TISE JOI:RNAL OF BIOLOGICAL C~~~MISTRY Vol.246, No. 10, Issue of May 25, pp. 30923102, 1971

Ptinted in U.S.A.

The Quaternary Structure of Proteins

Composed of Identical Subunits*

(Received for publication, October 29, 1970)

ETHEL J. CORIYISH-BOWDEN AND D. E. I~OSHLAND, JR.

From the Depadment of Biochemistry, University of California, Berkeley, Caliiornia 9/,720

SUMMARY

The possible structures which could arise on assembly of identical subunits into oligomeric structures have been made by assuming (a) that these are closed structures, (b) that whatever conformational changes occur on association are the same for each subunit, and (c) that there is no a priori reason for preferring a particular type of binding domain. When such an analysis is applied to dimers, trimers, and tetramers, it is seen that structures with heterologous binding sites are as likely as structures with isologous binding sites if thermodynamic considerations alone are relevant. Proteins with mixed isologous and heterologous binding sites are also found to be possible under some circumstances assuming possible values for subunit interaction constants. An analysis of the quantities of trimers, dimers, and monomers produced on dissociation of a tetramer shows that certain limits can be placed on the amounts of the dissociated species based on the possible protein structures. These limits therefore provide possible diagnostic tests for exploring protein design.

It is a major postulate of molecular biology that the study of the structure of a macromolecule will reveal insight into its mode of action. This has been amply proved in the case of DNA, and seems to be equally true in relation to proteins. One of the most interesting, and least understood, aspects of struc- ture-function relationships in proteins is the subunit assembly, which is relevant to the allosteric properties of simple enzymes, and to the properties of multicnzyme complexes catalyzing se- quences of reactions. Xonod, Wyman, and Changeux (1) pioneered in the treatment of the subunit structure of protein, describing the terms “isologous” and “heterologous” and present- ing t’he arguments for symmetry in proteins. Hanson (2) ex- tended these concepts from the point of view of a crystallo- grapher. Haschemeyer (3) has examined the subunit structure with electron microscopy, Klotz, Langerman, and Darnall (4) and Hoagland and Teller (5) from the standpoint of hydrody- namics. Last but certainly not least the classic x-ray studies of Perutz et al. (6) on hemoglobin and Adams et cal. (7) on lactic dchydrogenase are providing detailed descriptions of subunit

* This work was supported in part by grants from the National Institutes of Health and the National Science Foundation.

contacts. To augment these studies it seemed to us that a theoretical thermodynamic evaluation of the forces involved in the assembly would be desirable. Not only might such an analysis allow some predictions of the structures which are likely to be observed, but it would provide a means of extending the quantitative analysis of allosteric effects (8, 9). In order to simplify a complex subject, we have confined this discussion to proteins in which all the subunits are identical.

In the present approach, the possible ways in which identical subunits could come together have been csamined, and thermo- dynamic reasoning has been used to reduce the large number of possibilities. The properties of the remaining small number have been compared with properties reported in the literature. From these analyses it is seen that it is highly probable that proteins containing a small number of identBical subunits, i.e. dimers, trimera, tetramers, etc., will exist in a very limited num- ber of three-dimensional structures. Moreover, the analyses suggest techniques for distinguishing the types of structures which pertain to a given protein. These principles may t’hen provide a basis for analyzing the forces in more complex struc- tures.

THEORETICAL

In Fig. 1 schematic representations of some oligomeric pro- teins with two, three, and four identical subunits are shown. The identical peptide chains are shown as circles, although the peptide chain is not, of course, a perfect circle or sphere. To indicate the asymmetric nature of the peptide chain, the letters p, p, and r are used to identify certain loci on the subunit, which represent binding domains brtwcen subunits. The very large number of possible arrangements of subunits may be classified into “closed” and “open” arrangements (I), where closed ar- rangements are those in which every potential intersubunit- binding site is used for binding, and open arrangements are those in which there are vacant binding sites, as exemplified by Fig. 1, b, e, and j. In Fig. lj, it is seen that there are two p sites and two r sites which are used to form pr-binding sets, and two p and two T sites which are vacant. In practice this arrange- ment would be expected to be unstable, because the tetramer would associate to form higher polymers if pr were strong while it would dissociate into dimers if pr were weak. Thus, a con- siderable reduction of the number of possibilities is obtained by excluding all open structures from consideration. Then the remaining arrangements in Fig. 1 (a, c, d, f , g, h, and i) represent all of the possible planar closed arrangements of two, three, and

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four subunits. Konplanar arrangements, where each subunit can interact with more than two neighbors, are possible for tetramers and higher polymers, and will be discussed below.

For subunits arranged in a plane, with each subunit interact- ing with two neighbors, so that a ring is formed, there are two possible orientations for each subunit, “up” and “down”, which are represented in Fig. 1 by + and -. This distinction is not made for dimers, because there are many planes which can be defined as the plane of the molecule, and thus in the absence of a unique reference plane, the orientations of the subunits in a dimer cannot be defined.

Binding pairs may be like-like and like-unlike, which have been termed isologous and heterologous, respectively, by Monod et al. (I). Thus pp-, qq-, or rr-binding pairs would be isologous, while pq, qr, rp, etc., would be heterologous. For a dimer a closed structure can be achieved only with isologous binding (Fig. la) provided induced conformational changes leading to asymmetry are excluded. Such induced conformational changes occur when the two identical peptide chains of insulin interact to form a dimer as shown by the elegant studies of &4dams et al. (10). In that case the induced changes cause the two previ- ously identical chains to become different from each other and further aggregation may then be excluded by conformational changes in the exposed binding domains. Clearly this aspect of induced asymmetric conformational changes is extendable to the higher oligomeric proteins, but we shall exclude such con- siderations from the present article. This type of behavior is not observed in hemoglobin and lactic dehydrogenase and therefore, clearly, many proteins at least can be considered which do not show such an effect.

An open dimer of like subunits, in which conformational changes induced on association are the same, such as the pq

dimer shown in Fig. lb, will leave vacant binding sites. For trimers and higher polymers the binding may be either isologous or heterologous. Bn all heterologous trimer is shown in Fig. lc in which the subunits are arranged head to tail around a ring, each in the same + orientation. This type of arrangement is possible for any oligomer greater than a dimer, and the analogous tetramer is shown in Fig. If. All of these arrangements share some properties in common: (a) all the subunits are in identical positions, and are thus indistinguishable, (b) the molecule as a whole has an n-fold axis of symmetry, where n is the number of subunits. For Fig. 1, c and f, the 3- and 4-fold axes, respec- tively, are at right angles to the plane of the paper, through the geometrical center of the figure. For trimers, and all odd num- bered oligomers, no all isologous closed arrangements are possi- ble, and thus the “mixed” structure (defined as one in which isologous and heterologous structures are present) shown in Fig. Id is the only other trimer possible. In this structure, and in the mixed tetramers shown in Fig. 1, g and h, the subunits are not all in identical positions, and the molecule has no over-all symmetry. For oligomers with an even number of subunits, all isologous arrangements are possible. An example of this is the all isologous tetramer shown in Fig. li. In these arrangements, every subunit occupies an initially identical position, and the molecule has an n/2-fold axis.

