The phase problem

in protein crystallography

The phase problem

in protein crystallography

Bragg diffraction of X-rays

(photon energy about 8 keV, 1.54 Å)

Structure factors and electron density

are a Fourier pair

The problem is that the raw data are the squares of the modulus of the

Fourier transform.

That´s the famous phase problem.

In protein crystallography, there are several ways to get the phases:

• Molecular replacement

• Heavy atom methods

• Direct methods

• Non-standard methods

Mol A: GPGVLIRKPYGARGTWSGGVNDDFFH...Mol B: GPGIGIRRPWGARGSRSGAINDDFGH...

Mol A Mol B?

Molecular replacement

If we have phases from a similar model...

Amplitudes: Manx

Phases: Manx

Amplitudes: Cat

Phases: Cat

Amplitudes: Cat

we can use

Phases: Manx

...we can combine them with the experimental amplitudes to get a better model.

Patterson maps can be used to find

.... the proper orientation (rotation)

.... the proper position (translation)

for the search model.

The density map The Patterson map

í

iiiilkhlzkyhxifF )](2exp[

,,

j

jjjjlkhlzkyhxifF )](2exp[

,,

ji

jijijijilkhzzlyykxxhiffI

,,,

))]()()((2exp[

The Patterson map is the Fourier transform of the intensities.

It can be calculated without the phases.

The matching procedure requires a search in up to six dimensions

Luckily, the problem can be factorized into

• first, a rotation search

• then, a translation search

Flow chart of a typical molecular replacement procedure (AMORE)

xyzin1 (*1.pdb)

table1 (*1.tab)

hklpck1 (*1.hkl)

clmn1 (*1.clmn)

tabfunrotfun

(generate)rotfun (clmn)

hklin (*.mtz)

hklpck0 (*0.hkl)

clmn0 (*0.clmn)

rotfun (clmn)sortfun

rotfun (cross)}

rotfun (cross)

SOLUTRC

trafun (CB)

SOLUTTF

fitfun (rigid)

SOLUTF

pdbset

solution.pdb

Poor phases yield self-fulfilling prophesies

Amplitudes: Karlé

Phases: Karlé

Amplitudes: Hauptmann

Phases: Hauptmann

Amplitudes: Hauptmann

If Karlé phases Hauptmann, Hauptmann is Karléd!

Phases: Karlé

Heavy atom methods

?

Can we do X-ray holography?

Can we do holography with crystals?

In principle yes, but the limited coherence length requires a local reference scatterer.

For a particular h,k,l

FP

FH1

FH2

FPH1

FPH1

we can collect all knowledge about amplitudes and phases in a diagram

(the so-called Harker diagram)

• Normally, there´s the problem that different crystals are not strictly isomorphous.

• Thus, the best is a reference scatterer that can be switched on and off.

Absorption is accompanied by dispersion.

This Kramers-Kronig equation is very general:

Its (almost) only assumption is the existance of a universal maximum speed (c) for signal propagation.

Which elements are useful for MAD data collection?

7 keV

25 keV 0.5 Å

1.8 ÅK

LIII

26-46

64-

H He

Li Be B C N O F Ne

Na Mg Al Si P S Cl Ar

K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr

Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe

Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn

Fr Ra Ac Rf Ha

 

Lanthanides Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

Actinides Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr

 

The MAD periodic table

All phasing can be done on one crystal.

F1,2

F-1,-2

ab

F1,2 : scattering from b is phase behind

F-1,-2 : scattering from b is phase ahead

In the presence of absorption, Bijvoet pairs are nonequal.

dxdydzzyx eF

lzkyhxi

lkh

2

,,,,

dxdydzzyx eF

lzkyhxi

lkh

2

,,,,

zyxzyx ,,,,

FeF lkh

lzkyhxi

lkhdxdydzzyx

,,

2

,,,,

assuming

zyxzyx ,,,, with absorption:

Direct methods

?

Atomic resolution data

the best approach for small molecules

If atoms can be treated as point-scatterers, then

if all involved structure factors are strong

100 atoms in the unit cell

a small protein

The method is blunt for lower resolution or too many atoms.

Three-beam phasing

?

very low mosaicity data

an exciting, but not yet practical way

An example from our work

(solved by a combination of MAD and MR)

Metal ions

Can we tell from the fluorescence scans?

Normally yes, but not in this case!

Co

Zn

FeNi

Cu

Compton

Can we tell from the anomalous signal?

order in the periodic table: Fe, Co, Ni, Cu, Zn

2fo-fc map, 1.05 Å

anomalous map, 1.05 Å

anomalous map, 1.54 Å

Here´s the maps!

Quantitatively:

f“ (1.05 Å) = 1.85 0.05 f“ (1.54 Å) = 2.4 0.2

Thanks to my group, particularly S. Odintsov and I. Sabała

Thanks to Gleb Bourenkov, MPI Hamburg c/o DESY