Scaling, Phasing, Anomalous, Density modification, Model

building, and Refinement

What we’ve learned so far•Electrons scatter Xrays.

•Scattering is a Fourier transformation.

•Inverting the Fourier transform gives the image of the electron density.

•Waves have amplitude and phase. And we can’t measure phase.

•Inverting the Fourier transform without the phases gives the Patterson map, which is the map of all inter-atom vectors.

•Space groups are groups of symmetry operations in 3D.

•The Patterson plus symmetry gives us heavy atoms positions.

•Heavy atom positions plus amplitudes gives us phases.

What we can do.•Sum waves.

•Calculate the phase given atom position and scattering vector.

•Index spots on an X-ray photograph.

•Draw Bragg planes.

•Invert the Fourier transform using Bragg planes.

•Calculate cell dimensions from an X-ray photograph.

•Describe the symmetry of a crystal or periodic pattern.

•Convert a simple Patterson to heavy atom positions.

•Calculate heavy atom vectors given heavy atom positions.

•Solve for phases given amplitudes and heavy atom vectors.

From data to model

Collect native data: Fp Collect heavy atom data: Fph

Estimate phases

Calculate

Trace the map

Refine

Is the map traceable?

density modification?yes

no

From data to phases

native data: Fp heavy atom data: Fph

Calculate difference Patterson

Find heavy atom peaks on Harker sections

Solve for heavy atom positions using symmetry

Calculate heavy atom vectors

Estimate phases

From data to Patterson map

native data: Fp heavy atom data: Fph

Calculate difference Patterson

Find the best scale factor, k

Calculate Fdiff = k*|Fph| – |Fp|

From crystal to data

native data: Fp

Indexed film

Intensity, Ip(hkl) = F2

I is relative

Bigger crystal, higher I

Better crystal, higher I

Longer exposure, higher I

More intense Xrays, higher I

Because there is no absolute scale: Fp and Fph are on different scales

Internal scaling

What happens to the phase calculation if the scaling is off?

Radii are Fp and k*Fph

Scaling two datasets

h k l Fp

1 0 0 3233. 100.2 0 0 2028. 98.3 0 0 2179 88.4 0 0 ....

...

h k l Fph

1 0 0 1122. 50.2 0 0 1014. 49.3 0 0 1081. 44.4 0 0 ....

...

<Fp> = <Fph>*k, therefore: k = <Fp>/<Fph>

1st approximation: The intrinsic average amplitude of scattering is constant for different crystals. A simple scale factor k corrects for crystal differences:

Basic assumption

The total number of electrons in the unit cell is the same for each (isomorphous) crystal.

Note: isomorphous means “same space group, same cell dimensions.”

…when scaling two crystals.

Better scaling: Wilson B-factor

<I>=<F2>

X-axis=Two sine-theta over lambda = 1/d

5Å 3Å 2Åd = 20Å

water peak (ignored in slope calc)

Region of linear dependence of amplitude with

resolution. Slope =W= “Wilson B-factor”

Two sets of F’s might have different overall B-factors, because the

crystals may have different degrees of mosaicity So Wilson scaling is better than simple scaling.

<Fp> = <Fph>*k, k=W(1/d) + C

Low-resolution features (ignored in slope calc)

Averaged in resolution bins

Scaled separately

How good is scaling?

After solving the structure, we can go back and see how good the scaling was. Typically, error in scaling < 10%. In best cases < 2%.

Scaling error is worse if:

(1) crystals are non-isomorphous

(2) too many heavy atoms present (“basic assumption” is wrong).

Heavy atom difference Fourier

•The amplitude of Fh is only approximately Fph-Fp.

