STATISTICAL EXPECTATION VALUE OF THE DEBYE-WALLER FACTOR AND E(HKL) VALUES FOR MACROMOLECULAR CRYSTALS

If the unit-cell distribution of atomic mean-square displacement param eters B = 8 pi(2)[mu(2)] is assumed to be normal, with mean mu = [B] a nd variance sigma(2) = [(B-[B])(2)], the statistical expectation value of the Debye-Waller factor W-2 = exp(-2Bs(2)), where s = (sin theta)/ lambda, is [W-2] = exp[-2(mu-sigma(2)s(2))s(2)]. This result has been incorporated into procedures for scaling and normalizing measured Brag g intensities to their Wilson expectation values. The procedures can d etermine both isotropic mu(B) and sigma(B) and anisotropic mu(U-ij) an d sigma(U-ij) distribution parameters. Tests with experimental data an d refined structural models for several protein crystals show that the procedures yield reliable normalized structure-factor amplitudes for direct-methods applications, with values of R = Sigma(h)\\E(o)\ - \E(c )\\/Sigma(h)\E(o)\ averaging similar to 5%.