It is appropriate at this point to discuss qualitatively which arrangements can be excluded a prioti. Open structures are not considered in the present case since they would be expected to associate or dissociate into closed structures. It has been

Tetromers: I

P4, ++++ PP qq w*+++- PP qP w 04 ++-- pp49 PP qq t-+- I gq PP 44 Pr ++++

FIG. 1. Planar arrangements of two, three, and four subunits. All possible closed planar arrangements of these numbers of subunits are shown in the left-hand part of the figure. There are also an infinite number of possible open arrangements, with unsatisfied binding regions, and examples of these are shown on the right-hand side. The subunits are represented by circles, and the relative orientations (up or down) are indicated by the symbols + and - . These are omitted for the dimeric structures, because it is not meaningful to distinguish between orientations in this case.

argued that arrangements in which the subunits are not all in identical positions (e.g. Fig. 1, cl, g, h) are implausible, and would be selected against in evolution (1). There are examples, how- ever, of large proteins, such as the coat protein of the dahlemense strain of tobacco mosaic virus (II), where identical subunits are known to occupy nonidentical positions, and so the possi- bility that the same type of behavior may occur with smaller proteins cannot be ruled out. Therefore, in this paper we shall assume that any closed arrangement of subunits is permitt’ed, even such asymmetric and unesthetic arrangements as that shown in Fig. lh. The relative scarcity of odd numbered oli- gomers in proteins, compared with the abundance of dimers and tetramers, suggests that isologous binding is more likely than heterologous (1). To reason from this that isologous binding is required in all cases is dangerous, however, since proteins with odd numbers of subunits have been observed.

TETRAHEDRAL CASE

The tetrahedron represents a more complex problem than any of the planar structures considered above, because each subunit interacts with three instead of two neighbors. Each subunit has three possible binding sites, designated p, p, and T, which can interact with sites on other subunits. No assumptions will be made a priori about the specificity of these sites for other sites, e.g. it will not be assumed that p sites can only interact with other p sites, etc. Instead, numerical quantities will be assigned to the various possible interactions, allowing for all combinations of weak and strong interactions, and the most stable arrange- ment calculated for each set of assumed interactions. This is, of course, a generalizing rather than a restrictive assumption.

If the four subunits in a tetrahedral arrangement are placed so that every interaction site is in contact with another site, then each subunit can have three distinct orientations. Since

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(a) (b)

(c) 1s

pp,qq,rr, aa==

4 r P

A q’ P

q r r

P P

(dl I,H,

PP qq rr pq qr w aad

qAq qAp pj!Jq P r P q r r

(e) 12H4 (f) W4 (9) W4

PP qq qr2 rp, aaPP w rr Pq,v, ~QYY PP rr pq2qr2 44

p&LJp pJl& qAq

r q P r r P

(h) IH, (i) IH 5 (jl IH5

iv Pq qr3 rp aPya qq Pq qr rp, a@~ rr Pq, qr rp 40~

FIG. 2. Tetrahedral arrangements of four subunits. The ball representation in a is inconvenient for indicating the binding regions, and so the schematic representation shown in b is used instead. These two representations are equivalent. The ar- rangements shown in c to j represent all of the possible closed tetrahedral arrangements of identical subunits. The letters . CC, p, and y are used to indicate the orientations of the subunits, as explained in the text.

I Type of binding pair between each pair of subunits

there are four subunits, there are 34 or 81 possible arrangements. However, of these 81, 54 are mere rotations of the molecule as a whole from the other 27 possibilities. These remaining 27 may be analyzed as follows. The reference state is taken to be the all isologous arrangement shown in Fig. 2c, which will be desig- nated MUMY. In this designation, the letters a, ,8, and y are used to define the three rotational states of each subunit, and are analogous to the + and - used to define orientations in the planar arrangements. A new state is generated by rotating one subunit through 120”, keeping the other three unchanged, giving aa@ (Fig. 2d). By similar operations all 27 arrange- ments can be generated, and if this is done it is found that there are only eight which have different combinations of binding domains (shown in Fig. 2, c to j) The other 19 are all congruent with one or other of the eight shown.

Three different notations are shown in Fig. 2 to define the

various arrangements. The notation OI(YC@, etc., is useful to indicate how t,he various arrangements are generated, but it yields little information about the interactions. To do this the more complete notation pp qq rr pq qr rp acmp may be used, which is analogous to the pp qq pp qy/+ - + - of Fig. 1 nota- tion used for the planar structures considered previously. Fi- nally it is usually more convenient to use a notation in which all isologous interactions are given as I, and all heterologous interac- tions as H. Thus, the structure shown in Fig. 2d would be I&

It may be seen in Fig. 2 that the structure in Fig. 2c is Ig, in Fig. 2d is 18H3, in Fig. 2, e to g are IzH4, and in Fig. 2, h to j are IHS. Examination of the three which are designated IzH4 shows that they are distinguishable only by the arbitrary assignment of the letters p, y, and r to the binding sites, and that they are

identical in type. Similarly the three arrangements which are

designated II-Is are identical in type. Instead of eight different

tetrahedral arrangements, therefore, there are actually only four fundamentally different arrangements. Of the 27 possible ar- rangements, one is 16, eight are IaH+ six are I&, and 12 are

IHS. The possible tetrahedral arrangements are given in Table I.

TABLE I

Binding domains in tetrahedral arrangements

The eight possible ways of arranging four identical subunits to form a tetrahedron. The subunits are numbered from 1 to 4. Thus,

the column headed by 23 indicates the binding domains used to bind Subunits 2 and 3. Since the subunits are identical, the numbering is arbitrary, and the structures would be unchanged if, for example, the numbering of Subunits 1 and 2 were reversed. The labeling of the three different binding sites as p, p, and T is also arbitrary, and thus the three different arrangements given for IzH4 would not be distinguishable unless two or all three of them occurred as alternative arrangements for the same protein. Similarly, the three arrange-

ments given for IHs would only be distinguishable if more than one occurred for the same protein. The columns labeled I and H list, the numbers of isologous and heterologous bonds, respectively, in each arrangement. The right hand column shows the expected

frequency of each arrangement, if the frequency were purely st,atist,ical, i.e. if all possible binding arrangements were equally likely.

