DNA Circuit System and Charge Transfer Mechanism

doi:10.4236/eng.2013.510B077

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Engineering, 2013, 5, 381-385

http://dx.doi.org/10.4236/eng.2013.510B077 Published Online October 2013 (http://www.scirp.org/journal/eng)

DNA Circuit System and Charge Transfer Mechanism

Kunming Xu

College of Ocean and Earth Sciences, Xiamen University, Xiamen, China

Email: [email protected]

Received 2013

ABSTRACT

Based on the accurate identification of chemical structures as electric elements, we described charge transport mechan-

ism in a DNA molecule by a stepwise LC oscillatory circuitry, in which every base pair is a capacitor, every phosphate

bridge is an inductor, and every deoxyribose is a charge router. The circuit model agrees with established experimental

evidence that has supported super-exchange and hopping mechanisms so far. This alternative charge transport mechan-

ism through both strands of DNA matches the fidelity and reliability of its chemical structure.

Keywords: Conductivity; Capacitor; Inductance; Oscillatory

1. Introduction

The electrical conductivity of a DNA molecule is of pa-

ramount important in biological function. It has been

established that very fast charge transport can take place

over a short distance (~37 Å) and a slower transport may

propagate over a long distance (~200 Å) [1-6]. The re-

sults have led to two hypotheses of charge transport me-

chanism in DNA, one indicating super-exchange, or tun-

nel, through the sugar phosphate bridge between the

bound charge donor and acceptor for the short distance,

and the other suggesting charge hopping between dis-

crete bases through the DNA π-stack over long distance.

Although both mechanisms are widely regarded, they are

incompatible and ill-defined, giving the impression of

loose or uncertain charge transfer along nucleotides. As

genetic substance, DNA must possess well-defined elec-

trical behavior. Here we show a DNA double helix as a

stepwise LC oscillatory circuitry, in line with the expe-

rimental evidence. Electrical harmonic oscillations have

significant biological implications.

2. DNA Circuit System

The circuit model regards each base pair as an electric

capacitor because the base pair is composed of two hete-

rocyclic amines placed at a semi-conductive distance,

capable of storing opposite charges. The hydrogen bonds

between the base pair deliver electric pulse but prohibit

direct current between them, which is a typical characte-

ristic of electric capacitors. In the double helix, the phos-

phate bridges are twisted physically like both strands of a

rope under torsion. The wound phosphate bridge can be

treated as an electric inductor, capable of storing energy

as well. The rigid structures of the Watson-Crick parities

and of the pentose sugars help in forcing the torsion onto

the phosphate bridges mechanically. Charge transport

through the strands, like electric current through a coiled

wire, produces electric inductance. Hence DNA structure

is a circuitry composed of multiple oscillatory LC cir-

cuits (Figure 1).

The deoxyribose is a key electric element in the circuit,

which serves as a switch at the juncture (Figure 2). It has

Figure 1. DNA double strands (a) along with stepwise oscillatory circuits (b) and (c), where the dashed arrows indicate elec-

tric current directions and the deoxyriboses are represented by electric switches.

K. M. XU

382

Figure 2. Electric currents directe d by chiral carbons through

a nucleoside. The dotted arrows represent electric current

directions.

been stated that a chiral carbon center may transfer elec-

trons towards a selective pathway due to its asymmetric

polarizations [7]. Three chiral carbons in a deoxyribose

direct electric charges in a selective course: C1’ and C4’

are levorotatory, and C3’ is dextrorotatory [8]. Depend-

ing on whether the nucleotide base is positively charged

or negatively charged, the switch connects C1’ to C5’ or

connects C1’ to C3’ respectively. Direct charge flow

between C3’ and C4’ is choked by both chiral carbons.

The anti-parallel alignment of the pentose sugars on both

strands determines that electric current forms a stepwise

closed loop between every two adjacent base pairs.

