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Resolution: Implications in Refinement Swanand Gore & Gerard Kleywegt May 6 th 2010, 12-1 pm Macromolecular Crystallography Course

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Outline Intuitive idea of resolution – why higher order diffraction is better. Parameters, model, observations, refinement – more data is better. Observations, parameters, over-fitting in crystallographic refinement. Features that can be modeled at various resolutions. Refinement practices at low and high resolution.

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Idealized diffraction in 1D Images scanned from David Blow’s book h1 h2 h3 -h1 -h2 -h3

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Idealized diffraction in 1D Images scanned from David Blow’s book h1 h2 h3 -h1 -h2 -h3 Increasing resolution Assuming: B = 0 Occupancy = 1 Uniform scattering power in all directions. Phase angles = 0

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Idealized diffraction in 1D Images scanned from David Blow’s book Higher order diffraction Higher Fourier coefficients Higher frequency wave in real space Sharper signal Greater resolution

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What separation can be resolved? Images from B. Rupp’s book Image from Gerard’s ppt. Nominal resolution – The h-th order diffracted wave samples the lattice at interval of a/h. – a/h is the crystallographic resolution which is routinely quoted. – In tetragonal cell abc, diffraction hkl comes from planes separated by √[ (a/h) 2 + (b/k) 2 + (c/l) 2 ] – For tetragonal cell 100, 95, 90, and highest order diffraction 50, 52, 48, resolution is ~3.29. For non-orthogonal axes, corrections apply. Resolution intuitively means the least distance between objects below which they cannot be distinguished apart. – For 3D crystallography, it is ~0.92*d min, almost same as nominal. blob OO

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Atomic scatterers in 1D Images from B. Rupp’s book COCOCO Fourier Coefficients & Phases Resolution Filter Peaks get sharper as higher resolution Fourier coefficients are included.

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Occupancy and B factors Images from B. Rupp’s book Peaks get broader due to larger B factors and shorter due to lower occupancy.

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Data truncation Images from B. Rupp’s book Happens naturally due to B factors. Truncated data leads to incomplete reverse FT, causes ripples. Ripples around heavy atoms can ‘drown’ nearby lighter atoms. Ripples can seem to originate from real atoms. O at 0.5 occupancy?N’s at 0.5 occupancy?

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Diffracting duck in 2D Leaving out higher order diffraction data will reduce the detail retrieved through reverse transform. Leaving out lower resolution data will blur the boundaries. Randomly absent data is not too problematic for maps. – Doesn’t matter if R free set is / not used in map calc? Images from Kevin Cowtan’s website.

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Make everything as simple as possible, but not simpler….. ρ x = 1/V Σ F h exp (-2πi h.x + iφ h ) F h exp(i φ h ) =V Σ f i o i exp(2πi h.x) exp (-B i sin 2 θ/λ 2 ) Estimate phases. Model xyz, B, o. Model solvent. ….. Noise. Errors in data collection. Static, dynamic disorder. Images from B. Rupp’s book

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Choosing resolution cutoff B factors and scattering factors impose a natural cutoff on what can be observed. Reliability of measurement is indicated by S/N ratio and completeness Signal to noise ratio – A low SNR does not matter too much if proper maximum likelihood target is used to weigh in error estimates. – High matters when collecting data for phasing. Completeness – Low completeness in highest resolution shell does not confer a level of detail to the map as implied by nominal resolution – Effective resolution = d min. C -1/3 – Randomly or systematically missing data creates undesirable effects in reverse FT. – Completeness > 0.95 Number of reflections increases as cube of nominal resolution. – 2/3z π V UC / d min 3 – Not unique due to centro-symmetry and spacegroup symmetry Images from B. Rupp’s book

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Model and refinement Model is defined as a set of parameters and a set of functions over parameters, designed to explain observations Refinement – Is an algorithmic process of fitting a model to explain observations, by assigning optimal values to parameters. – Reduces the differences between observations and model-calculated values of observations A linear model in 2D consists of 2 parameters – Y = mX + c Some models are more accurate than others, depending on quality of refinement. Refinement is necessary when observations contain errors and there are enough observations to refine the parameters. c m 1 Observations Well-refined model Ill-refined model

