THE STOCHASTIC METHOD AND THE STRUCTURE OF PROTEINS By Linus Pauling Mr. President, Ladies and Gentlemen: It is a great honor for me to have been invited to speak at the opening session of the Thirteenth International Congress of Pure and Applied Chemistry, and I express my thanks to the officers of the Con- gress and of the International Union of Pure and Applied Chemistry for dD <—} a yy Se having extended the invitation to me and to, you SOF for pour courteous welcomes My subject today is the stochastic method and the structure of proteins. Many scientists have been interested in the question we of the way in which selentific discoveries are madee A popular idea is that scientists apply their powerful intellects in the straightforward, logi= cal induction of new general principles from known facts, and deduction of previously unrecognized conclusions from known principlese This method is, of course, sometimes used; but as-a—wmige the advances in knowledge that are made by it are less significant than those that -2- ern / conscious result from mental processes of another sort —- in large part subssasimus “ processese Henri Poincare, in his essay on methematical creation, iy 9S the said that knowledge of mathematics and of the rules of - he must also hogic is not enough to make a man e creative mathematicians kxpiftxafx knkukkiom ' é be gifted with an intugtion that permits him to select from among the infinite number of combinations .whtk mathematical entities already known, most of them absolutely without interest, those combinations which will lead to useful and interesting resultse He illustrated the role of the subconscious by describing his investigations of $OAhIf, the Puchsien h functions, which he had discovered while working at Caen. He left Caen we on @ geologic excursion, and for sometime, while traveling, made no on conscious effort to attack the problem; then one day, as he put his foot on the step of an omnibus, the idea suddenly came to him th t the trans- formations that he had used to define the Fuchsian functions were identical ja. ke with those of non-Euclidian geometry. He Nerified this conclusion, xtkmr-on hiea-returirto-caene -&.peried-ef non~productive effort, he... spent & few days et -the-seagi @nd-ene morning, while _walking on the | bintiyx The field of the determination of the structure of crystals by the x-ray diffraction tmekwkm method tm provides interesting il- lustrations of the ways that scientific progress is achieved. Work in this field consists in the solution of individual, largely unrelated problems = the determination of the mimmek atomic arrangement of individual crystals. If the atomic arrangement (the structure) is sufficiently simple from the observed x-ray diffraction pattern it can be determined/by straightforward, completely logical arguments. N The procedure developed by Mishikawa, Wyckoff, and Dickinson before 1920 consisted in the tabulation, with use of the theory of space groups, of all atomic arrangements compatible with the symmetry and size of unit required to account for the x-ray pattern, and the rigorous elimination with use of the observed x-ray intensities (especially of qualitative {nequalities in intensity of pairs of diffraction maxima) of all of these atomic arrangements except one, which was then accepted as the structure of the crystal. During the dozen years after the determination of the first crystal structures by W. Le Bragg and We He Bragg most structure -L= determinations were made in this waye It is well known that the electron distribution in a crystal can be expressed as a three~dimensional Fourier series in which the coefficients of the verious Fourier terms are proportional to the square roots of the intensities of the corresponding diffraction maximae The smrkex electron= va distribution function depends, however, on the phases of the Fourier generally avclicable experimentsl terms, and there is no/gmmsxaki method of determining these phasese For a crystal of even moderate complexity, such as an amino acid, the number of possible atomic errengements provided by the theory of space groups is so great that the exhaustive consideration of them and elimination of all but onaf ty use of the observed x-ray tneenatica/ cnn be cerried out even with the aid of electronic calculating machines, and the attack on the prakimmoafxhin problems presented by these crystels must be made in other wayse During the first few years of my activity as a research man I carried through a number of structure determinations by the rigorous method, in collaboration with my teacher Roscoe G. Rinkios: Dickinson and on pork” independently. I was deeply interested, however, in more complex substences —5= oy eee tid RortttrAduldty pore Ane than those that could be attached: n this way, and from 1928 on I Ny phil swewmeuseccotci~in applyingee different method of attack to many substances. One evening in 1933, after I had described the new method of erystal struc- ture