Isotope effect in tunneling and its influence on mutation rates

218

BIUTATION

RESEARCH

I S O T O P E E F F E C T IN T U N N E L I N G A N D ITS I N F L U E N C E ON MUTATION RATES*

PFA~I-()L()V I / ' ) W D 1 N

(~)ua~tum "l'heo~V Project, (,'~iversi O, ~ Fl(~rida, Gaim'svilb', Fla. (U.S.A.) and Qztanlmn Chemislrv (;roz@, Umversi&, qf Uppsala, Uppsala (Swede,s) (Received March 3oth, ~905)

In the W a t s o n Crick model of D N A , the genetic information is essentially contained in the proton-electron pair code associated with the hydrogen bonds between the base pairs. POLLAI{I) AND LEMKE H have carried out a beautiful and important e x p e r i m e n t on the influence of the proton mass on the m u t a t i o n rates, and have inperpreted their results as to the possible importance of the p h e n o m e n o n of proton t u n n e l i n g s u g g e s t e d b y t h e a u t h o P ~'. I n t h e t u n n e l - e f f e c t d i s c o v e r e d b y GAMOW~, -~ a n d C~URNEY AND CONDON a, a quantumomechanical particle having the mass m and energy E may penetrate a c l a s s i c a l p o t e n t i a l b a r r i e r I" V ( x ) , a n d t h e t u n n e l i n g p r o b a b i l i t y g p e r h i t is g i v e n r o u g h l y b y t h e f o r m u l a (see Fig. ~):

V (x)

I Xl

i x3

~ x

l:ig. ~. Tunneling out of a single potential well.

g ~ e-~%

K ~-~j~,

where h is Planck's constant, h = 6.625 • IO ~7 erg • sec. For a parabolic barrier, one gets the simplified formula: ff =

~

22~

" - ao ~/2ml~,,, 4 h

(2)

* A comment on tire paper " Rate of mutation to phage resistance in eH20 medium" by F.. POLLARD AND M. LEMKE (Mulation Research, 2 (~965) 2z3-2XT). I~ublished with pernaission of the authors. M~dation Research 2 (~965) 2~8-zz r

ISOTOPE EFFECT IN TUNNELING

2I()

w h e r e a0 is t h e " t u n n e l i n g d i s t a n c e " a n d V0 t h e t u n n e l i n g b a r r i e r for t h e e n e r g y E of t h e p a r t i c l e . L e t us first c o n s i d e r t h e t u n n e l i n g o u t of a single well. If t h e p a r t i c l e oscillates in t h e well w i t h t h e f r e q u e n c y Vl it is g o i n g to h i t t h e b a r r i e r Vl t i m e s per sec, a n d t h e t r a n s m i s s i o n p r o b a b i l i t y p e r sec ci will be Cl = vlg. T h e p a r t i c l e d e n s i t y ni in t h e w e l l will t h e n s a t i s f y t h e d i f f e r e n t i a l e q u a t i o n : dn, dt

--

C1~1'

(3)

w i t h t h e s o l u t i o n n d t ) - - nl(o) e x p { - - c l t } . A f t e r t h e " t u n n e l i n g t i m e " ~1 I/q, the p o p u l a t i o n has h e n c e d r o p p e d b y a f a c t o r I / e . Since t h e m a s s of t h e p a r t i c l e e n t e r s t h e E x p r e s s i o n s I a n d 2 for K , it is e v i d e n t t h a t t h e t u n n e l i n g t i m e s will be v e r y d i f f e r e n t for p r o t o n s a n d d e u t e r o n s a n d t h a t one c a n t h e r e f o r e e x p e c t a s t r o n g i s o t o p e effect. If t h e b o t t o m of t h e well is p a r a b o l i c a n d c h a r a c t e r i z e d b y t h e p o t e n t i a l a l ( X - - X o ) 2 / 2 , t h e f r e q u e n c y of t h e p a r t i c l e is r o u g h l y g i v e n b y t h e f o r m u l a va = ( a l / 4 n~ m)~ w h i c h is a q u a n t i t y p r o p o r t i o n a l to m-~. F o r a p r o t o n t h e f r e q u e n c y is of t h e o r d e r of m a g n i t u d e IO Ia to IO 1~ sec-% and, in t h e h y d r o g e n b o n d s , we shall use t h e f o r m e r figure. S u b s t i t u t i n g t h e c h a r a c t e r i s t i c d a t a for t h e p r o t o n a n d d e u t e r o n in E q s . I a n d 2, a n d e x p r e s s i n g ao in A a n d V0 in e V , one o b t a i n s P r o t o n : 1°log Cl = 1 3 - - 1 5 a0 ~ / V 0 ,

