"The height of the vector rods in the three-dimensional Patterson of haemoglobin"

Date:
1952
Reference:
PP/CRI/I/1/2
Catalogue details

Licence: In copyright

Credit: "The height of the vector rods in the three-dimensional Patterson of haemoglobin". Source: Wellcome Collection.

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    and spaced 10-5  apart, with their length parallel to X. Bragg, Howells & Perutz (1952) have shown that the |. F(0jM)|'s are consistent with such an arrange ment. Pauling, Corey & Branson (1951) have proposed a helical fold for the polypeptide chain known as the 3-7 residue arhelix. When so folded, the amino acid residues repeat at regular intervals of 1-5 Á along the chain direction. Perutz (1951a, b) has reported a reflexion of 1-5  spacing from planes perpendicular to the fibre axis in artificial polypeptides and fibrous proteins of the a-keratin type and also, though less clearly, in haemoglobin. The reflexion in haemoglobin is possibly weaker than in hair, and certainly weaker than in the artificial polypeptides (Perutz, un published). The above facts might suggest that the structure of haemoglobin consisted mainly of polypeptide chains folded into 3-7 a-helices, and packed side by side parallel to the X axis of the crystal. Rough calculations show that the vector rods to be expected from such a model would be more dense than those observed. The present paper describes an attempt to calculate rather more accurately the absolute vector density of the rods on the basis of certain simple models, and to compare the results with the observed density. A parallel attack on this problem has been made by Bragg, Howells & Perutz (1952), who, working in two dimensions on the X projection, have shown that the absolute value of |.F(063)| is only one-third of the value calculated for a model consisting entirely of straight and parallel chains extending throughout the length of the molecule. We shall use the terms 'coiled chains' or 'rods' to denote the coiled polypeptide chains, and 'vector rods' to denote the regions of high vector density they produce in a Patterson. In what follows 'haemoglobin' will refer to horse methaemoglobin in the usual monoclinic form. Method of calculation It is necessary to have a method which is sufficiently accurate without being too laborious. Simple methods involving merely counting vectors tend to be in accurate; complete structure-factor and Patterson calculations would be prohibitively lengthy. The method described here is a compromise. The basic idea is to reduce the three-dimensional calculation to a two-dimensional one. This is done by calculating the average vector height over a short length in the rod direction. The steps in the calculation are as follows: (а) The idealized model. —A crystal is taken to consist entirely of infinite lengths of a-helices packed in an infinite regular hexagonal array. The Patterson of an end-on projection of the chains is calculated. (б) The real model. —The haemoglobin molecule is considered to be made up of a certain number of lengths of «-helix distributed in a certain way. The electron density of the «-helix is assumed to be uniform in the rod direction; by counting vectors the ratio is found between the vector densities, in three dimensions, of the idealized and the real model at chosen points in their Pattersons. (c) The experimental data. —Consider the chosen point in the observed Patterson. The average value over a length of 3 Á on either side of it in the rod direction is evaluated. This is then compared with the calculated value. It is assumed that this averaging will compensate for the theoretical assumption that the electron density of the a-helix was uniform in the rod direction. The advantages of this method are that in step (a) the effects of side chains, heat motion, diffraction etc., can be easily and accurately allowed for. This is difficult to do if a vector-counting method is used throughout. Vector counting is then employed in step (6) to get the ratio of vector density between a known and an unknown case. Finally, since the observed three-dimensional Patterson has already been com puted by Perutz (1949), the comparison with obser vation can easily- be made. In what follows, attention will be mainly concentrated on the part of the three-dimensional Patterson of the haemoglobin crystal in the neighbourhood of the X axis, and averaged between X = +9 and +15 Â. This stretch is chosen because it avoids the region very near the origin, and because the vectors in it are mainly due to each short length of a-helix with itself, and thus the relative arrangement of the various lengths of «-helix in the molecule is not important here. All calculations will be made in terms not of the absolute height of the Patterson but of the height above the average (above zero as usually plotted with F 2 (000) omitted). It can be shown that this can be done, and the liquid in the crystal ignored, if that liquid has an electron density near the average electron density of the protein, as happens to be the case here (Wrinch, 1950). Finally, it should be stated that the above procedure can be looked at in a more rigorous manner by considering how a Patterson can be built up from the 'self-Pattersons' and 'cross-Pattersons' of parts of the unit cell, and by working throughout with a fictitious electron density given Dy (actual electron density) — (average electron density). Subsidiary calculations from this standpoint have shown, for example, that diffraction effects in this particular problem are probably negligible, and that the averaging process will not produce serious errors. Simple theory Consider a single cylinder of constant electron density Q e. 3 , of cross-sectional area A, and of total length L placed between X = —\L and X = +\L, where X + ¿2 -- [L-Uh+h)]e 2 Alk e. 2 Á~ 3 , all dimensions being in Angstrom units. Here kA is the cross-sectional area over which the Patterson vectors spread, and over which the average is taken. Consider such rods, but of infinite length in an infinite hexagonal lattice and so spaced that the vectors in Patterson space from one rod to itself do not overlap those between neighbouring rods; then the vector density, averaged over the volume covered by the vector rods, is L'e'A/k e. 2 Â~ 3 , where L' is the length of the nominal unit cell in the X direction. Since this vector density will be in dependent of X we can calculate it from the two- dimensional end-on Patterson projection. In practice the Patterson vectors between adjacent rods will overlap, but by comparing cases in which the overlapping effects are very similar we can avoid appreciable errors due to this. For the more complicated case of a rod whose electron density is a function of Y and Z, but inde pendent of X, the above expressions will still hold, as can be seen by projecting the electron density on to the X axis. Note that the contours of the vector density in cross-section (perpendicular to X) will have the same shape, though different absolute values, in the two cases for which formulae are given above. Thus the ratio of the values at corresponding points in these two cases will be the same as the ratio of the average values over the corresponding cross- sections. Thus the ratio of the above expressions— one referring to a model with rods infinitely long, the other to a model consisting of rods of limited length— may be used to obtain the ratio of the vector densities at the appropriate points. Results (a) Height of vector peaks A section through the Fourier projection of the a-helix, placed in a hexagonal lattice 10-4 Á apart, and viewed end-on, is given in Fig. 1. Note that it is volcano shaped. Each unit of ordinate represents 2-25 e. -2 for 27  length of chain. The corresponding Patterson projection is given in Fig. 2. Each contour represents an average vector density of 15 e. 2 Á -3 for a unit cell 27  long. The chain and the Cß atom have been given the co-ordinates listed by Pauling & Corey (1951) for the 18-resldue 5-turn a-helix, and the remainder of the side chains have been put in uniformly in the area outside a circle of radius 4 Â, with sufficient electron density to make the overall average density 0-43 e. -3 . The 'heat-motion' as sumed was that calculated for haemoglobin. It is rather stronger than the value estimated by Perutz, a being taken as 25 instead of 20, where a is defined by (iy = (/„) exp [—a?y 2 ] and r¡ is the reciprocal spacing in A -1 . The limiting circle was taken as 2-8 Â, to correspond with the haemoglobin data. The calculated height of the peak of the central vector rod, averaged over the chosen strip between X = 9 and X = 15 Â, is given in Table 1 for several models. The last column includes a very rough allowance for vectors between chains with different X co-ordinates, assuming a plausible model for the molecule, while no such allowance is made in the preceding column. The observed height, obtained from the three- dimensional data of Perutz, and put on an absolute scale by measurements of Perutz, is 230 e. 2 Á~ s . Table 1. Calculated Patterson peaks