determination to him f-tembiewhome—trrlew—FeMs, and contrastéd it with the rigorous method, Dre Karl Xamitetoerx Darrow suggested that I cal] it the stochastic method, and referred me to the introduction of this expression by the chemist Alexander Smith, who had written, in his "Inor- ganic Chemistry") 1909, page 142, the following: " ..e. When Miteherlich discovered that Glauber's salt gave a definite pressure of water vapor, he at once formed the hypothesis, that is, supposition, that other hydrates would be found to do likewise. Experiments showed this supposition to be correcte The hypothesis was at once displaced by the fact. This sort of hypothesis pxemt predicted predicts the probable existence of _ certain facts or connections of facts, hence.reviving a disused word, JS or TOLae TLeds 7 we call it a stochastic hypothesis (Greek stochastices, apt to dgvine the truth by conjecture). It differs from the other kind in that it professes to be composed entirely of verifiable facts and is subjected to verification In the stochastic method of treating very complex Nn ey as quickly as possible guessed crystals a pleusible structure is gumsmixwith the aid of hints provided i “debaske “he chs Bang on if, “hoe w Que a Koos cated ik iu rn ha ALL, 0 preneplie . . orb Lawless Re Q hse —_ Drea QAR OK Lyvte, wrtetulas wore AT, Weer 0 Q om. ath 3 ee | ae 64 sabe Pyne = ord vr fo ssaaao AWE hin and in che Lak nat 1) he take wre Vows iP yf fg uar aradscte dl =o an weer by the observed size of unit and space-group symmetry, end knowledge of general principles of molecular structure, and the stochastic hypothesis that this is the actual structure of the crystal thereupon is either veri- fied or disproved by the comparison of calculated and observed x-ray intensitiese As a rule, if ond /The agreement between observed calculated intensities is excellent the proposed structure may be accepted as the correct one. There is, however, always some possibility that the agreement is fortuitous. This was en- phasized by a discovery that Dr. M. De Skagm Shappell and I made in 1930, 1 during our study of the structure of bixbyite (Ma, Fe)203,, and the C- +i; wee ae! obs . vo “pe fae, modification of the rare-earth sesquitoxides. W. H. Zachariasen had in- vestigated these crystals, and had assigned positions to the 32 metal atoms in the unit cube. % 2 During our reinvestigation of the structure we ofa discovered that the space group 1,’ provides certain pairs of physically distinct arrangement which give exactly the same intensities of reflection of x-rays from all crystallographic planes, so that no unambiguous struc- NE gn SEAS \ ture determination ceet® be made by the consideration of x-ray intensities alone. This ambiguity, which has not turned out to be an important one in Ab practice, has been further investigated ty Ap Patterson. A I shall content myself with a few examples of the application of the stochastic method. The mineral enargite, Cu,4s8,, was found on x-ray investigation” to have an orthorhombic unit of structure with = a = 6646 R, b = 7043 &, and c = 6.18 &. This unit contains six copper atoms, two arsenic atoms, and L. Pauling and S. Weinbaum, Z. Krist., 88, 48 (1934). eight subfur atoms. The dimensions are closely similar to those of the hexagonal mineral wurtzite, WS: the dimensions of a double orthohexagonal unit of wurtzite, containing eight zine atoms and eight sulfur atoms, -B— are V3/= 6.65 g, 2a = 7.68 &, and ¢ = 6428 %. This suggests that the atomic crrans ements that shown in Figure 1, which results from replacing o one fourth of the zinc atoms in wurtzite by arsenicjand the remaining three fourths by copper atoms, in such a way as to give secre x-ray ad pestilaled groupse It was found that the calculated/intensities for }: structure well a RXXRARBMEMK Agree xrimunty with the observed Seat , and somewhat improved agreement is obtained by moving the sulfur atoms slightly closer to the 2.28 for arsenic atom and away from the copper atoms, giving the di stancesarsenic~ sulfur and 2.32 a for copper-sulfur. Although the structure depends upon thirteen parameters, there is little doubt that it is correct. 4 similar investigation with, however, different results, was J carried out for the mineral sulvanite.4 Sulvanite, Cu,VS,, has been found as a massive mineral in Burra Burra, Australia, and in cleavable ~ “ messes and a few small crystals near Mercur, Utah. It wesfound. to be a ha tS tf r =a : t i f % pe y cubic, with unit of structure with a = 52370 i, conteining three copper Ro cunning ck gale Se att? wth N vps atoms, one vanadian atom, and four sulfur atomse The fact that the uni na of structure of the cubic form of sinc sulfije, has a = 5eh2 z and contains strongly four zine atoms and four sulfur atoms # suggests that sulvanite has a similar structure, with copper atoms replacing three of the zinc atoms was noted, and vanadium the fourth. It im immediately mjfimexemiy however, thet the \ : nln 2 ade st