•The true difference |Fph - Fp| depends on the phase of Fh relative to Fp

Fph = Fp + Fh (vector addition)

Centrosymmetric reflections

•If the crystal has centrosymmetric symmetry, all reflections are centrosymmetric. Phase = 0° or 180°

•If the crystal has 2-fold, 4-fold or 6-fold rotational symmetry, then the reflections in the 0-plane are centrosymmetric. (Because the projection of the density is centrosymmetric)

For centrosymmetric reflections:

|Fph| = |Fp| ± |Fh|

This means the amplitude |Fh| is exact* for centrosymmetric reflections.

*assuming perfect scaling.

0-plane

RRRR Draw any set of Bragg planes parallel to the 2-fold. The projected density is centrosymmetric.

Therefore, phase is 0 or 180°.

Initial phasesThe most probable phase is not necessarily the “best” for computing the first e-density map.

weighted average, best phaseShaded regions are possible Fp and

Fph solutions.

Figure of merit

Figure of merit “m” is a measure of how good the phases are.

C is the “center of mass” of a ring of phase probabilities (that’s the “mass”). The radius of the ring is 1. So m=1 only if the probabilities are sharply distributed. If they are distributed widely, m is small.

Fbest(hkl) = F(hkl)*m*e-ibest

In class exercise: phase error

FP=5.00 =0.5

FPH1=5.50 =0.8 FH1=2.23 H1=-63.4°

FPH2=4.50 =0.9 FH2=0.50 H2=-164°

(1) Draw three circles separated by vectors FH1 and FH2.

(2) Draw circular “error bars” of width 2.

(3) Draw circle plot of Fp phase probabilities.

(4) Estimate the centroid c of probabilit.

(5) What is the Figure of Merit, m?

Anomalous dispersion

Heavy atom free electrons

bound electrons

Inner electrons scatter with a time delay. This is a phase shift that is always counter-clockwise relative to the phase of the free electrons.

Anomalous dispersion

Anomalous dispersion

SIR = single isomorphous replacement

Advantages: only one derivative crystal is needed. (fewer scaling problems)

Anomalous dispersion has a greater effect at higher resolution. (because the inner electrons are more like a point source)

Is the initial map good enough?

(1) The map is calculated using best.

(2) The map is contoured and displayed using {InsightII, MIDAS, XtalView, FRODO, O, ...}

(3) A “trace” is attempted.

Model building

e- density cages (1 contours) displayed using InsightII

Information used to build the first model:

Sequence and Stereochemistry

...plus assorted disulfide and ligand information.

Models are built initially by identifying characteristic sidechains (by their shape) then tracing forward and backward along the backbone density until all amino acids are in place.

Alpha-carbons can be placed by hand, and numbered, then an automated program will add the other atoms (MaxSprout).

Tracing an electron density map

sequence: AGDLLEHEIFGMPPAGGA

Can you locate the density above in the sequence?

Class exercise:

R-factor: How good is the model?

Depending on the space group, an R-factor of ~55% would be attainable by scaled random data.

The R-factor must be < ~50%.

Note: It is possible to get a high R-factor for a correct model. What kind of mistake would do this?

Calculate Fcalc’s based on the model. Compute R-factor

R=Fobs h( ) −Fcalc h( )

h∑

Fobs h( )h∑

What can you do if the phases are not good enough?

1. Collect more heavy atom derivative data

2. Try density modification techniques.

Density modification :

Fo’s and (new) phases

Map Modified map

Fc’s and new phases

initial phases

Density modification techniques

Solvent Flattening: Make the water part of the map flat.

(1) Draw envelope around protein part

(2) Set solvent to <> and back transform.

Solvent flattening

Requires that the protein part can be distinguished from the solvent part. BC Wang’s method: Smooth the map using a 10Å Guassian. Then take the top X% of the map, where X is calculated from the crystal density.

Skeletonization

(1) Calculate map.

(2) Skeletonize it (draw ridge lines)

(3) Prune skeleton so that it is “protein-like”

(4) Back transform the skeleton to get new phases.

