V. L. Lo

Iterative projection algorithms in protein crystallography. I. Theory

A general class of iterative projection algorithms is described and proposed as a tool for phasing in protein crystallography in order to improve the radius of convergence over that of conventional density-modification algorithms. Their relationship to conventional density modification is described. The common iterative projection algorithms, their convergence properties and their application to protein crystallography are described. These algorithms offer the possibility of protein structure determination starting with only information on the molecular envelope and low-order non-crystallographic symmetry.

Determination of molecular envelopes from solvent contrast variation data.

An algorithm is described for determining macromolecular envelopes from crystal diffraction amplitudes measured from a solvent contrast variation series. The method uses solvent contrast variation data that have been preprocessed to represent the structure-factor amplitudes of the envelope. The amplitudes are phased using an iterative projection algorithm that incorporates connectivity and compactness constraints on the envelope. The algorithm is tested by simulation on two protein envelopes and shown to be effective even in the absence of the very low resolution data, which are difficult to access experimentally.

Iterative projection algorithms in protein crystallography. II. Application

Iterative projection algorithms (IPAs) are a promising tool for protein crystallographic phase determination. Although related to traditional density-modification algorithms, IPAs have better convergence properties, and, as a result, can effectively overcome the phase problem given modest levels of structural redundancy. This is illustrated by applying IPAs to determine the electron densities of two protein crystals with fourfold non-crystallographic symmetry, starting with only the experimental diffraction amplitudes, a low-resolution molecular envelope and the position of the non-crystallographic axes. The algorithm returns electron densities that are sufficiently accurate for model building, allowing automated recovery of the known structures. This study indicates that IPAs should find routine application in protein crystallography, being capable of reconstructing electron densities starting with very little initial phase information.

Estimating Wind Velocities in Mountain Lee Waves Using Sailplane Flight Data

Abstract Mountain lee waves are a form of atmospheric gravity wave that is generated by flow over mountain topography. Mountain lee waves are of considerable interest, because they can produce drag that affects the general circulation, windstorms, and clear-air turbulence that can be an aviation hazard, and they can affect ozone abundance through mixing and inducing polar stratospheric clouds. There are difficulties, however, in measuring the three-dimensional wind velocities in high-altitude mountain waves. Mountain waves are routinely used by sailplane pilots to gain altitude. Methods are described for estimating three-dimensional wind velocities in mountain waves using data collected during sailplane flights. The data used are the logged sailplane position and airspeed (sailplane speed relative to the local air mass). An algorithm is described to postprocess this data to estimate the three-dimensional wind velocity along the flight path, based on an assumption of a slowly varying horizontal wind veloci...

Image reconstruction using symmetry

Symmetry provides a source of redundancy which can be exploited in image reconstruction. In particular, internal symmetries in molecules can help to compensate for the loss of Fourier phase information in macromolecular x-ray crystallography. Symmetry projections are incorporated into iterative projection algorithms for reconstruction of macromolecular electron densities from x-ray diffraction amplitudes from crystals. The effects of interpolation are studied and the algorithms are applied to reconstruction of an icosahedral virus.

Performance of Linear Block Codes Concatenated with Differential PSK

In this letter we investigate the bit and frame error rate performance of various linear block codes when concatenated with a differential phase shift keying modulator. The codes considered include single parity check, Hamming and product codes. We find block codes which significantly improve the frame error rate performance over previous results. An EXIT chart analysis is included.

Iterative projection algorithms for ab initio phasing in virus crystallography.

Iterative projection algorithms are proposed as a tool for ab initio phasing in virus crystallography. The good global convergence properties of these algorithms, coupled with the spherical shape and high structural redundancy of icosahedral viruses, allows high resolution phases to be determined with no initial phase information. This approach is demonstrated by determining the electron density of a virus crystal with 5-fold non-crystallographic symmetry, starting with only a spherical shell envelope. The electron density obtained is sufficiently accurate for model building. The results indicate that iterative projection algorithms should be routinely applicable in virus crystallography, without the need for ancillary phase information.

Characteristics of iterative projection algorithms

A brief description of various iterative projection algorithms and the relationships between them is given, along with some possible reasons for their ability to solve non-convex problems. An empirical model of their behaviour when applied to non-convex problems is also described.

REconstruction Of Macromolecular Envelopes From Crystal x-ray diffraction amplitudes

The envelope problem in macromolecular x-ray crystallography involves determining the boundary of a molecule from measurements of amplitudes of x-rays diffracted from a crystalline specimen. This represents a highly underdetermined image reconstruction problem with a large number of degrees of freedom. We regularize the problem by applying binary and connectivity constraints to the image, and seek the solution using the method of iterated projections. However, since the constraints are highly non-convex, the usual methods of generalized projections are not effective. We use the difference map projection algorithm and show that this is effective with simulated diffraction data from a protein envelope.

Aspects of binary image reconstruction from Fourier amplitude data

The reconstruction of a binary image from undersampled Fourier amplitude data is considered. Binary, connectivity, and compactness constraints are discussed and shown to be sufficient to enforce a unique solution. An iterative projection algorithm is described for a realistic crystallographic application.