J.Mol.Evol.5,327-332 (1975) © by Springer-Verlag 1975

Letter to the Editor

Symmetries of Genetic Code-Doublets H.-]. D A N C K W E R T S 1 and D. NEUBERT 2 1

Fachbereich Chemie der Universit&t Regensburg ~

2physikalisch-Technische Bundesan~talt, Institut Berlin Received September 4, 1974; March 28, 1975

Summary. The fact that 64 base triplets code only about 20 essential amino acids implies a strong degeneracy of certain base doublets. It is shown that the set of degenerate base doublets and the set of non degenerate base doublets are highly structured. A mathematical formalism is introduced which allows a systematic description of the consequences of an exchange of bases in a doublet. By this formalism it is shown that the two mentioned sets have in fact the same structure. Key words: Genetic Code, Degeneracy of ~, Symmetry of ~ - Base Triplets Base Doublets, Group Property of %, Transformation of

SYMMETRIES In

the

OF

genetic

GENETIC code,

CODE-DOUBLETS the

base

triplets

of

the m-RNA

determine

t h e a m i n o a c i d s o f t h e p r o t e i n . T h e f o u r b a s e s U, C, A a n d G o f m - R N A f o r m 64 d i f f e r e n t t r i p l e t s B I B 2 B 3 w h i c h in t u r n c o d e o n l y 20 e s s e n t i a l a m i n o a c i d s . A p p a r e n t l y , c e r t a i n d i f f e r e n t t r i p l e t s c o d e the s a m e a m i n o a c i d a n d it h a s b e e n n o t i c e d [I] that many amino acids are already fully determined by based o u b l e t s B I B 2 . T h i s d e g e n e r a c y of t h e automatically for a certain stability

genetic code provides of the genetic informa-

tion against natural and induced mutations. T h e p u r p o s e of t h i s c o m m u n i c a t i o n is to g i v e a s y s t e m a t i c d e s c r i p t i o n o f t h e r o l e o f d o u b l e t s in c o d i n g . E s p e c i a l l y , w e s h a l l d e a l w i t h t h e s t r u c t u r e o f t w o sets: T h e s e t MI c o n t a i n s as e l e m e n t s t h e d o u b l e t s f o r w h i c h t h e t h i r d b a s e in t h e t r i p let has no influence on the coded amino acid, the set M2 cont a i n s t h o s e d o u b l e t s w h i c h do n o t c o d e t h e a m i n o a c i d u n i q u e l y but require the knowledge of the third base. From the genetic ~Present address: Abt.f~r Biomedizinische Technik, Medizinische Hochschule Hannover 327

code

[I] the sets M I and M 2 are t a k e n

CC,

CU,

to be

(la)

M I = {AC,

CG,

UC,

GC,

GU,

GG}

(Ib)

M 2 = {CA, AA, AU, AG,

GA,

UA,

UU,

UG}

E v i d e n t l y , the d o u b l e t s of M I b e l o n g the f o u r f o l d d e g e n e r ate triplets, since e a c h d o u b l e t r e s u l t s in four d i f f e r e n t t r i p l e t s c o d i n g the same amino acid. In o r d e r to find out the s t r u c t u r e of M I and M 2 we introduce d o u b l e t - e x c h a n g e o p e r a t o r s (ai, aj), w h e r e the operators ai, i = 1,2,3,4, e x c h a n g e the four b a s e s A, C, U, G in pairs as d e f i n d e d in Fig. l a. B i o c h e m i c a l l y , the a i have the f o l l o w i n g m e a n i n g : I 6

is the unit oI~erator and does not p r o d u c e any e x c h a n g e e x c h a n g e s bases of p u r i n e - t y p e to p y r i m i d i n e - t y p e e x c h a n g e s b a s e s w h i c h can u n d e r g o h y d r o g e n bonds to c o m p l e m e n t a r y bases e x c h a n g e s a g i v e n p u r i n e a g a i n s t the o t h e r p u r i n e and a g i v e n p y r i m i d i n e a g a i n s t the other p y r i m i d i n e .

Mathematically, i.e. 2 (2)

1

the a i form an A b e l i a n

2 =

~

2 =

6

The d o u b l e t - e x c h a n g e (3)

(ai,

group

(Klein's

4-group)

2 =

y

=

1

operators

(ai, aj)

are now d e f i n e d by

a.)3 BIB2 = BIB ~

w h e r e a i o p e r a t e s on B I and aj on B 2 and w h e r e the d o u b l e t s BIB 2 and B~B~ are some c o m b i n a t i o n of two b a s e s f r o m A, C, U, G. W i t h i, j = 1,2,3,4, there are 16 d i f f e r e n t d o u b l e t - e x c h a n g e operators. The set M I of d o u b l e t s can be g e n e r a t e d by o p e r a t i n g w i t h s u i t a b l e d o u b l e t - e x c h a n g e o p e r a t o r s on a special, c o n v e n i e n t ly c h o s e n doublet, say AC. First, however, we g e n e r a t e a set Mo:

(4) (4a)

M

o

= { [ (1,1)u (e, 1)u (~,8)u (~,y)]

= {AC,

CC,

CG,

AC}

CU}

and then o b t a i n M I from M o by the o p e r a t i o n

328

A:

=C Y

W

a~=1 ~

CC CU CG[~C~

c~2=a o3= B c~4:y

MI

GC GU GG]~,

M2

UC I ~ ~ ~

c~ Fig.la. Definition of the exchange operators o

(5)

i

M1 =

F r o m M I the

(6)

M2 =

[(1,1)u(6,1)]

M o.

set M 2 is o b t a i n e d

b y the o p e r a t i o n

( ~ , ~ ) M I.

