THE STOCHkSTIC METHOD ANE THE STFXJCTUFJ3 3F PFtOTJZINS

By Linus Pauline;

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

upplied Chemistry, and I express my thanks to the officers of the Con-

gress and of the International Union of Pure and Applied Chemistry for
                           y$ c-7
                                           1 il I r? ,x'
having extended the invitation to me and to you
                          &  4F for your courteous

welcome.

@J subject today is the stochastic method and the structure

of proteins.  Many scientists have been interested in the question hf of

the way in which scientific discoveries are made. 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 principles. This

method is, of course) sometimes used; but- the advances in

knowledge that are made by it are less significant than those that


-2-

,X' . .
g' conscious

result from mental processes of another sort - in large part submmmsAmm8

processes0 Henri Poincare: in his essay on mathematical creation,

said that knowledge of mathematics and of the rules of

- he must also

logic is not enough to make a man a creative mathematician/ m&x

??? o ??                  ?
be gifted with an intuition that permits him to select from among the
           +

infinite number of combinations. mathematical entities already known,

most of them absolutely without interest, those combinations which will

lead to useful and interesting results. He illustrated the role of the

subconscious Q describing his investigations of ###&# the F'uchsian
                                          )

functions, which he had discovered while working at Caen. He left Caen

on a geologic excursion, and for somehime, while traveling,, made no
                           i

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

with those of non-Euclidian geometry. Hexerified this eene&u~~&J&on

spent a few deys ettheeea

13d.m ra~mbg, w)lil+ walking on the
                       _. -.-. -. -* . .._lI.__.* ,_.._,.


The field of the determination of the structure of crystals

by the x-ray diffraction &me- method is 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 mti atomic arrangement of individual

crystals. If the atomic arrangement (the structure) is sufficiently simple
     from the observed x-ray diffraction pattern
it can be determine& straightforward, completely logical arguments.

The procedure developed by Fllshikawa, 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 QUalitatim

inequalities 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 dosen years after the determination of the

first crystal structures by W. L. Bragg and W. H. Bragg most structure


determinations were made in this way,

It is well known that the electron distribution in a crystal can

be expressed as a thre&iimensional Fouri-er series in which the coefficients

of the various Fourier terms are proportional to the square roots of the

intensities of j;hm corresponding diffraction maxima. The hx electron-
       JNJ .'

distribution function depends,
           a      however, on the phases of the Fourier

          generally a,::Tlicable experiments1
terms, and there is no/m* method of determining these phases. For

a crystal of even moderate complexity, such as an amino acid, the number

of possible atomic errsngements provided by the theory of space groups is

so great that the exhaustive consideration of them and elimination of all

but one by use o
P       f      /
          the observed x-ray intensities  cannot be cerried out

even with the aid of electronic calculating ma&fines, and the attack on

the gmx$jrn problems presented by these crystals must be made in

other ways.

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. Bm  Dickinson f and L

independently.  I was deeply interested, however, in more complex substences


than those that could be a&@&&in this way, and from 1928 on I
                A

different method of attack to many substances.

One evening in 1933, after I had described the new method of crystal strut-

ture determination to himf'- and contrastdd it with
            u

the rigorous method, Dr. Karl E rdrk'l Darrow suggested that I

call 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",19@9, page l&2, the following: W o ooe When Mitcherlich
                                                                     .

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 correct.  The hypothesis was at once displaced by the fact. This

sort of hypothesis ~XB& WI& predicts the probable existence of

certain facts or connections of facts, hence reviving a disused word,
                  J
              ~T+vm-cr06

we call it a stochastic hynothesig (Greek             .;
                       &w&&&Jw, apt to d$vine the
                                                                       i

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
                              A

,            ,, /v

cj- -_  as quickly as possible3
       /J In the stochastic method of treating very complex

                             guessed
        crystals a plsusible structure is l-with the aid of hints provided



by the observed si.ze of unit and space-group symmetry, ti 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 ccmpariaon of calculated and observed x-ray

intensities.

