TY - CHAP

T1 - The use of wavelet transforms in low resolution phase extension

AU - Main, Peter

AU - Wilson, Julie C.

PY - 2001/6/1

Y1 - 2001/6/1

N2 - A method to extend low-resolution phases has been developed using histogram- matching, not only of the electron density, but also of histograms obtained from the different levels of detail in the electron density provided by its wavelet transform. Like Fourier analysis, wavelet analysis can be used to express an image (the electron density) in terms of a set of orthogonal functions. Unlike Fourier analysis though, the wavelet functions are localized in position as well as frequency. The wavelet transform of an image therefore provides a set of wavelet coefficients, each one giving the size of the contribution of the corresponding wavelet function to a particular position in the image. Using mathematical models to describe the different histograms we are able to predict the coefficients for an increased resolution and the inverse transform allows a new image to be reconstructed from these wavelet coefficients. The procedure alternates between real and reciprocal space so that the positions of new features in the map are also guided by the diffraction pattern. The method has been tried on a large number of model structures varying in size, solvent content and space-group. There is a build-up of errors as the calculation proceeds, but, starting with a good 10A map, we are currently able to produce new phases to about 6-7A with reasonable phase errors on all the structures tested. In most cases, the 10A map is little more than a mask roughly covering the molecule, whereas in the maps we obtain, secondary structure can often be identified. It is hoped that the addition of further information, such as knowledge of secondary structure, will improve the method and eventually allow phase extension to a resolution at which existing density modification techniques are effective.

AB - A method to extend low-resolution phases has been developed using histogram- matching, not only of the electron density, but also of histograms obtained from the different levels of detail in the electron density provided by its wavelet transform. Like Fourier analysis, wavelet analysis can be used to express an image (the electron density) in terms of a set of orthogonal functions. Unlike Fourier analysis though, the wavelet functions are localized in position as well as frequency. The wavelet transform of an image therefore provides a set of wavelet coefficients, each one giving the size of the contribution of the corresponding wavelet function to a particular position in the image. Using mathematical models to describe the different histograms we are able to predict the coefficients for an increased resolution and the inverse transform allows a new image to be reconstructed from these wavelet coefficients. The procedure alternates between real and reciprocal space so that the positions of new features in the map are also guided by the diffraction pattern. The method has been tried on a large number of model structures varying in size, solvent content and space-group. There is a build-up of errors as the calculation proceeds, but, starting with a good 10A map, we are currently able to produce new phases to about 6-7A with reasonable phase errors on all the structures tested. In most cases, the 10A map is little more than a mask roughly covering the molecule, whereas in the maps we obtain, secondary structure can often be identified. It is hoped that the addition of further information, such as knowledge of secondary structure, will improve the method and eventually allow phase extension to a resolution at which existing density modification techniques are effective.

M3 - Other chapter contribution

SN - 1 58603 080 9

VL - 325

T3 - Nato Science Series I : Life and Behavioural Sciences,

SP - 82

EP - 94

BT - Methods in Macromolecular Crystallography

A2 - Turk, Dusan

A2 - Johnson, Louise

PB - IOS Press

CY - Netherlands

ER -