A theory of the quaternary structure of dehydrogenases, dehydrogenating complexes, and other proteins

J. Theoret. Biol. (1965)8,

276-306

A Theory of the Quaternary Structure of Dehydrogenases, Dehydrogenating Complexes, and Other Proteins E. P. WHITEHEAD? Department

of Biochemistry,

University

College, London

(Received 16 June 1964) On the basis of a comparison of the molecular weights and other properties of dehydrogenases it has been suggested that they are all composed of subunits of molecular weights between 17,000 and 21,000. Only certain given numbers of subunits appear to be able to form enzymes. This fact is used to propose quaternary structures for the molecules. A physical model for the interaction between subunits is proposed according to which there is at most one specific point of interaction between neighbouring subunits. All other forces are unspecific and symmetrically distributed around the subunit. Structures are proposed for the cl-unit of glutamic dehydrogenase and the pyruvate and a-ketoglutarate decarboxylation complexes consistent with the proposed model and with the assumption that they have icosahedral symmetry. The dissociations, molecular weights, and compositions of these entities can be explained on this basis. It is shown that if virus shells having cubic symmetry are built according to the principles suggested here they will always have icosahedral or pseudo-icosahedral symmetry, and be based on the P = 1 or P = 3 icosadeltahedra, in aggreement with experimental findings to date. Assuming that the active site of a dehydrogenase spans two subunits, and using a slightly modified version of the model already suggested, the number of active sites per enzyme molecule can be predicted. The predictions agree fairly well with experimental evidence to date. Some aspects of in vitro complementation of mutant forms of dehydrogenases are discussed.

Introduction An understanding of the quaternary structure of proteins is undoubtedly an important part of the understanding of biological architecture generally. The purpose of the present paper is to present a new and detailed theory of quaternary structure, based mainly on the results of indirect physical studies of dehydrogenases. It has been possible to evolve a theory for this 7 Present address: Institut National Agronomique, France. 276

16, Rue Claude-Bernard,

Paris, 5,

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particular class of protein because of the large amount of experimental material available. However, many of the considerations discussed in this article may well be of wider application. The Subunits of Dehydrogenases

Table 1 lists the published molecular weights of enzymes, the criterion for whose inclusion in that table is that they catalyse the equilibration of NAD+ and NADH or NADP+ and NADPH with other substances. Inspection of this list, part of which is presented as a “spectrum” in Fig. 1

FIG. 1. Molecular weights of dehydrogenases up to 160,000. Abcissa represents mol. wt x 10w3. The height of vertical lines correspond to the number of enzymes observed with the given mol. wt (1 or 2). Broken lines represent subunits which have been obtained from some of the enzymes. TABLE

Name of enzyme

Source

Mol. wt

Malate Ox heart dehydrogenase Pig heart Malate dehydrogenase

15,00020,000 40,000 3 5,000

o-Mannitol1-phosphate dehydrogenase NADHcytochrome b, reductase NAWP) transhydrogenase Glutathione reductase “S” malate dehydrogenase

Aerobacter aevogenes

40,000

Calf liver microsomes

38,400

Spinach leaves

36,000

Rat liver

44,000

Beef heart

52,000

1

Mol. wt Binding I knc& sites/ mole subunits

Presumed number of subunits 1

References

Davies & Kun, 1957 Wolfe & Nielands, 1956; Pfleiderer & Hohnholtz, 1959 Liss, Horwitz & Kaplan, 1962

1

2

1 (FAD)

2

Strittmatter & Velick, 1958

2

Kleister, San Pietro & Stolzenbach, 1960 Mizo, Thompson & Langdan, 1962 Englard & Breiger, 1962

1 (GSH)

2 3?

278

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P.

TABLE

Name of enzyme

WHlTEHEAD 1 (continued)

Mol. wt Bmding knifm sites/ mole subunits

Presumed number of subunits

Source

Mol. wt

Pig heart

56,ooo

Beef heart

62,000

Pig heart

60,000

Germinating pe= Yeast

60,OCO

1 (FAD)

3

60,000X

X (FAD)

3x

Horse liver

73,00084,000

2

4

Rabbit Glycerolmuscle phosphate dehydrogenase

78,000

l-3

4

Yeast NADPHcytochrome c reductase Lactate Beef heart dehydrogenase “fraction A” NADH dehydrogenase

75,ooo

4

Theorell & Bonnisthen, 1951; Ehrenberg, 1957 van Eys, Nuenke & Patterson, 1958, 1959; Ankel, Bucher & Czok, 1959 Haas, 1944

72,ooO

4

Millar,

4

Glyceraldehyde phosphate dehydrogenase Diiydroerotic dehydrogenase NADPH, (acceptor) oxidoreductase Glyceraldehyde phosphate dehydrogenase

Mammalian muscles

76,000

Rao, Felton, Huennekens & Mackler, 1963 Elodi, 1958

Zymobacterium oroticum Yeast

75$00x

Malate dehydrogenase “M’ malate dehydrogenase Isocitrate dehydrogenase (decarboxylating) Glutathione reductase NADPH (acceptor) oxidoreductase Alcohol dehydrogenase

Glyceraldehyde phosphate dehydrogenase

Yeast

?

80,ooo90,ooo

Refere ----

Siegal & Englard, 1961 Siegal & Englard, 1961 Siebert, Dubuc, Warner & Plant, 1957

1 (FMN)

4

Mapson & Isherwood, 1963 Haas, 1955

1962

X(FAD)

4x

Friedmann & Vennesland, 1958

100,000105,OOtl

2 (FMN)

6?

Theorell & Akeson, 1956

120,000122,000

2

6

Taylor, 1951; Taylor & Lowry, 1956; Stockel, 1958; Velick, 1953 Elias, Garbe & Lamprecht, 1960

117,000

QUATERNARY

STRUCTURE TABLE

Name of enzyme

Source

A. aerogenes Sorbitol-6phosphate dehydrogenase Dihydrolipoic Eschericiu coli dehydrogenase pyruvate decarboxylating complex Pig heart Dihvdrolivoic dehydrogenase ketoglutarate decarboxylating complex NADHBeef heart cytochrome c reductase L-Lactate Beef heart dehydrogenase

Mol. wt

Glutamate Chicken dehydrogenase liver

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DEHYDROGENASES

1 (continued) Mol. wt Bindin Presumed f4 number sites/ k&n of mole subunits subunits

112,000~

References

.-____

6

Liss et al., 1962

112,000

2 (FAD)

6

Koike, Reed & Carroll, 1963

114.000 ’

41.000- 2 (FAD) 4.41000

6

Massey, Hofmann & Palmer, 1962

6

King & Howard, 1962

8

Nielands, 1952; Appella & Makert, 1961 Makert & Appella, 1961 Nisselbaum & Bodansky, 1961 Hayes & Velick, 1954; Bonnischen, 1953; Kagi & Vallee, 1960; Hersh, 1962 Fox & Dandliker, 1958