The set M o f all d o u b l e t s , t i o n the s t r u c t u r e

(7)

Fig.lb. Structure of the doublet set M

M = [(1,1)~(B,1

M = M I U M 2, has b y t h i s

][(1,1)u(~,~,)]

construc-

Mo

w h i c h c a n b e e a s i l y v e r i f i e d b y l o o k i n g at F i g . l b in c o m b i n a tion with Fig.la. From the complete transformation t a b l e (Figo2) it is s e e n explicitly, which transformations p l a y a s p e c i a l rSle: T h e operation [ ( 1 , 1 ) u ( B , 1 ) ] is the o n l y o p e r a t i o n p r o d u c i n g M I f r o m M o. E v i d e n t l y , (B,I) is the o n l y o p e r a t i o n b e s i d e the trivial operation (1,1) u n d e r w h i c h M 1 is i n v a r i a n t . N a t u r a l l y , a l s o M 2 is i n v a r i a n t u n d e r (B,I). F u r t h e r , we n o t i c e t h a t n o t o n l y (e,~) g e n e r a t e s M2 f r o m MI, b u t a l s o the o p e r a t i o n (y,~). H o w e v e r , if w e w r i t e (~,~)= ~(I,1) and ( y , ~ ) = ~ ( B , 1 ) , w e see t h a t (~,~) is t h e s i g n i f i c a n t o p e r a t i o n , the p o s s i b i l i t y of u s i n g a l s o (y,~) is a c o n s e q u e n c e o n l y of the i n v a r i a n c e of MI a n d M 2 u n d e r (B,1). Bec a u s e of Eq. (2), M I is g e n e r a t e d in t u r n f r o m M 2 by (e,~) : M I = (~,~)M2. The r e s t of the d o u b l e t - e x c h a n g e operators (oi, oj) t r a n s f o r m a l w a y s h a l f of M I i n t o M2 a n d h a l f of M2 i n t o M I . By i n v e r s i o n of Eq. (4) a n d Eq. (5) and m a k i n g u s e of Eq. (2) o n e o b t a i n s a r u l e to d e t e r m i n e , w h e t h e r an e l e m e n t B I B 2 bel o n g s to M 1 or to M2: f i r s t g e n e r a t e d o u b l e t s B'IB' 2 b y (8)

[ ( 1 , 1 ) u ( B , 1 ) ] [(1, I) u (~, I) u (~, B)u (~,y)] B I B 2 = B ' i B'

2

329

(1 ,c~) cG cutcc

(1,1) cc cu CGI~

(13') cu cc ~JcG

(113)

GC GU GGI~bc;C GG GUIGC GG~ . GCIGu GU GC~C,]GG ~C~.I,~~J~. UC ((~,y) (o~.13) (o,..o.) (0~.1) ~ CGLG_t~_CC CG1~. cc cu CUlCC ~ cG ~ . ~ ~.I uc GC[GU GG ~F/~,IGG GU GC GG~ GC GU GUIGC~ GG (13,v) (13.cd (B,13) ((~.I) GG GUlGC GG~ GCJGU Gu~ G G GC GU GGI~ UC I ~ ~ H~r~c~ cc cu CGJF~, CG CUlCC CG~ CC]CU CU CC~ ] CG cclcu CG

Cy.1)

('y,c~)

('y.!3)

GCGULG_U_EG_~~ , GI-GGGU GC GG~ AC~

~

~ ~ ~

cclcu CG" ~ C I %

ACI~ ~

('Y,Y)

1

GU GUIGC~

GG

~ 1 ~ ~ AC ~ l ~

CU CC CG~I~.~ CU culcc ~

i

CG

Fig.2. Transformations obtained by the operations (qi, oj) on M (unshaded doublets belong to M I, shaded doublets belong to M2).

If f o r o n e B'IB'2

of

the

B'IB' 2 h o l d s

= AC,

BIB 2 belongs

to M I

B'IB' 2 = CA,

BIB 2 belongs

to M 2.