As a rule, if

/$he agreement between observed calculated intensities is excellent
                4

the proposed structure may be accepted as the correct one. There is, however,

always some possibility that the agreement is fortuitous.  This was em-

phasized b a discowq that Dr. M. D. &~BH Shappell and I made in 1930, 1

during our study of the structure of bixbyite)(%, Fe),O,, and the C-
   & i.. 'I ii :       ..;, j '<
                               -p f A

modification of *the rare-earth m  W. H. Zachariasen had in-
                 ,

vestigated these crystals, and had assigned positions to the 32 metal atoms

in the unit cube. 32   During our reinvestigation of the structure we

2 W. H. Zachariasen, Z. Krist., 67, 455 (1928)

--e---w

L. Pauling and M. D. Shappell, Z. Krist., '75, 128 (1930).


discovered that the space group zh7 provides certain pairs of physically

distinct arrangemen

of x-rays fern all crystallographic planes, so that no unambiguous struc-

                      ? ; * :\
ture determination&&&be made by the consideration of x-ray intensities
             ?f\

alone.  This ambiguity, which has not turned out to be an important one in


practice, has been further investigated   "'Patterson .
                           /\

I shall content myself with a few examples of the application

of the stochastic method.

A

The mineral enargite, Cu3AsSq, W&S found on x-ray investigation'

to have an orthorhombic unit of structure with x 5 = 6.46 8, b = 7.43 $

and c - = 6.18 9. This unit contains six copper atoms, two arsenic atoms, and

---w-w

3

L. Pauline and S. Weinbaum, Z. Krist., 88, 48 (1934).

w-----w

eight suihfur atoms. The dimensions are closely similar to those of the

hexagonal mineral wurtsite, ZnS: the dimensions of a double orthohexagonal

unit of wurtsite, containing eight zinc atoms and eight sulfur atoms,


-8.

0
are 3= 6.65 f, 2e_ = 7.68 2 , and c, = 6.28 2. This,suggests that the
                              .
   $4
atomic arranement
         A s that shown in Figure 1, which results from replacing
                                                         J

one fourth of the zinc atoms in wurtzite by   w
                                arsenic and
                         A   the remaining


three fourths by copper atoms, in such a way as to $ive discr


                            x-ray
groups.  It was found that the cslculated/intensities for

structure       well
E- 'agree r3ms~Q with the observed %.%, and somewhat improved

agreement is obtained by moving the sulfur atoms slightly closer to the
                2.23 for

arsenic atom and away from the copper atoms, giving the distances Y arsenic-
                          d 'i

sulfur and 2.32 !! for copper-sulfur.  Although the structure depends

upon thirteen parameters, there is little doubt that it is correct.

A similar investigation with,
           t    however, different results, was

carried out for the mineral sulvanite. 4  Sulvanite, Cu3VS4, has been

w---e-

% . Pauling and R Hultgren, Z. Krist., 84, 204 (1933).
                                --.__.        !,`-

-0-0-a-

found as a massive mineral in Burra Burra, Australia, and in cleavable

mzsses and a few small crystals near Mercur, Utah. It&*& to be


         1                        i!
      -...---
       .Si,;'            -, &.J' 1~ <it
                                ,'       `A     ;
cubic, 4&h unit of structure with a = 5.370 1  @)
                           containing three copper
                    i\ -'-    f`;

         ,!q ,A
atoms, one vanadik atom, and four sulfur atoms.  The fact that the unit
            I-e Qu.bA;&2
of structure of the cubic form of zinc sulfi k e has ! =
                              h-   5.42 2 and contains

suggests that sulvanite has a

similar structure, with copper atoms replacing three of the zinc atoms

                            was         noted,
and vanadium the fourth.  It ix immediately :&mxuz#q however,   that the
                        ,-y $;!
                              P - F, I"

observed intensities differ greatly from those calculated for thishstruc-

ture; the stochastic method in this case has failed. There is, moreover,

no other plausible structure that suggests itself. Fortunately,

\q "62 c'\ &' ?/
%w`\ l'$'y<;q
7 J"       the crystal is si#ple enough to permit the rigorous method of structure

-;;"";/. "' y.
         p.`    determinstion to be applied.  Its application was found to lead to the
  e /  . i'

structure shown in figure 2.

This struchure is a surprising one - it puzzles me nearly as

much today as it did- twenty years ago, when it was discovered.  The

.

four sulfur atoms are in the same positions as for sphalerite, and the

three copper atoms are in the positions occupied by three of the four zinc

atoms in sphalerite.  The position of the fourth zinc atom ix remains,

                                                          r.
                         &.. c*;. : 3 ;  \
however, unoccupied; instead,    vanadium atom is -e,


-lO-

in

a  4
x position   n

the  same 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 atom.