Protein-like means: (a) no cycles, (b) no islands

Non-crystallographic symmetry

If there are two molecules in the ASU, there is a matrix and vector that rotate one to the other: Mr1 + v = r2

(1) Using Patterson Correlation Function, find M and v.

(2) Calculate initial map.

(3) Set(r1) and (r2) to ((r1) + (r2) )/2

(4) Back transform to get new phases.

R R

RR

What does a good map look like?

Before computers, maps were contoured on stacked pieces of plexiglass. A “Richards box” was used to build the model.

half-silvered mirror

plexiglass stack

brass parts model

Low-resolution

At 4-6Å resolution, alpha helices look like sausages.

Medium resolution

~3Å data is good enough to se the backbone with space inbetween.

The program BONES traces the density automatically, if the phases are good.

BONES models need to be manually connected and sidechains attached. MaxSPROUT converts a fully connected trace to an all-atom model.

Errors in the phases make some connections ambiguous.

Contouring at two density cutoffs sometimes helps

Holes in rings are a good thing

Seeing a hole in a tyrosine or phenylalanine ring is universally accepted as proof of good phases. You need at least 2Å data.

Can you see in stereo?

Try this at home. In 3D, the density is much easier to trace.

New rendering programs

“CONSCRIPT: A program for generating electron density isosurfaces for presentation in protein crystallography.” M. C. Lawrence, P. D. Bourke

Great map: holes in rings

Superior map: Atomicity

Rarely is the data this good. 2 holes in Trp. All atoms separated.

Only small molecule structures are this good

Atoms are separated down to several contours. Proteins are never this well-ordered. But this is what the density really looks like.

Refinement

•The gradient* of the R-factor with respect to each atomic position may be calculated.

•Each atom is moved down-hill along the gradient.

•“Restraints” may be imposed.

dRfactor

dr v i

*

What is a restraint?

A restraint is a function of the coordinates that is lowest when the coordinates are “ideal”, and which increases as the coordinates become less ideal..

Stereochemical restraints

bond lengths bond angles torsion angles

also...

planar groups

B’s

Calculated phases, observed amplitudes = hybrid F's

•Fc’s are calculated from the atomic coordinates

•A new electron density map calculated from the Fc's would only reproduce the model. (of course!)

•Instead we use the observed amplitudes |Fo|, and the model phases, c.

ρ(v r ) = Fobs

v h ( )

h.

∑ e−i 2π

r h •r r ( )+α calc

r h ( )( )

Hybrid back transform:

Hybrid maps show places where the current model is wrong and needs to be changed.

Difference map: Fo-Fc amplitudes

The Fo “native” map (Fo) differs from the Fc map (Fc) in places where the model is wrong. So we take the difference. In the difference map:

Missing atoms? (Fo-Fc) > 0.0

Wrongly placed atoms? (Fo-Fc) < 0.0

Correctly modeled atoms? (Fo-Fc) = 0.0

Q: Subtracting densities (real space) is the same as subtracting amplitudes (reciprocal space) and transforming. T or F?

2Fo-Fc amplitudes

The Fo map plus the difference map is

Fo where the differences are zero (the atoms are correct)

Less than Fo where the model has wrong atoms.

Greater than Fo where the model is missing atoms.

Fo + (Fo-Fc) = 2Fo-Fc

The “free R-factor”: cross-validation

The free R-factor is the test set residual, calculated the same as the R-factor, but on the “test set”.

Free R-factor asks “how well does your model predict the data it hasn’t seen?”

Rfree=Fobs h( ) −kFcalc h( )

h∈T∑

Fobs h( )h∈T∑

Note: the only difference is which hkl are used to calculate.

Why cross-validate?If you have three points, you can fit them to a quadratic equation (3 parameters) with zero residual, but is it right?

Observed data

R-factor = 0.000!!

calculated

Fitting and overfittingFit is correct if additional data, not used in fitting the curve, fall on the curve.