Arrangement -

-

-

Frequency of each type of binding pair Statistical factor

for total structure

12 13 14 23 ~~

If,. pp qq rr rr IaH,. . pp qq rp rr 12H4-a. . . pp qr rp rp IzHd-b. qq w Pq Pq I&-c. . rr Pq w qr IHS-a......................... pp qr rq rp IH,-b.. qq rp pr Pq IHs-c. rr Pq w w

-

24 34 --

qq PP w Pq qr w rp rr

Pq PP qp w rq rp pr Pq

_-

-

PP qq ~-

2 2 1 1 1 1 0 1 1 0 1 0 0 1 0 0

rr P4 qr

2 0 0 1 1 1 0 0 2 1 2 0 1 2 2 0 1 3 0 1 1 1 3 1

__-__ 0 6 0 1 1 3 3 8 2 2 4 2 2 2 4 2 0 2 4 2 1 1 5 4 3 1 5 4 1 1 5 4

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Two general points of interest may be made about the various tetrahedral structures. First, no all heterologous tetrahedral arrangement is possible; the most heterologous arrangement, IHs, contains one isologous binding pair. Second, the all isologous arrangement is the least probable if statistical factors only are considered; thus, if all of the different types of binding interaction were equally possible, the Is arrangement would be expected to occur only about once in 27 cases.

RESULTS

Quaqiitatioe EstimaGzbns-In order to determine which of the possible arrangements are the most likely to occur, two types of quantitative analysis have been carried out. In the first case, numerical values have been assigned to each of the possible bind- ing interactions, and the over-all binding energy of each arrange- ment obtained by summing the energies of the constituent inter- actions. By systematically studying the effects of changing the energies of the various interactions, it is possible to deter- mine what conditions are necessary to cause one arrangement to predominate in the equilibrium mixture. While this type of analysis gives insight into the way in which the different interac- tions determine the relative stabilities of the different arrange- ments, it has the drawback that it does not give information about the probability that the conditions to produce a predomi- nance of any particular arrangement will be found under evolu- tionary pressures and in fact this type of analysis has a possi- bility of an unconscious bias on the part of the investigator in selecting constants. Accordingly, a second type of analysis has also been used, in which the energies of the different interactions are assigned with a random number generator, and the fre- quencies in which I-arious arrangements predominate are deter- mined in a large number of trials.

For distinguishing between the four planar arrangements, I+ IIHH, IHIH, and I-Ih, the results from the systematic study were very simple and straightforward, and rendered the Monte Carlo analysis unnecessary. The planar arrangements, with their interactions and statistical factors are shown in Table II. It may be seen that if the pp, qp, and pg interactions were equally strong, so that only statistical considerations were applicable, then a mixture of 12.5y0 14, 5O70 IIHH, 25y0 IHIH, and 12.57, Hq would be expected. These are the maximum possible per- centages for the mixed arrangements IIHH and IHIH, i.e. there is no possible combination of values for the interaction energies of pp, qq, and pq pairs which would result in more than 50% IIHH or more than 25(r, IHIH.

Since the pp and py interactions always occur together, it is not necessary to consider their energies separately. Instead, it is sufficient to compare an averaged interaction energy I for the two isologous interactions, with the energy H for the heterol- ogous interaction pg. If I > H, then the 14 arrangement is preferred, and if H > I, then the Hd arrangement is preferred. In each case, the two mixed arrangements will be intermediate in stability, apart from statistical effects. In practice the statistical factors are too small to be significant. The largest statistical factor, 8/2, or 4, for the difference between IIHH and Iq or Hq, is equivalent to only 0.8 kcal at 298”K, distributed over the whole molecule. Thus, a difference of 0.4 kcal between I and 1-I is sufficient to outweigh statistical considerations. Since each interaction in a tetramer is likely to be of the order of several kilocalories at least, it is likely that in any example the int,eraction energies for isologous and heterologous bonds

TABLE II Binding domains in planar arrangements

The four possible planar arrangements of four identical sub- units. These arrangements are illustrated in Fig. 1,f to i. Since each subunit can be arranged in two orientations, + or -, with respect to the plane, there are 16 arrangements for all four sub- units. Two of these, +-+- and -+-f, result in the 14 ar- rangement, and thus this arrangement has a statistical factor of 2. The statistical factors for the other arrangements are obtained similarly.

Arrangement

Type of binding pair between each pair of

subunits

12 23 34 41

14.. PP qq PP qq IIHH. PP qq Pq Pq IHIH.. PP YP qq Pq I-I,. pq pq pq pq

FWplHlC~ of each type of Statistical

binding pair f ac;;kx&x

structure PP 94 lw I H

2 2 0 4 0 2 1 1 2 2 2 8 1 I 2 2 2 4

will differ by much more than 0.4 kca1. For this reason, it is extremely unlikely that the mixed arrangements will ever occur as predominant species. It is not possible to make any a priori

prediction of whether an isologous interaction should Ir;e greater or less than a heterologous one, so it is not possible to predict whether 14 or He will be the more probable in reality. However, we can definitely expect that these two arrangements will be the only two planar arrangements which need be considered as predominant species.

The analysis of the tetrahedral arrangements proved to be considerably more complicated, but some useful information was obtained from a systematic examination of the possibilities nonetheless. A summary of the conclusions from the systematic study is shown in Table III. It can be seen that it is theoreti- cally possible for every arrangement except the IjHs arrangement to predominate to the complete exclusion of all the other ar- rangements. Conditions are shown which would be suZicient to produce a 50, 90, 99, or 99.9% predominance of any arrange- ment (except IzH3). It is interesting to note that the values of w given in the table correspond in most cases to very small free energy differences. Thus, to produce an overwhelming pre- dominance of 99.9% 16, a ratio of only 20.3 between the interac- tion terms for isologous and heterologous interactions is needed. This corresponds to a difference of I .76 kcal between the two types of interaction, which should be readily obtainable in prac- tice. For the other arrangements the energy differences needed to produce an overwhelming predominance are greater, but still within reasonable expectation. The largest value of w shown in the table, 3997, corresponds to a free energy difference of 4.85 kcal at 298°K. The conclusions to be drawn from these results, therefore, are that the 16, IzH4, and II-15 arrangements must all be considered as possibilities for tetrameric proteins. The fourth arrangement, IsHs, in contrast seems to be a very unlikely one in practice, because according to this analysis it can only occur to a maximum of 390/c under any circumstance and then only as the major component of a complex mixture containing substantial amounts of every possible arrangement (20.7% Is, 39.0% 13H3, 6.0% for each of the three LB4 arrange- ments, and 7.47, for each of the three II-16 arrangements).