1 lO,OOO120,000 135,000

L-Lactate Beef heart 134,000 dehydrogenase (2 isozymes) L-Lactate Human 140,000 dehydrogenase heart i 4,000 Alcohol Yeast 140,000150,000 dehydrogenase

Glyceraldehyde Rabbit muscle phosphate dehydrogenase Glyceraldehyde Mammalian phosphate muscles dehydrogenase Glutamate Beef heart dehydrogenase

OF

34,000 + 2,000

8 8 36,000

138,000

4

8

3

8

143,000 250,000

430,000 It 2,000, 500,000

8 43,000, 30,00060,000

4

12

20

Elodi, 1958 Olson & Anfisen, 1952; Jirgensons, 1961; Frieden, 1962~ ; Ton&ins, Yielding & Curran, 1962 Frieden, 19626; Rogers, Geiger, Thompson & Hellerman, 1963

t Calculated from its sedimentation constant assuming the molecule to have the same frictional characteristics as D-mannitol-l-phosphate dehydrogenase.

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leaves little doubt that there is a definite clustering of molecular weights around certain values. Furthermore, the molecular weights are given by a formula : Mol. wt = nxM where IZ is an integer and M is a number lying in the range 17,000 to 21,500. Most of the estimates of molecular weight shown in Table 1 have been obtained from ultra-centrifugal measurements. A few of the values listed there have been obtained solely by measurements of flavin coenzyme contents. It is not suggested that the latter need be taken too seriously as evidence for the above formula (although they do agree well with it) and these values are not included in the “spectrum” of Fig. 1. In many cases, however, estimates of content of bound flavin or pyridine nucleotide coenzymes have given values of minimum molecular weights which are either the same as or exact divisors of the estimates obtained from physical measurements. These divisors are given in column 5 of Table 1. We may, therefore, summarize by saying that the combining weight of dehydrogenases C, is also given by a formula: C=n’xM

where n’ is a small integer. These facts suggest the theory that all dehydrogenases are made up of closely similar basic units, the units having a molecular weight which does not lie outside the range 17,000 to 21,500. (In subsequent discussion these hypothetical units will be termed “monomers”. The terminology of Fisher (1963) is not found very useful for the ideas presented here.) The higher molecular weights might be thought not to be useful as evidence for this theory since given the latitude allowed for possible values of&f, any molecular weight above 120,000 would satisfy the formula mol. wt = n x M. However, in three of these cases, the subunits satisfying the formula mol. wt = 2 x M have been obtained as a result of various treatments of the enzyme. Although subunits have not been obtained in any of the remaining examples, many of them are closely related to the enzymes for which subunits have been observed. The idea suggested here is by no means without precursors. Svedberg & Pedersen (1940) suggested that globular proteins occurred in discrete molecular weight classes. The mean molecular weight of each class was supposed to be a multiple of 17,600. However, as more information accumulated this empirical law appeared to become untenable (Bull, 1940). It has, however, recently been revived in a modified form by Wright (1962) who suggests that the molecular weights of enzyme fall into three geometric series whose first numbers are 12,000, 16,000 and 19,000. The series noted

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here for dehydrogenases does not, however, correspond to any of Wright’s series. The present author would prefer to suspend judgment on the validity of Wright’s treatment and would be prepared to deny significance to his geometric series;* nevertheless in looking for this kind of law attention is drawn to a significant fact: “the numbers were very unlikely to be of a random distribution, for not only were there unexpectedly large groups of enzymes with similar molecular weights, but also large spaces completely devoid of enzymes” (Wright, 1962). In the author’s opinion, the significant series will be seen most clearly if attention is first of all concentrated on establishing them for classes of proteins known to be related -such as the class of dehydrogenases. There is probably no such class for which as many measurements are available as for the dehydrogenases. However, a cursory inspection of data available for the kinases is enough to suggest a case for a subunit structure similar to that in the dehydrogenases (see Table 2). The likely significance of the large molecular weights will become apparent after discussion. TABLE

Enzyme

2

Source

Mol. wt

Arginine kinase

Lobster muscle

21.000

Myokinase Galactokinase

Rabbit muscle E. coli

21,000 20,000 (awox.) 43,000 81,000 120.000

Arginine kinase Creatine kinase 6 phosphofructokinase Pyruvate kinase

Crab muscle Rabbit muscle -

Glycerol kinase

Candida mycoderma

Rabbit muscle

230,000270,000 251,000

Reference Virden & Watts, personal communication Noda & Kuby, 1962 Sherman & Adler. 1963 Elodi & Szorenyi, 1956 Noda, Kuby & Lardy, 1954 Melnikova & Neifka, 1954 Warner, 195X; Morawiecki,

1958

Bergmeyer, Holtz, Kauder, Mollering & Wieland, 1958

To the author’s knowledge, direct evidence from published data of the physical existence of the postulated monomeric subunits exists in only two cases to date. The first such evidence comes from studies of dihydro-lipoic dehydrogenase. If a protein contains n subunits which are identical in their primary structure, then the number of different peptides obtained from a tryptic digest of the protein should be equal to the number of arginine and lysine residues in the protein divided by II. For hydrolipoic dehydrogenase tryptic digests indicate that II = 2 (Massey, Hofmann & Palmer, 1962).

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However, this enzyme can be split into subunits of molecular weight approximately 41,000 to 44,000 and if the molecular weight of the intact enzyme is 114,000 (Massey et al., 1962), it probably contains three of these units. It follows that at least one, and very plausibly all of the latter units must be regarded as containing two identical peptide sequences, two subunits of structure. Preliminary data reported by Fincham & Coddington (1963) suggests that the ultimate subunits of glutamic dehydrogenase (of Neurospora crassa) are at least significantly smaller than the subunits of the beef liver enzyme which are observed ultracentrifugally, so that the latter may not be the ultimate subunits of this enzyme. The true invariant underlying the approximate constancy of molecular weight of the postulated subunits may be the number of amino acid residues in the peptide chain of the subunit, in turn determined by the gene length. Assuming the molecular weight of rat liver lactic dehydrogenase, yeast alcohol dehydrogenase, glyceraldehyde phosphate dehydrogenase and beef heart “fraction A” lactic dehydrogenase to be 135,000, 145,000, 138,000 and 72,000, respectively, and the number of subunits to be 8, 8, 8 and 4, then the published amino acid analyses (Wieland & Pfleiderer, 1961; Wallenfels & Sund, 1959; Velick & Ronzioni, 1948; Millar, 1962) would give 149, 151, 161 and 159 as the number of residues per subunit. The differences between these numbers are certainly less than the uncertainty in their estimation. The notion that the number of amino acid residues in the peptide chain is invariant or nearly invariant is consistent with the ideas of Sorm (1962) on “permissible interchanges”. In view of accumulating evidence it seems likely to the author that all large protein molecules are composed of relatively small subunits. It may yet prove that such factors as the size of “messenger RNA” or that of the ribosome restrict the cell to manufacturing chains of a few surprisingly small pre-set lengths. The Geometry of the Quaternary Structure (A)

