Page 155 - Macromolecular Crystallography

144 MACROMOLECULAR CRYS TALLOGRAPHY

10.2 Fourier transforms and the independent equations as there are unknowns. Let
phase problem us therefore count the number of unknowns: there
are 2N unknown electron densities at the indepen-
We want to know the electron density that is deter-
dent grid points and 2N unknown phases. These 4N
mined by the measured structure factor amplitudes
unknowns are inter-related by 2N Fourier summa-
and their phases. The electron density at point x is
tions, so the system is underdetermined. Its solution
calculated by a Fourier summation:
clearly requires at least 2N additional equations.
2N
1 However, in view of experimental errors, 2N addi-
iφ(h j )−2iπx·h j
ρ(x) = F(h j ) e (1)
V tional equations are unlikely to be sufficient to solve
j=0
the phase problem. In practice, we can only expect a
In the above equation, ρ(x) is the electron density statistically meaningful solution if we include many
at x, while V is the volume of the unit cell, 2N is more equations and identify the solution that agrees
the number of relevant structure factors, and |F(h j )| most with all equations simultaneously. Further-
is the amplitude of the structure factor with Miller more, since Eq. 1 is non-linear in φ(h j ), we cannot
indices h j = (h j , k j , l j ) and a phase of φ(h j ). Note that expecttofindananalyticsolution. Hence, wehaveto
2N is determined by the size of the unit cell and the make initial guesses for the unknowns and improve
resolution. from there.
Equation 1 is a discrete Fourier transform. It is dis- Constraints effectively reduce the number of
crete rather than continuous because the crystalline unknowns while restraints add to the number of
lattice allows us to sum over a limited set of indices, equations in the system, without changing the num-
rather than integrate over structure factor space. The ber of unknowns. The effectiveness of a restraint is
discrete Fourier transform is of fundamental impor- partially determined by the number of independent
tance in crystallography – it is the mathematical equations the restraint introduces in the minimiza-
relationship that allows us to convert structure fac- tion.Also the discriminating potential of the individ-
tors (i.e. amplitudes and phases) into the electron ual terms between right and wrong models affects
density of the crystal, and (through its inverse) to the scale of the improvement. Clearly, a robust term
convert periodic electron density into a discrete set with a sharp minimum contributes much more to
of structure factors. phasing than a permissive term that hardly distin-
Even though the Fourier transform in Eq. 1 is guishes a wrong model from a correct one. Below we
discrete, ρ(x) is continuous, as it can be calculated summarize the general form of the additional infor-
for any grid point x. Obviously one could calcu- mation, constraints and restraints, which in practice
late Eq. 1 on an arbitrary fine grid, but in that case leads to a system of equations that is no longer
the density at any grid point is correlated to that of underdetermined.
its neighbours through interpolation. Since Fourier
transforms neither create nor destroy information,
the maximum number of uncorrelated, independent 10.3 Reciprocal space constraints
density grid points is limited to 2N, the number of
10.3.1 Friedel’s law
structure factors going into the summation of Eq. 1.
To conclude, if there are 2N structure factors, the In protein crystallography we assume that all elec-
corresponding electron density map has 2N inde- tron density is real, and does not have an imaginary
pendent grid points and there are 2N independent component. In reciprocal space this observation is
equations of type 1 relating the former to the latter. known as Friedel’s law, which states that a struc-
Intuitively, the correct application of restraints ture factor F(h) and its Friedel mate F(−h) have
and constraints to electron density should improve equal amplitudes, but opposite phases. The corre-
the phases. To quantify this notion, it is useful, spondenceofthesetwoassumptionsfollowsstraight
though unconventional, to proceed as if we are from Fourier theory and, in consequence, explicitly
solving a system of non-linear equations. A solu- constraining all electron density to be real is entirely
tion of such a system requires at least as many equivalent to introducing Nadditional equalities of