A special case of some interest is that in which one or more of the interactions between sites is so unfavorable (e.g. from steric

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3096 Quaternary Xtructuye 0~” Proteins

TABLE III

Vol. 246, To. 10

Relationship of strengths of subunit interactions to proportions of different tetrahedral arrangements Values of the subunit interaction terms sufficient to produce the specified relative concentrations of each of the tetrahedral arrange-

ments; e.g. if KPP = Kpp = K,, = UK, and KPp = KpT = K,, = K, where K can have any value (because it cancels from the expression for the relative concentration of any arrangement), then the 1, arrangement will form 50% of the mixture if w = 2.34,90% if o = 4.47, etc. The conditions given in the table are sufficient, but not necessary, to produce the specified relative concentrations. In order to simplify the conditions, certain restrictions are implied (e.g. K,, = K,, = K,, in the example given above) which are not necessary conditions

I Values of subunit interaction terms Value of w required for -

KW I

&, UK K UK K K K UK UK WK K K/m K K/w K K UK / !

99.9%

20.3 a

3997 3997 3997

32.6 32.G 32.6

99%

9.54 a

397 397 397

10.9 10.9 10.9

%J

K K CdK UK K K WK K

90%

4.47 u

37.1 37.1 37.1 3.87 3.87 3.87

2.34 n

5.00 5.00 5.00 1.72 1.72 1.72

Is.. I,Ha. 12H4-a.. I,Hb-b. 12H4-c. IH5-a.. II&-b. IHs-c..

WK CdK wK K UK K K/w K/w

a The ISH, arrangement cannot achieve a higher proportion than 39yo, which occurs when w = 1.62

than 95c/ of the IzH4 arrangement. This result suggest,s that

the I& arrangement m&IT be expected to be very Tare in practice, but that it cannot be ruled out as a possibility. As expected from the systematic analysis, the I& arrangement never oc- curred to any great extent, and only very rarely approached its theoretical limit of 39% ; only 0.1’2 of trials gave more than 35 yP of this arrangement. Since a mixture of, for example, IHS-a and IHs-b in equal proportions would be expected to have dif- ferent properties from a pure preparation of IHS-a or IHS-b, but the pure preparations would be indistinguishable from one another (because the labeling of the three sites as p, p, and r is arbitrary), the three different IH, arrangements were counted separately, but the totals are given in the table. Thus, a mix- ture of 58% a, 18% b, 3% c, and 21 T;, other components is given in the table as “more than 5Oc/I, II-I,” not as ‘(more than 75y0 IHs.” The IzHz arrangement was treated in the same way.

Binding Curves-If tctrameric proteins do in fact exist in sev- eral possible different arrangements, then it is to be expected that the different arrangements would display different, prop- erties which would permit identification of the arrangement. It would be expected, for example, that the I4 and Hq armnge- ments might show perceptible differences in the binding curves for ligands. The sequential model (9) can readily be extended to allow for the existence of different types of binding domain. Such an extension reveals that differences between the binding curves for the different arrangements do exist. These differ- ences can be used for diagnostic puTPoses, if the curves are c&Te- fully analyzed to obtain accurate values of the binding param- eters (12). Such an analysis is not possible if the protein follows the symmetry model of Monod et ctl. (1) since differences be- tween the subunit interactions do not enter the binding equations.

Association-Dissociation Equilibria--The various ways of arranging subunits to form tetramers may be expected to give rise to differences in the association-dissociation behavior of the protein, specifically in the probability of forming dimers or trimers as detectable intermediates between the monomeric and tet.rameric states. For example, in the case of the Hq arrange- ment, all of the subunit interactions are of the same type, pp,

exclusion or repulsion) as to be forbidden. The effect of this on the tetrahedral arrangements may be seen by examining Table I. For example, if one heteroloffhus interaction, e.g. pq, mere forbidden, this would exclude the 13113 arrangements, two of the three IZH4 arrangements, and all three IHS arrangements, leaving only Is and LHd-a as possible structures. Which of these two would predominate would depend on the relative strengths of (pp + 44 + 2 W) and (2 qr + 2 rp). The effect of forbidding pr or rp interactions would be similar. In con- trast, if one of the isologous interactions were forbidden, e.g. pp, this would exclude the IS and I3H3 arrangements, IzH4-a and -c, and IHs-a, leaving only I&d-b and IHS-b and -c as possibilities. Similarly, if any two heterologous interactions were forbidden only the 16 arrangement would be left as allowed, and if any two isologous interactions were forbidden, only the II-I5 arrangement would be left. In the rather implausible event that one isologous and one heterologous interaction were forbidden, only the IzH4 arrangement would be permitted, provided that a suitable pair of interactions were forbidden, i.e. (TY + pq), (pp -/- pr), or (44 + rp). In the course of evolution it is likely that arrange- ments which were weakly favored originally would become strongly favored today, so that in actual proteins existing today some interactions may well be impossible. Nonetheless at the time when the protein was first evolved there may have been very few restrictions on the structures which could be adopted.

The tetrahedral arrangements were also investigated by the Monte Carlo method, assigning values to the subunit interaction terms with a random number generator, according to a uniform probability of values in the range 0 to 4 for the logarithm of the term. This range of four orders of magnitude was selected in order that about 90% of the trials would result in a greater than 50% preponderance of a single arrangement. The results for 5000 trials are shown in Table IV, aad are in good agreement with the conclusions reached in the systematic analysis de- scribed above. The 16, IzH4, and IHS arrangements all occur in heavy preponderances in a significant number of cases. Of the trials, 15Y0 gave more than 95c/, of the Ig arrangement, and another 15% gave more than 950/;; of the IHs arrangement’. A much smaller number of trials, 2.6”/c of the total, gave more

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TAIILE IV

Percentage of given tetrahedral arrangements when strength of

binding pairs assigned by Monte Carlo probabilities

Results of 5,000 trials with random values for the subunit inter- action terms, KPP, K,,, et,c. For the logarithm of each term, there was a Imiform probability of values in the range 0 to 4, i.e. a range of I to 10,000 for the term itself. This range was selected in order that about 90% of the trials would result in a greater than 50% preponderance of one arrangement. It may be seen that in fact about 91% of the trials resulted in such a preponderance. The entries in the table show the percentage of trials which resulted in the preponderance shown for each arrangement. For example, 15.1% of the trials resulted in more than 05yo of the 4 arrange-

Arrangement

Is. I,H,. .:I:‘.::: : IsH,h III&.