Bade! apata 24 observed intensities differ greatly from those calculated for this, struc= ture; the stochastic method in this case has failed. There is, moreover, no other plausible structure that suggests itself. Fortunately, Wiig the crystal is sigple enough to permit the rigorous method of structure determination to be applied. Its applicetion was found to lead to the structure shown in Figure 2. This structure is a surprising one - it puzzles me nearly as much today as it aid Sey twenty years ago, when it was discovered. The four sulfur atoms are in the same positions as for sphelerite, and the three copper atoms are in the positions occupied by three of the four zine atoms in sphalerite. The position of the fourth zinc atom im remains, Mes PE however, unoccupied, instead, ** vanadium atom. is ~LOONOE -10~ xutoxthexsmakixgsxkiy ae in a positiongpn the seme side of the sulfur atoms as that toward which the sulfur-copper bonds extend, rather than the opposite side, which would complete the tetrahedral configuration of metal atoms about the sulfur atome This structure is satisfactory in that each metal atom is sur- rounded tetrahedrally by sulfur atoms. It is surprising, however, that the bonds formed by the sulfur atoms are not mkrx tetrahedrally directed. The explanation of the stability of this peculiar atomic arrangement may weak lie in the formation of/covalent bonds between the vanadium atom and the six surrounding copper atoms, and also in the Coulomb stabilization of this structure, relative to the structure similar to that of sphélerite, through the achievement of a small negative electric charge by the copper atoms and a small positive charge by the vanadium atoms, which would result Jf Q rvaidena lh bonds with the-fea- -oem® covalent character with the four-csttPireteng. One of the most complex crystal structures to have been discovered 5 is that of zunyite. Zunyite is an aluminosilicate that occurs as -l]~ colorless, transparent tetrahedra, about as hard as quartz. The mineral has been found in San Juan County, Colorado, intimately mixed with guiter- ¥ minke, a lead arsenate, and in Ouray County, Colorado, in an altered porphyrite, sharp crystals of zunyite have also been found with hematite as powder in pots from graves in Uaxactun, Guatemala. Its composition could not be determined from the chemical analyses until the x-ray studies had been made, which immixkax provided a determination of the formula weight. The cubic unit of structure was found to have a = 13.82 g, and to contain four molecules of composition 41135450,,(GH, F), Cle It was known in 1933 that in all silicates investigated up to that time & A tetra each silicon atom was surrounded sfkahedrally by four oxygen atoms, with OnR the silicon-oxygen distance about 1.60 g, and that aluminum atoms +ere usually surrounded octahedrally by six oxygen Stomsy whth aluminum-oxygen distance 1.90 &, although a tetrahedral environment of aluminum atoms ak > a> Jhttns Le Uw a Fad radia! “al ‘ océ e It was not easy to find a way of combining tetrahedra and octahedra by the sharing of corners and edges to give a © 52s wwe Ve ; cubic structure with the right dimensions, but finally a single structure -12- Crit); X-ray was discovered, and the calculeted/intensities for the structure were structure found to be in good agreement with experiment. In this skrmexkxe there in the crystal are complexes of twelve aluminum octahedra, which are held together/by sharing corners with one another, and also with a single aluminum tetra- hedron and a comolex of five silicon tetrahedra. This tetrahedron-octa~ hedron framework defines cavities which are occupied by chloride ionse The structures of many other minerals (topaz, mica, qaaolinite, A “fee Ret chlorite, swedenborgite, etce) were, determined by the stochastic method. A During recent years a number of complex intermetallic compounds have been successfully investigated xy in the same waye A fundamental structural principle for metals and alloys is that in general each atom surrounds itself by as many neighbors as possible. Some of the most in- teresting metal structures that are known have been discovered as a result of the recognition of the significance of polyhedra of the icosa— hedral groupe These polyhedra, which have five-fold axes of symmetry as well as three-fold axes and two-fold axes, cannot, of course, retain their symmetry elements completely in crystals, which cannot have Uru+f five-fold axes. However, it is found that them can be present in erystals oO -13- with only slight deformation. It is interesting that the icosahedral arrangement of twelve atoms around a central atom provides what may be called a closer packing than closest packing = that is, than closest packing of soheres of the same size. In the icosahedral configuration the central atom can be 10 percent smaller than the twelve surrounding atoms, which are still able to make contact with it.ixxeusumingxeexctiomz The most complex Sees mat