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Model and refinement A linear model in 2D – consists of 2 parameters : Y = mX + c – 1 observation, howsoever accurate, is not sufficient if model has 2 parameters Under-determined, over-fitted model Many models can be imagined – 2 distinct accurate observations are sufficient to determine the linear model Well-determined model – 3 accurate observations over-determine the model – But observations generally contain random error! Greater number of observations lead to error cancellation and more accurate model Model with too few params can lead to under-fitting Model with too many params can lead to over-fitting – Fitting to error too! Quality of modelling – Choice of model (linear, quadratic, higher polynomial?) – Quality of refinement (R value) Images from B. Rupp’s book

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Model and refinement In presence of errors, refinement quality does not indicate model quality – Well-refined model is of bad quality if it was fitted to erroneous observations. Hence, observations not subject to refinement are required to assess the accuracy. – R and “free” R – M1: 0.2, 0.3 – M2: 0.2, 0.4 – M1 better than M2 Free R and data/param ratio helps in comparing models with different number of parameters – MA: 0.2, 0.3. d/p = 15/2 = 7.5 Under-fit – MB: 0.1, 0.2. d/p = 15/3 = 5 optimal – MC: 0.01, 0.25. d/p = 15/10 = 1.5 Overfitting = Low d/p, high R free Images from B. Rupp’s book MA Occham’s valley M1 M2 MB MC

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A crystallographic model Biochemical entities – Biopolymers polypeptides, polynucleotides, carbohydrates – Small-molecule ligands (ions, organic) Crystallographic additives, e.g. GOL, PEG Physiologically relevant, e.g. heme, ions Synthesized molecules, e.g. a drug candidate – Solvent Coordinates, Displacement – Unique x,y,z – Partial, multiple, absent (occupancy) – Isotropic or anisotropic B factors – TLS approximation Crystallographic etc. – Cell, symmetry, NCS – Bulk solvent correction (Ksol, Bsol) 3hbq images made with pymol. http://www.cgl.ucsf.edu/chimera/feature_highlights/ellipsoids.png B factor putty from Antonyuk et al. 10.1073/pnas.0809170106 www.ruppweb.org/xray/tutorial/Crystal_sym.htm

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Quick note on NCS, TLS Non-crystallograpic symmetry – Molecule/s -> ASU -> locally-related ASUs -> Unitcell -> Crystal – Sometimes ASU can consist of multiple, nearly identical subunits. – The transformation operator between subunits is local and distinct from space-group operators. – Subunits need not be identical because they are in different environments, differences do not indicate problems! – This additional symmetry can be used in refinement (restraints, constraints) and validation. Translation-libration-screw – Overall anisotropy = lattice disorder + inter-molecular motions + intra- molecular rigid body motions within molecule + atomic anisotropy – Paradigm shift from atom-level anisotropy modelling to anisotropic movements of rigid bodies – 1d: a point (3) through which rotation axis (2) will pass + ratio (1) of rotation to translation on that axis = 6 – 2d: 2 points + 2 ratios + 2 orthogonal axes (3) + 2 more ratios = 13 – 3d: 3 points + 3 ratios + 3 orthogonal axes (3) + 6 more screws = 20(ish) – TLS group granularity can range from full domain to sidechain Images from Rupp book and Martyn Winn ppt

