Phase Problem in Crystallography Explained
A critical component in crystallography is the generation of an electron density map. The electron density map is important since it guides the creation of the protein model.
An equation for the calculation of the electron density (ρ(x,y,z)) is as follows:
Fhkl is the amplitude of the wave which is proportional to the square root of the intensity – this is measured during the crystallographic experiment as a reflection (or spot)
all hkl represent the measured location of the reflection on the detector
Vc represents the crystal volume
α is the phase
The diffraction pattern is collected with the spot positions and intensities being recorded. The intensity of the spots allow for the calculation of the amplitude. The volume of the crystal (Vc) can be determined from the spacing of the reflections. The component that is not recorded during crystallographic data collection is the phase (α).
Since the phase is not recorded it creates a problem hence the name “the phase problem” in crystallography. The challenge is to reconstruct the electron density map by approximating phases.
Tommy
May 5th, 2010 at 4:47 AM #
I was eager to read “Phase Problem in Crystallography Explained”, but it didn’t really provide me with an explanation of the phase problem. Nevertheless I enjoy reading your posts. They are often quite entertaining and informative. I especially liked the series you did on the falsified structures.
Sean
May 5th, 2010 at 9:10 AM #
Hi Tommy,
If you are eager to read a post about the phase problem then we are going to get along great. I am sorry this post didn’t provide an explanation. What else would you find helpful?
I tried to keep the post as straight forward as possible by not including the derivations.
AaronB
May 5th, 2010 at 2:48 PM #
Good straight forward explanation of a very important equation in X-ray diffraction. It may be of benefit to include why phase information is so dominant in calculating electron density (Kevin Cowtan’s Book of Fourier is a good reference). Also if we are going for the straight forward explanation, maybe including what is phasing would be helpful.
Tommy
May 6th, 2010 at 9:15 AM #
I’m sure your explanation of the phase problem is a good one. The problem is at my end. I just don’t know enough about crystallography and the math behind. I’m not a crystallographer. I just work with crystal structures. I guess ignorance is bliss for now.
Sean
May 7th, 2010 at 10:03 AM #
Hi Tommy,
If the explanation didn’t help you then let me try again without math.
Think of yourself moving slowly along in a boat.
Waves are approaching and colliding with the boat.
You can see of how big the waves are (amplitude) and how far the waves are spaced apart (wavelength).
Now imagine you are out at night and you reach your hand into the water to feel the waves that have bounced off of the side of boat. In crystallography, we are interested in waves that bounce off of a crystal.
Even with your hand in the water it is hard to tell whether you are feeling the top, middle or bottom of the wave. If you were asked the exact position of the wave (phase) it would be very difficult.
In this case, it is difficult by sense of touch to predict the phase of a wave. In crystallography, we don’t measure it at all.
Tommy
May 7th, 2010 at 11:38 AM #
Thanks for your efforts in elucidating the phase problem. I hope it’s not like throwing pearls to pigs.
I never understood, how you can solve the phase problem, if you don’t do molecular replacement. Maybe I’ll read about that in another future post of yours.
Thanks for taking your time to explain.
Sean
May 9th, 2010 at 8:19 PM #
No problem, AaronB had the right idea. I will work on putting together a post on the subject. Thanks again for the comment.
BR
May 13th, 2010 at 2:06 PM #
There should be a minus sign in front of the entire exponent [ ]. Image 9.15 of BMC visualizes the phase problem.
http://www.ruppweb.org/garland/gallery/Ch9/pages/Biomolecular_Crystallography_Fig_9-15.htm
Sean
May 13th, 2010 at 2:42 PM #
Hi Bernhard – thanks for the pointing that out! (fixed it)
Tommy
May 14th, 2010 at 5:25 AM #
Thanks Bernhard. I just ordered your book from Amazon.
Sean
May 14th, 2010 at 9:14 AM #
Great, I am glad to see the two of you were able to connect!
Morten
May 18th, 2010 at 4:27 AM #
I take issue with the line “The electron density map is important since it guides the creation of the protein model.”
It makes it sound like the model is more important than the electron density, when they are roughly equally important (in my mind at least) and the electron density is actually what we measure in the xray crystallographic experiment.
Sean
May 18th, 2010 at 9:19 AM #
Hi Morten,
‘important’ sounds like ‘more important’? I agree that both play a role.
Viz
May 26th, 2010 at 5:23 PM #
Here’s the explanation for the phase problem: The structure factor , F_hkl, is a complex number (z = a + bi), and intensity, the raw data in any x-ray crystallographic experiment, is the square of the absolute value of F_hkl. In math we know that this will eliminate the imaginary part of the complex number. But the phase information (of the diffracted waves) is contained within the imaginary part, as in z = a + bi = |z|exp(ix). This is how the phase information is lost during x-ray data collection (of spot intensities).
Sven Hovmöller
April 29th, 2011 at 10:29 AM #
The easiest way to understand the phase problem (and many more things) in crystallography is via electron crystallography. That is because in the electron microscope you can get BOTH an enlarged image of your crystal and the diffraction pattern. While an electron diffraction pattern looks very similar to the (precession) X-ray diffraction pattern of the same crystal, the EM image is a differen matter! An EM image can be an enlargment of a million times, and then individual atoms are seen! This works for inorganic materials, such as metal oxides, that can resist the high electron dose. For proteins it is harder, but taking great care you can still get (almost) atomic resolution EM images also from proteins. If you now calculate the Fourier transform (FT) of an EM image, it will look just like the diffraction pattern, although the spots in the FT don’t go so far out from the center.
Now to the phases: An image of a crystal is a sum of several sets of lines (black/white/black…) in all directions. Crystallographers would say these lines are projected Bragg planes. Each such set of lines gives rise to just a single spot (well, and its Friedel pair) in the diffraction pattern. From the position of that spot you know the direction and spacing of the Bragg plane. If the spot is strong it means this set of black/white lines are strong (high amplitude). BUT – you don’t know how all these sets of lines are positioned, relative to each other. That is the phase problem in crystallography. In an EM image, on the other hand, you see the fringes and when you calculate the Fourier transform of the EM image, you get also the phases! Numerically in degrees – just read out from the FT! This was first realized by Aaron Klug at the MRC in Cambridge around 1965 and he got the Nobel prize for this (and more) in 1982. Read more at http://en.wikipedia.org/wiki/Electron_crystallography.
In August 2011 Oxford University Press will publish a textbook on Electron Crystallography.