Laboratory of Macromolecular Crystallography

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This is a review of the works carried out in LMC of the IMPB RAS. Information on other papers in this field may be found in the original papers listed below.

Development of an ab-initio procedure for solution of the crystallographic phase problem starting from low resolution.

(1990-2002)

       The aim of this project was to develop an ab-initio procedure for structure factor phasing. The term "ab-initio" is reserved here for the methods, in which the starting information is the structure factor magnitudes and knowledge of a general type, not connected with any additional experimental studies of an object.

       The traditional goal of the first stage of revealing a macromolecular structure is to find the function r(x,y,z), which presents the electron density distribution in the crystal of a studied object. This function is periodical in the three space directions and may be presented as a three-dimensional Fourier series

(1)     

      In crystallography, the complex coefficients F(h,k,l)exp[ij(h,k,l)] are referred to as structure factors while real values of F(h,k,l) and j(h,k,l) are called magnitudes and phases, respectively. In a standard X-ray experiment one can only determine the magnitudes F(h,k,l). The problem of restoring the phase values is called the phase problem of X-ray crystallography. Obviously, to solve it, some additional information on the studied object is required. Once approximate phase values have been found, they (together with experimental magnitudes) may be used to calculate an approximate density distribution by formula (1). If only a finite number of structure factors are used to calculate the series (1) one says that the Fourier synthesis of a finite resolution is calculated. The synthesis resolution depends on the number of structure factors used. The more terms in the series (1) are used, the more fine details may be recognized in analysing this synthesis.

      A general scheme of the ab-initio procedure developed may be presented as:

• a lot of phase sets are generated randomly;
• for every generated phase set some "selection criterion" value is calculated and the sets resulting in good enough values of this criterion are selected for further analysis;
• the selected phase sets are grouped in "clusters" of close sets;
• average phase values are calculated for every cluster.

       Obviously, the success of this procedure depends on the selection criterion used. Different criteria were tried as a working tool in this procedure.

       A Fourier synthesis histogram is a set of frequencies, which reflects how frequently a particular value of the electron density may be found in the Fourier synthesis. This histogram has a specific asymmetric shape for properly phased syntheses and is close to gaussian distribution for badly phased ones. A method was suggested to predict the true histogram for a protein crystal with as yet unknown three-dimensional structure. The closeness of the predicted histogram to that obtained from a trial phase set may be used as a selection criterion.

       Connectivity properties of high density regions provide with another selection criterion . For every Fourier synthesis a region may be defined which consists of the crystal cell points possessing of maximal density values. For a properly phased macromolecular syntheses these regions consist of a small number of connected pieces. In contrast, badly phased syntheses show a lot of small "drops". The number of connected components in the region having the highest synthesis values may serve as a selection criterion.

       Statistical likelihood. Every phase set results in a Fourier synthesis (calculated with these phases and observed magnitudes) and then the region of the highest values of this synthesis may be defined. The latter may be interpreted as a hypothetical molecular envelope. The likelihood corresponding to this envelope (and to the phases used to define the envelope) is defined as the probability to have the magnitudes calculated from an atomic model equal to the observed magnitudes if the coordinates of the model atoms are chosen randomly inside the envelope.

       Few Atom Models. The approach discussed is based on the hypothesis that a low-resolution Fourier synthesis may be approximated by a small number of "broad" gaussian functions. These functions may be considered as some huge pseudo-atoms or "blobs". Below we call such approximations Few Atom Models (FAM). The phases calculated from these blobs may be used as a reasonable approximation for low-resolution phases. In favourable cases even a one-blob approximation may provide a dozen of rather good phases. The problem is to define the coordinates of suitable centres of these blobs. For every FAM one can calculate the corresponding structure factors. The closeness of the calculated magnitudes to the corresponding observed values reflects to some extent the model quality and correspondingly the quality of the calculated phases.

       As it was found in numerous tests with a variety of selection criteria and macromolecular crystals, none of the known criteria allows identifying the correct phase set unambiguously. Usually the best variant in the population does not have the best value of the selection criterion. On the contrary, the best selection-criterion value may correspond to a totally wrong phase set. Nevertheless, such criteria are useful. There exists a statistical tendency that good variants have better criterion value than bad variants. To exploit this tendency, we formulate our task not as the one of finding the variant with the best criterion value, but rather as one of selecting all variants with a reasonable criterion value. It must be noted that the best variants may be lost in this process and some wrong variants may be retained. Nevertheless, this procedure increases the concentration of good variants in the selected population in comparison with the initial population. The following averaging of selected variants allows getting a reasonable start solution.