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Counting parameters Average-case parameters – Per atom 4 params 3 params for coordinates 1 param for isotropic B factor – No hydrogens, 1 water per residue – 8 atoms per residue – N * 8 * 4 = 32 N Increasing the parameters – 6 params per atom for anisotropic B factor (>2x) – Refining occupancy (1.25x) or multiple occupancy – Hydrogens modeled explicitly (8 per residue) (2x) – Multiple models (M x) Reducing parameters – 20 params per TLS group 5 groups: 20 * 5 groups of 40 res each = 100 => 32 * 200 to 100 (1/64 x for 200 res protein) – Strict NCS (1/n x for n-fold) 1clm, calmodulin, 1.8Å 1132 protein atoms + 4 Ca + 71 waters = 4828 xyzB #unique reflections = 10610 Data / params = 2.2 Restraint counts taken from: http://ccp4wiki.org/~ccp4wiki/wiki/images/9/9f/Winn_prague09_data_parameters.pdf 1exr, calmodulin, 1Å 1467 protein atoms with alt conf + 5 Ca + 178 waters 9900 anisotropic B + 316 occupancy = 15166 params #unique reflections = 77150 Data / params = 4.6 1h6v, 3Å 6 TLS groups = 120 params 22514 protein atoms + 552 ligand atoms + 9 waters xyzB = 92300 (residual) #unique reflections = 69328 (5% free) d/p = 69328/92300 = 0.7

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Data to parameters ratio r = (number of unique reflections) / (number of parameters) – Graph for a calmodulin 1up5, ~2500 atoms, xyzB – r < 1, i.e. under-determined for d min < 2.5Å – Reflections-based refinement is possible only for r > 10, i.e. resolution approaching 1Å! – But most PDB entries have r ~ 2-5 There must be more observations provided to refinement than only the reflections – Reflections = observations specific to a particular MX experiment – But there are other more general observations applicable to any MX refinement – Covalent geometry, steric clashes, …. (Graph by Konrad Hinsen, 2008) Image from B. Rupp’s book

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Observations to parameters ratio Observations = reflections + constraints and restraints based on well-known features of macromolecules o/p > d/p – Tricky to estimate the difference due to dependences, but generally sufficient to make refinement possible – 1exr: 1Å, 22732 restraints Bonds, angles, planarity, chirality… – o/p = (22732 + 77150) / 15166 = 6.1 > 4.6 = d/p Bungee jumper RElaxation = REstraint Hangman CONvict = CONstraint Images from Gerard’s slides Restraint counts taken from: http://ccp4wiki.org/~ccp4wiki/wiki/images/9/9f/Winn_prague09_data_parameters.pdf Energy length

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Observations to parameters ratio o/p > d/p for 1h6v at 3Å – Restraints (including NCS) = 209378 – o/p = (209378+69328)/92300 = 3 – d/p = 0.7 < 3 = o/p 2 components of refinement residuals – Data-based Changes model (xyzB..) to reduce Fo ~ Fc – Knowledge-based Changes model (xyz) to take values of geometric features towards idealized values – Q tot = w x Q x + Q geom – Small w x : greater stress on geometric correctness Low resolution, low d/p – Large w x : model deviation from ideal geometry High resolution, high d/p Restraint counts taken from: http://ccp4wiki.org/~ccp4wiki/wiki/images/9/9f/Winn_prague09_data_parameters.pdf

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Greater d/p => more detail (given decent phases) Image from http://www.crystal.uwa.edu.au/px/alice/projects/SCOA_atomic.html, 1mxt Images from Rupp’s book 0.95Å

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Lower d/p => lower detail decent phases often not available Pics of 2g34, 1z56 with coot using EDS maps 2g34, 5Å 1z56, 3.9Å

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Lower d/p => lower detail Pics of 2bf1 with coot using EDS maps 2bf1, 4Å

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All resolutions not equal… From Gerard’s slides and Phil Evans

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Levels of detail interpretable at various resolutions Protein FeatureResolution (Å) Helix9 Sheet4 Main chain3.7 Aromatic sidechains3.5 Small sidechains3.2 Sidechain conformations2.9 Carbonyl, peptide2.7 Ordered waters2.7 Central dimple of aromatic ring2.4 Correct stereochemistry at Ile CB2.2 Proline pucker2.0 Individual atoms1.5 Nucleic Acid FeatureResolution (Å) Double helix20 Single strand12 Stacked base pairs4 Phosphates3.5 Purine or pyrimidine?3.2 Individual bases2.7 Ribose pucker2.4 Individual atoms1.5 From David Blow’s book Orbitals and bonds (beyond 1Å)!

