Entropy, likelihood and phase determination

Initial results with unknown protein structures are also impressive. Joint imposition of maximum-entropy and solvent-flatness conditions leads to substantial map improvement by two different algorithms, including one procedure called maximum entropy solvent flattening (MESF) [[11]xA multisolution method of phase determination by combined maximization of entropy and likelihood. I. Theory, algorithms and strategy. Bricogne, G. and Gilmore, C.J. Acta Crystallogr. A. 1990; 46: 248–297Crossref | Scopus (100)See all References, [12]xEntropy maximization constrained by solvent flatness: a new method for macromolecular phase extension and map improvement. Xiang, S., Carter, C.W Jr, Bricogne, G., and Gilmore, C.J. Acta Crystallogr. D. 1993; 49: 193–212Crossref | PubMedSee all References] and another that has not been described in detail [[13]xCrystallographic determination of the active site iron and its ligands in soybean lipoxygenase L-1. Minor, W., Steczko, J., Bolin, J.T., Otwinowski, Z., and Axelrod, B. Biochemistry. 1993; 32: 6320–6323Crossref | PubMedSee all References, [14]xThe 2.2 a crystal structure of transducin-α complexed with GTP-γS. Noel, J.P., Hamm, H., and Sigler, P.B. Nature. 1993; 366: 654–663Crossref | PubMedSee all References, [15]xThe crystal structure of the bacterial chaperonin GroEL at 2.8 a. Braig, K..., and Sigler, P.B. Nature. 1994; 371: 578–586Crossref | PubMed | Scopus (894)See all References].The real promise of these methods, however, lies in their ability to use the native amplitudes to determine new phases directly, by the Bayesian process of hypothesis generation, likelihood scoring and statistical testing. Two recent structure determinations illustrate how using MICE in this way can transform otherwise unsolvable structures into solved ones. The first of these was Bacillus stearothermophilus tryptophanyl-tRNA synthetase (TrpRS) [[10]xOvercoming non-isomorphism by phase permutation and likelihood scoring: solution of Bacillus stearothermophilus tryptophanyl-tRNA synthetase. Doublie, S., Xiang, S., Gilmore, C.J., Bricogne, G., and Carter, C.W Jr. Acta Crystallogr. A. 1994; 50: 164–182Crossref | PubMed | Scopus (20)See all References, [16]xTryptophanyl-tRNA synthetase crystal structure reveals an unexpected homology to tyrosyl-tRNA synthetase. Doublie, S., Bricogne, G., Gilmore, C.J., and Carter, C.W Jr. Structure. 1995; 3: 17–31Abstract | Full Text | Full Text PDF | PubMed | Scopus (137)See all References]. The phase problem, intensified by a serious loss of isomorphism upon derivatization, was ultimately solved using isomorphous (selenium–sulfur) differences between native and selenomethionyl-TrpRS. MESF alone was unsuccessful because of phase errors in the starting basis set. Strong, unphased reflections were therefore identified and their phases permuted according to full and incomplete factorial sampling designs [17xProtein crystallization using incomplete factorial experiments. Carter, C.W Jr and Carter, C.W. J. Biol. Chem. 1979; 254: 12219–12223PubMedSee all References[17]. Nodes were scored using the LLG, and phase indications were then estimated jointly for all permuted reflections by least squares with t-testing. A similar permutation was carried out for five different binary choices of how to calculate and describe the molecular envelope. The resulting phase improvement made it possible to locate nine out of ten selenium atoms in an isomorphous difference Fourier map and thereby incorporate these isomorphous differences into the final experimental phase determination.The combined multiple isomorphous replacement with anomalous scattering (MIRAS) phases produced a map with a clearly defined solvent boundary and secondary structures, but with poor continuity and side-chain definition (Figure 2aFigure 2a). MESF, permutation of seven more phases, and phase recombination produced a significantly improved map (Figure 2bFigure 2b) in which all main-chain atoms and virtually all side chains have well-defined density, and which closely resembles the final {2|Fobs| - |Fcalc|,φcalc} map (Figure 2cFigure 2c).View Large Image | View Hi-Res Image | Download PowerPoint SlideView Large Image | View Hi-Res Image | Download PowerPoint SlideView Large Image | View Hi-Res Image | Download PowerPoint SlideFigure 2Electron-density maps from the tryptophanyl-tRNA synthetase (TrpRS) phase determination. (a) The best experimental (MIRAS) map, calculated with phases based on a gold-mercury double