Suppose in the closed circuit (Figure 1(b)), charges

stored in capacitor C1 transport through the phosphate

inductors L1 and L2 to reach capacitor C2 under the

routing of the deoxyriboses. The anti-parallel alignment

of the pentose sugars on both strands determines that

positive charges transfer along th e strand in the direction

of 3’ → 5’ while negative charges transfer along the

strand in the direction of 5’ → 3’. Electric current forms

a closed circuit so that by Kirchhoff’s loop rule, we have

voltage drop relationship of

112 2

dI dI

IR LIRL

Cdt Cdt

=+++ +

, (1)

where I is electric current around the circuit unit, Q1 and

Q2 are charges stored in base pair capacitances C1 and C2

respectively, and R1 and R2 are representative resistors

along the path. Let

RR R+=

, (2)

LL L+=

, (3)

111

CC C

, (4)

12 0

QQ Q+=

, (5)

then upon differentiation on both sides, Equation (1) be-

comes

dd 0

I II

t+ +=

. (6)

This second-order differential equation is a damped

harmonic oscillator. Depending on the relative values of

L, R and C, the system may be over-damped, critically

damped, un der-damped, or simple harmonic. When

24/R LC<

, the circuit property falls into the two latter

categories with a current function of

cos( )

I Iet

−

, (7)

where

is the initial curren t and

is angular veloc-

ity of the oscillation with a value of

LC L

= −

. (8)

The circuit system transfers electric charges from a

base pair to another in stepwise oscillatory processes, i.e.,

from capacitor C1 to C2 (Figure 7.1b) and then from ca-

pacitor C2 to C3 (Figure 1(c)) along the double helix.

Simple as it is, this model is in line with experimental

evidence [1-6] so far established concerning DNA con-

ductivity and reconciles both super-exchange and mul-

ti-step hopping mechanisms. The LC circuit is robust and

classical in physics, yet revolutionary in chemistry and

biology. This interdisciplinary analysis produces a result

of general interest in biological physics and may have

potential influence in molecular electronics.

3. Charge Transfer Mechanism

It has been reported that the rate of electron transfer

within a short distance decreases exponentially with in-

creasing distance [2-5]. Such a phenomenon corresponds

to the under-damped condition of the circuit system due

to the relatively low (

24/R LC<

) resistance response of

DNA backbones to the artificial introduction of voltage

drop between base pairs in the experiments. However,

when R value is considerable, the exponential signal of

electric current prevails over the sinusoidal cycle in Equ-

ation (7) and vanishes within a few oscillatory cycles.

Since the distance between the b ase pairs of B-DNA is a

fixed value of 3.4 Å and each stepwise oscillatory cycle

takes a certain interval of time for that distance, we may

replace the time variable in Equation (7) with a distance

variable

r∆

. In considering that electric current is a

measure of electron transfer rate k, Equ a tion (7) is equiv-

alent to the Marcus correlation [9]

−∆

∝

(9)

where

values between 0.1 and 1.4 Å−1 have been

estimated for the double helix [2-5]. The dramatic dif-

ference can be ascribed to the uncertainty of resistance R

that is sensitive to various experimental conditions. The

presence of considerable R value along the circuit is be-

cause the phosphate bridges are under persistent high

voltage drop induced artificially in experiments so that

inductances partially manifest as resistances in the cir-

cuit.

K. M. XU

383

Based on frequency value of 10−10 s−1 measured by

Fukui et al. [2], we simulate the under-damped oscilla-

tion with parameters of C = 0.02 pF, L = 0.01 μH, and R

= 100 Ω in Equation (8). Assuming three radical cations

are initially generated to trigger charge migration from a

base pair to anot her t hrough the ste pwis e oscill atory cycles,

the curre nt funct i on of Equation (7) is calculated to be

0.005

5.4cos(0.071 )

Ie t

−

, (10)

where electric current is in the unit of nA and time in ps

(Figu re 3) . Suppos e at t = 0, capacitor C1 carries charges

in the polarity as shown in Figure 1(b), the deoxyribose

switches (S1 to S4) will route the charges in the dotted

arrow direction. After one oscillatory cycle at t = 90 ps,

the charges reach capacitor C2 so that switches S3 and S4

change their connections. The charges at capacitor C2

will then be transferred to capacitor C3 in the next step

(Figu re 1(c)). The amplitude of the electric current along

the double helix decreases exponentially with each oscil-

latory cycle in the under-damped situation (Figure 3).

Because each oscillatory cycle takes 90ps for charges to

migrate 3.4 Å along DNA strands, the β value in Equa-

tion (9) is 0.13 Å−1 in this case.

