Det Kongelige Danske Videnskabernes SelskabBiologiske Meddelelser, bind 22, nr. 3

Dan. Biol. Medd. 22, no. 3 (1954)

POSSIBLEMATHEMATICAL RELATION

BETWEEN DEOXYRIBONUCLEIC ACID AND PROTEINS

BY

G. GAMOW

Københavni kommission hos Ejnar Munksgaard

1954

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Bibliografisk forkortelse Abrévialion bibliographique

Dan. Vid. Selsk. Overs.

Dan. Hist. Filol. Medd.Dan. Hist. Filol. Skr.

Dan. Arkæol. Kunsthist. Medd. Dan. Arkæol. Kunsthist. Skr.

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Det Kongelige Danske Videnskabernes SelskabBiologiske Meddelelser, bind 22, nr. 3

Dan. Biol. Medd. 22, no. 3 (1954)

POSSIBLEMATHEMATICAL RELATION

BETWEEN DEOXYRIBONUCLEIC ACID AND PROTEINS

G. GAMOW

Københavni kommission hos Ejnar Munksgaard

1954

Prinled in Denmark Bianco Lunos Bogtrykkeri A-S

»Die Methoden der Polypeptidsynthese ge­statten den Aufbau langer Ketten mit vielfachen Variationen in der Reihenfolge. Es ist drum kein blosses Spiel mit Zahlen, wenn man die gege­benen Möglichkeiten berechnet.«

( E m il F is c h e r , Sitzungsber. der Kgl.Preuss. Akad. der Wiss., p. 990, 1916.)

It was recently show n by W atson and C rick (1) th a t the m ole­cules of Deoxyribonucleic Acid (DNA), which constitute ch ro ­

m osom e fibers of living cells, are form ed by double sequences of four basic com pounds (Adenine, Thym ine, Guanine, and Cyto­sine) held together by two parallel sugar-phosphate chains. Since, according to that scheme, Adenine can p a ir only w ith Thym ine, and Guanine only with Cytosine, one h a lf of th a t double sequence of bases is com pletely determ ined by the other half. T hus, if we denote the four bases by figures 1, 2, 3, 4 (a m athem atician w ould p refer 0, 1, 2, 3), all hered itary properties of any living organism should be characterized by a long number ( “ num ber of the b east” ) w ritten in a four digital system, and containing m any thousands of consecutive digits. The num bers describing two different m em ­bers of the sam e species m ust be very sim ilar to each other (though not quite identical, unless they belong to a p a ir of iden t­ical tw ins), w hereas the num bers representing the m em bers of two different species m ust show larger differences. Since the n u m ­ber of all possible arrangem ents of four elem ents in sequences of several thousand is incredibly large*, we m ust conclude that all living organism s represen t only a negligible fraction of all “ m athem atically possib le” form s of life. For exam ple, it is ex­trem ely unlikely th a t any organism w hich ever lived on the su r­face of the earth was represented by such fam iliar num bers as n , or [/3 written in the four digital system. W a tso n an d C rick

* For a chain which is, for example, 10,000 steps long, the number of possible arrangements is 410>000 = 106>000, which is much (much!) larger than the number 1076 representing the number of all atoms in the Universe within range of the 200" telescope at Palomar Mountain Observatory.

1 *

4 Nr. 3

Table I. The list of Amino acids.

1. G/utemic acid 14. Phenylalanine2. Leucine 15. Cystine3. Aspartic acid 16. Histidine4. Serine 17. M ethionine5. Lysine 18. Tryptophane6. Glycine 19. Cysfeic acid7. Va/ine 20. H ydroxyproline8. Proline 21. (N orvaline)9. Aryinine 22. (H ydroxy glutam ic acid)

10. A/anine 23. (A sparagine)11. Threonine 24. (G lutam ine)12. /soleucine 25. (C annine)13. Tyrosine

also suggest a very plausible m echanism by w hich the replication of DNA m olecules m ay take place, w ith a provision for occasional m utations, i. e. the change of some digits in the original “ num ber of the b east” .