SMALL

ENZYMES

Examination of the molecular weight c‘spectrum” of Fig. 1 reveals that the number of groupings of molecules of similar weights is fewer than might be expected. In other words, in the formula mol. wt = 12x M, not all values of n are permissible. The permissible values of IZ can be accounted for theoretically. The data suggests that the permissible values of n are 1, 2, 3, 4, 6, 8 12, and a questionable value which we shall assume for the moment to be 20. The last five numbers are equal to the number of vertices of the regular polyhedra (platonic solids). This has led the author to suppose that the molecules consist of subunits arranged in structures where each monomeric

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subunit is at the vertex of a regular figure-the figures in question are the regular polyhedra for the last five figures; for the example where n = 3 the figure is an equilateral triangle; for y1= 2 a straight line. The postulated arrangements are illustrated in Fig. 2.

n=8

n= I2

@@

FIG.

2. Proposed quaternary structures of dehydrogenases for values of n from 2 to 20.

The necessity for these arrangements simple postulates : (1) each subunit is identical, subunits are concerned :

would follow from a very few

in so far as the binding forces between

284

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(2) there are no speczjic sites on any subunit for binding other subunits. The binding potential of each subunit has cylindrical symmetry. The binding forces are all directed towards the same hemisphere; (3) the attractive forces are such that each subunit seeks to surround itself with the maximum possible number of other subunits. In the explanation of these postulates which follows, the physical symmetry of the subunits will be referred to as “pseudo-symmetry” and likewise the symmetry of a structure when the subunits are considered as structureless objects with cylindrical symmetry will be referred to as the “ pseudosymmetry” of the structure. Each subunit, being identical, will exist in an identical environment if possible. Postulate 2 allows that this is possible. The only structure consistent with these postulates are those whose pseudo-symmetries are the symmetries of regular figures. The axes of cylindrical pseudo-symmetry of the units will pass through the centre of the figures. It is necessary to postulate cylindrical symmetry of binding, if we are concerned to maintain the belief that monomeric subunits in different dehydrogenases are basically similar. In the different figures proposed the units may have threefold, fourfold or fivefold pseudo-symmetries. But these symmetries are mutually incompatible unless we suppose that there is a 60-fold pseudo-symmetry which is intrinsically unlikely, and is practically the same thing as cylindrical symmetry anyway. If we are to maintain that all subunits are similar we can hardly assume that different ones contain different numbers of specific binding sites with incompatible pseudo-symmetries. We are compelled therefore to adopt the idea that the binding potential has cylindrical symmetry. Postulate 3 is introduced because three-dimensional bodies obeying postulates 1 and 2 can be arranged with each unit in an identical environment in an infinite number of ways in structures whose pseudo-symmetries are the symmetries of regular polygons-such as that of Fig. 3 with five subunits. This would be consistent with all values of n. We have maintained that no n other than those proposed above can occur and are therefore obliged to add a postulate which will exclude the regular polygons other than the example where n = 3. To see this more clearly consider the forces which act in the mutual binding of bodies which we will assume for the purpose of argument to be prolate ellipsoids (see Fig. 4). Let us suppose the unspecific forces of attraction exist between the ends X of the units and unspecific forces, probably but not necessarily of repulsion, between the ends Y. Which of the possible associations would be more stable depends on the balance between the two types of force. For instance, increasing strength of repulsive forces between the

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285

3, Imaginary “polygonal” arrangement of five subunits, It is proposed that arrangements like this do not exist except when n = 3.

FIG. 4. Illustration

of suggested physical forces between two monomeric subunits.

Y ends of the units should tend to favour association of low n. The repulsive forces may be supposed to set a maximum on the angle 4 in any given case. For a given “polygonal” arrangement, q5takes a defined value of 360”/n = I$~ say, where n is the number of subunits. But for any & > 60” there exists a polyhedral arrangement for which 4 < +1. The point is that in a polygonal arrangement, a unit is in contact with only two other subunits. But if the repulsive forces would permit such an arrangement the above theorem shows that they will also permit a polyhedral structure in which each unit is in contact with more than two other units. The attractive forces would therefore ensure that the latter structure be formed. Conveniently, however, the above argument does not hold when 4 = 60” which corresponds to iz = 3, a triangular arrangement, but also to the tetrahedral arrangement T.B. 19

286

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fl = 4. Thus our postulates are sufficient to account for the restriction of n to the quoted values. It seems likely that in the associations where n = 1, 2, 3, repulsive forces between the Y ends of the subunits are relatively large. They are largest of all where n = 1, too large to allow an association to form. Where n = 2, they are not so strong, but strong enough to stop 4 exceeding 60”. Where n = 3, (b = 60” which is exactly the same value for 4 as in the tetrahedral arrangement n = 4. In the latter arrangement, however, the Y ends are closer together than in the triangular structure. Presumably which arrangement is favoured in any particular case, depends on whether or not the gain in potential energy of Y-repulsion due to the closer approximation of the Y ends in the tetrahedral structure is more than the 3/2-fold greater loss due to the extra attraction of the X-ends. Presumably also, the reagents which cause splitting of dehydrogenases into subunits which must be supposed to contain two monomeric units (Kagi & Vallee, 1960; Hersh, 1962; Jirgensons, 1961; Freiden, 1962; Appella & Makert, 1961; Massey et al., 1962) (see also Table 1) in some way decrease the attractive forces of binding relative to the repulsive forces. In the light of the preceding discussion, we can see that it is possible that one of the units in contact with a given unit is bound at a specific point. Subunits can still be disposed at regular intervals around a given subunit if, so to speak, a starting point is fixed. Concerning the postulate of the identity of the units, it is in fact clear that they are in most cases not strictly identical. If we assume that an active site of a dehydrogenase is borne by one unit it then follows that the units are not identical because there are in many cases fewer active sites than subunits, hence some of them differ. from others in not bearing active sites. This argument is, however, not conclusive since factors of orientation of the subunits to be discussed below may play an important role here. More direct evidence on the primary structure of dehydrogenases is obtained from tryptic digest work as explained previously. In lactic dehydrogenase and several glyceraldehyde phosphate dehydrogenases peptide sequences are repeated four times in the molecule (Cahn, Kaplan, Levine & Zwilling, 1962; Harris & Perham, 1963), and in dihydrolipoic dehydrogenase twice (Massey et al., 1962). We therefore have to conclude that there are three types of subunits in the last-mentioned enzyme and two in the other cases. So long as we maintain, however, that the molecular weights of these units are approximately the same, and that they are identical in their binding function the theory is not affected. It is not suggested that the theory proposed here is the only possible explanation of how subunits could come to be arranged in the pro-

