Macromolecular structure refinement

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Macromolecular structure refinement

• Garib N Murshudov
• York Structural Biology Laboratory
• Chemistry Department
• University of York

2
Contents

• Purpose of and considerations for refinement
• Prior information Dictionary of ligands
• Prior information B value How to deal with
  them
• Conclusions and future developments

3
Purpose

• Optimal fit of the model to the experimental data
  while retaining its chemical integrity
• Estimation of errors for the refined parameters
• Improvement of phases to facilitate model
  building (automatic e.g. ARP/wARP or manual)
• Give deviation from chemistry and experiment to
  aid analysis of the model

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Considerations

• Function to optimise
• Should use experimental data
• Should be able to handle chemical information
• Parameters
• Depends on the stage of analysis
• Depends on amount and quality of the experimental
  data
• Methods to optimise
• Depends on stage of analysis simulated
  annealing, tunneling, conjugate gradient, second
  order (normal matrix, information matrix, second
  derivatives)
• Some methods can give error estimate as a
  by-product. Second order methods give error
  estimate.

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Function

• Probabilistic view
• Chemical information prior knowledge
• Fit to experiment - likelihood
• Total function - posterior

• View from physics
• Internal energy
• External energy
• Total energy internal external

Gibbs distribution Probability of the state of
the system is
Bayess theorem Probability of the system (x)
given experiment(x0)
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System describing treatment of the experiment
Internal energy or Prior probability
External energy or likelihood
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Function likelihood and prior

• Likelihood describes fit of model parameters into
  experiment. There are few papers describing
  various aspects. E.g.
• Murshudov, Vagin, (1997) Acta
  Cryst. D53, 240-255
• Pannu, Murshudov, , Read (1998)
  Acta Cryst D5, 1285-1294
• Prior Should include our knowledge about
  chemistry, biology and physics of the system
  Bond lengths, angles, B values, overall
  organisations

Dodson
Dodson
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Chemical information Two atoms ideal case

• Distance between atoms 1.3Å. B values 20 and
  50
• Thin lines single atoms
• Bold line - sum of the two atoms

P
X
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Chemical information Phe at two different
resolutions
2 Å and High mobility

• 0.88 Å

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Monomer library
Macromolecules are polymers. They consist of
chemical units (monomers). Monomers link with
each other and form polymers. When they make link
they undergo some chemical reaction. Links
between monomers must contain chemical
modification also
ALA
SER
CYS
CYS
PHE
THR
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Monomers and links
ALA-SER
ALA
SER
All atoms Atom types Charges Bonds Angles Planes T
orsions Chiral volumes
All atoms Atom types Charges Bonds Angles Planes T
orsions Chiral volumes
Modifications of monomers Change, add, delete
atoms, atom types, angles, planes, torsions,
chiral volumes Bond Angles Torsions Planes Chiral
volumes
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Schematic view of library organisation
Monomers are independent units. Modification can
act on them. Links can join two monomers. Links
may have modification also
Monomers
Modifications
Links
Modif.
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Dictionary Plans

• Finish mutual test of Feis program and
  dictionary
• Improve values using CSD and quantum chemical
  calculations
• Input formats SMILE, MDL MOLFILE
• More automation of links and modifications
• More chemical assumptions
• Better links to other web resources (e.g. sweet,
  disacharide data base, corina, prodrg, msd/ebi)
• More monomers and links???
• Adding more knowledge like frequently occurring
  fragment, most probable rotamers
• etc

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B values

• B values are important component of atomic models
• They model molecular mobility as well as errors
  in atoms
• Distribution of B values is important for proper
  maximum likelihood estimation
• If estimated accurately their analysis can give
  some insight into biology of the molecule
• Note Protein data bank is very rich source of
  prior information. But one must be careful in
  extracting them

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Modeling of B values TLS

• TLS model of atomic B values assumes that they
  depend on position of atoms (as implemented in
  REFMAC)
• U Uind T r x L x rT rT x S ST x rT
  A(r)
• Effect of this on electron density
• This linear equations must be solved to calculate
  electron density without TLS

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B values Intuition and Bayesian

• B values are variances of Gaussians
• B values cannot be negative!!!!!
• Larger mean B larger variation of B
• Inverse gamma is natural prior of variances (It
  is used in microarray data analysis and can be
  used in X-ray data processing)
• Assumption B values of macromolecules have
  inverse Gamma distribution.

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B distribution Inverse gamma

• Inverse gamma distribution
• We can assume that to some degree ? is constant
  for all proteins.

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B distribution Mean vs variance

• Values of sqrt(?) vs indices
• 5000 of proteins are included.
• Proteins are sorted according
• to resolution.
• average value of ? is
• around 7

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B distribution 500 higher than 1.5A resolution
structures

• sqrt(?) vs indices for 400
• structures.

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B distribution Theoretical and from PDB

• B values of four proteins
• after normalisation by
• standard deviation are
• pooled together.
• Remaining parameter
• of the IG is estimated using
• Maximum likelihood

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One PDB Not very good example

• Histogram of B values
• for one protein.
• Red histogram of B values
• Blue parameters fitted
• using these B values
• Black ? 6.7 (average
• for all high resolution
• proteins)

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Use of B distributions

• Restraints on individual B values. It will allow
  refinement of B values reliable at medium and low
  resolutions
• Better restraints on differences between B values
  of close atoms.
• Detection of outliers (low B value potential
  metal, high B value potentially wrong)
• For normalisation of structure factor
• For improved Maximum likelihood estimation
• For map improvement

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Conclusion and future perspectives

• Dictionary of monomers and links have been
  developed and implemented
• B value distributions look like IG.
• Analysis of B value distribution for solvent is
  needed
• Future
• Proper B value restraints
• Global and local improvement of dictionary
• Restraints to external information (small
  fragments)
• Twin, psuedotranslational (etc) refinement
• Inversion of sparse and full (Fisher information)
  matrix to estimate reliability of the parmaters

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Acknowledgements

• Alexey Vagin
• Andrey Lebedev
• Roberto Steiner
• Fei Long
• Dan Zhou
• Najida Begum
• Mark Dunning
• Gleb Bourinkov
• Alexander Popov
• YSBL research environment
• Users
• CCP4
• Wellcome Trust, BBSRC, EU BIOXHIT project

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And of course!!!!