It is well know n, however, that, while the chrom osom es are responsible for carrying all (or, at least, m ost of) the hered itary in form ation from the paren ts to the progeny, the actual w ork of the growing and the developm ent of any organism are carried out by enzym es w hich catalyze various biochem ical reactions in the cytoplasm of the cell. In fact, while a chrom osom e can be com ­pared w ith the fde cabinet in the d irec to r’s office of a large factory w here all the b lueprin ts are stored, the enzymes p lay the role of engineers, forem an, and w orkers constituting the m ajor p a rt of the entire outfit. It follows that there m ust exist a very precise m echanism w hich shapes the enzym es produced in any given organism , exactly according to the hered itary in form ation carried by chrom osom es. The enzym es, being proteins, possess, however, an entirely different constitution than DNA m olecules, and are form ed by long sequences of m any different am ino acids. The num b er of different am ino acids w hich partic ipates in the struc­ture of proteins is usually taken as 20, although actually there m ay be a few m ore. In T able I these am ino acids are listed in o rder of their relative abundance in proteins.

Nr. 3 5

If one assigns a letter of the alphabet to each am ino acid, each protein (and , in particu lar, each enzym e) can be considered

R g ' f .

Vyh)fg bails ;b a s e s .

J^Jack ba Hs : Cli'nino oc ic(s

Avvows • H b

T.p+id*as a long word based on an alphabet w ith 20 (o r som ew hat m ore) different letters.

It was recently suggested by the au thor (2) that a sim ple re la ­tion between the order of bases in chrom osom e fibers, and the

6 Nr. 3

order of am ino acids in the corresponding enzymes, can be estab ­lished by considering basic geom etrical features of the W atson and C rick m odel of a DNA m olecule.

In fact, as can be easily seen from Fig. 1, the helical natu re of the DNA m olecule gives rise to d iam ond-shaped configurations form ed by four bases. The two bases at both ends of the h o ri­zontal diagonal can be only A denine-Thym ine or G uanine-Cyto­sine pairs, w hereas the u pper and lower corners of each “ d ia ­m ond” m ay be occupied by any of the four bases. T hus, the total n um ber of different “ d iam onds” is given by the num ber of d if­ferent triple com binations of four elem ents, and one can easily calculate th a t it is equal to 20. These 20 different “ d iam o n d s” are show n in Table II, w here the num erals 1, 2, 3, and 4 stand for Adenine, Thym ine, Guanine, and Cytosine, respectively. E ach type of “ d iam o n d ” is denoted by a letter of the alphabet*. The idea is that the am ino acids partic ipating in the structure of p ro ­teins are those whose residues fit into the various d iam ond-shaped cavities form ed by the com binations of four bases in the DNA molecule. Since the sequence of bases determ ines in a un ique w ay the sequence of d iam onds, we have here a m echanism for the production of a specific set of enzym es by each particu la r ch ro ­m osom e. According to this point of view, DNA molecules act as highly specialized catalysts, arranging the am ino acids from the surround ing m edium in well defined sequences, an d holding them in that position long enough to perm it the am ino group of each of them to com bine w ith the hydroxyl group of its next neighbour.

The proposed schem e is quite consistent w ith the existing data on the linear dim ensions of polynucleotide and polypeptide chains. In fact, the distance between the P-atom s in the DNA m olecule (along the sp iral line) is about 7Å, while the distance betw een two Ca-atom s in an extended polypeptide chain is 3.6 Å. If one tries to construct a polypeptide chain on the convex side of the DNA helix, one gets two am ino acids for each “ d iam o n d ” in the poly­nucleotide chain , w hich is certainly not correct. However, con ­structing the sam e chain on the concave side of the DNA helix, one gets the correct relationship of one am ino acid for each dia-

* The assignment of letters in Fig. 2 is different from that given in the author’s original article (Ref. 2). The new assignment associates more probable diamonds with more abundant letters of the alphabet in the English text, which makes the “words” made up by various diamond sequences more pronounceable.

