Title: Anisotropic twicing for single particle reconstruction using autocorrelation analysis,
Abstract: The missing phase problem in X-ray crystallography is commonly solved using
the technique of molecular replacement, which borrows phases from a previously
solved homologous structure, and appends them to the measured Fourier
magnitudes of the diffraction patterns of the unknown structure. More recently,
molecular replacement has been proposed for solving the missing orthogonal
matrices problem arising in Kam's autocorrelation analysis for single particle
reconstruction using X-ray free electron lasers and cryo-EM. In classical
molecular replacement, it is common to estimate the magnitudes of the unknown
structure as twice the measured magnitudes minus the magnitudes of the
homologous structure, a procedure known as `twicing'. Mathematically, this is
equivalent to finding an unbiased estimator for a complex-valued scalar. We
generalize this scheme for the case of estimating real or complex valued
matrices arising in single particle autocorrelation analysis. We name this
approach "Anisotropic Twicing" because unlike the scalar case, the unbiased
estimator is not obtained by a simple magnitude isotropic correction. We
compare the performance of the least squares, twicing and anisotropic twicing
estimators on synthetic and experimental datasets. We demonstrate 3D homology
modeling in cryo-EM directly from experimental data without iterative
refinement or class averaging, for the first time.