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Credit: Dissertation in Draft: Chapter III. Source: Wellcome Collection.
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![W. COCHRAN, F. H. C. CRICK AND Y. VAND the atoms in a helical structure. We have to consider how\he contributions of a number of sets of atoms on helices of different radii, and which may start off with thèyfirst atoms not at x = r, y = 0, z — 0, but at X = r ôos 9 o, y = r sin <p ,z = z, are to be combined. The transform of a discontinuous helix, starting with the last-mentioned coordinates, is ^ exp [t(— n<p-\-2nlzlc)] . This follows from , the fact that the displacement of the first point to z in a cell of length c corresponds to a multiplication of the transform by the factor exp [2jiilz/c]. The rotation of the helix through an angle <p, to bring the first point to the coordinates given above, results in its transform being rotated in the same direction, and by the same amount. A point then at (R, ip, Ç) obviously came from (R, ip—<p, £)■ Hence if a particular term in the series for F had the form J n (2nRr ) exp [m(y+f^)]> ^ now becomes J n {2nRr) exp [i(nip—ncp+^nh^-2nlz/c)] . (7) In poly-y-methyl -l -glutamate there are ten atoms per residue, and each chain consists of sets of identical atoms occurring at the points of ten different dis continuous helices. It follows that the structure factor F c of one such unit, for I = 1 for example, is given by F C (R, ip, 1/c) = — :Rfj) exp [i{— l(ip+i¡7t)+7q> i +27iz ¡ /c}']+ io fjJ^ZnRrj) exp [¿{ll(y)+^)—llç> ? +2wz ? /c}]+,/.. The general expression is F C (R, ip, l/c) = 2!2J fi J ni^ Rr i) ex P [*{»(v>— I e }] ■ ( 8 ) For the purposes of computation it! would be useful to graph the functions C„ = cos (nip)J n (2j[Rr), S n — sin ( nip)j/ n (2nRr) . If we write (7) in the form' J n (2jcRr) exp [in(ip + £)],/where e = q)-\~27ilz/nc , then (7) becomes C n (ipA-e)-\-iS n {ip-\-s). If, for example, one prepares a contour map of C„ against cylindrical coordinates (2jiRn;ip), one can then place over it a grid whose intersections correspond to the reciprocal- lattice points for the value of r appropriate to a particular set 7 of atoms. By turning this grid to the angle e (which one has to compute) one can read off C n {ip+ e)/for all the reciprocal-lattice points. This process/can be repeated for each atom, and the contributions summed; similarly for S n (ip+e). This'is particularly valuable when the phases of the Fourier components are required. The theory can easily be extended to cover cases where there is more than one chain per lattice point by considering a chain displaced from the origin to the point (x 0 , y 0 , z 0 ) and turned about its axis by an angle rp 0 . The contribution of this chain to the trans form is obtained by multiplying the general expression (7) by a factor exp [2jti(hx 0 ¡a-\-ky 0 lb+lz 0 lc)] exp [— in(p 0 ] . It is interesting to note that the helix which con sists, chemically, of one polypeptide chain, is in fact only one of the possible solutions which are consistent with the general helical arrangement. For example, a discontinuous right-handed helix which has p= 1-5 Â, and makes 5 turns in 27 Â may be regarded as two separate but intertwining left-handed discontinuous helices, each with p — 3-0 Â and making 4 turns in 27 Â. Such structures will generally not be stereo- chemically feasible. Conversely, if one has to consider a structure Which actually does consist of several chains intertwined, it is convenient for computation to imagine the residues, however they may be con nected chemically, to be associated with a single 'primitive' helix, which is chosen as the one for which both the z translation and the angle of rotation be tween successive residues have the smallest values. All calculations can be made in terms of this one helix, 'using the theory given above. 5. Application to poly-y-methyl-l-glutamate Ás we have seen, we may imagine the infinite poly peptide chain as made up from a number of sets of atoiris, each set consisting of atoms occurring at inter vals p on a helix of axial spacing P and radius r. The number of sets of atoms composing one chain will be equal to the number of atoms per residue of the polypeptide,; each set will in general occur on a helix of different Radius, and if we take one helix as a standard, the/others will in general be rotated and translated relative to the first. This helical con figuration (defined by P and p) of every set of atoms is in itself enough^ to enable us to make general predictions about the intensities of the X-ray re flexions to be expected from such a structure—there is no need for detailed -assumptions about the exact relative positions of atoms belonging to the same residue to be made. This is a situation which occurs very seldom-in X-ray analysis ; usually a crystat- ^brilcíúre problem must be solved in detail before anything at all ^an'be said aboüt the atomic arrange ment, and ' conversely only when the structure jis known completely can the intensities of the X-ray reflexions he üalculated^fe-this-ease, the basis of the predictions is that reflexions to which all sets of atoms make only a small contribution will be abspnt, whereas](https://iiif.wellcomecollection.org/image/b1817467x_PP_CRI_F_1_3_0006.jp2/full/800%2C/0/default.jpg)