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Rules of thumb at all resolutions for model-building and refinement Start with few parameters and slowly enrich the model Be very conservative till a majority of backbone is identified and produces stable refinement Prioritize: Backbone > side-chains > small-mols > waters Be aware of prevalent modeling practices at your resolution Whole model contributes to quality of region of interest. Use similar structures for comparison and copying. Use quality criteria often.

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Low resolution refinement Low resolution structures offer great biological insights. – Mainly for complexes e.g. 70S ribosome at 7Å, SIV gp120 envelope glycoprotein at 4Å – Large complexes generally diffract to lower resolution. Components may have physiologically relevant conformations only in complexed states. High impact – In absence of better resolution, low resolution data must be used. – Low resolution does not have to mean low quality! Basic guidelines for model building and refinement. – Low d/p => Be cautious of biasing the model – Make extensive use of information in addition to reflections – Use as few parameters as possible – Increase params only when confident Images from Karmali et al. Acta Cryst. 2009.

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Low resolution refinement Build model with fewer parameters – Mainchain-only model – Constrain B factor values to be isotropic and constant. – Full occupancies only. – TLS to model anisotropic motions of rigid domains. – Strictly constrained or restrained NCS to reduce params many-fold – No waters or small molecules, use only ‘bulk solvent’

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Low resolution refinement Model cautiously – Initial tracing Build regions that are likely to be seen clearly Good packing, low B factors, bulky group, electron-rich groups core, mainchain, helices, big sidechains, bases, phosphates – Sequence registry Beware of register and topology errors Guess sequence register from bulky sidechains Extend the register by trial and error Check sequence register with a homologous structure Truncate to Gly wherever unsure of residue identity From Gerard’s slides

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Low resolution refinement – Try copying fragments from other high resolution structures when there is clear homology – Treat ligands extra-carefully Copy high-quality observed conformation or predicted low energy conformation Restrain tightly unless there is density and other clues to deviate Axel T. Brunger et al. 2009. Acta Cryst D 65 128–133 X-ray structure determination at low resolution.

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Low resolution refinement density modification tools Expected solvent density – define solvent boundary – followed by solvent flattening / flipping, histogram matching Images B. Rupp’s book and from Acta Cryst. (2003). D59, 1881-1890. The phase problem. G. Taylor Brunger 2006, Low resolution crystallography. Acta Cryst. https://wasatch.biochem.utah.edu/chris/tutorial/Density_Modification.pdf a ppt on DM

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Low resolution refinement density modification Averaging maps of NCS-restrained copies Image from B. Rupp’s Brook. unger 2006, Low resolution crystallography. Acta Cryst. https://wasatch.biochem.utah.edu/chris/tutorial/Density_Modification.pdf a ppt on DM

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Low resolution refinement density modification B-factor sharpening – High-resolution reflections get attenuated most by B factors – Application of negative B factors can artificially up-weigh high-res terms to obtain greater detailed but possibly noisier map Brunger 2006, Low resolution crystallography. Acta Cryst. https://wasatch.biochem.utah.edu/chris/tutorial/Density_Modification.pdf a ppt on DM

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Low resolution refinement Refinement techniques – Rigid body refinement A fragment is constrained to be internally rigid, has only 6 degrees of freedom B factor is isotropic and constant Powerful first step of refinement needing only low resolution data Arbitrary rigid fragments (high quality helices, high- resolution domain structures) can be optimized for location and orientation relative to each other to yield better phases and maps – Torsion angle refinement Bonds, angles, chirality, planarity not variables, only torsion angles are refined Protein is divided into rigid subgroups to sample thoroughly a limited conformational space Higher radius of convergence, reduced overfitting Image from Schwieters, C.D. & Clore, G.M. (2001) Internal coordinates for molecular dynamics and minimization in structure determination and refinement. J. Magn. Reson. 152, 288-302 Nice tutorial at http://speedy.st-and.ac.uk/~naismith/workshop/torsion.pdf See Axel Brunger’s papers on torsion angle refinement

