Helix diffraction theory

Both in theory and in practice there exist eight gluco-pyranose homopolymers, and some of the molecular conformations of three of these, i.e. cellulose and amylose (l.,2., 3,4), and (1+3)-8-D-glucan (5.,6.,.7) have been established by x-ray analysis. Although (1+3)-a-D-glucan is among the five homopolymers previously unsolved by x-ray diffraction, possible chain conformations were predicted with computers to be an extended ribbon (8.,9.) a single helix (9.), or a double or triple helix (10). 

X-ray diffraction studies (68-78) of the possible conformations available in hyaluronic acid in solution indicate that helices packed together in various arrangements are possible. It is therefore appropriate to relate the present analysis to the theory of the helix-to-coil transition. In the model developed by Zimm and Rice (79) for helix-to-coil transitions in charged macromolecules, the fraction, 6, of intact hydrogen bonds in the helix is given by ... 

The complete solution of the problem has to take into account the effect of multiple reflexions. This can be done by setting up difference equations closely similar to those formulated by Darwin< > in his dynamical theory of X-ray diffraction. For the purposes of this theory we shall regard the liquid crystal as consisting of a set of parallel planes spaced P apart. Each plane therefore replaces the m layers per turn of the helix. We ascribe a reflexion coefficient —iQ per plane for right-circular light at normal incidence. Assuming a kinematical approximation for the m layers, Q is given by... 

The team in Milan had to face two major scientific problems assign the structure of the new polymer and explain its genesis. The first part could be accomplished with a clever analysis of the X-ray diffraction patterns, recognized by Corradini [20 and references therein] as compatible with the 3i helical conformation anticipated by Bunn [21] for a hypothetical stereoregular hydrocarbon with the configuration that we now name isotactic after the suggestion by Natta s wife [11] (Fig. 3). 1 am not a believer in Lukacs reflection theory [22], but 1 find it a fascinating coincidence that Watson and Crick published their fundamental work on the a-helix of DNA at about the same time [23]. 

It has already been pointed out that Moffitt s theory predicts a negative bo for right-handed helices and a positive one for left-handed helices. However, in view of the incompleteness of the theory, it is necessary to have an experimental demonstration of the relation between the sign of bo and the screw sense of the helix. Elliott and Malcolm (1958) have analyzed the X-ray diffraction pattern of fibers of poly-L-alanine in the -helical form and have concluded th t the helix is right handed. Since optical rotation measurements on solid films of poly-L-alanine give bo = —475°, it would seem to follow that a negative value of bo corresponds to a right-handed screw sense (Elliott et al., 1958). 

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