1. Diffraction intensity2. Patterson map

Lecture 6

2-1-2006

Calculation of the electron density (x,y,z)

We already know:

This is equivalent to:

Because:

The reverse is also true:

or

This two Fourier syntheses form the mathematic foundation for all crystallographic calculation

F is a Fourier synthesis of

is a Fourier synthesis of F

Physical meaning of Fourier synthesis

Scattering ---First Fourier Transform

Focusing -----Second Fourier Transform

An simple example of Fourier transform

+

+

=

f=2

f=3

f=5

By superimposing three cosine waves, we have

If we have more higher frequency terms, we have

Symmetry in diffraction pattern

• A diffraction pattern possesses at least the same or higher symmetry than the crystal– Remember that reciprocal lattice rotates with the

crystal itself. • All diffraction patterns have a center of

symmetryI(h,k,l) = I(-h, -k, -l)These two reflections are called a Friedel pair.

• From diffraction intensity, we should be able to determine space group symmetry in most cases (symmetry of diffraction pattern, systematic absences)

Symmetry in diffraction pattern: a center of symmetry

Therefore,

Symmetry in diffraction pattern: a center of symmetry- electron density is a real quantity

The implication is:

Which means is real, as expected for a physical quantity.

Because

Space groups with screw axes, glide planes, F, C, I centered lattices will have systematic absences.

Effect of the size of the unit cell on the diffraction intensity

V is the volume of unit cellVcr is the size of crystal is wavelength

Experimental data look like this …

h k l F

0 1 2 25

1 0 3 14

-1 2 3 14

-2 1 2 11

1 2 1 10

3 -2 0 23

3 1 3 19

h,k,l are miller indices,F is structure factor amplitudeNo phase information.

Intensity statistics: Rmerge

• Only a small number of reflections are collected on each photograph• To construct the reciprocal space, we need to merge reflections

from different frames.

• Measurement error is quantitatively described by Rmerge

Where hkl are miller indices of a particular reflection, and i is the ith measurement of that reflection

• Rmerge varies from a few percent in low resolution ranges to over thirty percent in high resolution ranges

R        =             I  (hkl) -           /           I  (hkl)

Intensity statistics: what are the important quantities

Intensity statistics: Wilson plot

• After data merging, the first thing to do is to calculate the Wilson plot• Wilson provide an estimation of temperature factor

Because

we have

If ri ≠ rj, 2i(ri-rj)S will vary from 0 to 2, resulting in 0 value for I(abs, S)

Therefore,

and we know

then we have

With some adjustments, then we have

Intensity statistics: Wilson plot• After data merging, the first thing to do is to calculate the Wilson plot

• Wilson provide an estimation of temperature factor

Slope => B factorY intercept => absolute scale C

However, for 2i(ri-rj)S to vary from 0 to 2S has to be at least 1/3Å-1

What is phase problem?

Electron density can simply be calculated as:

However, we can only measure I(hkl), and we can only get |F(hkl)|=sqrt(I). The phase information (hkl) is missing.

Structure determination is all about phase determination.

Methods for phase determination

Multiple isomorphous replacement (MIR)• Use heavy atom to perturb diffraction intensities• At least one native crystal and two crystals soaked in two heavy atom solutions must be available. • Need no information about the unknown structure.

Multiwavelength anomalous dispersion (MAD)• As X-ray wavelength approaching absorption edge, anomalous scattering occurs – Friedel pairs are no longer equal in intensities.• Only one crystal is needed but multiple data sets must be collected at three different wavelengths. • Se-Met protein is usually needed. Excellent electron density map.

Molecular replacement (MR)• Only one native crystal is needed. One homologous structure must be available. Quick and simple.

Hg Pt

U

Se

(S Se)

The first crystal structure of a protein molecule

• 1962: Max Ferdinand Perutz and Sir John Cowdery Kendrew win the Nobel Prize in Chemistry for their studies on the structures of globlular proteins.

• The structure of myoglobin was solved by MIR.

(Max Perutz, 1914-2002)

Patterson functionIn 1934, Patterson published a paper suggesting that the Patterson function:

giving rise to a map showing interatomic vectors (not individual atomic positions, though)

The three variables u, v, w vary from 0 to1 within the crystal unit cell.

No phase information is needed for Patterson synthesis

Pattern map shows intermolecular vectors

There are a total of N2 vectors, of which N are located at the origin, and N2-N are distributed throughout the volume of the unit cell.

If atom i contains Zi electrons and atom j contains Zj electrons, the corresponding vector rij will have a weight proportional to ZiZj.

Patterson map has a center of symmetry.

The space group of patterson map may be derived from crystal space group by adding a center of symmetry and losing translational elements associated with screw axes or glide planes.

Pattern map shows intermolecular vectorsBy definition:

Because:

we have:

Pattern map shows intermolecular vectors -another example

Patterson maps often have serious overlapping of peaks

1. Peaks are generally broad2. N2 –N peaks in total

1-D

2-D

Simple structures can be solved from Patterson maps

Space group Pm: (x,y,z) and (x,-y,z)

There will be a set of vectors between symmetry-related atoms at:(0,2y, 0) in Patterson map.

Simple structures can be solved from Patterson maps

• Space group symmetries usually give rise to concentrated vector points in the forms of Harker lines or Harker planes

• Example: space group P21

– General positions are: (x,y,z), (-x, y+1/2, -z)

– Interatomic vector for symmetry-related atoms is: (2x, ½, 2z)

– There will be a lot of vector on v=1/2 section in Patterson map

– v=1/2 is called Harker section

– From Harker peaks, we can obtain x and z coordinates for every atom

Patterson map in protein crystallography

• Locate heavy atom positions for multiple isomorphous replacement method or multi-wavelength anomalous dispersion method

• Rotation and translation function searches for molecular replacement method