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Low resolution refinement Solving multiple times – Try to automate as much as possible the process of model building and refinement, and then repeat it – Consensus substructures are more reliable, average them – Regions with differences are unreliable, remove them – Gives an idea of precision Gradual increase in number of parameters – Mainchain -> bulky sidechains -> sequence register -> other sidechains – Finally known small mol binders with known binding site can be modelled if reasonable density appears Validation – Keep track of Ramachandran and sidechains rotamers – Remove unlikely parts of mainchain and sidechain – Do not restrain Rama distribution or sidechains to rotamers during refinement, it may give false validation results Read what others are doing for low resolution – e.g. Axel Brunger’s literature, CCP4 & phenix tools, CCP4bb Images from wikipedia and Furnham et al. Structure 2006.

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High resolution refinement High resolution structures provide atomic insights – Packing, binding – Flexibility – Enzyme mechanisms – Hydration Basic guidelines for model building and refinement – High d/p => Be cautious of under-fitting! – Make greater use of data than in low res case – Make as detailed a model as possible, esp of interesting regions – Check all empty density critically

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High resolution refinement Allow model to deviate from geometry when data is strong – Weight on xray term can be slowly increased to reveal any unusual geometry without risking model bias Use automation to fit biopolymers – Trace secondary structure automatically, in coot or with phenix tools – Trace mainchain and build sidechains using programs, e.g. with buccaneer, warpNtrace, Rapper – Do this multiple times to identify regions requiring manual attention Validation tools: can they indicate the information content of macromolecular crystal structures? EJ Dodson et al. Volume 6, Issue 6, 1998, 685-690. Image from Terwillinger et al. papers in Acta Cryst D on automatic chain tracing.

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High resolution refinement Explain all unoccupied density – Is it due to ligands? Build expected ligands (including MX additives) Search unexpected small-mols E.g. coot or phenix ligand tools – Is it due to multi-conformer sidechains? – Is it water? Images from B. Rupp’s book and Terwillinger et al. Acta Cryst. 2005.

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High resolution refinement Build waters – Peak-pick semi-automatically to form a reasonable hydration network with sidechains Model hydrogens – When difference density is visible Image from B. Rupp’s book Atomic resolution crystallography reveals how changes in pH shape the protein microenvironment. Lyubimov et al. Nature Chemical Biology 2, 259 - 264 (2006)

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High resolution refinement Verify correct sidechain orientations of NQH – Manually or automatically flip NQH sidechains to improve h- bonding – Model more sidechain conformations if necessary Use non-standard atomic scattering models – At subatomic resolution, model electron density with nonspherical multipolar model, or model bonds as scatterers Image from B. Rupp’s book Afonine et al. Acta Cryst. (2007). D63, 1194–1197 Jelsch et al. PNAS 2000 97 7 3171.

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High resolution refinement Even in high res, maintain order of adding detail to avoid overfitting – bb > sc > ligand – Anisotropy, multiconformers, waters, hydrogens Image from Antonyuk S V et al. PNAS 2005;102:12041-12046 Image from David Blow’s book. Invest more parameters around the regions of interest – multi-conformers – Anisotropy – waters near active site – Possibility of multiple ligands – Releasing constraints / restraints

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Summary Resolution is the least distance between Bragg planes with observable reflection. Two atoms closer than resolution cannot be observed distinctly using data at that resolution. Resolution dictates the detail revealed by electron density maps. – Low resolution => low detail – High resolution => high detail Parameters in the model must be chosen to suit the resolution. Over-fitting can be detected using R free and data to parameter ratio. Knowledge-based constraints and restraints augment experimental data to make refinement possible. Geometric target is weighted more than crystallographic data at low resolution. Model is allowed to diverge from ideal geometry at high resolution. Greater detail should be modelled at higher resolution to make best use of data.

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Acknowledgements Alejandro & IPMont MX organizers Sameer Velankar, Jawahar Swaminathan (EBI) Online resources – Kevin Cowtan – Rupp web – Randy Read’s course – Various papers and images therefrom – Martyn Winn’s ppt at on data to params Books – David Blow – Alex McPherson – Bernhard Rupp

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