W0194

Direct Methods of Non-uniform Distribution and their Mathematical Problems. D.Y. Guo, Hauptman-Woodward Medical Research Institute. 73 High St. Buffalo, NY 14203

Key steps to solve the mathematical problems of non-uniform distribution are given as follows. As well known, for uniform distribution <cos(2[pi]Hr_k)> = 0, and <cos²(2[pi]Hr_k)> =1/2, provided that r_k's are randomly distributed in whole unit cell [1]. However, For protein structures, waters in solvent regions are distributed in the between protein molecules, and the position parameters r_k for both water and protein atoms, or globs, are actually not uniformly distributed. Then we have mathematical problems: what are the values of <cos(2[pi]Hr_k)> and <cos²(2[pi]Hr_k)> over these restricted position parameters r_k?

It is easy to see the problems by some ideal and extreme cases which can show the theoretical differences between uniform and non- uniform distributions. Let V, Vp and Vs be the volumes of unit cell, protein molecule and solvent, respectively. Then, it is not difficult to find the problems that 1. If Vs/V is very small, <cos(2[pi]Hr_k)>_ 1 and also <cos²(2[pi]Hr_k)> _ 1 for solvent position parameters. 2. If Vp/V is very small, there are the same problems for protein atoms.

This mathematical problem has been solved for the cases with Vs/V in a range from 0.2 to 0.7, which are generally satisfactory for most protein structures. Then the theoretical routine [1] can be applied even for protein structures.

[1]. H. A. Hauptman, Crystal Structure Determination, The book of 1972.

Support from NIH grant GM-46733 is acknowledged.