In v ivo, we believe that natural electron transport from

a base pair to another should incur trivial electric resis-

tance. Even if there is certain electric resistance along the

strands, the thermal energy produced by resistors would

immediately be absorbed by both energy storage com-

ponents of the base pair capacitor and the phosphate in-

ductor. Hence electric resistance can be neglected. Let C

= 0.02 pF, L = 0.05 μH, and R = 5 Ω, the circuit declines

into a series of LC oscillators that transfer charges step

by step along the strands harmonically at a slower pace.

The frequency of the harmonic oscillation is slower than

that of under-damped oscillation as can be predicted

from

formula under the relatively high value of in-

ductance and trivial resistance. It takes about 200 ps for

electrons to transfer from a base pair to the next. But the

amplitude of the electric curren t remains almost the same

in each oscillatory cycle. A comparison of under-damped

oscillation and simple harmonic oscillation can be found

in Figure 3.

In the stepwise LC circuits, electric current is defined

as positive when a base pair capacitor is being charged

and negative when discharged in the next cycle down the

chain. The stepwise oscillations agree with the evidence

that has supported hopping mechanism through the base

π-stack [1-6], but electric current through the sugar phos-

phate bridges is more reliable than the haphazard migra-

tion by hopping across the base rungs. Charges stored in

capacitor C1 transport through both strands to capacitor

C2, and will continue to move towards C3 in the similar

sinusoidal manner but at a lagging phase of

in the

cycle. During the processes, positive holes move in the

Figure 3. Model predictions for charge transfer of DNA

circuit by fast stepwise under-damped oscillations in vitro

(black curve) in contrast to slower stepwise simple harmon-

ic oscillations in vivo (gray curve) where each cycle spans

3.4 Å in distance. Charges undergoi ng under-damped osc il-

lations have a hasty speed but cannot reach a long distance.

direction of 3’ → 5’ on one strand while electrons flow

in the direction of 5’ → 3’ on the other strand in good

synchronization. The mechanisms for charge transport

through long and short distances are the same. It takes

more cycle s of osci ll ations f or char ges t o transport through

longer sequences of base pairs. Without considering the

effect of sequence dependence, traveling time is propor-

tional to distance.

From Equation (8), we kno w that

is codetermined

by inductance L and capacitance C when R is negligible.

Assuming constant inductance for all nucleotides, the

harmonic frequency would be determined by two neigh-

boring capacitance values in the closed circuit, such as C1

and C2, which are specific to the base pairs. And this is

perhaps the most subtle part of the story for it indicates

that charge transfer rate is sequence dependent [3-5,10].

Because nucleotide bases have ionization potentials in

the order of G < A < C < T and electron affinities in the

order of C < T < G < A (disregarding negative sign) [11],

it takes the least amount of energy to charge G:C base

pair in the polarity of +G:C− and requires the highest

amount of energy to charge +T:A− capacitor. This means

that the capacitances of the base pairs are in the order of

+G:C− > +A:T− > +C:G− > +T:A− polarities. Thus there

are four distinct capacitance values depending on the

base pair and polarity.

Since

is determined by two neighboring capacit-

ances in series, it may take eight possible values. It is

predicted that charge transport along the strands will pro-

duce various frequencies reflecting the identity of the

bases. In other words, the pattern of charge transport is

precisely controlled by the gene sequence. The +G:C−

capacitor has a higher capacitance than other base pairs

and carries higher amount of charges in experiments whe-

reas +T:A− has a lower capacitance than other base pairs

K. M. XU

384

and is the limiting step or bottleneck in charge transfer

along the DNA strands [6,9]. However, at long (+T:A−)n

sequences the size of the bottleneck remains the same so

that the sequence distance dependence vanishes [9]. Be-

cause +G:C− has a relatively high capacitance, it is easy

to trap charges, so it is likely to become the end point of

a charge transport [6,9]. In this regard, the circuit model

prediction agrees with experimental resu lts completely.

Oscillatory current through the double helix is likely to

have physiological significance. For example, if a base

pair is mismatched at a certain position, then the oscilla-

tory rhythms would be broken. Proofreading enzyme that

scans the DNA sequence continually might easily locate

the trouble point by the abnormal electric signal. Fur-

thermore, bio molecules are inherentl y unstable. Only con-

stant flow of energy prevents them from being disorga-

nized. It stands to reason that the incessant charge vibra-

tions in the genetic substance are vital for living organ-

isms to maintain the integrity of the gene sequence.

From organic chemistry perspective, the base pair is

capable of storing considerable amount of charges in ei-

ther polarity by at least two conceivable mechanisms.