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posed arrangements. It is, however, certainly one of the simplest possible. The molecular weight of chicken liver glutamic dehydrogenase is unfortunately not well enough established to enable one to say with certainty that it should contain 20 subunits. Assuming that the molecular weight of a dimeric subunit of this enzyme is the same as that obtained for bovine liver glutamic dehydrogenase, 43,000 (145) the value of 430,000 (Freiden, 1962) for the molecular weight of the chicken liver enzyme agrees rather well with a value of 20 for ~1, but the estimate of 500,000 4 30,000 (Rogers, Geiger, Thompson & Hellerman, 1963) does not agree so well. The only likely structures for this enzyme other than a “dodecahedral” structure with 20 subunits, are the one with 24 subunits and octahedral (432) symmetry, or one with twelve of each two different types and tetrahedral (23) symmetry. The latter structures are more in harmony with the conception of proteins as assymmetric units with rather specific binding points. The author believes that the structure of this enzyme is the “dodecahedral” one. The recent electron-microscopic observations of Horne & Greville (1963) of the p-unit of glutamic dehydrogenase by no means disagree with the present theory, although those authors suggest a different interpretation. They found that the majority of particles in the electron micrograph appeared to have a threefold symmetry, and roughly the shape of an equilateral triangle. An association of 12 subunits having the “icosahedral” structure suggested here would naturally rest on a surface with three particles touching the surface and an axis of threefold symmetry or pseudo-symmetry pointing vertically upwards. In this position the negative staining technique could give the sort of image published by Horne and Greville. Three roundish subunits are in fact visible in some of the pictures. The concept of the attractive forces between protein subunits as a uniform unspecific attraction rather than a highly specific interaction between particular groups in an assymmetric structure gains support from a recent X-ray diffraction study of apoferritin (Harrison, 1963). This molecule with 20 subunits has the “dodecahedral” structure predicted here for chicken liver glutamic dehydrogenase in which each subunit is surrounded by three others, symmetrically spaced. Although Harrison explains this by assuming similarly spaced specific areas of attraction on the subunit, it is clear that the structure could also be explained by a theory of the present type. Large Molecular

Weight Dehydrogenases Complexes

and Dehydrogenating

In our discussion so far we have attempted to give an account of dehydrogenases with molecular weights up to about 500,000. The molecular weight 19-a

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of bovine glutamic dehydrogenase in this account was taken as about 250,000 (n = 12) since a particle with this molecular weight is enzymatically active. However, the molecular weight of this enzyme is frequently observed to be “about 1,000,000” and there is an equilibrium between the two forms which is dependent on conditions of pH, etc. (Olsen & Anfisen, 1952; Freiden, 1958; Kubo, Iwatsabo, Watari & Scyama, 1959; Wolff, 1962). The two types of particle probably have identical enzymic activities (Fisher, Cross & Macgregor, 1962a). Under different conditions the enzyme can be dissociated into units of molecular weight approximately 43,000 (Jirgensons, 1961). The tendency has been to interpret these phenomena in terms of the concept of “levels of structural organization” (Fisher et al., 1962a; Fisher, MacGregor & Power, 19623). According to this interpretation, a number of small subunits would come together by some kind of force to form a unit of molecular weight 250,000 (p-unit). The b-units can themselves associate, by means of a different kind of force, to form a larger unit (the a-unit) in which the internal structure of the p-units remains the same. In what follows, a different theory, which leads to a certain further unification, will be proposed. The structures which have already been discussed exhaust the possibilities whereby identical particles may be surrounded by y particles and have y-fold rotational pseudo-symmetry. If larger associations are to be formed then the environment of each particle must be less symmetrical. Supposing the particles themselves or their environments have no symmetry, their packing should obey the principles laid down by Crick & Watson (1956) according to which n may be a multiple of 12, in an association with tetrahedral (23) symmetry, 24 in one with octahedral (432) symmetry or 60 in one with icosahedral (532) symmetry. The molecular weight of the a-unit of bovine glutamic dehydrogenase is certainly consistent with n = 60. It is therefore suggested that it has icosahedral symmetry. The a-unit has generally been assumed to contain four p-units (Frieden, 1962). The formulation n = 60 demands that it be composed from five p-units. In view of this, the announcement that the a-unit does indeed dissociate into five units (Fisher et al., 19623) is of great interest. So too is the report (Fisher et al., 1962b) that under other conditions the a-unit splits into two units (n = 30). Dissociations of the a-unit into five units, and also into heavier units clearly cannot be explained by a “levels of organization” idea. These dissociations would also be difficult to explain on the basis of an a-unit with tetrahedral or octahedral symmetry. It may at first sight seem curious that a particle with cylindrical symmetry should exist in an environment without rotational symmetry as the subunits of a protein with true icosahedral symmetry must do. It should be realized, however, that they can be placed in an environment which is quite similar

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to one with threefold symmetry if they are placed as if with their axes of physical symmetry one-third of the way along each edge of a regular icosahedron (see Fig. 5). This structure is a reasonable one even if there is more

FIG. 5. Suggested structure for cc-unit of glutamic dehydrogenase viewed along a threefold axis. Each subunit makes “equivalent” contacts with two of its three neighbours.

than one type of subunit, in which case it would not possess true icosahedral symmetry. (In the remainder of this section all subunits of a given enzyme will be treated as if they were identical for convenience of discussion. The falsity of this assumption does not affect the substance of the discussion.) On the basis of the assumption of cylindrical symmetry of subunits and icosahedral symmetry of the a-unit, dissociations of the a-unit into 45 or cq’3 units are easily conceivable. The a + fl dissociation could take place and the new structure formed without any loss of contact between monomeric subunits of each new particle. An 42 unit is difficult to accept on this basis; it would have to be a structure with pseudo 532 symmetry in which each subunit was situated as if at the mid-points of the edges of a regular icosahedron and surrounded by four other particles, but with twofold rotation pseudo-symmetry. Glutamic dehydrogenase is the largest ‘csimple” dehydrogenase to have been studied. However, the particles which catalyse the oxidative decarboxylation of the a-keto acids, pyruvate and a-ketoglutarate a-ketoacid + CoA-SH + NAD’ + acylCoA + CO, + NADH + HC (1) are of great interest in connexion with the present structural hypotheses. These “enzyme complexes” have been isolated from animal sources and from Escherichia coli (Jagannathan & Schweet, 1952; Schweet, Katchmann, Both

290

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& Jagannathan, 1953 ; Koike, Reed & Carroll, 1960). It has been presumed that there are as many as four different enzymes in the complex catalysing the reactions (Reed, 1960) (brackets denote tightly bound species). pyruvate f (thiamine-P-P) (acetaldehyde-TPP) (acetyl-S-lip-SH)

