Macromolecular structure refinement

Garib N Murshudov

York Structural Biology Laboratory

Chemistry Department

University of York

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

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

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.

Function

Probabilistic viewChemical information – prior knowledge

Fit to experiment - likelihood

Total function - posterior

View from physicsInternal energy

External energy

Total energy = internal + external

Gibbs distribution: Probability of the state of the system is:

externalinernal EEE

kTxEKxP

))/()(exp()(

Bayes’s theorem: Probability of the system (x) given experiment(x0)

));(ln)(exp(ln );()( );( 000 xxPxPNxxPxPNxxP

System describing treatment of the experiment

Internal energy orPrior probability

External energy or likelihood

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

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

Chemical information: Phe at two different resolutions

2 Å and High mobility0.88 Å

Monomer library

ALA

CYS

PHE

SER

CYS

THR

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

Monomers and linksALA SER ALA-SER

All atomsAtom typesChargesBondsAnglesPlanesTorsionsChiral volumes

All atomsAtom typesChargesBondsAnglesPlanesTorsionsChiral volumes

Modifications of monomers:Change, add, delete atoms, atom types, angles, planes, torsions, chiralvolumes

BondAnglesTorsionsPlanesChiral volumes

Schematic view of library organisation

Monomers

Modifications

Links Modif.

Monomers are independent units. Modification can act on them. Links can join two monomers. Links may have modification also

Dictionary: Plans

• Finish mutual test of Fei’s 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

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

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

)(),()(

parameters withplayingafter space reciprocal In

)(),()(

0

0

hFhkhTkF

ydyxyxTy

R

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.

B distribution: Inverse gamma

Inverse gamma distribution:

We can assume that to some degree is constant for all proteins.

2/ ))2()1(/(1 )),1(/(1

or

//1 / ,//1

properties with

)) /(1exp(),;(

2B

222

21/B

2221/B

1

BB

BB

BBBIG

B

B distribution: Mean vs variance

Values of sqrt() vs indices5000 of proteins are included.Proteins are sorted accordingto resolution.average value of isaround 7

B distribution: 500 higher than 1.5A resolution structures

sqrt() vs indices for 400

structures.

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

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)

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

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

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

And of course!!!!