(4) D e u t e r o n : 1°log c, = 1 2 . 8 5 o 5 - - 2 1 . 2 1 3 a0 ~ / V 0 . T a b l e I gives a c o m p a r i s o n b e t w e e n t h e t u n n e l i n g t i m e s za = I/Ca for t h e t w o p a r ticles. T h e effect is e n o r m o u s , a n d it is this large effect POLLARD AND LEMKE 11 h a v e b e e n l o o k i n g for a n d f o u n d m i s s i n g in t h e i r e x p e r i m e n t . TABLE [ TUNNELING

TIMES

a 0 X, V 0 \ V I T H

ao ~,/ Vo

T 1 FOR

go IN

A,

THE

PROTON

AND

THE

DF;UTERON

AS A FUNCTION

OF THE

PARAMICFER

V 0 IN er

Proton r~

Deuteron v I I 0 --13 s e e

_

0

I0

13

0.1

I0

11.5 SOG

I0

11 s e e

0.2

I0

10

see

I0

9

SeC

0. 3

10 -8.5

see

IO

6

sec

0. 4

IO -7

sec

1o4 sec

0. 5

I0

5.5

sec

10

0.0

I0

4

sec

1. 3

0. 7

io

2

sec

2 lllin

0.8

io

1

sec

o.9 i.o 1.I 1.2 1. 3

3 sec 2 gain

I-4

1. 5 1.6 1. 7 1.8 1.9 2.0

see

.53 I n i n

i.i6 days 1.22 months 3 years 1oo years 3000 years io ~ years 3 " 1°6 years lO6 years 3 " IO~ years

3.7

2

SCC SOC

h

2o days 8 years 10a years I. 3 - IOa year 1. 7 • lo 7 years 2.2 - IOU years

Mutation Research 2 (1965) 218 22i

220

P . - ( ) . 1.(J~VDI N

However, in connection with the h y d r o g e n bonds in D N A , it should be observed t h a t the p r o t o n is s i t u a t e d in a double-well (See Fig. 2) a n d t h a t it m a y r e t u r n to the original position due to "back-tunneling". This oscillatory c h a r a c t e r of the particle over the entire double-well has to be s t u d i e d in order to u n d e r s t a n d the isotope effect properly. I n d e x i n g the l e f t - h a n d well b y ~ a n d the r i g h t - h a n d well b y 3, one obtains b y generalizing Eq. 3 the following equations for the particle d e n s i t v : d~t a

dl

I (

- c~z~ i c3~¢a,

dn~

(5)

~: [~1 'ill -- C3 ]13,

dt

'

'

where (?1 ~ Pig a n d ca : v~g. If the initial condition is u~(o) = a~ a n d ~a(o) =:-: o, one gets the explicit solution:

Ill@ ) =

Ca{[ 1

~.

thai

e

c~-] ca

c~@ca

.

Cla 1

t:lg/1

]~(~)

C l - ] C3

['1"~ C:I

(c,~c~)t

(~,) e (m+e~)t,

1

Fig. 2. ( ) s c i l l a t i o n s in a d o u b l e w e l l a s s o c i a t e d

with the tunneling

ciIcct.

The t i m e r~.a --~ ( q t ca) -~ m a y be d e n o t e d as the " r e l a x a t i o n t i m e " of the double-well, a n d one should observe t h a t this q u a n t i t y is m a s s - d e p e n d e n t to the same e x t e n t as the single tunneling-time. F o r t < ~,a, the isotope shift would also show u p in the particle d i s t r i b u t i o n . However, for t >> ~,.% the q u a n t i t i e s nt a n d ~¢a a p p r o a c h quickly the c o n s t a n t values

]~ 1 --~

~,~a~ -"

~'~+ ~a

' ~]~a ~

~,~ a, -

" ~

(7)

~, -5 ~'~

which satisfy the simple relations ~ : n a = fa:Vl. This m e a n s t h a t , for t ~ ~.a, the p a r t i c l e d i s t r i b u t i o n is essentially i n d e p e n d e n t of the transmission coefficient g and i n d e p e n d e n t of the p a r t i c l e mass m. R e c e n t calculations on the G - C base pair indicate ~Vl~tlalio:z Research 2 ( ~ 9 6 5 ) 2 r 8 - 2 2 1