Table II.

8 Nr. 3

m ond-shaped cavity (3 .) T hus, we m ay conclude th a t enzyme mole­cules grow “ on the inside” o f the chromosome helix, as indicated schem atically in Fig. 1.

Inspecting the 20 d iam onds shown in Table II, one notices that 12 of them (m arked by asterisks) can exist in two bilaterally sym m etrical forms. Thus, for exam ple, the letter H , w hich is given

in the table as ^3 • - • 4^, can also exist in the form ^4 * ^ • 3^.

If one considers the two bilaterally sym m etrical form s as two d if­ferent entities, one should raise the num ber of different d iam onds to 32. However, inspection of the structural form ulae of am ino acids indicates that, w ith possibly a few exceptions, the b ilatera l sym m etry of the d iam onds is of no im portance. In fact, there are only th ree or four am ino acids whose residues possess no b ilateral

Table III.Possible com bination of “ d iam o n d s” .

A, E, I, O associates with

A, D, E, F, G, H, I, L, M, N, O, P, U, V

D, G, H, N associates with

A, B, C, D, E, G, H, I, K, N, O, R, S, T

L, M, P associates with

A, E, I, O, L, M, P

K, S, Tassociates with

D, G, H, K, N, S, T

B, C, Rassociates with

D, F, G, H, N, U, V

F, U, V associates with

A, B, C, E, I, O, R

Nr. 3 9

sym m etry around the Ca-bond. The existence of these particu la r am ino acids m ay account for the “ excess over 20“ of the total num ber of am ino acids w hich can be used by DNA molecules for the process of protein-(enzym e)-synthesis.

If the theory of DNA protein correlation described on the p re ­vious pages is correct, there m ust exist a w ay of establishing a un ique correspondence between the am ino acids, given in Table I, and the d iam onds given in Table II. This correspondence should be based on the expected intersym bol correlation between various d iam ond-shaped cavities provided by the polynucleotide helix, and the observed intersym bol correlation between the am ino acids as arranged in polypeptide chains.

The form er correlation can be easily established by consider­ing various possible com binations of the d iam onds listed in the table. Thus we find, for exam ple, that the letter A can be associated with the letters M, N, G, etc., bu t cannot be associated with the letters B, C, R, etc. The com plete list of possible “ pairs of d ia­m onds“ is given in Table III.

W e notice th a t eight d iam onds (A, D, E, G, H, I, N, O) have a m uch larger “ affinity” for other d iam onds th an the rem aining twelve. The table also shows that the six letters (B, C, F, R, U, V) can occur only singly, the twelve letters (A, D, E, G, H, I, K, L, M, N, O, S) can occur in pairs, and the rem aining two (P and T) can repeat consecutively an unlim ited num ber of times.

The em pirical in form ation on the order in w hich various am ino acids occur in protein molecules is, how ever, very m eager. The only protein, for w hich the sequence is knowm, is insulin w hich was studied by F. S ang er and his collaborators (4). The “ w ords” representing two insu lin chains, knowm as A and B, are :

Gly-Isol-Val-Glu-Glu-Cy-Cy-Ala-Ser-Val-Cy-Ser-Leu-Tyr-Glu-Leu-Ast-Tyr-Cy-Asp

andP he-V al-A sp-G lu-H is-L eu- C y-G Iy-Ser-H is-Leu-V al-G lu-A la-

Leu-Tyr-Leu-V al-Cy-G ly-G lu-A rg-G ly-Phe-Phe-Tyr-Thr-Pro- Lys- Ala.

The two series are 2 t and 30 term s long, respectively; they m ake use of 16 different am ino acids listed in Table I.