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 m 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 sphalerite,

through the achievement of a small negative electric charge by the copper

atoms and a small p



from the formation

atoms, which would result
   % Lb\4i.&&,.&/,
bonds with W

-covalent character@&-

One of the most complex crystal structures to have been discovered

is that of cunyitee5  Zunyite is an aluminosilicate that occurs as

-------

   5 L. Pauling, 2. Krist., 84, 44.2 (1933).
---em  -I


-11.

colorless, transparent tetrahedra, about as hard as quarts.  The mineral

has been found in San Juan County, Colorado, intimately mixed with guiter-

9-i
?A
n te, 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 j&x&xx provided a determination of the formula

weight. The cubic unit of structure was found to have a, = 13.82 2,

and to contain four molecules of composition ~ll,Si5020(gI, F)18CL

It was known in 1933 that in all silicates investigated up to that time
       i-h          tetra
each silicon atom WM surrounded m&xhedrally by four oxygen atoms, with

the silicon-oxygen distance about 1.60 2, and that aluminum atoms-

usually surrounded octahedrally by six oxygen atoms, whth aluminum-oxygen

distance 1.90 4, although a tetrahedral environment of aluminum atoms 6. 'A4 I
        ID&j ~*..L:~hjglt~& I)$ I
                                i'
                                .  It was not easy to find a way of combining

tetrahedra and octahedra Qr the sharing of corners and edges to give a
                      T
                       YbAe4.J. g )
cubic structure with the right    A but finally a single structure


-12-

                        x-ray
was discovered t\ and the calculated/intensities for the structure were

                                        structure

found to be in good agreement with experiment. In this ntrpandloDI 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 complex of five silicon tetrahedra. This tetrahedron-octa-

hedron framework defines cavities which are occupied by chloride ions.

    The structures of many other minerals (topaz, mica, qbaolinite,
            k;sisf                      6

chlorite, swedenborgite, etc.) were determined by the stochastic method.
                    d\

During recent years a number of complex intermetallic compounds

have been successfully investigated 4 in the same way. 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 poljrhedra of the icosa-

hedral group.  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
             if&t
five-fold axes.  However, it is found that Uq can be
                    t


-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 spheres 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.*

  .J.    The most complex

& 53. ': v-4,

-dm!~t metal structure ?e-+-+-& ia that
                     6

G

atoms is then surrounded by twenty atoms,

dodecahedron &, H each of these twenty atoms thus lying directly out

from the center of ma on&f the twenty faces of the icc$shedron.  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 BE which lie at the corners of a rhombic triacontahedron.

The next shell a&&xmaxmpndx consists of sixty atoms, each directly out

from the the center of a triangle which forms one half of one of the


-14-

                           n/
thirty rhombs bounding the ti hombic triacontahedmnj Xxxb*ese sixty

atoms lie at the corners of a truncated icosahedron, which has twenty

hexagonal faces and twelve pentagonal faces. Twelve additional atoms

1. --

are then located out fromjthe centers of twelve of the twenty hexagonal
                            /
     very large

faces.  TheLcomFlexes  are then onndensed together,with their centers at

the point of a body-centered cubic lattice, in such a way that each of, ._
          Y( 3,      t !
                                   CT                               4 ;
                %,fi' `z \f ~"~~~J\,VAh~   '
                                             &.b L (I :         .-.f>
                                                $. 3'     I ,.
                                                                           : ,"

the seventy-two outer atomsAis shared between two complexes; each.;then  I F i i !", '%
                                      ,&.,;, i. .'- *".  A:'~ -3.3
contributes thirty-six atoms per lattice point, which with the m

CS@ZUS of forty-five atoms / gives eighty-one atoms per lattice point,

or 162 in the unit cube.

The comnlexity 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 r&q&~

intermetallic compound with the simple formula NaCdz is to be attributed

to the same cause.  I have now been working on the problem of the struc-

              ?
ture of this crystal for thirty years*  and the structure still remains

--w--w
`8 L. Pauling, J. Am. Chem. Sot., 45, 2777 (1923).
                   --"II ___,,_, __-._ *.,
-a-----


-15-

undiscovered.  The crystal is cubic, with the edge of the unit cube slightly

over 30 4.  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 that the stochastic method will ulti-

mutely be successful.
mutely be successful.

csk Dr.
csk Dr.


    d let's put in a
    d let's put in a


    3 figurecif they
    3 figurecif they

The power of the stochastic method is illustrated by its recent

application to the p roblem of the configuration of polypeptide chains in

proteins.  The history of this application is illuminating also in showing

that an investigator who strives to anply 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          with Dr. alfred Mirsky, had

formulated a general theory of the structure and process of denaturation
  1
of proteins  , I spent several months in an unsuccessful effort to apply the

stochastic method in the discovery of an acceptable structure for a keratin.