First, both pyrimidine and purine are heterocyclic rings

composed of carbon and nitrogen atoms. On the one hand,

nitrogen is more electronegative than carbon for attract-

ing higher electron density in covalent bonds with carbon.

On the other hand, nitrogen atom has lone pair electrons

to share with carbon under electron deficiency. The com-

bination imparts great flexibility to the nucleotide bases

for either holding or releasing electrons. Second, the he-

terocyclic rings are aromatic that possess diamagnetic

ring current. In the circuit, aromatic ring current may re-

duce the charge saturation of the nitrogenous base through

electromagnetic effect, and as a result increase the elec-

tric capacitance of the base pair. Both properties enable a

base pair to be a good bipolar capacitor. The recognition

of a base pair as a valid capacitor provides a sharp insight

into molecular electronics.

A base pair is an electric capacitor. Alternate current

passes through th e hydrogen bonds. There are oscillating

waves between each pair of nitrogenous bases within

DNA nucleotides. Hydrogen bonding is a dynamic elec-

tronic action that delivers messages between the two

bases, helping to balance the strain in a DNA double he-

lix. Hydrogen bonds serve as the gateways to deliver

messages. In this way, we describe the hydrogen bonds

as a portal between the pair of bases. This dynamic in-

terpretation is contrary to traditional static model of hy-

drogen bonds. Since the e xpressi on of any forces r equires

message exchanges, it is the wave information conveyed

between pairs of bases that makes hydrogen bonds strong

enough to hold the double helix of DNA together. With-

out the wave message, the atoms on both sides of hydro-

gen bonds would be detached and the organization of

DNA woul d start to u nravel.

In the stepwise oscillatory circuit, a capacitor must be

charged up to a threshold potential for positive charges to

overflow throug h the ether bond of the deoxyr ibose. And

this is mediated by the ether bond shown in Figure 4 as

an electric switch. The oxygen atom would preferably

use a 2px and a 2py orbitals to form covalent bonds with

carbon atoms in the ether bridge. A full electron octet is a

stable configuration of the oxygen atom because eight

outer electrons are in a complete circulating cycle [8].

Electric current through the oxygen atom must comply

with the circulation direction (Figure 4). Like a gas-

filled tube, once a path is channeled by a positive thre-

shold potential, the ether switch will remain on until the

base pair is over-discharged to apply a negative potential

of the same magnitude to the ether bond, thereby flipping

the oxygen state from Figure 3 (ON) to 3 (OFF). To

open the channel again requires the base pair to be re-

charged up to the initial threshold potential to flip the

state over again. In the ON state, C1’—O covalent bond

is 2py-2py overlap while C4’—O covalent bond is 2px-

2px overlap; in the OFF state, the nature of the covalent

bonds is reverse (Figure 4). The chirality of C3’, C1’

and C4’ determines that electric current is unidirectional

through the ether bond and the phosphodiester bonds

while the base pair is being charged and discharged cyc-

lically. Such a mechanism ensures that the oscillatory

current is in strict stepwise process. It has been found

that charge migration in DNA is an ion-gated transport

depending on the hydrated counter-ions and configura-

tions [12]. But we believe that in vivo the circuit is gated

by the dependable ether bond instead of by random ions

coming from the environment.

Finally, we wish to present a general molecular model

for phosphate in DNA backbone based on the electron

circulations within oxygen atoms. The central phospho-

rus atom is bound by four oxygen atoms whose electron

circulations are linked together forming four loops like a

coiled wire, each loop involving an electron octet (Fig-

ure 5). Electric current passing through the molecule

Figure 4. Atomic switch to control the flow of positive

charges (dashed arrow) through the ether bond by flipping

between two interconvertible states of contrary electron

circulation (solid arrows) within the oxygen atom.

K. M. XU

385

Figure 5. Schematic diagram of electron flow via the conca-

tenated electron circulations within a phosphate bridge for

electric induction corresponding to its chemical structure.

inevitably incurs considerable induction due to the con-

voluted electron flow, which explains why phosphate is

an inductor and perhaps why ATP can serve as a physio-

logical energy source when broken into ADP + Pi even

though the energy of the phosphoanhydride bond is not

so large. The convolution collapse is the key factor in re-

leasing a large amount of inductive energy. In DNA the

electrical tension of phosphate chains must equilibrate

with the mechanical torsion of the strands.

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