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ABSTRACT
Refinement of atomic models is a necessary step in solving the macromolecular structure by X-ray diffraction methods. Nowadays, high automation and well-developed interfaces give a possibility to use the most popular refinement programs as black boxes. Nevertheless, working with complex objects requires an understanding of the internal structure and principles of operation of these programs and critical assessment of the results of refinement. In this review, we discuss the basic principles of the organization of refinement programs and the history of their improvement and development, as the studied objects became more and more complicated. The discussions are kept at the level of basic mathematic knowledge avoiding unnecessary formalism and too detailed expressions.
Acknowledgements
The authors thank J. Helliwell for an informal and very valuable help with preparation of this manuscript and G. Andersen, J. Cavarelli and B. Klaholz who influenced preparation of this review. AU acknowledges the infrastructure of Instruct-ERIC and that of French Infrastructure for Integrated Structural Biology FRISBI [ANR-10-INBS-05].
Disclosure statement
No potential conflict of interest was reported by the authors.
Subject index
algorithmic differentiation, derivatives 27,96
alternative conformations 32,57
anisotropic displacement 34,60,92
anti-gradient 15,54,97
atomic coordinates 10,18,27,43,62
atomic displacement parameter, ADP 33,39,47,62
atomic model 4,8,17,18,34,47
bulk solvent 34,43
Cartesian coordinates 90
conjugate directions 54,97
constraint 21,23,40,43,58
convergence 14,22,30,45,54,55,98
convolution 33,53,95
coordinate errors 19,46,49,62
cryo-electron microscopy, cryoEM 52,61
crystal imperfections 41,91
crystallographic map 27,44
data-to-parameters ratio 21
deformation density 44
direct space 90
discrete Fourier Transform 25,95
dynamic disorder 31
electron density distribution 8,31,52,90
experimental data 11,14,30,38,45,47,51,61
exponential model (solvent) 36
fast differentiation algorithm 26
Fast Fourier transform 25
finite difference 24,96
flat mask model (solvent) 36
Fourier series 91,94
Fourier synthesis, syntheses 35,52,91
fractional coordinates 7,90
gradient 15,24,26,54,97
gradient methods 54,97
harmonic motions 32,40
high resolution 20,24,32,59
hydrogen 37,60
ideal crystal 30
incomplete model 46
independent atoms 31,44
intensity (of structure factor) 6,7,9,42,45,50,91
interatomic scatterer 44
isotropic (displacement parameter) 32,44
joint refinement 60
least-squares 9,52
low resolution 17,35,40,43,59
magnitude (of structure factor) 6,17,45,47,92
maximum likelihood 46,58
Miller indices 6,91
minimization, optimization method 10,13,15,30,54,97
ML parameters 49
ML (crystallographic) target 47
model errors 9,19,47
model validation 28,61
multicopy 57
multiple conformations 28,32,45
multipolar model 43
neutron refinement 8,59
occupancy 31,34,44,58
optimization problem 9,11,14
parameter values 9,28,43
quadratic approximation 48,54,99
real space (refinement) 51
reciprocal space 6,7,53,91
refinement program 4,24,50,54,56,63
reflection 5,7,20,29,44,45,91
restraint 21,29,34,43,46,59,60
R-factor 11,17,20,29,58
Rfree-factor 29,58
riding hydrogens 37
rigid group 39,43,96
second-order methods 99
simulated annealing 55,57
solvent molecules 34,37
static disorder 31
statistical parameter 48
steepest descent 15,30,54,98
structure factor 6,12,24,31,34,37,38,45,91,95
subatomic resolution 43,60
target function 10,12,13,23,45,51
test data set 28,49
TLS modelling 38
TLS refinement 40
torsion angle 23,43
twinning 41
uncertainties 26,32,37,40,91
X-ray 5,45
Notes
1. It is important to note that after refinement of free atoms, especially for models with large starting errors, some model atoms can be swapped since diffraction data do not take atomic labels into account. From that point of view, a formal calculation of mean coordinate errors measuring the distance between positions of the equally labelled atoms in two models gives confusingly large numbers, unless positional errors are small for two sets of unlabelled atoms.
2. Random displacements, probability of which is described by the Gaussian law.
3. Exact conditions and speed of convergence are out of scope of this work.
Additional information
Notes on contributors
Alexandre G. Urzhumtsev
Alexandre Urzhumtsev is a full professor at the Université de Lorraine, Nancy, and a researcher at the IGBMC (Institut de Génétique et de Biologie Moléculaire et Cellulaire) à Illkirch, France. After the Kolmogorov’s school-internat and the Faculty of Computational Mathematics at the Moscow State University (1978), he started to work in the Research Computing Center of Russian Academy of Sciences in Pushchino. He got his PhD in X-ray macromolecular crystallography in 1985 (Institute of Crystallography, Moscow). In 1991 he moved to France. He got his Habilitation in the University of Strasbourg in 1997. His main research interests are development of computational methods and programs for macromolecular structure solution, refinement and validation, as well as solution of ‘difficult’ structures. He works on refinement problems for more than 40 years.
Vladimir Y. Lunin
Prof. Vladimir Y. Lunin is a Chief Researcher in the Keldysh Institute of Applied Mathematics of Russian Academy of Sciences. He graduated from the department of Mechanics and Mathematics of Lomonosov Moscow State University and got PhD in differential equations and mathematical physics in 1977. He was awarded the degree of Doctor of Science in Crystallography (Habilitation) by Shubnikov Institute of Crystallography RAS in 1992. Since 1976 he headed the computer support and software development for biological crystallography in the Research Computing Center of Russian Academy of Sciences in Pushchino (from 1992 Institute of Mathematical Problems of Biology, from 2016 the department of Keldysh Institute of Applied Mathematics RAS). In the end of 70s, together with Alexandre Urzhumtsev, he started to work on macromolecule refinement problems. Another branch of his scientific interest is the low-resolution ab initio phasing in biological crystallography and single particle studies.