BioPHYSICAL chemistry

Phase determination

In a diffraction experiment, the intensity of each diffraction peak, that is

|F(hkl)|^2 , is measured. To determine the electron density, the value of the

amplitude and phase of F(hkl) is needed. Unfortunately,the phase cannot

be determined directly from the intensity because the structure factor is a

complex number and the phase associated with the structure factor is not

determined in the experiment. There are three approaches that can be

used to overcome this phase problem.

Molecular replacement

In the case that the structure of the protein crystallized is unknown, but

the structure of a related protein has already been determined, then the

approach of molecular replacement can be used. The structure of the related

protein is first modified to make a structural model that resembles the

unknown protein. Based upon a comparison of the sequences, regions are

removed that are found in the related protein but not in the unknown

328 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY

S=ha* +kb* +lc*

(db15.9)

r=(x/a)a+(y/b)b+(z/c)c

where (a,b,c) and (a*,b*,c*) represent unit vectors in real and reciprocal space, respectively.

The expression for the structure factor (eqn db15.6) can now be written in terms of the

position and diffraction vectors:

(db15.10)

A more general form of this relationship treats the electron density of the diffracting object

as a continuous object rather than as a sum of discrete atoms. In this case, the summation

is replaced by an integral and the discrete atomic scattering factors are replaced by the elec-

tron density at the location:

(db15.11)

This last equation represents a Fourier transform from real space to reciprocal space. Jean

Baptiste Joseph Fourier lived in France during the French Revolution and discovered Fourier

transforms as a method of relating two functions by integral transforms. These transforms

are used in many applications, including the conversion of the temporal response of timed

nuclear magnetic resonance (NMR) pulses into a chemical-shift spectrum (Chapter 16).

FS() ()=∫ρre^2 πi()rS⋅ dV

FS()= ( ) ()

=

⋅

∑f hkl ei

i

N

irSi

1

2 π