-+

(acetaldehyde-TPP)

+ (lip S,)t --t (acetyl-S-lip-SH) + CoA-SH -

+ CO2

(2)

+ (TPP)

(3)

acetyl-S-CoA + (lip(SH),)

(lip(SH),) + NAD + Gi5 (lip S,) + NADH

+ H’

(4) (5)

and a corresponding series of reactions in the ketoglutarate complexes. The E. coli pyruvate complex has been resolved into three components (Koike, Reed & Carroll, 1963), a “carboxylase” which catalyses reaction 2, a “lipoic reductase-transacetylase” (LRT) which catalyses reactions 3 and 4, and a dihydrolipoic dehydrogenase (DHLD) which catalyses reaction 5. The pig heart a-ketoglutarate complex has been partially resolved (Massey, 1960). The complexes can be reconstituted under appropriate conditions simply by mixing the separated fractions. The lipoic reductase-transacetylase of the E. coli pyruvic complex can combine with either the pyruvic carboxylase or the dihydrolipoic dehydrogenase or with both, but the carboxylase and the dehydrogenase cannot combine with each other in the absence of lipoic reductase-transacetylase (Koike et al., 1963). The reconstitution experiments of Massey which show that the cc-ketoglutarate complex contains approximately equal weights of the three types of enzyme together with the fact that this complex contains equimolar amounts of bound TPP, lipoic acid and FAD are strong indications that it contains equal numbers of subunits of each type of enzyme, the subunits of each enzyme being approximately the same molecular weight. A complex of icosahedral symmetry containing three different types of assymmetric units must have 60 of each type of unit. Thus the assumption of icosahedral symmetry for this complex would enable us to explain both the proportion of the different enzymes present and (as will be shown presently) its sedimentation constant. We do not have any independent evidence of the molecular weight of the carboxylase subunits, but on the assumption that one mole of lipoic acid is bound per subunit, the molecular weight of the lipoic reductase-transacetylase subunit should be 20,000 to 30,000 (Koike et al., 1963). What account can we now give of the pyruvate complexes ? There is no evidence for more than three different types of enzyme in the complex. The t Abbreviations used: acetyl-lip-SH, S-acetyldihydrolipoic acid; DHLD, dihydrolipoic dehydrogenase; lip Sz, lipoic acid; lip SHz, dihydrolipoic acid; LRT, lipoic reductase transacetylase; TPP, thiamine pyrophosphate.

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maximum number of x different types of particle that can associate in a complex with icosahedral symmetry if each particle of a kind is to be related by symmetry is 60x. Yet the molecular weights of the pyruvate complexes are at least twice those of the cc-ketoglutarate complexes. Consequently not all of the subunits of a given kind can be related by icosahedral symmetry. To explain this we must invoke the theory of “quasi-equivalence” which was proposed by Caspar & Klug (1962) to explain how virus shells could be constructed out of more than 60 units. Caspar and Klug note that multiples of 60 identical assymmetric units may be assembled, such that non-symmetry related units are in near-identical environments, in a shell of icosahedral symmetry. As they remark: “Molecular structures are not built to conform to exact mathematical concepts but, rather to satisfy the condition that the system be in a minimum energy configuration.” The theory of quasi-equivalence enables us to account for the fact that the ratio of bound lipoic acid to FAD which is 1 : 1 in the a-ketoglutarate complex (Koike et al., 1960; Massey, 1960) is 3 : 1 in the pyruvate complex (Koike et al., 1960). As Caspar & Klug (1962) explain, of the polyhedral surfaces (with symmetrical dispositions of vertices) into which it is possible to fold a plane net, still maintaining the nearest neighbour pattern, that which involves the least strain is the folding of a triangular net into a polyhedron with icosahedral symmetry. The polyhedra whose faces are all equilateral triangles and which have icosahedral symmetry (“icosadeltahedra”) have 20T facets given by T = Pf 2, where f is any integer and P can only take certain values 1, 3, 7, 13, 19 . . . (of which only the values 1 and 3 have been shown to occur in icosahedral viruses). T = 3 and T = 7 icosadeltahedra are drawn in Fig. 6. The number of quasi-equivalent subunits is 6OT. Thus, if a complex contains 60 units of a given kind the next highest number of units of that kind that a complex can contain is 60 x 3 = 180. What evidently happens in the cases under study here is that in pyruvate decarboxylation complex the number of LRT units as compared to that in the ketoglutarate complex is increased by the minimum factor 3, but the number of dihydrolipoic dehydrogenase units rem.ains the same. The number of carboxylase particles are probably also increased threefold in the pyruvate complex, since in the reconstitution experiments of Koike et al. (1963), carboxylase, LRT and DHLD combine in the ratio of approximately 3 : 3 : 1 (by weight). It is to be noted that 180 particles can be in quasi-equivalent environments, only in respect of their interactions with each other and with 180 particles of a different kind-they cannot in general be in quasi-equivalent interactions with only 60 particles of another kind. Quasi-equivalence would be attained if each dihydrolipoic dehydrogenase particle is symmetrically surrounded by three LRT particles and also symmetrically surrounded by three car-

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(a)

(b)

FIG.

6. (a) 7’ = 3 icosadeltahedron.

(b) and (c) Two mirror-image icosadeltahedron.

forms of the T = 7

boxylase particles and the DHLD particles themselves have, physically speaking, cylindrical symmetry. It is therefore predicted that the DHLD particles will be found in the complex at the centres of symmetry of the triangles of a T = 3 icosadeltahedron, i.e. one-third of the way along each icosahedral edge (the same positions postulated for the axes of subunits of glutamic dehydrogenase previously). The sedimentation constant of the DHLD-LRT complex is consistent with its containing 60 LRT particles and 20 DHLD particles. In this structure the DHLD particles would have to be placed upon threefold icosahedral axes and the LRT particles disposed about them with threefold symmetry. Thus, the LRT particles could surround the DHLD particles in the same way in the LRT-DHLD complex as they are postulated to do in the entire pyruvate complex. Given the local symmetries of the whole complex we need only to suppose that the carboxylase particles in the complex are at a greater distance than are the LRT particles from the DHLD particles, to embody the facts that LRT can complex with either carboxylase or DHLD, but carboxylase cannot complex with DHLD in the absence of LRT. For we can then only envisage structures in which there is no contact between DHLD and carboxylase particles. The same supposition also enables us to explain how, when carboxylase is removed from the complex it decomposes into a smaller particle of 26.8 S but when