In s u m m a r y , w e h a v e d e m o n s t r a t e d complete symmetry between the s e t s M I a n d M 2 u n d e r t r a n s f o r m a t i o n s a f f e c t e d by the doublet-exchange operators (oi, oj). E s p e c i a l l y , we have shown: a) M I a n d M 2 a r e i n v a r i a n t b y o p e r a t i n g w i t h (8,1) on BI, b u t no o p e r a t i o n o n B 2 l e a v e s M I or M 2 i n v a r i a n t . Thus B 2 carries more information t h a n B I a n d B 2 is t h e r e f o r e m o r e i m p o r t a n t for the s t a b i l i t y of M I a n d M 2 t h a n B I. A c h a n g e of B I w i t h r e s p e c t to its h y d r o g e n b o n d p r o p e r t y d o e s n o t c h a n g e t h e r e s u l t i n g a m i n o a c i d s if all d o u b l e t s of e i t h e r M I o r M 2 a r e affected. Reversing supposition and conclusion, M I and M 2 may be def i n e d as t h o s e d o u b l e t s e t s of 8 e l e m e n t s w h i c h a r e i n v a r i a n t under the (6,1)-transformation. Then experience s h o w s t h a t MI a n d M 2 are f o u r f o l d a n d l e s s t h a n f o u r f o l d d e g e n e r a t e respectively. 330

b) T h e o p e r a t i o n (~,~) t r a n s f o r m s M I i n t o M 2. T h i s i n d i c a t e s that purine- and pyrimidine-type bases are distributed in a well determined o r d e r o n t o the b a s e s B I a n d B 2 a n d t h a t t h i s order determines, w h e t h e r the t h i r d b a s e c a r r i e s i n f o r m a t i o n or not. By Eq. (8) it is s e e n t h a t f o r s p e c i a l d o u b l e t s t h i s different o r d e r is s i m p l y t h e r e s e r v e o r d e r of t h e b a s e s . T h e r e a r e a l s o o t h e r w a y s to l o o k at M I a n d M 2. F r o m Fig. Ib o n e s e e s e.g., t h a t M I is C a n d G d o m i n a t e d a n d M 2 is A and U dominated in the f o l l o w i n g sense: M I is d e t e r m i n e d by B I = C or G, M 2 is d e t e r m i n e d b y B I = A or U; B2, h o w e v e r , is d e c i s i v e also: B 2 = C b e a t s B I = A o r U, B 2 = A b e a t s B I = C or G. S i n c e C a n d G a r e e q u i v a l e n t w i t h r e s p e c t to t h e h y d r o gen bond property, o n e m i g h t s a y t h e r e f o r e , M I is h y d r o g e n bond dominated a n d s i m i l a r l y , M2 is d o m i n a t e d by t h e h y d r o g e n bond complement property. O n e m a y t r y to f i n d s i m p l e r c r i t e r i a . If o n e d e f i n e s e.g. a set S I consisting of d o u b l e t s c o m p o s e d of b a s e p a i r s w i t h t h r e e h y d r o g e n b o n d s , S I = {CC, CG, GC, GG}, a n d a s e t $2 consisting of d o u b l e t s w i t h two h y d r o g e n b o n d s , S 2 = {UU, UA, AU, AA}, t h e n S I is a s u b s e t of M I a n d $2 a s u b s e t of M 2. H o w e v e r , in t h i s c a s e t h e r e is s t i l l a s e t S 3 = { a l l o t h e r doublets} f o r w h i c h t h e r e is no a n s w e r to the q u e s t i o n , if t h e d o u b l e t s p e c i f i e s an a m i n o a c i d u n i q u e l y or not. Evidently, the t r a n s f o r m a t i o n properties of M I a n d M2 a r e independent of t h e a r r a n g e m e n t of t h e g e n e t i c code. T h e t a b l e of F i g . l b h a s b e e n c h o s e n f o r c o n v e n i e n c e of r e p r e s e n t a t i o n , b u t t h e s a m e r e s u l t s w i l l be o b t a i n e d b y a p p l y i n g d o u b l e t - e x change operators o n t o the t a b l e of " b e s t a l l o c a t i o n s " prop o s e d b y C r i c k [2~. It h a s b e e n p o i n t e d o u t b y W o e s e e t al. [31 t h a t q u e s t i o n s r e l a t e d to t h e e x p l a n a t i o n of the g e n e t i c c o d e "are a l m o s t certainly c l o s e l y a l l i e d to the a n s w e r s to q u e s t i o n s regardi n g the g e n e t i c c o d e ' s e v o l u t i o n " . In fact, c o d e g e n e r a t i n g equations l i k e t h o s e g i v e n in Eq.(4) to (7) p r o v i d e a formalism for describing the genetic code's evolution if i n t e r p r e t e d as b e i n g e x e c u t e d in time. Of c o u r s e , t h e r e a r e m a n y t y p e s of i n f o r m a t i o n , e.g. s t a t i s t i c a l information like that given by Roberts [41, w h i c h h a v e to be t a k e n i n t o a c c o u n t in the search for a consistent description of the r o l e o f d o u b l e t s in t h e g e n e t i c code. It m a y be n o t i c e d , t h a t the n u m b e r of d o u b l e t s in M I a n d M 2 is b o t h 8. O n l y d u e to t h i s p r o p e r t y t h e g i v e n s y m m e t r y relations between M I and M 2 are possible. U n l e s s it c a n be proven that the fourfold degeneracy of exactly 8 doublets h a s no biological significance, w e s u g g e s t to a d o p t the e x i s t e n c e of t h i s f a c t as a h y p o t h e s i s in f u t u r e i n v e s t i g a t i o n s .

331

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