Totals.

Proportion of trials giving more than given percentage of species at left

50% 7% 90% ____

20.0 17.3 0.0 0.0 9.2 4.2

35.4 I 22.6

64.7 ~ 44.1

95% T

99% 99.9%

23.7 o.oa

18.1 49.5

91.3

15.1 0.0 2.6

15.1

32.8

11.3 0.0 0.2 5.9

17.4

6.ti 0.0 0.02 0.5

7.2

a In accordance with expectation, no trials gave more than 50y0 IaH,, 5.2y0 gave more than 10% IJI3, 0.9% gave more than 25%, and 0.1% gave more than 35%.

b The three different 12Hz arrangements were counted sepa- rately, but the results were added together for inclusion in this table. The differences between the three arrangements were statistically insignificant (as expected). The three different IHb arrangements were treated in the same way.

so one would expect different amounts of dimer to be formed, from those in the 14 case where cleavage of pp bonds might be easier t,han qq bonds.

The concentrations of monomer, dimer, trimer, and tetramer for an associating protein, M4, may be related in general by the association constants, Kz, KS, and Kg, as in Equations 1 to 3.

(Mz) = K~(il/Jl)~ (1)

(MS) = Ks(M# (2)

(Mb) = K4(Jfd4 (3)

A set of constants, fi, to represent the fraction of monomer units present in the form of M’~, may also be defined, as in Equations 4 to 7, where the free monomer concentration is shown as m.

fl = m/ZiKimi (4)

f2 = 2 Kzm2/2iK~mi (5)

f3 = 3 K3m3/FZiRimi (6)

f4 = 4 K<m4/Zi&mi (7)

It is possible to use these expressions to show how the composi- tion of t,he mixture of oligomers varies from 11~ for a particular set of v&es of the association constants.

I n practice it is more convenient to measure total protein concentration, than free monomer concentration. If the equilib- rium constants are known, the total protein concentration can be calculated from Equations 1 to 3, and the results may be displayed as shown in Fig. 3, for the case where Ka/K,z = 0.1,

TOTAL PROTEIN CONCENTRATION IN MONOMER UNITS

FIG. 3. Composition of a mixture of monomer, dimer, trimer and tetramer at various concentrations of total protein, for a protein with KS/K? = 0.1 and KP/K~~ = 0.1, where Kz, K,, and 1‘4 are the association constants from monomer of dimer, trimer, and tetramer, respectively. The shapes of the curves are charac- teristic of the ratios Ka/K22 and Kq/Kz3 and not on the absolute values of the constants. For this reason the total protein con- centration is expressed relative to KZ and not in absolute units. The composition of the mixture is expressed in terms of the per- centage of monomer units present in each form. The maximum amount of dimer possible with this combination of constants is 54.1% at a total protein concentration of 3.45 KS, and the maximum amount of trimer is 13.37, at a concentration of 40.9 Kz. The dashed line represents the number average molecular weight, expressed in monomer units.

K4/Kt3 = 0.1. It can be seen that with this set of equilibrium constants the maximum value offs is 0.541, when m = 3.45 Kz and the maximum value of f3 is 0.133, when ?n = 40.9 K,.

It is inconvenient to draw a separate set. of curves for every combination of equilibrium constants, but it is possible to calcu- late the maximum values of f2 and ~“3 in the following way. 13~ differentiating Equation 5 with respect to m and setting djJ/dm

to zero it may be shown that maximum values of jZ and f3 occur when the values of m satisfy Equations 8 and 9, respectively. Each of these equations has a unique real root, if KS, KS, and Kq

are all positive, and thus can be solved unambiguously in all physical meaningful situations. Then

Maximum fz: 8 K4~~~3 + 3 IL& - 1 = 0 (8)

Maximum f3: 2 K4m3 - Knm - 1 = 0 (9)

are all positive, and thus can be solved unambiguously in all physically meaningful situations. Then the maximum values of fi and f3 may be calculated from Equations 5 and 6. If this is done in general, the results may be expressed as “maps)) for fi and f3 as shown in the top line of Fig. 4. In Fig. 4a the con- tours show the maximum value of f~ while the maximum values of f3 are shown in Fig. 4b. The situation illustrated in Fig. 3 is shown by the point l , and it may be seen that the position of the point, relative to the contours is in agreement with Fig. 3.

In order to use the maps shown in Fig. 4, a and b, it is necessary to know what, restrictions the various arrangements place on the values allowed for K3/Kz2 and Kd/K.j. To illustrate how this is done the 14 arrangement will be used as an example (cf. Table V). Since it is arbitrary which site is p, and which 4, no loss of generality is involved in the assumption that the pp

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-2 80% //

Quaternary Structure of Proteins Vol. 246, No. 10

I I I I I I I 1 I I I

r

‘\ ’ ‘\ 10% Dimer ‘\. ‘\\

‘\ 2-

'I '\

'\ '\

-8 -6 -4 -2 0 2

Log (KdK; 1 FIG. 4. Amounts of dimer and trimer possible with various rangement of subunits, located within the shaded arm. The 10%

arrangements of subunits. a, dependence of the maximum and 90% contours for dimer (solid lines) and trimer (broken lines) amount of dimer possible on the values of Ka/K2 and Kh/Ksa, are taken from a and b. A short portion of the 307, trimer contour where Kz, K,, and Kq are the association constants for the forma- is included to show that no more than 307, trimer is possible tion of dimer, trimer, and tetramer, respectively, from monomer. with this arrangement. cl, permitted region for the H4 arrange- 0 represents the situation shown in Fig. 3, where K3/K22 = 0.1 ment. The location of the 30% contours for dimer and trimer and Kq’K23 = 0.1. The location of the point in relation to the show that no more than 3070 trimer, and barely more than 307, 50% and R070 contours shows that a little more than 50% dimer dimer, are possible with this arrangement. e, permitted region is possible with this combination of constants, in agreement for the IS arrangement. This region is similar to the region with the maximum of 54.17, for the dimer curve in Fig. 3. b, shown in c for the 14 arrangement, and is subject to the same limit contours for trimer. o represents the same situation as in a and of 3070 trimer. f, permitted region for the IH, arrangement. in Fig. 3, and its location between the 10% and 2Ooj, contours is in The location of the 30% dimer contour shows that barely more agreement with the maximum of 13.37, for the trimer curve in than 307, dimer (actually 31.5%, a limit which also applies to the Fig. 3. c, values of K3/K2 and Kh/KG permitted for the 14 ar- H4 arrangement) is possible with this arrangement.