xntaxmetaiiiexesopeunis metal structure to-peve-been-determined is that 6 of MB50(Zns at) 9° In this crystal a central atom is surrounded by - a ThTw b aonell Le * » Nebare 4 105-7 (185.2), an icosahedron of twelve slightly larger ome This group of thirteen Brg 7 ¥), atoms is then surrounded by twenty atoms, at the corners of a pentagonal dodecahedron i, m each of these twenty atoms thus lying directly out from the center of wx oneof the twenty faces of the icogshedron. The next twelve atoms lie out from the centers of the pentagonal faces of the dodecahedron; this gives a complex of forty-five atoms, the outer thirty-two of ga which lie at the corners of a rhombic triacontahedron. The next shell mixatamexeomtx consists of sixty atoms, each directly out from the the center of a triangle which forms one half of one of the -l4- wv thirty rhombs bounding the yhombic triacontahedron§ Eueix these sixty atoms lie at the corners of a truncated icosahedron, which has twenty hexagonal faces and twelve pentagonal faces. Twelve additional atoms. - are then located out from the centers of twelve of the twenty hexagonal very large faces. Thefeommlexes are then opndensed together,with their centers at the point of a body-centered cubic lattice, in such 4 way that each of | ‘ fe . ‘ ‘ { 4 i . 4 a ‘ ae a war tke ada drpandt Bh LC gu Isr, g the seventy-two outer atoms ,is shared between two complexes; each, then f ok Sf? 4 ae A : F ( a 1 tie & of pert BA Ae LN we he contributes thirty-six atoms per lattice point, which with the centrei cometes of forty-five atoms J gives eighty-one atoms per lattice point, or 162 in the unit cube. The complexity of this structure may be attributed to the dif- ficulty of fitting complexes with icosahedral symmetry into a crystal with cubic symmetry. It seems not unlikely that the complexity of the mknupiex intermetallic compound with the simple formula NaCd, is to be attributed to the same causee I have now been working on the problem of the struc- ture of this crystal for thirty years,* and the structure still remains Fr, Pauling, Je Am. Chem. Soce, 45, 2777 (1923). -15- undiscovered. The crystal is cubic, with the edge of the unit cube slightly over 30 &. This unit contains about 384 sodium atoms and 768 cadmium atoms. The attempt to determine its structure by rigorous methods would, of course, be hopeless; but I think thet the stochastic method will ulti- mately be successfule We ‘sk Dre Bergnati ‘if “neha Bn ux a | figure to” Tilustrate ‘Structure a \ a \ \ and iX the ‘dia let's put in a! fefence Figure 4 an Nate the figure. ey afr have 2 er 3 tiguressis | they have See smo , tteure. | The power of the stochastic method is illustrated by its recent application to the p roblem of the configuration of volypeptide chains in proteins. The history of this application is illuminsting also in showing that an investigator who strives to avply the method must have confidence in himself. In 1937, after I had become interested in proteins, and had carried out a number of experimental studies of their properties (especially the magnetic properties of nenoglobiit) and, with Dr. alfred Mirsky, had formulated a general theory of the structure and process of denaturation of srotetned I spent several months in an unsuccessful effort to apply the stochastic method in the discovery of an acceptable structure for a keratine 98 Ee Mirsky and L. Pauling, Proce Nate Acad. Scie, 22, 439 (19360. oC Em, Pauling and C. D. Coryell, Proce Nat. Acad. Sei, 22, 210 (1936). ~16- It was possible at that time to predict that the amide group in 4 peptide would be planar, because of the resonance of the double bond between the two positions C-0 and C-N, and it was possible to predict the interatomic distances and bond angles, essentially as given in Figure §, with reasonable confidence. In addition it was recognized that a stable configuration aval would $e—dne—thns—pesmabeed the formation of hydrogen bonds between the N-H group and the oxygen atom of the carbonyl group, with N-Hee-0 distance h iv] approximately 2.80 %. No reasonable configuration was found in this 27+ t x investigation, however, and in consequence the possibility wes considered that the structural parameters of the polypeptide chain might be sig- nificently different from those predicted from information obtained through Mirrnrert Ue. M the determinationf of the structures of somewhat déetenbiy~reteted sub- stancese At that time no structure determination had been made of any amino acid, simple peptide, or other simple substance G related to proteinse My coneceefir teen Robert B. Corey ool had oums- dine al ‘+2 years earlicg made x-ray photographs of several proteins par Re ke Ge Men ence en Wy _Wrekorl) had-ettempted-te-analyze hium, and I concluded that we should em= bark upon a program of precise structur@ determination of these simple on | Jom steadily substances. This program has been/under way since 1937, and has led precise to the/determination of the