.." ..--I
,-' -1
,/"     -w--w
9 % E. MLrsky and L. Pauling,
cc%. Pauling and        Proc. Nat. Acad. Sci., 22, 439 (19369.
          C. D. Coryell, Proc. Nat. Acad. Sci., 22, 210 (1936).
       ~-----


-16-

It was possible at that time to predict that the amide group in a peptide

would be planar, because of the resonance of the double bond between the

two positions C-O 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
,&,g,
would - the formation of hydrogen bonds between the

N-H group and the oxygen atom of the carbonyl group, with N-H*=*0 distance

approximately 2.80 2.                       9
          No reasonable configuration was found in this ..QSb~:tA
                        cy

investigation, however, and in consequence the possibility was considered

that the structural parameters of the polypeptide chain might be sig-

nificantly different from those predicted from information oktained through
                                                        I
   J           L .&%'flr-; L .h
the determination  of the structures of somewhat m sub-

stances.  At that time no structure determination had been made of any

amino acid, simple peptide, or other simple substance closely related to

    proteins. ?+& colleque  rofessor Robert B. Corey   ti            -t ..,.
                                                 who had &      5,



                                                                           ----.. . . .
               --- ^ ."_,-"-  Fe
,dfi.'? 2. yenrs earlie; made x-ray photographs of several    3' 1
              proteins &[ with R. W. G.
                 (
       -...-...- ._..                                                                . _
  Wyckof$                    and I concluded that we should em-

      .

bark upon a program of precise structurCI determination of these simple


-1%

                    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. tr

Through these investigations it waa found that the planarity of the amide

group and the interatomic distances and bond angles

of the simple substances and can confidently be expected to apply also

to the polypeptide chains in :xoteins.  In addition, hydrogen bonds with

N-B*.. 0 distance 2.79 i 0.12 !! have been found to be PM universally

present.

Even after this program was well under way, and WRWI&~EXX&~~,S

it was recognized that the structural parameters of the polypeptide chain

Gould 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 -


718 -

might well indicate a corresponding structural complexity. Then one day

in March 1948, while I was h--at my home in Oxford (where I was

                                      again

serving as Eastman Professor) @        a cold, I decided/to

attack the problem of the configur&tion of polypeptide chains, for the

first time in eleven years.  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

-I-sv --I. ',  .__ -J& (&&#yA&>e
   asymmetric element (xx&xBsx not its
          4         -9 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 helix.

helical

I attempted accordingly to find/configurations of polypeptide chains in-

volving planar amide groups with known dimensions, such that suitable Szm%&~&

hydrogen bonds were formed.  Within an hour, with the aid a
                                A  pencil and

a piece of paper, I had discovered a satisfactory helical structure. It

did not, however, explain B 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 Dr. H. R. Branson, a young man interested in the application of

mathematics to p chemical problems, that he make a search for other satisfadtory
             only e                          .
helical configurations. He found/one &~!-a F
                      2
                           . /

B and in 1951 a description of the two helixes, the a helix

     40
was published. '

a-----

@a"
I- L, PauUng, R. B. Corey, and H. R. Branson, jclaritperar
Nat. Acad. Sci., 37,'205 (1951).         w PC'
)". ^-_r . ,. --..          vfi

--s-w-       f+F i  6)
                +a .)
      Although the a heli "1\ which has about 3.6 amino-acid residues

per turn and a pitch of about 5.4 !? , did not explain in an obvious way

the skm principal feature of the x-ray diagram of the a-keratin proteins,

P

a strong meridional reflection with spacing 5.15 A, 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


-2o-

P

origin of the 5.15 A meridional reflection seemsAt have been solved
                        " I/

by the simultaneous suggestion by F. H. C. Cr!.ck, 46 and Professor

     SW
Corey and me' that in the a-keratin proteins the a helixes are twisted

about one another.             46
           A detailed structure is that shown in Figures 1 anda;

       a-
it involves yt" seven-strand cable of a helixes, with two additional

a helixes in the interstitial positions.

-e-----w

I' @% F. h. C. Crick, Nature, 170, 1882 (1952)`

\2/~~" t                                         -
  L. Pauling and R. B. Corey, Nature, 171, 59 (1953).