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Issue of May 25, 1971 A. J. Corn&h-Bowden and D. E. Koshland, Jr. 3099

TABLE V

Association constants for the 14 arrangement Derivation of the association constants for dimer, trimer, and

tetramer for the 14 arrangement of subunits in the tetramer. The stronger of the two types of interaction in t)he molecule is defined as pp, and the weaker as qq.

Oligomer Structure Association constant

Monomer 0 P 9

Dimer

Trimer

Tetramer

co PP 4 4

P P 4 q F 4

P

P P q q 83 9 q

P P

KPP %

2 2 KPP Kw

interaction is stronger than the qq, and thus that the dimer is predominantly a pp dimer. It follows from the definitions of the association constants that KS/K2 = K,,/K,,, and K4/ K23 = Kip/Kpp. Since K,, < K,, (because the qq interaction is weaker than the pp interaction), K3/Kz3 depends on the limits placed on K,,: the lower limit must be 1, because if K,, were less than 1 the tetramer would not ring-close. Thus, the range of values for K4/Kz3 is from K3/Kz2 to infinity, where K3/Kz2 is not greater than 1. This limit restricts the Iq arrangement to the shaded area in Fig. 4c. In the same way permissible regions can be defined for the Hd, 16, and IH5 arrangements, as shown in Fig. 4, d to f. In the case of the IHs arrangement it is as- sumed that the heterologous interaction which occurs three times (qr in the case of IH6-a, see Table I) is the strongest one present. This is not a necessary assumption, but it is a plausi- ble one, and was found to be true for the majority of combina- tions of constants which gave rise to the IH5 arrangement in the Monte Carlo analysis discussed above. The effect of this as- sumption is that the top border for the shaded area in Fig. 4f may be drawn somewhat low.

Examination of the shaded areas for the different arrange- ments shown in Fig. 4 shows that while they are not so well separated that analysis of the association-dissociation properties would permit a clear definition of arrangement in all cases, there are nonetheless considerable differences in properties to be ex- pected. For example, the 14 and 1,~ arrangements cannot give more than 30% trimer under any conditions, but can readily give high proportions of dimer. In contrast, the IHs arrange- ment can give high proportions of trimer, but ba.rely more than 30% dimer. The Hq arrangement can give barely more than 30% dimer, and no more than 30% trimer, under any conditions,

T.~BLE VI

Associafion-dissociation properties of tetrameric arrangements

Comparison of the maximum amounts of dimer and trimer pos- sible with four different tetrameric arrangements of subunits. In deriving the results for I,, I&, and 16, no assumptions were made beyond that implied in the whole of this analysis, namely that the strength of any given interaction is the same regardless of the arrangement of the rest of the molecule, e.g. that a pp interac- tion has the same strength regardless of the number of pp, pq, or qq interactions in the rest of the molecule. For the IHs arrange- ment the additional assumption is made t,hat the particular inter- action which occurs three times is the strongest interaction in the molecule. If this assumption is not. made, there is no limit on the amount of dimer possible for the IH, arrangement. In the case of the IzH4 arrangement there is no similarly plausible assumpt,ioll which can be made about the relat,ive strengths of t,he interactions in the molecule, and for this reason the IzH4 arrangement is not included in the table. The IHIH, IIHH, and IaH, arrangements, which are mentioned in the text, are also omitted from the table because there is no expectation that any of these three arrange- ments would form t,he predominant species for any protein.

Observation

Including tetramer Maximum dimer < 319&. Maximum dimer > 319$. Maximum trimer < 30%. Maximum trimer > 30%.

Excluding tetramer Maximum dimer = 377, Maximum dimer < 377,. Maximum dimer > 37yG.

I4

Yes Yes Yes NO

Yes NO

Yes

Consistent with

H4

Yes No Yes No

Yes No No

IS

Yes l-es Yes No

Yes Yes Yes

IHa

Yes NO

Yes Yes

Yes Yes NO

and thus in practice it would be unlikely that either dimer or trimcr would be detected for this arrangement. The differences between the permitted areas for the 14 and I6 arrangements are so slight that no qualitative difference in properties is to be ex- pected. This is logical since 14 is a special case of Is: the Iq ar- rangement is merely an 16 arrangement in which the rr interac- tions are so weak as to be negligible. The I,H, arrangement has not been included in this discussion, because no plausible assumptions can be made which would simplify the analysis sufficiently to be useful. The diagnostic tests resulting from this analysis are shown in Table VI.

A monomer-dimer-trimer system is considerably simpler to analyze than a system which includes tetramer as well, and further information about the arrangement of subunits can be obtained if the tetramer can be excluded from consideration. In practice this would be possible if the equilibrium between oligomers could be frozen, so that separation was possible, or if the equilibrium were slow enough to determine the relative amounts of monomer, dimer, and trimer without interference from tetramer. (In the case of arginine decarboxylase, for example, the equilibrium appears to be slow enough for this type of experiment to be feasible (13).) It is clear that in this simpler system the proportion of monomer can range from lOOn., at low protein concentration, to zero at high protein concentra- tion, and that trimer can also range from zero at low protein concentration to 100% at high protein concentration. The proportion of dimer, however, is considerably more interesting, because it is restricted by the nature of the interactions. By

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3100 Quaternary Xtructwe of Proteins Vol. 246, Xo. 10