Any attem pt to establish a one-to-one correspondence b e ­tween twenty am ino acids of Table I and twenty “ d iam o n d s” of

10 Nr. 3

Table II m ust face the fact that the num ber of possible assign­m ents is im m ensely large, being given by : 20! = 2.3 • 1017. H ow ­ever, due to highly restricting intersym bol correlation rules given in T able III, such an attem pt becomes possible. T hus, for ex­am ple, there are only sixteen different possible cases in w hich a double letter is followed by another double letter. These sixteen cases b reak up into four groups in such a way that the m em bers

of the sam e group can be transform ed into each other by a sim ple in ternal perm utation . T hus, in order to explain the sequence:

Glu-Glu-Cy-Cyin insu lin A, one m ust try only four different possibilities (A-A- N-N; L-L-M -M ; K-K-T-T, and T-T-S-S). In each of these four cases, we also know the letter-assignm ent of the fourth am ino acid to the right from Cy-Cy, since it is also a Cy. This restricts the choice of the letter for Val since it m ust precede both Glu and Cy.

Nr. 3 11

Proceeding in this way, it is possible to “ dec ipher” the first eleven places in insu lin A as:

P-O-D-K-K-T-T-N-C-D-T- (cf. Fig. 2).The choice of nine letters from the first K to the last T is

un ique, except for the above-m entioned in terna l perm utations. However, the next step runs into a difficulty. The 12th am ino acid is Ser, w hich was a lready assigned the letter C. But, accord­ing to Table III, the letter C cannot follow the letter T. T hus, at least in this particu la r case, the correspondence between am ino acids and “ d iam o n d s” cannot be established, although one comes ra ther close to it.

It is possible that the difficulty encountered in “ deciphering” the structure of insulin in term s of “ d iam o n d s” is due to over­sim plification of the situation in the proposed form of the theory. For exam ple, it is not im possible th a t some of the “ d iam o n d s” can accom m odate m ore than one am ino acid, or that som e am ino acid can fit into m ore than one diam ond. The answ er can be given only by actually constructing a m odel of a DNA molecule from an atom ic-m odel-kit, and com paring it w ith the m odels of various am ino acids.

But it looks m ore likely that insulin is not a good case for test­ing the validity of the proposed theory. In fact, the theory pertains p rim arily to “hered itary p ro teins” , i. e. the proteins whose struc­ture is directly and com pletely determ ined by genes in the chro­m osom es. Thus, for exam ple, it should apply directly to the su b ­stances p rim arily responsible for colour vision, or coagulation of blood w hich are subject to strict laws of heredity. It is not at all certain that insu lin falls into this category of organic proteins, especially because diabetes, a sickness connected with insulin deficiency, does not seem to possess hered itary characteristics.

U nfortunately, however, we have no inform ation concerning am ino acid sequence in any other protein.

A som ew hat different app roach to the problem can be provided by the study of relative abundances of different am ino acids in various proteins. One would, in fact, expect that different types of “ d iam onds” are affected in different ways by the variation of Adenine-Thym ine to Guanine-Cytosino ratio. Suppose that, w ithin a particu la r group of living organism s, the abundance of the first pa ir of bases varies between the limits of ( A — A X ) and

12 Nr. 3

(X + A X ), so th a t the abundance of the second pair lies w ithin the limits (1 — X ± A X ) . It is easy to see that, in such a case, the expected variability of different types of “ d iam onds” will be different. The “ d iam o n d s” defined entirely by the num bers of the

first p a ir («-type), such as, for exam ple, or ^1 ^2 ^ , will

appear w ith the relative probability X 3 (since there are only three

free choices), w hereas the d iam onds of the type ^3 jj 4^ or ^3 ^ 4^

(¿-type) will have the relative probability ( 1 — X )3. The dia-

Fig. 3.

m onds of the m ixed type, such as I 1 - 2 j (/5-type), w ith the excess

of first p a ir of bases, or such as ^3^4^j (y-type) w ith the excess

of second p a ir of bases, will have relative probabilities given by X 2 ( l — X ), an d X (1 — X )2, respectively. These four functions are plotted in Fig. 3 in respect of X. W e notice that, in case that X varies, let us say, w ithin the limits 0.65 ± 0.05, the relative probability of «-type diam onds varies in ra th er w ide lim its, w hereas the probability of /i-type d iam onds rem ains alm ost con­stant because the corresponding curve passes through a m axi m um near X = 0.65. This theoretical result m ay be correlated w ith observations (5) w hich seem to suggest that som e of the am in o

