Phasing

Recap

A Fourier Series describes the series of waves which can be summed to reproduce some initial function.

A series of waves are summed one-by-one to eventually approximate a step function. (video from 3blue1brown)

The nature of diffraction is such that the constructive interference of the diffracted quanta can be easily converted into a piece of a 3-Dimensional Fourier Transform. This relationship is through the Ewald Sphere projection, and the structure factor equation.

The waves of the Fourier Transform, just like any other waves, have three core properties:

Frequency
- The frequency of each wave is determined by its diffraction angle off the subject, or the location of the specified peak on the detector
Amplitude
- The amplitude of each wave is recorded by the intensities of the peak on the detector
Phase
- The phase of the waves is lost during data collection

When we record diffraction patterns, we cannot record the phase of the quanta. This problem is compounded by the fact that the phase contains most of the information to reconstruct the structure.

The Phase Problem

And so, one of the largest issues a crystallographer faces is known as the phase problem. This problem arises from the fact that no detector has been built that can determine the phase of an incoming X-ray. This is largely because the wavelength of X-rays is so small that each degree in phase is far too minuscule to determine. It is for this reason that structure factor amplitudes may be represented as circles rather than vectors, the information describing the direction of the vector is lost.

Here we see the result of the wavelets of four different atoms constructively interfering to form a structure factor in reciprocal space. This example implies that the phase information of the wavelets was recorded, which is not the case in practice.

In reality, we only know $|F|$ , not it’s direction/phase. There are several methods used to get around this missing information.

This applies to all structure factors in the model fourier transform.

Direct Methods

The best way to summarize Direct Methods is that it essentially a slightly better than random guess-and-check based on the fact that there are constraints as to what an electron density map (or electrostatic potential map, if your incident quanta are electrons rather than photons) can look like. Direct methods has been revolutionary in the field of crystallography and consequentially earned Herbert Hauptman and Jerome Karle the 1985 Nobel Prize in Chemistry.

Isomorphous Replacement

Molecular Replacement

Molecular replacement is the most self-explanatory method of the three. Now that crystallography has a long history and many, many structures solved, it is possible to refer back to previous structures with similarities to the target. Substituting the known structure provides a partial solution to the real-space structure of the target. Since the structure factors (including their phases) can be derived from the real-space structure, you will inherently gain phase information from this substitution.

There are two interrelated problems you may quickly notice with this method:

This method requires prior structures to be solved, which at some point must not have used molecular replacement
There is an obvious bias in terms of solving the structure using prior data. Some may even question whether this would count as “solving” the structure.

https://people.mbi.ucla.edu/cascio/LECTURES/M230B/phasing_4b.html

phasing_v4b.html

StructFactor.mp4