arguments analogous to those used above for the more complex case, it may be shown that in the monomer-dimer-trimer system the proportion of dimer reaches a maximum when 1 - 3 Kam” = 0, i.e. m = l/2/3 I&, and that for any values of Kz and Ks, the maximum proportion of dimer is given by Kz/(d3 K3 + K?). In the simplest case, when KS = Kz2, as would be expected for fragments of the Hq tetramer, this expression has a value of 0.366, or 36.6c,‘,. It is easy to see how this value would be dif- ferent for fragments of the ot’her tetrameric arrangements: in the Iq cage, it would bc expected that KS < K?, because of the two types of interaction present in the trimer, pp and qq, the stronger would be present in the dimer. Thus, for this arrange- ment the maximum amount of dimer possible would be at least’ 36.61 C. In the 16 arrangement, there are three interactions in the trimer, because it is a triangular fragment of a tetrahedron. Of these three, only the strongest is present in the dimer. Thus, in this case no prediction about the relative magnit’udcs of KS and KS can be made, because the strongest of the three interac- tions may be either stronger or weaker than the sum of the two tryo weaker interactions. Consequently, in this case the maxi- mum proportion of dimcr may be either greater or less than 36.6~~4. Finally, in the IHj case, the trimer cont,ains three iden- tical pq interactions, and thus Ka > K,2. Consequently in this cast t,he maximum proportion of dime< cannot be greater than 36.65. These conclusions provide useful diagnostics for dis- tinguishing between arrangements, and are included in Table VI. It is of particular interest that a distinction between 14 and 1, is possible in principle, in contrast to the results obtained IThen the tetramer was included also. This Tvould be useful if s-ray data established beyond doubt that a particular protein existed in an all isologous form, but left in doubt the question of whether the third type of interaction in fact’ contributed to the structure. X situa,tion similar to this exists in the case of hemoglobin since the x-ray data indicate clearly that cc@1 and a& contacts exist and contribute to the structure, but leave doubt as to whether the ,&3 and particularly the CXOL contacts are significant.

uIscussIo~ / I

Thermodynamic considerations of possible ways in which identical subunits might come together in an oligomeric protein lead t,o the conclusion that a limited number of protein designs Tvill be found in nature. Three assumptions are made in this analysis: (a) that the species involved are closed structures in the sense that they do not polymerize further,l (b) that the binding domains are interchangeable in the sense that a protein containing binding domains p, q, and r will allow pp-, qq-, rr-, pq-, pr-, qp-, and qr-binding pairs, and (c) that there are not induced conformational changes of the insulin type which lead to asymmetry in like binding domains. An insulin-like situation could be treated similarly with the dimer instead of the monomer as the unit of association. Based on such assumptions and allowing the binding pairs to have any thermodynamic values, a number of interesting conclusions are reached.

1 This statement means that we shall consider here only the binding processes leading to the native physiologically relevant, struct,ures. Chymotrypsin, for example, polymerizes at very high concentrations of protein but the association constant is small and the protein is clearly designed to be a monomer under nhrsioloeical conditions. Similarly a tetramer like aldolase Light form loose associations at very high concentrations which require new binding domains but we are concerned here with the associations leading to t,he closed tetramer.

Only one type of stable dimer will exist and two types of stable trimers are possible. One trimer would be predominant when isologous interactions, i.e. pp, or pp, are stronger than heterologous interactions, i.e. pg, and the other trimer would predominate when the opposite is true. For tetramers arranged in a planar structure in which each subunit interacts with two neighbors, only the all isologous and all heterologous species (14 and H.J could be predominant species. There are no values for t)hc strengths of the subunit-binding pairs which can give more than 500/;, of the mixed species, IIHH or more than 25% of the mixed species IHIH. For a tetrahedral structure in which each subunit interacts with three other subunits, each of the ar- rangements Is, I&, and IH, could be predominant to the ex- tent of 99.97; of the specaies present given the appropriate values for the subunit-binding pairs. The only conceptually possible structural form which could not be predominant would be the 13H3 form which could not exceed a maximum of 39:/,. Even when it achieved 39c/; of the existing species, this form would be present in a complex mixture and therefore it seems unlikely as a possible protein arrangement in a biological system.

TL\-o interesting features emerged from these analyses. One was bhat there was no evidence for an advantage between isolo- gous flersus heterologous binding pairs in protein design. Neither by systematically assigning values to the subunit interaction terms nor by choosing these randomly by a Monte Carlo method was t,here any indication that 14 should predominate over Hd in the planar arrangement of subunits or that 16 should pre- dominate over IHj in the tetrahedral arrangement. A second interesting feature was that small differences in the strength of binding made large differences ill the predominance of a given structure. Thus, a 20.fold increase in isologous interaction terms relative to heterologous interaction terms is sufficient to

convert the I6 species from a minor constituent to 99.977 of the observable species of a proDein. This is a change of only l.i6 kcal and indicates that small changes in this interaction param- eter can lead to large changes in the obserl-ed species.

It, is, of course, not necessary to assume that all of the sites are interchangeable. If a large group were projecting from one site so that a certain type of interaction, e.g. pp, were impossible, one can calculate the resultant limitations on potential arrange- ments. If one iiologous interaction, e.g. pp, is excluded, only t,he II-IS and 12Hz species would be allowed in the tetrahedral arrangement and only El4 in the planar. I f a single heterologous int,eraction were excluded, e.g. pq, only the IS, 12H4, and 14 ar-

rangements would be possible. X very unfavorable interaction between the sites, such as might be caused by juxtaposition of like charges, has already been accommodated in the calculations since they assume any difference in energies of the binding domains. It is of some interest, however, that the 12114 species, which is improbable on purely statistical grounds (it occurred as more than 95(x, of the total protein in only 2.67;) of the cases), is the only form which could exist if one isologous and one heterologous interaction were excluded because of steric hin- drance.

The conclusions reached in this thermodynamic analysis therefore are in contrast to t\To previous examinations of similar closed protein structures. The classic analysis of Monod et al. (1) deduced that isologous bonds were more probable than hcterologous largely on the basis of the mamler in which such binding pairs might arise. On the other hand Klotz et al. (4) esamined the same proteins in a logica manner and favored

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Issue of May 25, 1971 A. J. Cow&h-Bowden and D. E. Koshlancl, Jr. 3101

Protein

Pyruvate carboxylase Inorganic pyrophosphatasc DNA-dependent RNA-PO

lymerase Protocollagen proline hy

droxylase “Fraction I protein” Dihydrolipoyl transsucci

nylase Hemocyanin Orosomucoid

Arginine decarboxylase Erythrocruorin Glutamine synthetase Aspartate decarboxylase Apoferritin Dihydrolipoyl transacety

lase Chlorocrnorin

“Protein” Human erythrocyte 40 Hemocyanin n!rollusks 30n

TAHLF: VII Structure of ploleins from electron microscopy

S0lllXe Ko. of

subunits Arrangement of subunits

Chicken liver mitochondria 4 Square Escherichia coli 6 Hexagon E. coli 6 Hexagon

Chick embryos 6 P-1) Hexagon (or stack of four hexagons)

Chinese cabbage leaves Bovine kidney mitochon-

dria Arthropods Human urine

8 Cube 8 Cube

8n

8 (n)

Cubic monomer “Ball”, (or “chain” of indefinite

length) E. coli 10 Pentagon of dimers Eumenia crassa, etc. 12 Stack of two hexagons E. coli 12 St,ack of two hexagons” Alcaligenes faecalis 12 Distorted icosahedron Horse spleen 20 Pentagonal dodecahedron Bovine kidney 20 Pentagonal dodecahedron

Spirographis spallamanii 36 Stack of two hexagons, each having three srlbunits at each vertex

Stack of four decagons Stack of 3 to 12 layers, each cow

sisting of a decagon surrounded by a pllckered icosagon

No. of neighbors

2 2 2

2 (3-4)

3 3

30 30 (2)

3c 3 3 5 3 3

3-4

All isologous possible?