March, 24, 2003
V.Lunin

Publications

The full texts of papers

Lunin, V.Yu., Urzhumtsev, A.G. & Skovoroda, T.A. (1990). "Direct low-resolution phasing from electron-density histograms in protein crystallography". Acta Cryst., A46, 540-544.


Lunin, V.Yu. (1991). "Use of the electron-density-syntheses histograms or solving of the phase problem in protein crystallography". Preprint, Pushchino Research Center, Pushchino, Russia.


Lunin, V.Yu. (1992). "The use of statistical properties of the electron density Fourier syntheses for the solution of the phase problem in protein crystallography". Resume of Dr.Sci.Theses, ONTI PNC RAN, Pushchino, Russia. (In Russian)


Lunin, V.Yu. (1992). "The use of statistical properties of the electron density Fourier syntheses for the solution of the phase problem in protein crystallography". Dr.Sci.Theses, Institute of Crystallography RAS, Moscow, Russia. (In Russian)


Lunin, V.Yu. (1993). "Electron-Density Histograms and the Phase Problem". Acta Cryst. D49, 90-99.


Lunin, V.Yu., Lunina, N.L., Petrova, T.E., Vernoslova, E.A., Urzhumtsev, A.G. & Podjarny, A.D. (1994). "On the ab-initio solution of the phase problem for macromolecules at very low resolution. The Few Atoms Model method". Joint CCP4 and ESF-EACBM Newsletter on Protein Crystallography, 30, 37-44.


Lunin, V.Yu., Lunina, N.L., Petrova, T.E., Vernoslova, E.A., Urzhumtsev, A.G. & Podjarny, A.D. (1995). "On the ab-initio Solution of the Phase Problem for Macromolecules at Very Low Resolution: the Few Atoms Model Method". Acta Cryst., D51, 896-903.


Volkmann, N., Schlunzen, F., Urzhumtsev, A.G., Vernoslova, E.A., Podjarny, A.D., Roth, M., Pebay-Peyroula , E., Berkovitch-Yellin, Z., Zaytzev-Bashan, A. & Yonath, A. (1995). "On ab-initio phasing of ribosomal particles at very low resolution". Joint CCP4 and ESF-EACBM Newsletter on Protein Crystallography, 31, 23-32.


Urzhumtsev, A.G., Vernoslova, E.A. & Podjarny, A.D. (1996). "Approaches to Very Low Resolution Phasing of the Ribosome 50S particle from Thermus thermophilus by the Few-Atoms-Models and Molecular-Replacement Methods". Acta Cryst., D52,1092-1097.


Urzhumtsev, A. (1996). "Developpement de methodes et logiciels pour la determination de structures macromoleculaires par radiocristallographie. Applications e differents projets". Synthese d'activite scientifique, Strasbourg, 1996.


Podjarny, A.D., Urzhumtsev, A.G. & Lunin, V.Y. (1997). "Model based low resolution phasing". Iin Direct Methods for Solving Macromolecular Structures, ed. S.Fortier, NATO ASI Series C, Vol.507, 421-431.


Urzhumtsev A.G., Lunin V.Yu. & Podjarny A.D. (1997). "Low resolution crystallographic images". In "Recent Advances in Phasing", ed. By K.S.W.Wilson, G.Davies, A.W.Ashton & S.Bailey, Proceedings of the CCP4 Study Weekend, University of York, 3-4 January, 1997, 207-214.


Podjarny, A.D. & Urzhumtsev, A.G. (1997). "Low resolution phasing". In Methods in Enzymology, Academic Press, San Diego., C.W.Carter, Jr., R.M.Sweet, eds. 276A, 641-658.


Lunin, V.Yu., Lunina, N.L., Petrova, T.E., Urzhumtsev, A.G. & Podjarny, A.D. (1998). "On the Ab initio solution of the Phase Problem for Macromolecules at Very Low Resolution. II. Generalized Likelihood Based Approach to Cluster Discrimination". Acta Cryst. D54, 726-734.