structures of crystals of half a dozen amino acids, several simple peptides, and several other simple substances (such as acetylglycine) closely related to proteins. smcocresxtixetithisxnerk Through these investigations it was found that the planarity of the amide x dor group and the interatomic distances and bond angles ane gpreserved in all of the simple substances and can confidently be expected to apply also to the polypeptide chains in vwroteins. In addition, hydrogen bonds with N-H-e-0 distance 2.79 + 0012 g have been found to be pxw universally present. Even after this program was well under way, and te=stimermtans it was recognized that the structural parameters of the polypeptide chain would be reasonably well predicted, there was delay in the application of the stochastic method to the problem of the structure of proteins. This delay probably resulted from the failure of the earlier effort and from the feeling that the chemical complexity of the proteins ~ their construction from about twenty different kinds of amino-acid residues < =s18 - might well indicate a corresponding structural complexity. Then one day in March 1948, while I was lgukimusimeibed at my home in Oxford (where I was aes again serving as Eastman Professor) g@ recuperatw from a cold, I decided/to attack the problem of the configuration of polypeptide chains, for the first time in eleven yearse It occurred to me to make a search for the simplest configurations - those in which all of the amino-acid residues are structurally equivalent. The most general operation that converts an asymmetric element ,(smekxmax not its muaxmkiomex} mirror image) is a rotation about an axis combined with a translation along the axis. The repetition of this general operation automatically leads to a helixe helical I attempted accordingly to find/configurations of polypevtide chains in- volving planar amide groups with known dimensions, such thet suitable netineat O° hycrogen bonds were formed. Within an hour, with the aid,a pencil and a piece of paper, I had discovered a satisfactory helical structure. It did not, however, explain setbefeeteniay the details of the x-ray diagram. of hair and other a-keratin proteins, and nothing more was done along these lines for some months. -19— After my return to Pasadena Professor Corey and I suggested to Dre He Re Branson, a young man interested in the application of mathematics to » chemical problems, that he make a search for other satisfactory only «war helicel configuretions. He found/one, wkiuancdoone sone add+ttenet—sebtefercctory - conftpurethen, and in 1951 a description of the two helixes, the a helix o and the Ynelix, was publi shede® Oz. Pauling, Re Be Corey, and H. R. Branson, xxxienxxfhemodoexyx Proce Nate “head. Seis, 37,'205 (1951). Pee Although the a helizg which has about 3.6 amino-acid residues per turn and a pitch of about 5.4 x, did not explain in an obvious way the xa principal feature of the x-ray diagram of the a=-keratin proteins, 2 & strong meridional reflection with spacing 5015 4, its predicted x-ray pattern was found to be in excellent agreement with the observed pattern for synthetic polypeptides. Moreover, the general similarity in appear- ance of the x-ray diagram for the a=keratin proteins and that for the synthetic polypeptides gave reasonably strong support to the assignment of the a helix to these fibrous proteins also. The problem of the -20~ ~ origin of the 5.15 A meridional reflection seems ato have been solved aT] : a, %0 by the simultaneous suggestion by F. H. C. Crick, and Professor Ali? Corey and me* that in the a-keratin proteins the a helixes are twisted about one another. A detailed structuréis that shown in Figures / and f: aA it involves op seven-strand cable of a helixes, with two additional a helixes in the interstitial positions. \! 405 p, it. GC. Crick, Neture, 170, 1882 (1952) Ve ur . - ity, Pauling and Re B. Corey, Nature, 171, 59 (1953). Reasonably convincing evidence has also been obtained, largely through the work of Perutz and Kendrew, who have studied hemoglobin and myoglobin, and of Riley and Arndt, who have prepared radial~distribution ar a met many mgd pris, p apd tte! y oad curves of a number of proteins, that/globular proteins contain, seguenten with the configuration of the a helix. In hemozlobin and myoglobin these a~helix segments lie in approximately parallel orientation to one anothere -2]1- No detailed information tuxmsxyetx has so far been obtained about the way in which the polypeptide chains make the transition from one a-helix segment to another. Silk fibroin is shown by its x-ray diagram to have a structure which repeats in the distance 7.00 g along the fiber axis. The antiparallel chain pleated-sheet structure, shown in Figure fH, is predicted to have the identity distance 7.00 R along the fiber axis, and in other respects it accounts satisfactorily for the x-rey diagram of silk fibroin; it can be confidently accepted as representing this proteine