One of the difficulties associated with the X-ray study of biological structures arises from the fact that such structures, while not in general unorganized “powders,” are nevertheless usually built up of numerous submicroscopic individuals of continuously varying orientation: in the typical biological “fibre,” for example, the imperfectly crystalline molecular aggregates all lie with one and the same crystallographic direction either approximately parallel to the fibre-axis or spirally inclined at some approximately constant angle to it; but subject to this limitation there may be present within the compass of the X-ray beam all orientations up to the maximum possible consistent with axial symmetry. This means that though we may not be condemned to work in the least profitable field of X-ray technique, that of the completely random “powder photograph,” yet we are debarred from the full geometrical advantages to be derived from operating with a single macroscopic crystal. Speaking briefly, the main trouble lies in the difficulty or impossibility of measuring sufficient inter-directional
angles
to define the molecular arrangement without ambiguity. Sometimes it is possible to draw very plausible conclusions, or even conclusions almost certainly correct; but in others the diffraction effects are so ill-defined as to preclude altogether the use of direct geometrical argument, and compel us to fall back on indirect reasoning based on evidence from various sources, including comparative photographs of related structures. The X-ray investigation of proteins in particular is a many-sided enquiry of this nature, for the diffraction effects are susceptible of interpretation only in relation to other physical and chemical data. The X-ray photographs then serve to give form, so to speak, to such data—to provide the three-dimensional framework necessary to build them into a coherent whole. Papers I and II in this series show how, working along these lines, it has been found possible to derive the basic features of the keratin molecule or complex, both in its unextended form (α) and in its extended form ((β), and to apply the proposed model to the interpretation of the long-range elasticity and other characteristic properties of mammalian hairs. The structure of β-keratin may be described most simply as that of a flat “polypeptide grid,” in which a succession of fully extended main-chains are bound side by side through linkages, both electrostatic and co-valent, between certain of their side-chains; while that of α-keratin (the normal equilibrium form) may be thought of as derived from (β-keratin by a regular folding of the main-chains in planes transverse to the side-chains. By this means the length of the molecule in the direction of the main-chains is reduced to approximately one-half (the average distance apart of the side-chains is decreased from rather less than 3·4 A to about 1·7 A), while the average separation of the main-chains in the plane of the side-chains (the plane of the “grid”) remains roughly constant (9·8 A). In the β-keratin crystallites the grids are piled one on top of another with the main-chains parallel and separated by a distance of 4·65 A.
A geometrical argument for a theorem of G. E. Welters