, México
2
Facultad de Ingeniería, Arquitectura y Diseño, Universidad Autónoma de Baja California. Carretera
Tijuana-Ensenada Km 106, 22800 Ensenada, Baja California, México
Autor de correspondencia: Daniel García Flores, Facultad de Ingeniería, Arquitectura y Diseño
Universidad Autónoma de Baja California. Carretera Tijuana-Ensenada Km 106, 22800 Ensenada,
Baja California, México, daniel.garcia.flores@uabc.edu.mx
Recibido: 26 de Junio del 2019 Aceptado: 19 de Septiembre del 2019 Publicado: 30 de Septiembre del 2019
Resumen. - El ADN ha presentado a través de experimentos una gran variabilidad en términos de
sus características electrónicas. Han demostrado que puede adquirir el comportamiento de un
conductor, semiconductor o aislante, lo que lo convierte en un buen candidato para replicar en
dispositivos electrónicos a escala mesoscópica. En el presente trabajo, el coeficiente de transmisión
cuántica se calcula para cadenas de ADN de varias longitudes con el uso del procedimiento de
diezmado y renormalización, dentro de la aproximación de unión estrecha y la teoría de dispersión
de Lippmann-Schwinger. Se obtuvieron perfiles de transmisión de energía, lo que ayudó a inferir las
propiedades de transporte electrónico del sistema. Además, también se calculó la relación corriente-
voltaje para una cadena de 30 pares, y se comparó con los resultados experimentales de Porath et al.
Los resultados muestran las características de los semiconductores de la molécula, y un parecido con
el trabajo de Porath, mostrando la calidad del procedimiento y el modelo utilizado.
Palabras clave: ADN; Transporte electrónico; Corriente eléctrica; Transmitancia.
Abstract. - DNA have presented through experiments great variability in terms of its electronic
characteristics. They have shown that it can acquire the behavior of a conductor, semiconductor or
insulator, making it a good candidate for replicating at the mesoscopic scale electronic devices. In
the present work, the quantum transmission coefficient is calculated for DNA chains of various lengths
with the use of the decimation and renormalization procedure, within the tight binding approximation
and the Lippmann-Schwinger scattering theory. Transmission-Energy profiles were obtained, which
helped to infer electronic transport properties of the system, Additionally, the current-voltage relation
for a 30-pairs chain was calculated as well, and compared with the experimental results of Porath et
al. Results show the semiconductor characteristics of the molecule, and a resemblance with the work
of Porath, showing the quality of the procedure and the model utilized.
Keywords: DNA; Electronic transport; Electrical current; Transmittance.
Revista de Ciencias Tecnológicas (RECIT). Volumen 2 (3): 118-123.
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ISSN: 2594-1925
1. Introducción
Deoxyribonucleic acid (DNA) is well known for
being the molecule where life is codified.
Advances in nanotechnology have opened
opportunities to study more about the molecule in
another context: as an electric component. Recent
experiments have shown that DNA can emulate
electronic components (conductor,
semiconductor, insulator, etc.) at the mesoscopic
scale, depending on the conditions the experiment
and environment [1]; taking into account that the
transport occurs in a monomodal way at the
central chain of nitrogenous pair bases [2], making
it a great candidate for molecular electronics.
DNA also provides complexity in its structure. To
calculate electronic transport, we have simplified
the problem by describing the system through
effective parameters by the means of the Green
Function technique [3] for the renormalization of
it, with a model that will carry its electronic
characteristics. We focus on finding the relation
between the transmittance T—i.e., the amount of
ingoing charge versus the outgoing, and the
induced energy E, to obtain T(E) profiles. These
profiles will help us to infer the electronic
properties of the system. Additionally, we have
calculated deformations in the system given an
external factor, with the purpose of locating
optimal energy intervals where transmittance
keeps its maximum value, if the system is strained.
To see the effectiveness of the theoretical analysis,
we also calculated the I-V curve and compared it
with the experimental results of Porath et al [4],
where they measured electrical current through
DNA wires.
2. Methodology
For the purpose of simplifying the molecule and
to obtain an analytical result—based on [3], we
made use of the decimation and renormalization
procedure within the tight binding approximation,
on a model that could interpret the behavior of the
transport in the intern and outer column (of
nitrogenous base pairs and sugar phosphates,
respectively). The fishbone model (Figure 1),
previously utilized [5], was chosen for this.