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DHLD is removed a large (49.7 S) particle remains (Koike et al., 1963). The latter operation can be performed and a new stable structure formed by a slight contraction of the structure to one in which each LRT subunit would make contact with three others. Removal of carboxylase, however, would give rise to an obviously unstable structure which could not be stabilized in this way but which could dissociate to re-form a smaller stable structure. Reference to the possible structure of the complex illustrated in Fig. 7 will clarify these points. Specification of the symmetry and the local symmetries of the complexes does not, of course, define the structures. However, the author finds the following argument, using which tentative structures can be proposed, attractive. It seems unlikely that the forces operating between the different types of subunit will be different in kind from those operating between the dehydrogenase subunits. If the forces are thus unspecific and if the particles are all of about the same size and weight it is not unreasonable that their positions should all be related quasi-equivalently, approximately at least. We notice that the total number of particles in the a-ketoglutarate decarboxylation complex is 3 x 60 and in the pyruvate decarboxylation complex 7 x 60, which are values permitted by the principle of quasi-equivalence. Now if all the particles have a physical cylindrical symmetry, they will all be situated one-third of the way along the edges of the triangles of an icosadeltahedron. One set of 60 of these positions in the T = 7 deltahedron coincides with the positions already postulated for the DHLD particles in the pyruvate complex. The structure of this complex would then be that of Fig. 7, or its mirror image (see below) which is entirely consistent with all of our discussion above. When DHLD is removed from the complex the particles cannot be related quasi-equivalently, though each particle of a given enzyme can be so related. The requirement that all particles be quasi-equivalently related cannot be as exacting as the one that all particles of a given kind be quasi-equivalently related. The electron microscopic appearance of the complex will no doubt depend on its precise structure. If the structure is based on the T = 7 icosadeltahedron as proposed and, as is reasonable, subunits cluster into hexamers and pentamers the total number of “morphological units” (i.e. distinct clusters distinguishable by electron microscopy at low resolution) would be 72, presenting the appearance of Fig. 8 (or its mirror image; P 3 7 icosadeltahedra have the property of existing in two forms which are mirror images of each other (see Fig. 6). The skewness of the complex would in principle be apparent from the arrangement of “morphological units” if the clustering is as described above. It would not be surprising, from the point of view of the present theory, if both mirror-image

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FIG. 7. Schematic illustration of proposed structure of pyruvate decarboxylating complex. Black spheres represent dehydrogenase subunits, shaded spheres carboxylase subunits and white spheres lipoic-reductase transacetylase subunits. The structure here is viewed along a threefold axis. Fivefold axes pass through the holes formed by five carboxylase particles.

FIG. 8. Predicted appearance of pyruvate decarboxylation complex if clustering of subunits into pentamers and hexamers occurs. Note that the skewness of the complex is observable in this picture.

forms were found-whereas this would be unlikely if there are highly specific mutual binding sites on assymmetric subunits.) Other types of clustering than into pentamers and hexamers are also conceivable, for instance clustering of carboxylase subunits into hexamers and pentamers and of dehydrogenase and LRT particles together into tetramers. For a discussion of the clustering of subunits in icosahedral shells see Caspar & Klug (1962).

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By a similar argument to that given for the pyruvate complex, the structure of the a-ketoglutarate complex should be that of Fig. 9. On removal of DHLD from the complex an intact carboxylase-LRT complex appears to remain (Massey, 1960) so that DHLD must correspond either to the subunits shown in black in Fig. 9 or those shown cross-hatched, more probably the former.

FIG. 9. Proposed structure of or-ketoglutarate decarboxylation complex. White spheres represent lipoic reductase transacetylase subunits; it is probable that the black particles are dehydrogenase subunits and the shaded ones carboxylase subunits. (See text for explanation.) The structure is here viewed down a threefold axis. Fivefold axes pass through the holes formed by five black particles. Twofold axes pass through each of the points of contact of the shaded spheres with one another.

It may be anticipated that removal of one enzyme from the structure (shown in black) would have an intact complex of 60 particles of each of the other two enzymes, that removal of another (shown cross-hatched) would probably cause breakdown of the rest of the structure into smaller units which might be 1 : 1 complexes of the remaining two enzymes, and that no complexes could be formed in the absence of the third enzyme (shown in white). The most likely arrangement is that the black particles are DHLD, the white LRT and the cross-hatched, carboxylase. With clustering into hexamers and pentamers 32 morphological units would be observable in this complex. There remains one set of facts which is not at lirst sight consistent with our account of the complexes. The estimated molecular weight of the ketoglutarate complexes are: 2 x lo6 for pig heart complex (Sanadi, Littlefield & Bock, 1952), 2.4x lo6 for the E. coli complex (Koike et al., 1960) and of the pyruvate complexes, 4 x lo6 for the pigeon breast muscle (Schweet et al., 1953) and 4.8 x IO6 for the E. coli complex (Koike et al., 1960). Particles of these weights are too small to contain 180 and 420 particles with molecular

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weights in the region of 20,000. However, if the particles are essentially hollow shells, as they are in the above account, the molecular weights deduced from their hydrodynamic behaviour would be under-estimates. That the postulated numbers of subunits are reasonable may be seen from the following calculation. The sedimentation constants of thin shells should be proportional to the square root of their masses (as opposed to the two-thirds power of their masses for solid homogeneous bodies). The sedimentation constant for LRT is 21.2 S (Koike et al., 1963) and a molecular weight of about 1.6 x lo6 is assigned to it. This particle we would assume to contain 60 subunits. Then, a hollow particle containing N times as many subunits should have a sedimentation constant approximately 21.2 x N” S. Sedimentation constants expected on this basis are compared with experimental values in Table 3. What looks at first like a difficulty turns out to be, if anything, confirmation of our theory. TABLE 3 Species cr-ketoglutarate decarboxylation particle Pyrnvate decarboxylation particle LRT-carboxylase complex LRT-DHLD complex

Theoretical sedimentation constant 21.1

Experimental sedimentation constant

3% = 36.7 S 37.0 S (Koike et al., 1960) 37 S (mean) (Sanadi et nl., 1952) 21.2 X 7% = 58s 58 S (Schweet et al., 1952) 56-63.7 S (Koike et al., 1960) 21.2 x 6’h = 52 S 49.7 S (Koike et al., 1963) 21.2 x (4)x = 245 S 26.8 S (Koike et al., 1963) x

Four points are worth making about the above theory. One is that the explanations we have been able to give of the dissociation behaviour and other properties of glutamic dehydrogenase and the a-keto acid decarboxylating complexes are not possible unless the sizes of the protein subunits are approximately what we postulated in the first section of this paper. The second is that we have found it necessary to re-introduce our original concept of cylindrical symmetry of binding forces of dehydrogenase subunits in what is, perhaps, a slightly unexpected context. Third, quite simple factors such as size and shape of subunits and the magnitude of their unspecific forces may be sufficient to explain the apparent specificity manifested in the combination properties summarized above of the three constituent enzymes of the decarboxylating complexes. Fourth, and quite generally, the only ratios of two different kinds of particles present in an association with icosahedral symmetry strictly consistent with the principles outlined here

