Quantum tunnelling in DNA

Lbume 8: number 1. %. :. ,.

CHEMICALPH&CS LETTERS

1 January 1971

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:

.

‘QUANTUM

TUNNBLLING

IN-DNA

.‘_

B. R. PARKER and 51 VAN EVERV Physics Depatiment, Idcho State Universily, PocateZZb. Idaho 85201, USi Received 26 November

1970

A calculation of the energy;y‘leveIs for the DNA base units A-T and G-C. along with the forms G-T, .GC excited and G-C anion.. was made. The lifetime of the proton on each of the levels and tunnelling time through the barrier at each level was then determined. Finally, the distribution and tunnelling of the protons on the levels for various densities of radiation was considered. Graphs were plotted which showed thnt considerable tunnelling could take place for radiation in resonance with the levels.

11 ~~TRODUGTION _: It is now a well established fact that mutations can be induced by exposure to radiation. LWdin has recently shown that these mutations may,be a result of a shift in position of the hydrogen .bond protons in the base units of DNA. This relocation of the proton is assumed to proceed through what is known as quantum tunnelling [I]. The proton in the hydrogen bond linkage is locked in position by. a double well potential [Z]. ‘The shape of this potential has recently been calculated. Due to the length and difficulty of the calculation, hcwever, several approximations were used so the potential curve cannot be con-

sidered as highly accurate [S]. Fig. 1 shows the shape of the potential for the tauromeric base combination G-T. Of particular interest to us is the central barrier between the two wells. In this case the proton is assumed to be located in the left well, and under normal circumstances will be on the lowest level. It cannot tunnel through to the right hand well from this level but must be excited to level four. The tunnelling probability at level four is quite small: however, at the upper levels it may become quite large due to the fact that the barrier is much smaller there. Normally the proton does not populate these levels, but may be excited to them if radiation of proper frequency is applied to the system.

Volume 8. number 1

PHYSICSLETTERS

CHEWICAL

The energy levels and tunnelling times at each level were calculated for the normal base pairs. A-T and G-C, and also for the forms G-T, G-C excited, G-C anion and one referred to as A-T crosscut. The potential cross sections were taken from the papers of Harris and Rein [3] and Lunnell and Sperber 141. A-T crosscut is curve E in the A-T well in the paper of Rein and Harris. The lifetimes of the proton on the various levels was also calculated so that it eouId be compared to the tunnelling time at this level.

I Jmlnry 1971

by suitable analytio functions. Each curve had to be broken up into severai sections, and each section fitted separately. Roth polynomial and exponential functions were tried but polynomiat ones proved to be most satisfactory: The WKJ3 method was then used to obtain the enem levels. The following formula was obtained, & d : tan(j k du + $1 tan(j- kdx+ &E) = em25 4. a

c

(1)

where

2. . CALCULATION OF THE ENERGY LEVELS The first step in obtaining the energy levels was that of approximating the potential curves

in agreement with L6wdin. The limits on the integral to the left are those of the end points of

Fig. 2. The energy Ievels for A-T ground state.

x.. .o

. lb

2

,.

L.3

.

.I*f*lC:~ cl,

1

1.t

la

_

I%

2.2

Fig. 3. Tde energy levels of G-C ground state. , 95

1 Janiary ^ -. T&kL -. iif&.irngs and tunnekng tinieg of proton it y&ous

: .” .

.._

.,..

‘: Bnseunit

.,.

4-T ground state _‘.

:.

G-C groundstate

Energy

of.

Ieve!.(

'-_ ,

levsls.of

each b&e unit..

iife bf proton ,. .: Tuhnhng time on levet'(sec) at level (set)

.

0.304 0.817

0.1714 x10:3

l.ZS2

0.1650 x1O-2

0.3463x~O-~

-1.353

0.3366x1O-3

q.101~x10-4

1.728

0.7336 ~10-2

0.7950x10-8

1.805

0.1158 x10-?