ISOTOPE EFFECT IN TUNNELING

22I

t h a t t h e r e l a x a t i o n t i m e s for t h e p r o t o n s as well as t h e d e u t e r o n s a r e v e r y s h o r t in c o m p a r i s o n t o t h e r e p l i c a t i o n t i m e of D N A , so t h a t t h i s n o r m a l l y s e e m s t o a p p l y . T h a n k s t o t h e t u n n e l effect, t h e s y s t e m h a s t h e n also e n o u g h t i m e t o r e a c h a t h e r m a l e q u i l i b r i u m o v e r t h e d o u b l e - w e l l a n d , in t r e a t i n g t h e B o l t z m a n n d i s t r i b u t i o n , t h e r e is f i n a l l y a v e r y s m a l l i s o t o p e s h i f t s h o w i n g u p d e p e n d i n g o n a s l i g h t d i f f e r e n c e in e n e r g y l e v e l s for t h e p r o t o n a n d t h e d e u t e r o n . T h i s s m a l l s h i f t is p r e s e n t l y b e i n g s t u d i e d in U p p s a l a . W e h a v e h e r e s t u d i e d a s i n g l e p a r t i c l e i n a d o u b l e - w e l l w h e r e a s , i n a r e a l base. p a i r , o n e h a s t o c o n s i d e r a t l e a s t t w o p r o t o n s in a q u a d r u p l e - w e l l a n d t o t a k e i n t o a c c o u n t t h e p o l a r i z i n g effect of t h e p r o t o n s o n t h e m o b i l e e l e c t r o n s of t h e b a s e p a i r a n d i t s effect o n t h e p o t e n t i a l s u r f a c e s . T h i s p r o b l e m a n d i t s c o n n e c t i o n w i t h t h e q u e s t i o n of t h e s p o n t a n e o u s a n d i n d u c e d m u t a t i o n r a t e s is n o w b e i n g t r e a t e d in a series of p a p e r s b y LADIK ~,5, REIN AND HARRIS 12 i~. A m o r e d e t a i l e d s u r v e v of t h e e n t i r e p r o b l e m is also g i v e n e l s e w h e r e I°. T h e s e r e m a r k s d o i n n o w a y d i m i n i s h t h e i m p o r t a n c e of t h e e x p e r i m e n t c a r r i e d o u t b y POLLARD AND LEMKE n , a n d we o n l y w i s h e d t o p o i n t o u t t h a t t h e r e m a y also b e o t h e r f a c t o r s t o b e c o n s i d e r e d in m a k i n g t h e p r o p e r i n t e r p r e t a t i o n . ACKNOVCLEI) GMENT T h i s w o r k w a s s u p p o r t e d i n p a r t b y t h e C a n c e r I n s t i t u t e of t h e U.S. N a t i o n a l I n s t i t u t e s of H e a l t h u n d e r C o n t r a c t C A o685O-Ol w i t h U p p s a l a U n i v e r s i t y . RI{FER1LNCES I GAMOW, (;., Zur Quantentheorie der Atomkernes. Z. Ph3,sih, 51 (1928) 204. 2 GAMOW, G., Zur Quantentheorie der Atomzertrtimmerung. Z. Physih, 52 (I928) 5 Io. 3 GURNF.Y, R. \V. AND E. (!. CONDON, Quantum mechanics and radioactive llisintegration. Phys. I~ev., 33 119-,9) 127. 4 LADIK, J., Possible interpretation in the mutagenic effect of ultraviolet radiation. J. Theoret. Biol., 6 (I964) 2oI. 5 I.ADIK, J., Some new results in the quantmn-mechanical calculation of I)NA. In B. PULLI~IAN, Electronic ~qspecls ofBiochemist~ql,, Acaclemic Press, New York, 1964, p. zo 3. () L6WDIN, P.-O., Proton tunneling in DN.\ and its biological implications. Rev. Mod. Phys., 35 (I963) 7-'4. 7 L6WDIN, P.-O., I~ler~¢alional Science a~d Technology, CIlnover-Mast Publ., New York, May ~963, p. 64. 8 1,6WmN, P.-O., Effect of proton tunneling in DNA on genetic information and t/roblems of mutation, aging, and tumors. Biopo!vmers Syrup., ~ (1964) 101. 9 L6wD~y, P.-()., Some aspects on DNA replication; incorporation errors and proton transfer. In B. PULLMAN,Eleclronic .4 spects of Biochemistry, Academic Press, New York, I904, p. ~67. lO L6WDIN, P.-O., Quantum genetics, Adv. Quantum Chem., 2 11965). I I ~:~OLLARD,E. AND M. f.EMKE, Rate of mutation to phage resistance in zH~O mediuni. 3lulatio~t Research, 2 (I965) 213-217 . 12 REIN, l{. AND F. HARRIS, Studies of hydrogen-bonded systems. 1. The electronic structure and the lhmble-well potential of the N - H . . . H hyltrogen bond of the guanino-cytosine base pair..[. Chem. Phys., 4 ~ (~964) 3393. 13 REIN, R. AND F. HARRIS, Proton tunneling in radiation induced mutation. Scie~ce, 1 ~ (~964) 649. 14 1{~;i~, R. AND F. HARRIS, Studies of hydrogen-bonded systems. II. Tunneling in tautomeric equilibria in the N H . . . N hydrogen bond of the guanine-cytosine base pair. J . Chem. Phys. 42 (~965) -,~77. 15 REIX, R. ANn F. HAR~RIS, Studies of hydrogen-bonded systems. 11 I. Potential energy surface, tunneling, and tautomeric equilibria in the N - H . . . N and O . . . H - N bonds of the g u a n i n e cytosine base pair. Preprint QB 23 , Uppsa]a, Q u a n t u m Chemistry Group (i965), nntlublished. 3lutation Research, 2 (1965) 218-z 21