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are 1 : 1 and 1 : 3. Certain other ratios are possible given various slight relaxations of those principles. The only ratio consistent with highly specific binding sites on the subunits is 1 : 1. The Generality of the Foregoing Principles This discussion has been restricted to the dehydrogenases because of the availability of relevant physical data concerning this class of enzymes. However, there is no reason why the principles elaborated here should not apply also to other types of protein. We have already suggested that they may be true of pyruvate carboxylase and LRT. Electron microscopy of erythrocruorins (Levin, 1963) has revealed structures which probably have dihedral symmetry and which must be organized on principles different from those proposed here. On the other hand, the structure of apoferritin (Harrison, 1963) is fully consistent with these ideas. Our postulate physical cylindrical symmetry is of help in explaining some features of the construction of the protein shells of viruses. Thus although Crick & Watson (1956) pointed out the theoretical possibility that these shells could have tetrahedral or octahedral as well as icosahedral symmetry, neither of the first two symmetries have been observed in any of the ten species of virions that have been demonstrated to have cubic symmetry (Horne & Wildy, 1961; Caspar & Klug, 1962). It has been pointed out (Caspar & Klug, 1962) that quasi-equivalence can be realized with much less strain in systems with 532 symmetry than in the other two systems. However, an icosahedral (7’ = 1) virion probably containing 60 subunits all equivalently related is known (Hall, Maclean & Tessman, 1959; Tromans & Horne, 1961) and there is no reason, on these grounds, why structures containing 24x subunits of x different kinds related by 432 symmetry or 12x related by 23 symmetry should not exist. As discussed earlier it is in harmony with the ideas of this paper, but hardly likely on the hypothesis of specific binding sites, that all subunits even of different kinds in a quaternary structure should be quasi-equivalently related. This assumption, together with Caspar & Klug’s arguments, enables us to explain why tetrahedral arrangements with more than 12 or octahedral ones with more than 24 subunits are unlikely. This leaves the question of associations of 12 and 24 subunits. We have discussed how units can be arranged in certain positions in a structure with 532 symmetry such that the environment of each unit is nearly symmetrical. A corresponding structure with octahedral symmetry would be much more strained. Twelve subunits could be arranged with 23 symmetry without strain but according to our present theory the structure would in fact have pseudo-532 symmetry. A

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“core” protein of carnation mottle virus may indeed have this structure (Markham, Frey & Hills, 1963). Only icosahedral viruses based on the P = 1 and P = 3 deltahedra have been discovered and it is suspected that for some fundamental reason they are never based on P > 7 (skew) deltahedra (Caspar & Klug, 1962). If the forces between subunits are symmetrical and the units are consequently arranged each third of the way along the edges of the deltahedral facets, then it does make sense to say that, in general, the skew classes are more strained than the non-skew classes. In the non-skew deltahedra two of the three “bonds” which each pseudo-symmetrical subunit makes to others will be practically equivalent but in a skew deltahedron all three bonds will be more different from each other. Consequently a structure based on a skew icosadeltahedron will be more strained than one of comparable size based on a nonskew deltahedron. It may seem that this explanation is in contradiction with our proposal that the pyruvate decarboxylating particle might be based on a P = 7 deltahedron. However, this complex contains more than one kind of particle and the above argument does not necessarily apply when one unit makes contact with two or three different sorts of particles as do the carboxylase and LRT particles, respectively. Molecular

Symmetry and the Relation of Quaternary Structure to Enzymic Activity In our discussion of small molecular weight dehydrogenases above the structural symmetry of the molecules, i.e. the molecular symmetry when the subunits are considered as asymmetric units rather than featureless objects, was unspecified. It is a fact that in each structure with less than 20 subunits, all the subunits could permissibly be related by exact rotational symmetry: by cyclic symmetry where n = 2,3, dihedral symmetry where n = 4,6,8 and tetrahedral symmetry where n = 12. (This does not mean that the facts are best explained on the basis of specific binding sites.) However, in no case yet known is the number of substrate or coenzyme binding sites equal to the number of hypothetical subunits. This is enough to establish that the subunits are not all related by symmetry. Moreover, we have already had occasion to note that some of these enzymes contain more than one type of subunit. By qualifying the idealizations made thus far concerning the forces between subunits it is possible to offer a tentative theory which explains a number of these facts. Why should the subunits not all be related by symmetry where possible? The existence of non-related subunits would be necessary if the active site

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of the enzymes were to span two subunits. Work on “in vitro complementation” of mutant forms of dehydrogenases (Glassman & Maclean, 1962; Fincham & Coddington, 1963) and also other enzymes (Woodward, 1959; Loper, 1961; Schlesinger & Levinthal, 1963) may be interpreted as signifying that in some cases the co-operation of two subunits in a quaternary structure is necessary for enzymic activity. Suppose, for argument’s sake, that a subunit possessed two specific sites-A and B-and that in an active enzyme site A of one subunit must be in contact with site B of another. If the mutual orientation of the sites on a given subunit is arbitrary, then these two subunits in contact cannot be related by symmetry. On these assumptions alone, a dehydrogenase with an even number of subunits would be expected to contain half as many active sites as subunits-half the specific A and B sites would be unused because unable to make contact with complementary sites. But this appears not always to be the case, for instance, when n = 6 or 12, the enzymes only contain one active site per three subunits. Some additional physical restriction on symmetry is therefore necessary. A reasonable one can be suggested if we suppose that in addition to the specific binding points already postulated there are further slight deviations from cylindrical symmetry of the attractive or repulsive forces between subunits. In this case it is more or less implicit in what has already been said about these forces that a given attractive area of the surface of a subunit will prefer to make contact with the same area of another unit were possible. A general rule which meets this condition is that the molecular symmetry will be such that as many subunits as possible will be related to a neighbour by twofold symmetry. For example, in an enzyme with six subunits, where the choice is between three-symmetry and two-symmetry (see Fig. 10) the latter will be preferred. The predictions made on this basis are detailed in Table 4. It is seen that in accounting for the number of active sites (strictly speaking, binding sites) in the various enzymes (cf. Table 1) the theory has considerable, though not complete success. The numbers in column 3 of Table 4 are the number of non-symmetry related classes (whose members are related by symmetry) of subunit, i.e. the number of different environments if the subunits are identical. Since according to our present theory half or more of the specific A and B sites do not affect enzyrnic activity they could be modified without impairing the survival value of the enzyme. Hence the evolution by some mechanism of gene duplication and translocation similar to that suggested by Ingram (1961) for the haemoglobins, of as many different types of subunit as shown in column 3 would be understandable. This expectation agrees with the experimental results for some known cases where n = 6 and n = 8, discussed in the first section of this paper. However, the number of active sites per molecule is, for our theory, a more