Here α
1,4,7
represents the base pair atomic sites of
the intern column, meanwhile α
2, 3…,9
the sugar
atomic energy sites of the outer. On the other
hand,
is the bond energy from the site i to the
site j, taking into account the tight binding
approximation to first neighbors. Terminals 1 and
2 are molecular, widely used in experiments [6].
From this the renormalization equations of the
system can be obtained, with aid of the Lippmann-
Schwinger equation transmittance T is calculated,
that will yield T(E) profiles, where electronic
properties of the system can be identified, and
later used to calculate current through it and
compare with the experiment of Porath [4]. Porath
et al measured current through a 10.4nm long,
double-stranded poly(G)-poly(C) DNA molecule,
which is about 30 base pairs; specific base pairs
were not taken into account, just atomic energy
sites.
Figure 1. DNA’s fishbone model
2.1.Renormalization procedure
We start with the Greenian matrix (G
mn
) version
of the discretized Schrödinger equation, which
takes the form:
(1)
where E is the energy,
are the atomic
energy site m (bond energy from site m to site k),
is the Kronecker delta, and
the Green
function given the points k, n. The problem is
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ISSN: 2594-1925
solved in parts; first the particular equation for a
base pair and its respective sugars, and then the
junction of renormalized sites to form effective
dimers. The proposed model for the particular
case of a base pair can be seen in Figure 2, where
the superscripts z and b tag whether it is a sugar or
a base, respectively. Expanding (1) for the 1, 2 and
3 considered molecule sites, we obtain:
(2)
where represents the renormalized energy site.
We got rid of the bond energy subscripts, which
we have tagged with a superscripts z since it is
pair-sugar bond energy (and later b for pair-pair
bond energy), and we indistinguish the site
energies, as we assume, they have the same
magnitude.
Figure 2. A base pair with its respective sugars connected to
molecular terminals
The process is illustrated in Figure 3.
Figure 3. Renormalization procedure of a DNA base pair
Once obtained the renormalization equation, we
may acquire the equations for an effective dimmer
that contains information of any numbers of base
pairs; equations for a trimer and a dimer shall be
obtained, as seen in Figure 4.
Here,
are the respective energies given by (2)
and
, with
defined as a deformation
coefficient. The latter will help us to parameterize
changes in specific bonds that will later represent
deformation in the system.
(a)
(b)
Figure 4. Wires of (a) two and (b) three base pairs, each site of the
dimer (trimer) represents a base pair dimer per se.
Expanding (1) for the dimer we obtain:
(3)
(4)
(5)
and for the trimer:
(6)
(7)
(8)
where
is the dimensionless
reduced energy [3],
,
and
, but since we don’t have
different inter-base-base bond energies, then
. It is clear to see that, if
, the results of
[3] are recovered. With these equations we may
build a cable constituted of any number of base
pairs, substituting the sites and bond energies on
themselves recursively.
2.2 Transmission coefficient
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For calculating the transmission coefficient of the
system, we used the Lippmann-Schwinger
equation in its discrete form:
(9)
here,
and
are the perturbed and free
wave function, respectively,
is the
unperturbed Green function of the system, which
for an atomic wire is [7]:
(10)
and V is the potential, which is the scattering
potential for an electron propagating through the
atomic wire with a dimer impurity (see Figure 5),
that is:
(11)
whence, is the bond energy,
and
are the
reduced site and bond energy, respectively, and
are the wave function states on those sites,
as illustrated in figure 5.
Figure 5. Scattering potential for an electron propagating through
the atomic wire
For a free particle entering the system, equation
(9) can be rewritten as:
(12)
with being the transmission coefficient. Solving
(12) for the transmittance
yields:
(13)
where
(14)
(15)
With (13) we can obtain the
profiles. For
that, we may set the particular dimensionless
values for the atomic sites of
, the
bond energies and
,
this being noted in experiments: base-sugar bond
energy doubles base-base bond energy [4], and a
bandwidth of
.
