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 the Few Atom Model method for solution of the phase problem in protein crystallography.

(1994-1998)

      This project was developed in collaboration with the Laboratory of Biological Structures of the Institute of Genetics and Molecular and Cellular Biology (IGBMC) (Strasbourg, France, PI Prof. A.D.Podjarny). The project was aimed at generalising the procedure of low-resolution ab-initio phasing, which was initially developed to manage the information on Fourier syntheses histograms [1].

      The traditional goal of the first stage of determining a macromolecular structure is to find the function ρ(x,y,z), which presents the distribution of electron density 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[iφ(h,k,l)] are referred to as structure factors while real values of F(h,k,l) and φ(h,k,l) are called magnitudes and phases, respectively. In a conventional 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, some additional information on the studied object must be attracted to solve this problem. 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) they say 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. Low-resolution syntheses are calculated with the use of a small number of structure factors having the smallest indices hkl. Such syntheses allow one to locate the molecules in the crystal cell and to have the first information on the molecule shape.

      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 the 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. Nevertheless, a straightforward search for blob positions, which optimise some formal criterion (e.g. maximise the magnitude correlation coefficient) often leads to a false optimum and the corresponding phases do not match the true ones. As an alternative to the global optimization, a procedure of Monte Carlo type may be suggested [3, 4, 8], which includes the following steps:

a large number of FAMs are generated randomly and a set of structure factors is calculated for each generated model;
if the correlation of the calculated and observed magnitudes is large enough, then the phases corresponded to this FAM are announced "admissible" and stored for further analysis;
the selected phase sets are grouped into "clusters" of close phase sets;
the average phase values are calculated for each of isolated clusters.

      A small number of sets of the averaged phases (one set for one cluster) are considered as alternative (for the moment) solutions of the phase problem. It should be noted that an essential feature of cluster analysis and phase averaging is an alignment of phase sets in accordance with the set of admissible shifts of the origin [2,6].

      Sometimes cluster analysis reveals only one significant cluster and averaging in this cluster provides a unique (for the current step of the study) solution of the phase problem. If several clusters have been found, the problem arises to choose the best one. For each cluster the averaged phases (together with the observed magnitudes) may be used to calculate Fourier synthesis and to determine a molecule envelope on the basis of this synthesis. The likelihood based choice may be used as a way to choose out of alternative envelopes. In brief, the idea of the likelihood based choice may be explained as follows. Let some region in the crystal unit cell be specified. We hope that this region is an approximate envelope of the molecules, i.e. the majority of atoms are located in this region. The soundness of this hypothesis may be checked with the following statistical test. Let the atomic coordinates be chosen randomly in the tested region and the structure factors corresponding to this random model be calculated. The values of these structure factors are random variables and we may inquire what the probability for the calculated magnitudes to coincide with the observed ones is. One may believe that this probability will be high if the tested region does contain the most part of real atom positions. On the contrary, if the region has nothing to do with the real molecules envelope, then the atoms placed in this region will hardly ever reproduce the observed magnitudes. The probability for the calculated magnitudes to be equal to the observed ones will be low in this case. The probability discussed is nothing else than a statistical likelihood corresponding to the hypothesis that the set of experimental magnitudes may be considered as the magnitudes calculated from the model generated randomly inside the tested region. If several alternative regions are suggested as candidates for a possible molecule envelope, then the likelihood may be calculated for each of them. The choice of the region which provides the maximal likelihood is just the realisation of the maximal likelihood principle, which is widely used in mathematical statistics. In practice, the value of the likelihood corresponding to a particular envelope may be estimated by a specially designed computer simulation procedure of Monte-Carlo type.

      Statistical modelling and likelihood based methods may be used for solving of some other crystallographic problems too.

      The approach developed was used in low resolution phasing of ribosomal T50S particle [5, 7, 10, 11, 13] and low density lipoprotein [12].

      The main results concerning the development of this approach were summarised in N.Lunina's PhD thesis [10, 11].

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. & Woolfson, M.M. (1993). "Mean Phase Error and the Map Correlation Coefficient". Acta Cryst., D49, 530-533.


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 Crystallograph, 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.


Lunin, V.Yu. & Lunina, N.L. (1996). "The Map Correlation Coefficient for Optimally Superposed Maps". Acta Cryst. A52, 365-368.


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 Crys., D52,1092-1097.


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


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. These, 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., 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.


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.