0.2032x10-.8

2.108

0.1780X10-l

0.1217~10-~'

2.211

0.3656 x1O-2

0;2467~1O-l~

:- 6.136 6.453

0.7136 x 1O-3

0.797

0.5979 x 10-z

1.160

0.3155x10-1

1.507

0.8114x10-l

-0.3558xlO-l"

1.546

0.5848X1O-2

0.1369x10-10

1.750

0.4311x10-1

0.2629x10-12

0.851

0.1521 x 1O-3

0.4930

0.963

0.6458x 10-4

0.6141X10-1

1.366

0.1241x10-2

0.5544x10-4

1.515

0.3259x10-3

0.4712x10-5

1:848

0.3173x10-2

0.302

G-C first

excited state

G-C anion

2.025

0.1124xLO-~

0.2281x10-' 0.1455x10-6

2.658

0.5586

0.2031x10-12

0.203 0.418

0.6797

x 1O-4

0.2802x lO-7

0.690

0.5890

x10-3

0.1427x10-'

0.910

0.3173x10-2

0.3431x10-11 0.1686x13-7

C.386

0.1411 x10-5

0.441

0.2822X1O-4

0.7528.X 16'

0.673

0.7890 x

0.1657x10-9

0.716

0.3956x~O-~

0.8193x 10Llo

0.958

0.4442.x1&2

-0.2121x10-11

0.1577 x10-2

0,1111x10-11

s.004 G-T

1O-3

6,321 .--1~612

.0X.77 x1p-3 i o.lYlo5-~~lo-

.2:029_

_o.~obo. .xio$

0,380

$3658

x

107.

0'1288 &t-' .

1&i

Volume 8, number 1

CREMXAL PHYSICS LETTERS’

1 January 1971

Table 1 (continued) aaseunit

G-T

Energy

of level (ev)

Life of proton on level (see).

Tunnettingtime at LeveLfsec)

2.795

0.1419x10-2

0.126.i Xl02

3.212

0.8105x10-2

0.2796X10-l

3.364

0.2620x10-2

0.3287~10-~' '

3.766

0.1526x10-l

0.1343x10-4

3.910

0.4605x10-2

0.1977x 10-5

4.287

0.2648X10-l

0.1481x10+

4.422

0.7923x10-2

0.2715 X LO+

4.768

0.4318X10-'

0.3811x L&O

4.888

0.1379x10-1

0.9149x lo-l1

5.173

0.5019x10-1

0.3536x10-12

the level in the left hand well; those on the other integral are the limits in the right hand well. The computer was used in conjunction with this equation to find and plot each of the energy levels in the well. The results are shown in figs. 1 to 3 *. Once the energy levels were obtained, the tunnelling time through the central barrier at the position of each level was calculated using the formula 7

=

These wavefiinctions had to be joined appropriately across the region of the turning points. At these positions they went to infinity. They also had to be normalized and checked for orthogonality. A plot of the actual wavefuntions used (fig. 1 for G-T) showed that they were sufficiently ortliogonal for our puiposest. The lifetime of the protons-on the upper levet were then obtained using these wavefunctions in conjunction with

(ug)-1) 7

where

-l =(asJlv,3/c3)~Etmn,

H

g = exp{-2 J[2m(V-

where E,/@]1’zd4

and

The results are given in table 1. v = 1.0013 x lo-12/d2

;

d is one half the distance between the turning points. The results are given in table 1. Next, the lifetime of the particle on the vrrious levels .wa.s calculated. For this the wavefunctions were required. Standard WKB wavefunctions of the form *pl = K-1/2exp(-JKdx), @2 = 2K-z/2 COS(~Kdx-v/4),

3. TUNNELLtNG AS A RESULT OF RADIATLON Referring to fig. I we see that tu&eLling can take place in both directions through the central barrier. As the proton population builds up to the right of the barrier as a result of tunnelling. tunnelling back through the barrier from the right to the left weit takes place. The tunnelling rates in the two directions are given by

(3)

Cl

= vlg;

C2’02g.

From these we define an effective

‘where K =-[Znt(V-E,,@]‘/” were used

* When the uppermost level was very close to the $eak of‘the central potential it was very-difficult to obtain .- it titicurately~hen~ein a~numberof cases it was left ,c-‘Off. ..

,

71g= (q+c2)4.

(5) tunnelIing’time ((3 *

UIw$n has shown 121 that the effective number tunneling through the b,arrier pn a given level for / t
Voluirie i. number 1

-.