derivative, a lead acetate derivative, and isomorphous differences from selenomethionyl-TrpRS. Main-chain continuity is lacking in many regions. (b) The centroid map after MESF and direct phase determination for seven intense reflections for which the experimental phases were unreliable. (c) The final {|2Fobs| - |Fcalc|,φcalc} map. Maps in (b) and (c) are virtually indistinguishable. (Reproduced from [10xOvercoming non-isomorphism by phase permutation and likelihood scoring: solution of Bacillus stearothermophilus tryptophanyl-tRNA synthetase. Doublie, S., Xiang, S., Gilmore, C.J., Bricogne, G., and Carter, C.W Jr. Acta Crystallogr. A. 1994; 50: 164–182Crossref | PubMed | Scopus (20)See all References[10] with permission.)Human chorionic gonadotropin (HCG) [18xCrystal structure of human chorionic gonadotropin. Lapthorn, A.J..., and Isaacs, N.W. Nature. 1994; 369: 455–461Crossref | PubMed | Scopus (677)See all References[18] provided a second example of crucial map improvement by phase permutation. This fascinating structure consists of two interwoven subunits with very similar antiparallel β-sheet structures whose interior cores are crosslinked by five and six disulfide bonds, respectively. As disulfide and main-chain directions were nearly indistinguishable, map interpretation involved a very large number of potential ambiguities in the part of the structure that normally provides the most confident tracing (Figure 3aFigure 3a). Tracing the chain correctly therefore required unambiguous connection of surface loops, normally the least confident part of the map. Conventional solvent flattening of the MIR map failed to define the surface loops (Figure 3bFigure 3b). Application of MESF again provided insufficient map improvement, so phases were permuted for seven reflections with large re-normalized structure-factor amplitudes. The resulting phase choices improved the density of surface loops sufficiently to permit an unambiguous chain tracing and refinement of the structure (Figure 3cFigure 3c).Figure 3Electron-density maps from the human chorionic gonadotropin phase determination (kindly provided by A Lapthorn and N Isaacs). (a) The best experimental map. (b) The map following conventional solvent flattening. (c) The map after MESF and direct phase determination for seven intense reflections. Only this map shows the continuity of the surface loop in the upper right.View Large Image | View Hi-Res Image | Download PowerPoint SlideAn encouraging aspect of these two case studies is how few reflections were necessary to overcome otherwise difficult problems. For TrpRS, 35 reflections were phased directly. Of these, 28, including one basis set reflection whose incorrect phase led to a large re-normalized structure-factor amplitude, were phased prior to locating the selenium atoms. Seven others, phased to improve the best MIRAS map (Figure 2aFigure 2a), were responsible for some of the improvement between the maps in Figure 2aFigure 2a and Figure 2bFigure 2b. Post hoc comparison with phases from the refined model showed that all three centric reflections were phased correctly, and that the mean phase error for four acentric reflections phased by quadrant permutation was 27°. The incomplete factorial design thus provided surprisingly accurate interpolation between the four sampled phase values. Only one permutation was done for HCG, without trying to identify incorrect basis-set phases. Nevertheless, adding seven new reflections to the basis set with phases inferred from likelihood scoring improved side-chain density in crucial regions of the map.These successes were achieved working with difficult structures for which conventional methods had failed to provide interpretable maps. Thus, they confirm the relevance to macromolecular crystallography of the theoretical principles and computational procedures developed by Bricogne. The central tools of the Bayesian paradigm —using maximum entropy as the basis for joint probability distributions, maximum-entropy extrapolation as the basis for likelihood scores and for identifying the most important missing phase information, and sparsely sampled factorial designs —all function as predicted, even with moderate resolution (3.0 a) data. The work to date represents only the first steps towards use of the Bayesian paradigm in the more challenging contexts envisioned by Bricogne. Future applications should benefit a wider group of macromolecular crystallographers and may ultimately permit ab initio phase determination.