It is probable that the B-keratin proteins (such as stretched hair) have a closely similar structure, with, however, the skaggexmi polypeptide chains axkiernakingxin axkeniationxwitipodks SM in parallel, rether than antiparallel, orientation. The x-ray diagram of collagen and gelatin is characteristic of these proteins, showing that they have a structure different from that of a= the proteins of the/keratin classy and also different from that of silk and the B-keratin proteins. The stochastic method was used in the formula~ 1D tion of a structure for collagen and gelatin, which, however, has since been found to be in disagreement with some of the features of the x-ray diagrame Despite much effort that has been expended on the problen, « Pauling and Re B. Corey, Proce Nate Acad. Seis, 37, 272 (1952) -22- no satisfactory structure for collagen and gelatin has ¥ yet been founde The problem of the structure of collagen and geletin may be used to illustrate an important aspect of the stochastic method. The first step in the application of this method is to make a gmexsx hypothesis - a guesse The second step is to test the hynothesis, by some comparison with experiment. In general the test cannot be sufficiently thorough to provide weet” root that the kxpokeksis hypothesis is correct = it may easily be shown that the hypothesis is incorrect, My through the AT discovery of 4 significant disagreement with experiment, agreement on A a limited number of voints cannot be accepted as verification of the hypothesis. In order for the stochastic method to be sieniniomt / the principles used in formulating the hypothesis must be restrictive enough Rrra. to make the hypothesis itself essentially unique; in other words, every investigator who makes use of this method should be allowed one guesse If he were allowed many guesses he would sooner or later make one that was not in disagreement with the limited number of testify points, but 04) there would then be little justification for sccepting that teathes as correct. I may, however, contend that Professor Corey and Tashoule demon 5 Dado Toa VUAly collagen, b, & y a a: é ‘ : . y _ 2 y bogs ; A +2h, an Bron » f pe” 7 “so £L 1 i f A r ' ff fe eh aol } ‘ab 7 LAset- fy BIO f Ay . fa J o - Ay iL he. 64. Anes’ BA tan ol prac wn Opes | ' j i wT & Ata : a ye prdae - /$- maken Legends for Figures Cuz,4sS,. Fige 1. The structure of enargite,/ The lerge circles re- present sulfur atoms, the smal circles copper atoms, and the small shaded circles arsenic atomSe Fige 2e The structure of sulvanite, Ou,VS,. Fige 3. A portion of the structure of zunyite, Aly 5ig0,,(H, F)jg01- 410, groups are pr represented by octahedra, and SiO, and A410, groups by tetrahedra, the last being marked Al. Smaller spheres represent oxygen atoms, larger spheres chloride ionse Groups of five tetrahedra and twelve octahedra preserve their icentity in the structure. Er peel —_ Fige 2. Diagrammatic representation of the configuration we of the volypeptide chain in the a helix. FigerSe The a helix. Fig. Ye 4t the left, a compound a helix - an a helix whose axis describes a helical configuration. The diameter, shown as about 10 a, ~25— includes the volume occupied by side chains as well as the main chains of the protein. Center, a 7=strand a-cable. In the proposed structures of proteins of the a~keratin type these watkes cables are packed together, with compound helixes as bhown at the left in the interstices. At the right, a 3~strand rope of a helixes. Fige % 4 cross section of the a-keratin structure, showing the 7-strand a-cables AB, and the interstitial compound helixes C. The orientation of the cross-section of the cable changes with coordinate along the fiber axis. The central cable is shown in the most unfavorable orientation for the interstitial a helixes. The protein chains are not so nearly circular in cross section as indicated in the drawing, and space is filled more effectively than is indicated. Fige 3. Drawing representing the antiparallel-chain pleated-sheet structure. Fige 4. The structure of the intermetallic compound MBao(ZnzAl)/ ge The six drawings from left to right in the top row and then left to right in the bottom row, have the following significance: a central atom surrounded twelve atoms at the points of by/a nearly regular icosahedron;afx the icosahedral group of thirteen atoms twenty atoms at the points of surrounded by/a pentagonal dodecahedron; the complex of thirty-three atoms surrounded by twelve atoms at the corners of & icosahedron, the outermost shell of sixty atoms at the corners of a truncated icosahedron, plus twelve atoms out from the centers of twelve of the hexagons of this polyhedron; packing drawing showing the complex of forty-five atoms plus an outer shell of sevanty-two atoms; the structure of the crystal, in which these complexes located about the points of a bodyecentered cubic lattice share all of the seventy-two atoms of the outermost shell with neighboring complexes.