Low residual in the “test set” justifies the fit.

residual≠0

cross-validation=Measuring the residual on data (the “test set”) that were not used to create the model.

The residual on test data is likely to be small if

is large.

a line has 2 parameters

dataparameters

parameters versus dataExample from Drenth, Ch 13:

Papain crystal structure has 25,000 reflections.

Papain has 2000 non-H atoms

times 4 parameters each (x, y, z, B)

equals 8000 parameters

data/parameters = 25,000/8000 ≈ 3 <-- this is too small!

restraints are “data”

bond lengths

bond angles

torsion angles

Bond lengths, angles, etc. are “measurements” that must be fit by the model. The true “residual” should include deviations from ideal bond lengths, angles, etc.

In practice, residual in restraints (e.g. deviations from ideal bond lengths, angles) is very low. This means that restraints are essentially “constraints”.

planar groupsvan der Waals

constraints reduce the number of parameters

bond lengths

bond angles

Bond lengths, angles, and planar groups may be fixed to their ideal values during refinement (“Torsion angle refinement”).

Using constraints, Ser has 3 parameters, Phe 4, and Arg 6.

There are an average 3.5 torsion angles per residue.

Papain has ~700 torsion angle parameters.

data/parameter=25,000/700≈35 planar groups

radius of convergence

total residual

parameter space

...=How far away from the truth can it be, and still find the truth?

radius of convergence depends on data & method.

More data = fewer false (local) minima

Better method = one that can overcome local minima

Molecular dynamics w/ Xray refinement

MD samples conformational space while maintaining good geometry (low residual in restraints).

E = (residual of restraints) + (R-factor) dE/dxi is calculated for each atom i, then we move i downhill. Random vectors added, proportional to temperature T.

The simulated annealing MD method:

(1) start the simulation “hot”

(2) “cool” slowly, trapping structure in lowest minimum.

“X-plor” Axel Brünger et al

Phase bias, and how to fix it.

The model biases the phases.

The effect of phase bias is local to the errors.

To correct a part of the model, we must first remove that part .

An “OMIT MAP” is calculated. The phases for an omit map are derived from a partial model, where some small part has been omitted.

Omit maps

This residue has been removed before calculating Fc.

2Fo-Fc density = Fo + (Fo - Fc) = The native map plus the difference map.

Two inhibitor peptides bound to thrombin. The inhibitors were omited from the Fc calculation. (stereo images)

FÉTHIÈRE et al, Protein Science (1996), 5: 1174- 1183.

The final modelMonday, January 29, 2001 Structure Explorer - 1AAJ Page: 1

http://www.rcsb.org/pdb/cgi/explore.cgi?pid=9090980799705&pdbId=1AAJ

Structure Explorer - 1AAJ

Summary Information Summary InformationView StructureDownload/Display FileStructural NeighborsGeometryOther SourcesSequence Details

SearchLite SearchFields

Title: not availableCompound: Amicyanin (Apo Form)

Authors: R. C. E. Durley, L. Chen, L. W. Lim, F. S. MathewsExp. Method: X-ray DiffractionClassification: Electron Transport

Source: Paracoccus DenitrificansPrimary Citation: Durley, R., Chen, L., Lim, L. W., Mathews, F. S., Davidson, V. L.: Crystal structure

analysis of amicyanin and apoamicyanin from Paracoccus denitrificans at 2.0 A and 1.8 Aresolution. Protein Sci 2 pp. 739 (1993)[ Medline ]

Deposition Date: 09-Apr-1992 Release Date: 31-Oct-1993

Resolution [Å]: 1.80 R-Value: 0.155Space Group: P 21

Unit Cell: dim [Å]: a 28.95 b 56.54 c 27.55 angles [¯]: alpha90.00 beta 96.38 gamma 90.00

Polymer Chains: 1AAJ Residues: 105

Atoms: 905HET groups: HOH

© RCSB

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Other data commonly reported: total unique reflections, completeness, free R-factor