Yes Ii Yes 15 Yes 16

Yes (no)

X0

No

No NO* (yes)

Yes< No No No No No

No

No No

17

18 19

20 21

13 22 23 2-l 25 10

26

27 20

1 Re- erence

a In each “monomer.” * Assuming a cubic arrangement for the “ball.” c A reasonable arrangement which is not ruled out by the electron microscope pictures would be a [pp ‘1 qle ring in Tvhich the Fp

dimer units were so closely bound that they were unresolved in the photographs. d Some tendency was observed to produce extended stacks of 2 n, hexagons with alternately closer and 17.ider spacing between hexa-

gons.

cyclic structures that would involve only heterologous bonds. The present analysis which allowed all possibilities within the framework of closed structures and assigned bond strengths in a systematic way or by Monte Carlo probability methods leads to the conclusion that there is no reason from a thermodynamic viewpoint to prefer isologous bonding over heterologous bonding. Either can lead to stable structures and neither can lead to con- sistently more probable or more exclusive structures. This does

not mean that all of these structures must exist. However, there is no thermodynamic reason to select for one type over another and this would suggest that many types would be formed.

The protein species that have been observed with the electron microscope are summarized in Table VII. I f the structures have been correctly assigned by the electron microscopist as shown in this table, it is clear that heterologous binding must exist,.

The further finding that not all of these proteins are dimers or tetramers and the observations of stable proteins with an odd number of subunits all argue for the existence of heterologous binding domains. Whether the heterologous binding domains contribute as much to the structures in terms of thermodynamic stability as the isologous binding, of course, must await further testing. The decamer of arginine decarboxylase studied by Boeker and Snell (13) may be two pentagons sandwiched on top of each other. I f this is true, heterologous interactions must be

present. It might also be a pcAnt,amer of closely packed dimers in which case two types of isologous interactions could be pres- ent. In structures such as hexagons, however, subunit armnge- merits which are all isologous are as reasonable thermodynam- ically as all heterologous structures alt,hough both are cyclic (cf. Fig. 1). It is of interest that the electron microscope has so far not answered the question of predominance of isologous versus hcterologous binding because either type is compatible wit,h most of the structures observed so far.

Furthermore it is not necessary to assume all subunits are in the same environment. In the dahlemense strain of tobacco mosaic virus a small change in the size of each subunit from t’he normal strain makes it impossible for the normal packing to oc- cur correctly and in order to approximate the normal packing subunits are arranged in 98 similar but nonidentical ways. The icosamer apoferritin is thought to have 20 subunits placed at the vertexes of a pentagonal dodecahedron, which is significant because it is impossible for asymmetrical units to occupy identical environments if they are so placed. A similar structure has been assigned for dihydrolipoyl transacetylase (19).

Since there may be a number of ways of arranging subunits for oligomers of three or more subunits, it is of interest to exam- ine experimental techniques which could identify these structures. The thermodynamic analysis discussed above can be extended to predict, the types of species which should occur under dis-

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3102 Quaternary Structure of Proteins Vol. 246, No. 10

sociating conditions for different protein arrangements. Such an analysis was presented extensively above for the case of the tetramer and similar analyses can easily be carried out for other structures. It was shown (in Fig. 4, c to f) that the existence of a stable structure, e.g. Iq or IS or Hh, required certain relations for the binding pairs which then placed limits on the association constants for dimer, trimer, and tetramer. From this limitation one can calculate which species are “allowed” for certain experi- mental observations, e.g. observing 40% dimer in an ultra- centrifuge experiment. Some of these limitations are sum- marized in Table VI. In this table the experimental experiment postulated is that in which a protein is successively diluted and examined for the maximum amount of dimer and trimer. I f no more than 31y0 dimer is formed under any conditions, all the species shown in the table are possible. However, if greater than 3157, dimer is observed under some conditions, then the species I& and IH6 are automatically excluded. Similarly less than 30~~ trimer is compatible with all four structures but greater than 30~‘,, trimer is compatible only with the IHE structure. Consultation of Fig. 4 easily shows what additional molecular species would be present for each of the predominant tetrameric forms. Furthermore, if tetramer can be separated from the lower molecular weight species, analyses of the proportions of monomer, dimer, and trimer can give *insight into the subunit arrangements. The delineation of subunit structure is of interest not only in such assembly processes but also because the de- tailed binding domains are needed for the full understanding of allostcric interactions (9). Detailed calculations can then be made in any particular experiment to predict dissociation be- havior compatible with the subunit arrangement postulated.

It might well be asked whether all the species which are thermodynamically possible would still exist in nature. The subunit-binding sites which exist today are highly specific and have undoubtedly undergone improvement over evolutionary time. The possibility that the binding domains can freely interchange as postulated in the calculations therefore seems improbable for many of the existing proteins. Nevertheless it would seem probable that in an early evolutionary period bind- ing domains were less specific and arose when an exposed hy- drophobic area interacted N-ith other hydrophobic areas. It would be logical in that case that more than one domain could interact with other domains and that some would be more favora- ble than others. Mutations could occur to increase the strength and specificity of the interactions and eventually one would ex- pect the presently observed highly specific sites. If such a path for evolution of subunit interactions occurred, the thermody- namic analyses described here would suggest that both isologous and heterologous binding pairs would be formed in the beginning and evolutionary descendants of both types would be found today. If only one type is found, this would suggest that other nonthermodynamic factors were involved in the selection proc- ess. Some mixed structures might even occur as minor species present in thermodynamic equilibrium with more predominant

structures and this would present a form of isomorphism in quaternary structure, although the protein is composed of identi- cal subunits. Depending on the groups exposed in such a struc- ture, the minor species might even be electrophoretically dif- ferent from the predominant species and might be observed at some future time. Possibly the faint bands observable on elec- trophoresis of isoelectric focusing of some proteins composed of apparently identical subunits might be an indication of such unusual arrangements.

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Athel J. Cornish-Bowden and D. E. Koshland, Jr.The Quaternary Structure of Proteins Composed of Identical Subunits

1971, 246:3092-3102.J. Biol. Chem. 

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