---se  --

Reasonably convincing evidence has also been obtained, largely

through the work of Peruta and Kendrew, who have studied hemoglobin and

myoglobin, and of Riley and Arndt, who have pr?pared radial-distribution
                          a- ;; +(.-f
                         maw    & h&l`. f
curves of a number of proteins,         @O$.$ pr;:.L& &J$?
               that/globular proteins contain
                     4-

with the configuration of the a helix.  In hemoglobin and qyoglobin these

a-helix segments lie in approximately parallel orientation to one another.


-2l-

No detailed information ixxsosxy&x 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 4 along the fiber axis.  The antiparallel

chain pleated-sheet structure, shown in Figure T , is predicted to have

the identity distance 7.00 t 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 protein. It is probable

that the p-keratin proteins (such as stretched hair) have a closely similar

structure, with, however, the x&qqgmr& polypeptide chains &x~mtingxiX

HI&zB&&~s&&~ in parallel, rather 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 class, and also different from that of silk

and the p-keratin proteins.  The stochastic method was used in the formula-
                        13
                              a--

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

diagram.
- z-2-    Despite much effort that has been expended on the problem,

   L. Pauling and R. E. Corey, Proc. Nat. Acad. Sci., 37, 272 (1952)
-a---


-22-

no satisfactory structure for collagen and gelatin has % yet been found.

The problem of the structure of collagen and gelatin 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 gmmxzrx hypothesis -

a guess. The second step'is to test the hypothesis, by some comparison

with experiment.  In general the test cannot be sufficiently thorough to
   J-L -w&AA
provide & proof that the 3qrpuWf9 hypothesis is correct - it

may easily be shown that the hypothesis is incorrect,

discovery of a significant disagreement with experiment, agreement on
                           h

a limited number of points cannot be accepted as verification of the

hypothesis.  In order for the stochastic method to be significant / the

principles used in formulating the hypothesis must be restrictive enough

to make the hypothesis itself essentially unique; in other words, e

investigator who makes use of this method should be allowed one guess,

If he were allowed many guesses he would sooner or later make one that

was not in disagreement with the limited number of test   points, but


there would then be little justification for accepting that

I may, however, contend that Professor Corey


                                    ---ai -3
?           b         collagen, bi



                         -.  @i?@Jv


Legends for Figures

                Cl+AsSq.
Fig. 1. The structure of enargite,/ The large circles re-

present sulfur atoms,     N-
       the small~circles copper atoms, and the small

shaded circles arsenic atoms.

Fig. 2.  The structure of sulvanite, Cu,VS,.

Fig. 3.  A portion of the structure of zunyite, "~,3Si~20(~, F)#.

AlO groups are w represented by octahedra, and SiO, and AlO, groups w

tetrahedra, the last being marked Al.  Smaller spheres represent oxygen

atoms, larger spheres chloride ions.  Groups of five tetrahedra and

twelve octahedra preserve their ic'entity in the structure.

  Fip.2. Diagrammatic representation of the configuration
. ..--` A

of the polypeptide chain in the a helix.

Fig.&  The a helix.

Pig. . 7  At the left, a compound a helix - an a helix whose

axis describes a helical configuration. The diameter, shown as about 10 f,
                                                                           J


-25.

includes the volume occupied by side chains as well as the main chains

of the protein.  Center, a 'irestrand a-cable. In the proposed structures


of proteins of the a-keratin type these ISX%~EE cables are packed together,

with compund helixes as hhown at the left in the interstices.   At the


right, a j-strand rope of a helixes.

    F%& A cross section of the a-keratin structure, showing


the 7-strand a-cables AB6 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.

fig* f-  Drawing representing the antiparallel-chain pleated-sheet

structure.


   Fig. 4. The structure of the intermetallic compound Hg32(Zn,U)qq.
The six drawings)from left to right in the top row and then left to tight
in the bottom zow)have the following significance: a central atom surrounded
twelve atoms at the points of

by/a nearly regular icosahedron;Pfrr the icosahedral group of thirteen atoms

twenty atoms at the points of

surrounded by/a pentagonal &decahedron; the comyllex of thirt~three atawa

surrounded by twelve atoms at the corners of cqicosahedron;  the outermost

shell of sixty atoms at the corners of a truncated ioosahedron, plus twelve

atoms out from the centera of twelve of the hexagons of this polyhedron;

packing drawing showing the complex of forty-five atoms plus an outer shell

of eevanty-tuo atoms; the structure of the crystal., in which these complexes

located about the points of a body-centered cubic lattice share all of the

seventy-two atoms of the outermost shell with neighboring complexes.