Nr. 3 13

acids show w ider variations in different proteins than some others. It w ould be interesting to undertake a special investigation by com paring the relative abundances of various am ino acids in the cytoplasm of cells w ith the ratios of base-pairs in the correspond­ing nuclei.

It is the au th o r’s p leasan t duty to express thanks to Dr. F. H. C. Crick, and also to the m em bers of the Cyclotron L ab o ­ra to ry (D .T . M.) of the Carnegie Institution of W ashington, for helpfu l discussion of the problem s.

Note added in the proof (June, 1954):Since this article was sent to print, several alternative attempts were

made to decipher the sequences of amino-acids in protein molecules in terms of base-sequences in the molecules of nucleic acids. Although no finite solution of that basic problem has as yet been found, a number of interesting possible relationships came to light. The work in this direction is now being continued, and will be reported in due time.

The George Washington University.Washington, D. C.

U. S. A.

References.(1) I. D. Watson and F. H. C. Crick, Nature 171, 964 (1953).(2) G. Gamow, Nature 173, 318 (1954).(3) This remark is due Dr. F. H. C. Crick.(4) F. Sanger and H. Tu p p y , Biochem. Journal, 49, 481 (1951);

F. Sanger and O. P. Thompson, Biochem. Journal, 53, 336 (1953).(5) Information theory in Biology. Edited by H. Quastler , University

of Illinois Press (1953).

In d lev e re t til se lsk ab e t den «5. fe b ru a r 1954. F æ rd ig f ra try k k e r ie t d en 4. se p te m b er 1954.

Det Kongelige Danske Videnskabernes SelskabBiologiske Meddelelser

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1. Bohr, Hans H.: Measurement of the BloodjVolume by Meansof Blood Corpuscles Labelled with P32. 1950 .......................... 2.00

2. Larsen, Poul: The Aspects of Polyploidy in the Genus Sola­num. II. Production of Dry Matter, Rate of Photosynthesis and Respiration, and Development of Leaf Area in some Di­ploid, Autotetraploid and Amphidiploid Solanums. 1943 ___ 4.50

3. Westergaard, M.: The Aspects of Polyploidy in the GenusSolanum. III. Seed Production in Autopolyploid and Allo­polyploid Solanums. 1948.............................................................. 2.00

4. Hagerup, O.: Thrips Pollination in Calluna. 1950.................... 1.505. Hagerup, O.: Rain-Pollination. 1950.............................................. 1.506. Jensen, P. Boysen: En Metodik til Undersøgelse af Landbrugs­

planternes Vandøkonomi og Stofproduktion. Mit deutscher Zusammenfassung. 1950.................................................................. 3.00

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8. Børgesen, F.: Vauqhaniella. A New Genus of the Dictuotaceae.1950 ..................................................................................................... 1.00

9. Christiansen, H.: A Tetraploid Larix Decidua Miller. 1950 . . . 1.5010. Jensen, P. Boysen: Untersuchungen über Determination und

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11. Børgesen, F.: Some Marine Algae from Mauritius. Additionsto the Parts Previously Published.il. 1950......................... 5.00

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ma L. Studies on the Activity of Insects III. 1949 ................. 3.005. B orgesen, F.: Some Marine Algae from Mauritius. Additions

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Printed in D enm ark. Bianco L unos Bogtrykkeri A-S.

Id: 244Forfatter: Gamow, G.Titel: Possible Mathematical Relation between Deoxyribonucleic Acid and Proteins.År: 1954ISBN:Serietitel: Biologiske Meddelelser Serienr: B 22:3 SerienrFork: B Sprogkode: Eng