PN: X-Ray Crystallography

Light can diffract, which means it bends around edges of similar dimension as its wavelength.

In x-ray crystallography, we use x-rays which have nm dimension wavelengths so they can bend around protein lattices.

Fourier transformation

When two x-rays a and b shine on two atoms in a crystal that have a distance of d, beam b must travel an additional length Δx.

This distance can be described with

Δx = d sin(θ)

with θ being the wave angle.

Bragg’s Law says that

2Δx = 2d sin(θ) = nλ

with λ being the wavelength and n any integer or half integer.

When n is 1,2,3…, we have constructive interference.

When n is 1.5, 2.5, 3.5, …, we have destructive interference.

A path difference as described by Bragg’s Law leads to a phase difference and constructive/desctructive interference between two x-ray beams.

Interference is what we see in the diffraction pattern of an x-ray measurement which we can fourier transform to obtain an electron density map.

When d is small (the atoms are close together), you need high angles to resolve them, according to Bragg’s Law (d and sin(θ) are inversely proportional)

—> small angles: low resolution

—> high angles: high resolution

• virtual planes in the crystal that fulfil Bragg’s Law

  • atoms that are located in a Bragg plane cause constructive interference and are visible in the diffraction pattern

  • atoms that are not located in a Bragg plane cause diffractive interference and are not visible in the diffraction pattern

The thinner the “slices”, the higher the resolution.

Each spot of a diffraction pattern has three structure factors - one for the frequency, one for the amplitude, one for the phase of the wave.

• frequency - distance of Bragg planes

• amplitude - intensity

• we are missing the phase information (offset of the Bragg planes from the origin)

We need both structure factors to measure the electron density map.

• Molecular replacement

  • you know a structure of a related/similar protein and you can get the phases from the similar parts

• add heavy atoms

• SAD/MAD

The R factor describes how well your model fits the crystallographic data.

When you build and refine a model, a part of the data should not be included in the modeling.

The R-factor of this data is called Rfree and describes how well the model you got from the rest of the data fits the data you did not use to build the model.

This is used to avoid overfitting - both R and Rfree need to get lower during refinement and need to be about the same.

(overfitting: overinterpreting noise)

2fo-fc maps show the electron density map for a protein.

fo-fc maps show disagreements between the observed map and the calculated map (+3sigma: usually green, where atoms are missing), (-3sigma, usually red, where atoms are too much)

Miller indices are used to describe the location of the Bragg planes.

How to calculate Miller indices for a plane:

1. Find intersects with the x-, y-, and z-axis (when a plane does not cross an axis, this value is infinite)

2. Take the reciprocal value (1/x, 1/y, 1/z).

3. Multiply all reciprocal values with the smallest number possible so that you have integers again.

solution: 5,2,5

Instead of having the crystal in a capillary, we keep it rotating in a loop cooled with liquid nitrogen.

The maximal possible solution is the distance between two Bragg planes. This is calculated via:

with h,k,l being the Miller indices

and a being the lattice parameter

RNA and DNA are chiral and have a handedness.

To enable chiral molecules to have one of 230 space groups, we need to have racemic mixtures of them.