CHEMXAL

ivj - =c;t - q(c1_+ *i0

c2)

r2;i + . : -

PHYSICS

(7)

I

when NiO is. the original number on the dth level on the left side’of -the barrier. For f >> 712 the corresponding formula is

03) Combining these expressions state expression,

with the steady

Bk, k-i1 pvk, k+l ti

(‘I

- k=l wk, k+l + Bk, k+l PVk, k+l

where B&. k+l. Ak;k+l are Einstein coefficients and‘ouk &I is the density of radiation, then gives ari expression for the number of protons tunnelling through at a given level compared to the number in the ground state (for the deepest well). assuming the molecule is being radiated. The expressions obtained are for f Cc 712 3

= kcI (-$$@&Il-.

&+c2)f2/2~, V

(10)

where the (k+ 1) index has been omitted for brevity. and for I >>-‘12

4.. DISCUSSION The energy levels in the double well system are doublets except for the part of the lowest well below the minimum of the other well. of particular importance are the lowest levels, ffom which tunnelling can take place. When the radiation level is low most tunnelling takes place at these levels. At higher radiation levels, however; the upper levels become important and tunneiJ.ing is just as likely to occur there. The lifetime of the proton on a given level was com-pared to the tunnelling time on each level. In most cases it turned out that the lifetime was longer than tlne’correspondixig tunnellirg time. -This was particularly true for the upper levels. For example, in the A-T ground state the lowest tunnelling level haa a lifetime associated with it of 0.165 x 10-2 set while the corresponding tuni nelIing time is 0.3463 x 10-4 sec. On the upperr&st level, h&ever, the corresponding figures., are 0.3656 x 10~2 set aiid 0.2467~ lo-11 sec. -These lifetimes .may seem too,long. In%large molecule like DNA, however, lifdmes of .tbis

:

LETTERS

lJanuary1971

order are expected ,Also,- it may appear as if the Lunnelling probability is particularly large which conflicts with the fact that-we know that both A-T and G-C are very stable. Their stability, however, is associated with the fact that under-normal circumstances (no radiation) the.probability of the proton being on, for example, the 3rd level in A-T is particularly small (the ratio of the number of particles on level 3.compared to level 1 under normal conditions is approximately 10-16). The probability of it being on the. lowest tunnelling level in G-C under normal conditions is even lower. The .introduction of radiation into the system esxcites the particles to the upper levels. Fromzhe ,upper levels they can tunnel much easier. It should be mentioned at this point, however, that .%is radiation may change the overall base unit co an excited state in which case the potential well shape can change quite considerably. Fig. 4 shows how radiation effects the base units. A complete set of graphs was plotted but only selected ones are included in the report. In the graphs related to tunnelling two cases were considered, namely times f Cc 712 and 6” 712. The curves resulting from the various plots in this case (N2/i@ versus p) were similar, .in that they all had the same general shape. A typical one is shown in fig..4 (A-T for t c< 712). Level 3, which is the lowest tunneiling level for t - lo-IO set, shows that at the relatively low radiation density of 16-16 W/m3 the ratio of tunnelled particles to particles in the ground state is quite low. As the radiation density is increased this ratio increases rapidly but finally levels off at about 7 x 10-5.

level 7 “-”

1

,

lewd6

lo-"

16'" 16'0

level 3

t

=I6”

c

p(w/m~

Pig. 4. A.plot

of NS/*

for. the ground et&e,of A-T for

-- tcc7,

Volume

8. pumber

1

CHEMICAL

PHYSICS LETTERS

. For t y‘ 712 the curves again have the same general-shape but are higher up on the graph and level off or saturate at higher values.

I Janunr:;

and Dr. S. Lunnell for permission potential curves.

1971

to reproduce

REFERENCES ACKNOWLEDGEMENT

111 P.-O. LBwdfn. Rev. Mod. Phys. 35 (L963) 721. (21 P.-O. La\\din. Advances in qunntum chemistry (Academic Press, New York). (31 R. Rein and F.E. Hnrris. J. Chem. Phys. -43 (19GS) 4115. [41 S. Lunell nnd G. Sperber. Preprint QB32 (19tiG) Quantum Chemistry Group. University of Uppsnla. Slseden: J. Chem. Phys. 46 (1967) 2119.

The authors are grateful to Dr. J. Ryan for several helpful suggestions and to T. Rupp for help with the computer programming. They would aiso like to thank J. D. Son for his help

the project.

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on

Thanks are extended to Dr. F. Harris

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99 ..