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 parameters B = 8 pi(2)[mu(2)] is assumed to be normal, with mean mu = [B] and 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 Bragg intensities to their Wilson expectation values. The procedures can determine both isotropic mu(B) and sigma(B) and anisotropic mu(U-ij) and sigma(U-ij) distribution parameters. Tests with experimental data and 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%.