2.3 Current through the system
The current can be calculated with [8]:
(16)
Here,
is the Transmission-Energy profile for
a 30-base pair DNA wire,
is the
Fermi function, and are the Planck constant
and the electron charge, respectively. The room
temperature is set
to make
resemblance with the experiment [4].
and
are the electrochemical potentials of the
electrodes, given by [8]
and
, where
and are the
equilibrium Fermi energy and the applied voltage,
respectively. The parameter describes how the
applied voltage is divided across the electrodes
and the molecule. We then set the parameters of
the system to real experimental values
:
a fixed
Fermi energy
, and
instead
of
or
of other works [1].
3. Results
In figure 6 and 7 are presented the
profiles
of and base pair chains.
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Figure 6. Transmission through 2, 3 and 4 base pairs
Figure 7. Transmission through 10 base pairs
We see from the former a well-defined gap for
energies between and where transport
does not occur, and resonances ( ) depending
on the number of base pairs in the chain. This is
important since it can be interpreted as a
semiconductor, where a minimum of energy shall
be applied for conduction to take place. Figure 8
shows different 10 base pairs
plots where
deformations profiles are applied, being:
is
unperturbed,
and
, representing an increment on the
central part of the wire;
and
, representing a
decrement on the central part of the wire. We see
that gaps are generated between resonances, at
low energies for the profiles
and
, and at
higher for
and
.
(a)
(b)
(c)
(d)
Figure 8. Transmission onto the deformation profile (a)
, (b)
,
(c)
, (d)
in comparison with
.
Figure 9 shows the curve for the theoretical
result of (16) in comparison with the experimental
data of Porath et al [4].
Revista de Ciencias Tecnológicas (RECIT). Volumen 2 (3): 118-123.
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ISSN: 2594-1925
Figure 9. Current through a 30-base pair wire
The proposed model makes a well approximation
within the gap interval, and then it saturates at
high voltages. Saturation might be given by the
short bandwidth utilized for first neighbors.
4. Conclusions
We see that, for each added base pair resonances
appear, and the gap is more defined. These results
allow us to infer that, given a deformation on the
system, there will be intervals where
transmittance keeps its maximum value. These
intervals are optimal for building molecular wires
that, although deformed, can transport a desirable
amount of charge. Also, where a gap will be
generated, whose application may be as molecular
sensors that can detect a resonance shift. The
curve makes a good resemblance with the
experimental data. This shows the effectiveness of
the model besides the simplification taken. Future
work involves sequence engineering for building
wires with different nitrogenous bases, exploring
decoherence within the system and tight binding
for second neighbors.
Referencias
[1] X. Li and Y. Yan, "Electrical transport through
individual DNA molecules", Applied Physics Letters, vol
79, no. 14, pp. 2190-2192, 2001.
https://doi.org/10.1063/1.1407860
[2] S. Roche, "Sequence dependent DNA mediated
conduction", Physical Review Letters, vol. 91, no. 10, pp.
108101-108104, 2003.
https://doi.org/10.1103/PhysRevLett.91.108101
[3] K. Sulston and S. Davison, "Transmission of
renormalized benzene circuits", 2015. arxiv: 1505.03808v1.
http://doi.org/10.4204/EPTCS.151.11
[4] D. Porath, A. Bezryadin, S. de Vries, and C. Dekker,
"Direct measurement of electrical transport through DNA
molecules", Nature, vol. 403, pp. 635-638, 2000.
https://doi.org/10.1038/35001029
[5] D. Klotsa, R. Römer, and M. Turner, "Electronic
transport in DNA", Biophysical Journal, vol. 89, no. 4, pp.
2187-2198, 2005.
https://doi.org/10.1529/biophysj.105.064014
[6] X. Xiao, B. Xu, and N. Tao, "Measurement of single
molecule conductance: Benzenedithiol and
benzenedimethanethiol", Nano Letters, vol. 4, no. 2, pp.
267-271, 2003. https://doi.org/10.1021/nl035000m
[7] S. Davison and M. Steslicka, Basic Theory of Surface
States. Clarendon Press, 1992.
https://global.oup.com/academic/product/basic-theory-of-
surface-states-9780198518969?cc=mx&lang=en&
[8] S. Datta, Electronic Transport in Mesoscopic Systems.
Cambridge University Press, 1997.
https://doi.org/10.1017/CBO9780511805776
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