Lunina, N.L. (1998). "Computational approaches to the solution of the low resolution phase problem in macromolecular crystallography". Resume of Ph.D. Theses, ONTI PNC RAN, Pushchino, Russia. (In Russian)


Lunina, N.L. (1998). "Computational approaches to the solution of the low resolution phase problem in macromolecular crystallography". Ph.D. Theses, ITEB RAS, Pushchino, Russia. (In Russian)


Lunin, V.Y., Lunina, N.L. & Urzhumtsev, A.G. (1999). "Seminvariant density decomposition and connectivity analysis and their application to very low resolution macromolecular phasing". Acta Cryst. A55, 916-925.


Petrova, T.E., Lunin, V.Y. & Podjarny, A.D. (1999). "A likelihood-based search for the macromolecular position in the crystalline unit cell". Acta Cryst. A55, 739-745.


Lunin, V.Y., Lunina, N.L., Petrova, T.E., Skovoroda, T.P., Urzhumtsev, A.G. & Podjarny, A.D. (2000). "Low-resolution ab initio phasing: problems and advances". Acta Cryst. D56, 1223-1232.


Urzhumtsev, A.G., Lunina, N.L., Skovoroda, T.P., Podjarny, A.D. & Lunin, V.Y. (2000). "Density constraints and low-resolution phasing". Acta Cryst. D56, 1233-1244.


Petrova, T.E., Lunin, V.Y. & Podjarny, A.D. (2000). "Ab initio low-resolution phasing in crystallography of macromolecules by maximization of likelihood". Acta Cryst. D56, 1245-1252.


Lunin, V.Y., Lunina, N.L. & Urzhumtsev, A.G. (2000). "Connectivity properties of high-density regions and ab initio phasing at low resolution". Acta Cryst. A56, 375-382.


Lunin, V.Y., Lunina, N.L., Petrova, T.E., Skovoroda, T.P., Urzhumtsev, A.G. & Podjarny, A.D. (2000). "Low-resolution ab initio phasing: problems and advances". Acta Cryst. D56, 1223-1232.


Urzhumtsev, A.G., Lunina, N.L., Skovoroda, T.P., Podjarny, A.D. & Lunin, V.Y. (2000). "Density constraints and low-resolution phasing". Acta Cryst. D56, 1233-1244.


Petrova, T.E., Lunin, V.Y. & Podjarny, A.D. (2000). "Ab initio low-resolution phasing in crystallography of macromolecules by maximization of likelihood". Acta Cryst. D56, 1245-1252.


Lunin, V.Y., Lunina, N.L., Ritter, S., Frey, I., Berg, A., Diderichs, K., Podjarny, A.D., Urzhumtsev, A. & Baumstark M.W. (2001). "Low-resolution data analysis for low-density lipoprotein particle". Acta Cryst., D57, 108-121.


Petrova, T.E. (2000). "Using of the maximum likelihood principle in the solution of the phase problem in macromolecula crystallography". Resume of Ph.D. Theses, Puschino, Russia. (In Russian)


Petrova, T.E. (2000). "Using of the maximum likelihood principle in the solution of the phase problem in macromolecula crystallography". Ph.D. Theses, ITEB RAS, Puschino, Russia. (In Russian)


Lunin, V.Y., Podjarny, A.D. & Urzhumtsev, A. (2001). "Low-resolution phasing in macromolecular crystallography". In : Advances in Structure Analysis, CSCA, Prague, Czech Republic, R.Kuzel & J.Hasek, eds., 4-36.


Urzhumtsev, A., Podjarny, A. & Lunin, V.Y. (2001). "Ab initio phasing starting from low resolution". Invited article. Euroconference on Phasing, 23-27 June 2001, Martina Franca, Italy, 4.35-4.40.


Lunin, V.Y., Urzhumtsev, A. & Bockmayr, A. (2002). "Direct phasing by Binary integer programming and its use for envelope determination". CCP4 Newsletter on Protein Crystallography, 40_12.


Lunin, V.Y., Urzhumtsev, A. & Bockmayr, A. (2002). "Direct phasing by binary integer programming". Acta Cryst. A58, 283-291.