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FIG. 10. Possible symmetries and dispositions of active sites in a hexamer. Heads and tails of arrows represent components of an active site. Components of active sites which cannot combine with a complementary component are not shown (see text for explanation). (a) and (b) Structures with twofold symmetry (two active sites). (c) Structure with threefold symmetry (three active sites). TABLE

4

No. of subunits

Predicted symmetry

No. of types of subunit

No. of active sites

2 3

None None 2 2 222 222 52

2 3 2 3 2 3 2

1 1

4

6

8 12 20

2 2 4 4 10

Structures with the predicted symmetries for 6, 8 and 12-mers are illustrated in Figs. 10, 11 and 12, respectively.

significant parameter than the number of types of subunit of distinct primary structure, since it follows more directly from the symmetry arguments. It is not impossible that in some cases all the subunits may be identical. Apoferritin, in which, since it has 20 subunits in a “dodecahedral” arrangement, the subunits cannot all be related by symmetry (and according to Harrison (1963) probably has two unrelated classes of 10 subunits related

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by 52 symmetry, in harmony with the present theory) is nevertheless composed of only one type of subunit. It may seem that we have, in this section, made a departure from the idea of unspecific binding, but in fact we have only assumed the one specific binding site per subunit which was admitted as a possibility from the start. The other deviations from cylindrical symmetry are thought of as being small and the assumption which leads to the prevalence of twofold axes is very much in harmony with the idea of unspecific forces. Some of the predictions made in this and the following section could not be made without assuming that the subunits had nearly cylindrical pseudo-symmetry. Some Further Applications of the Symmetry Theory According to our theory, if the cc-unit of glutamic dehydrogenase had true icosahedral (532) symmetry it could not have activity. The predicted molecular symmetry for this molecule is 52. This could be consistent with 10, 20 or 30 active sites. If, however, it is assumed that the orientations of the subunits are such that they approximate as nearly as is possible, consistent with specific binding at active sites, to icosahedral symmetry-a most plausible assumption considering that, in the ideal structure of Fig. 5,

FIG. 11. Structure of an octamer with 222 symmetry (four active sites). subunits

are not pseudo-symmetrically

disposed

about a given unit and

that each subunit is ideally related to one neighbour by twofold symmetrythen the expected number of active sites is 20. This is in satisfactory agreement with the findings of Fisher et al. (1962a) that the LX-and p-units of glutamic dehydrogenase have identical activities when NAD+ is used as coenzyme. According to the above considerations both contain one active site per three subunits. It is further to be noted that, since only 10 entities T.B. 20

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can be related by 52 symmetry, there must, according to this theory, be two sets of equal numbers of somewhat different binding sites in the or-unit. This also appears to be the case (Frieden, 1959a, b, 1961). Our theory predicts that all simple 3pt-mers possess one active site per three subunits, one-third of the subunits playing no direct role in the catalysis. This concept is useful in explaining some seemingly paradoxical observations on the in vitro complementation of mutant enzymes. When increasing

FIG.

12. Structure of a dodecamer with 222 symmetry (four active sites).

quantities of an extract of one xanthine-dehydrogenase less drosophila (ry) are added to a fixed quantity of a similar extract from another such mutant (ma-l), the rate of recovery of xanthine dehydrogenase’activity was proportional to the quantity of ry extract added until twice as much of it was present as of ma-l extract; beyond this no faster recovery was obtained, However, on addition of ma-l to ry extract, recovery of activity also took place until there was a twofold excess of ma-l over ry (Glassman & Mclean, 1962). These two observations can be reconciled if we suppose that the hybrid molecules formed have the formulae (AB,), and (A2B), (where A and B are the monomers of the mutant enzymes) and that both are equally active enzymically (furthermore the inactive forms of the hybrids must recover activity at an equal rate). Our present theory leads one to expect that the activity of a 3n-mer would be independent of the nature of the one-third of inactive partners. The wild type xanthine dehydrogenase is no doubt also a 3n-mer. The physical properties of glutamic dehydrogenase of Neurospora crassa (Fincham & Coddington, 1963) indicate that it is probably comparable to the p-unit of bovine glutamic dehydrogenase, thus falling into the class

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of 3n-mers. But, in contrast to the previous case, glutamic dehydrogenase activity reaches the maximum when one of the complementation partners (the am1 protein) is present in about twice the concentration of the other (the am3 protein) whichever component is held fixed (Fincham & Coddington, 1963). Here, however, it is probably significant that enzymic activity can be induced in the am3 protein by a special treatment. We might then represent the defect in the am3 protein by the kink in the “tail” of the subunit in Fig. 13 and assume that the specific “head-tail” linkage can still be formed, but gives different catalytic properties from the wild-type enzyme. Then it is obvious from Fig. 13 that if the probabilities of the am3 “head” linking with the am’ or am3 “tail” are about equal, the experimental findings

”

FIG. 13. Non-geometrical illustration of proposed scheme of complementation for glutamic dehydrogenase of N. crussa. Top line: the pure am1 and am3 proteins. Middle line: the (am3) (am1)z hybrid. Bottom line: the two (an?) (ar~P)~ hybrids. As before the components of active sites are represented by heads and tails. The kink in the tail of am8 represents a defect in that component of the active site. 20-z

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are accounted for. Although, for a fixed quantity of am1 there are twice as many hybrids when [am31 = 2[am1] than when [am”] = &[am’], in the former case half of them are inactive so that maximum activity is reached when [am”] = *[am’]. It will not now seem paradoxical that the optimum ratio is with the protein with potential activity (am”) in the smaller proportion. Behaviour more like that of the fly xanthine dehydrogenase might be expected with other mutant forms of glutamic dehydrogenase. It is to be noted that in the case of alkaline phosphatase of E. coli, which physicochemical evidence strongly indicates is not a 3n-mer (Rothman & Byrne, 1963) the optimal ratio for complementation of mutant forms of the enzyme is 1 : 1 (Schlesinger & Levinthal, 1963; Garen & Garen, 1963) as would be expected for such cases. In this last section it is the use of symmetry arguments rather than any specific hypotheses which the author wishes more to stress. Conclusion Modern methods for elucidation of three-dimensional structures will no doubt be applied to the proteins discussed in this paper and should settle the validity or otherwise of the theories advanced here. Whatever the results, it seems clear that explanations of the kind of phenomena considered here must invoke symmetry arguments of the present type, perhaps with different, or additional, detailed hypotheses. The author hopes to have shown the usefulness of such arguments in provisional interpretations of experimental data, even in the absence of detailed structural information. The author is greatly indebted to ProfessorDorothy Wrinch who first stimulated his interest in quaternary structure. He thanks Dr A. P. Mathias who read the manuscript, Miss B. Wrightman who typed it, and Mlle A. Giupponi and Mlle L. Palmenty for assistancewith the preparation of someof the diagrams.Thanks are also due to the Medical ResearchCouncil for a personalgrant while this work wasdone. REFERENCES ANKEL,H.,

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