USING MAXIMUM-ENTROPY METHODS TO EXPLOIT MEASURED TRIPLET PHASES FOR THE STRUCTURE DETERMINATION OF SMALL PROTEINS
Edgar Weckert, Kerstin Hölzer, Ralf Müller, Klaus Schroer, Johannes Zellner
Institut für Kristallographie, Universität Karlsruhe (TH), 76128 Karlsruhe, Germany
Keywords: Three-beam interference, experimental phase determination, Maximum-entropy method
It has been shown during the last years that triplet phases can be determined from crystals of small proteins by means of three-beam interference experiments [1]. These experiments with crystals from a known protein structure (tetragonal lysozyme) revealed that it is possible to measure enough triplet phases (>800) in order to obtain a first model for the protein which can be used as start for least square refinement [2]. In using measured triplet phases for structure determination it has to be considered that (i) triplet phases have to be connected among each other to obtain single phases which are necessary for fourier analysis and (ii) especially for protein crystals only three-beam cases of reflections with large structure factor moduli give rise to a significant interference contrast. Like in a standard Direct Method approach the experimentally determined triplet phases can be connected to obtain single phases by building a suitable phasing tree starting from some origin fixing and e.g. semi invariants reflections, for which single phases can be determined directly in favourable cases. In general the experimental effort to connect each branch of this phasing tree with sufficient reliability will be large. Therefore, it is of advantage or even necessary in some cases at a certain stage of the structure determination to introduce symbolic phases for some reflections with large structure factor moduli. This concept, however, requires a possibility to determine the phases of these symbols by means of other e.g. statistical methods. The time required for the measurement of a single triplet phase is relatively long compared to a simple intensity measurement. About five triplet phases from a small protein structure can be determined per hour at an ESRF bending magnet beamline under good conditions. Therefore, in the course of the experiment it is important to measure the phases of triplets that will fix phases of reflections most important for the structure determination at that stage.
Inspired by the work of others [3,4] it has been tried to accomplish these tasks by developing a method using Bayesian statistics in this context. During this approach tentative trail phases will be assign to the symbolic phases for example by the well known magic integer method [5]. This will give hypothetic phase sets containing all the measured triplet phases. For each of these trail phase sets an entropy maximization (MEM) of the 'electron density' has to be calculated. The concept of likelihood [3] can then be used as figure of merit to determine the most likely set of symbolic trail phases. Hereby the extrapolating capabilities of a maximum entropy density on the moduli of so far unphased reflections is exploited. On the other hand unphased reflections with large structure factor moduli, that are badly predicted by a fourier inversion of the MEM electron density, are the best candidates for further measurements if an interpretation of the electron density is not yet possible.
The crucial step within this approach has been the development of a suitable entropy maximization algorithm. Two different approaches will be presented and compared with respect to their applicability for the structure determination by measured triplet phases and the calculation of an intrinsic positive electron density for accurate structure analysis.
Results will be presented for the structures of small molecules and a known small protein as well as a first attempt to an unknown protein structure.
This project is funded by the German Minister of Education and Research (BMBF) under contract No. 05 647 KA 5.