7. Structural
resolution
If you do not see a menu on the
left, please, use this
link. 

In previous chapters, we have seen how Xrays interact
with periodically structured matter (crystals), and the implicit question that we have raised
from
these earlier chapters is:
Can we "see" the internal
structure of crystals?
or said in
other words:
Can
we "see" the atoms and molecules that build crystals?
The
answer is definitely YES!



Molecular structure of a pneumococcal
surface enzyme 
Molecular packing in the crystal of a
simple organic compound, showing its crystallographic unit cell. 
Geometric
details showing several molecular interactions in a fragment of the
molecular structure of a protein. 
As the examples above demonstrate, Crystallography can show us the
structures of very large and
complicated molecular structures (left figure) and
how
molecules pack together in a crystal structure (center figure). We can
also see every geometric detail, as well as the
different types of interactions, among molecules or parts of them
(right figure).
However, for a better
understanding of
the fundamentals on which this response is based, it is
necessary
to introduce some new concepts or refresh some of the previously
seen ones...
In previous chapters
we have seen that crystals represent the organized and ordered
matter, consisting of associations of atoms and/or molecules,
corresponding to a natural state of it with a minimum of energy.
We also know that crystals can be described by repeating units in the
three directions of space, and that this space is known as direct
or real
space. These repeating units are know as unit
cells (which also serve as a reference system to describe
the atomic positions). This direct
or real
space,
the same in which we live, can be described by the electron density, ρ(xyz),
a function defined in each point of the unit cell of coordinates
(xyz), where, in addition, operate symmetry
elements which repeat atoms and molecules within the cell.
Unit cell (left) whose threedimensional
stacking builds a crystal (right) 

Motifs (molecules, etc.) do repeat
themselves by symmetry operators inside the unit cell.
Unit cells are stacked in three dimensions, following the rules of the
lattice, building the crystal. 
We have also learned that Xrays
interact with the electrons of the
atoms in the crystals, resulting in a diffraction
pattern, also know as reciprocal
space, with the properties of a lattice (reciprocal
lattice) with a certain
symmetry, and where we also can define a repeating cell (reciprocal
cell). The "points" of this reciprocal lattice contain the
information on the diffraction intensity.

Interaction
between two waves scattered by electrons. The resulting waves show
areas of darkness (destructive interference), depending on the angle
considered. Image taken from physicsanimations.com. 

One of the hundreds
of diffraction images of a protein crystal.
The black spots on the image are the result of the cooperative
scattering (diffraction) from the electrons of all atoms contained in
the crystal. 
Through this cooperative scattering
(diffraction), scattered waves interact with each other,
producing a single diffracted beam in each direction of space, so that,
depending on the phase
differences (advance or delay) among the individual
scattered
waves, they add or subtract, as shown in the two figures below:
Composition of two scattered waves. A
= resultant amplitude; I
= resultant intensity (~
A^{2})

(a)
totally in phase (the total effect is the sum of both waves) 
(b)
with a certain difference of phase (they add, but not totally) 
(c) out
of phase (the resultant amplitude is zero) 
Between the two mentioned spaces (direct
and reciprocal)
there is a holistic
relationship (every
detail of one of the spaces affects the whole of the other, and
viceversa). Mathematically
speaking this
relationship is
a
Fourier transform that
cannot directly be solved, since the diffraction
experiment does not allow us to know one of the fundamental magnitudes
of the equation, the relative
phases of the diffraction beams.
All this could be summarized using the following scheme:

Outline on basic crystallographic concepts: direct and
reciprocal spaces. The issue is to obtain information on the
left
side (direct space) from the diffraction experiment (reciprocal space).

ELECTRON
DENSITY
To
know (or to see) the internal structure of a crystal involves solving a
mathematical function that defines the "electron density;" a
function that is defined at every point in the unit cell (a basic concept of the
crystal structure introduced in another chapter).
The function of electron density,
represented by the letter ρ,
has a certain value at each point within the unit cell given by the
coordinates (x,
y,
z),
referring to the unit cell axes. At those points where this function
takes maximum values (estimated in terms of electrons per cubic
Angstrom) is where atoms are located. That means that if we are able to
calculate this function, we will "see" the atomic structure of the
crystal. 

Formula
1. Function defining the electron density in a point of
the unit cell given by the coordinates (x, y, z)
 F(hkl)
represents
the resultant diffracted beams of all atoms contained in the unit cell
in a given direction. These magnitudes (actually waves), one
for
each diffracted beam, are known as structure
factors.
Their moduli are directly related to the diffracted intensities.
 h,
k, l are the
Miller indices of the diffracted beams (the reciprocal points) and Φ(hkl)
represent the phases of the structure factors. In reality we have
limitations due to the extent to which the diffraction pattern (= the
reciprocal space = the number of structure factors) is observed, that
is, that the number of structure factors is finite, and therefore the
synthesis will be approximate only and may show some truncation effects.


Appearance of a zone of the electron
density map of a protein crystal, before it is interpreted. 
The same electron density
map after its interpretation in terms of a peptidic
fragment. 
The equation above (Formula 1)
represents the Fourier
transform between the real
or direct space (where the atoms are, represented by the
function ρ) and the reciprocal
space (the Xray pattern) represented by the structure
factor amplitudes and their phases.
Formula 1 also shows the holistic character of
diffraction,
because in order to calculate the value of the electron density in a single
point of coordinates (xyz) it is necessary to use the
contributions of all structure factors produced by the
crystal diffraction.
When the unit cell is centrosymmetric, for each atom at
coordinates (xyz) there is an identical one located at (x,y,z).
This implies that Friedel's law
holds [F(h,k,l) = F(h,k,l)] and the expression of the electron
density (Formula 1) is simplified, becoming Formula 1.1. And the phases of the structure factors
are also simplified, becoming 0 ° or 180 °...
Fórmula
1.1. Electron density function in a point of coordinates (x, y, z)
in a centrosymmetric unit cell.
It is important to realize that the quantity and quality of information
provided by the electron density function, ρ, is very dependent on the
quantity and quality of the data used in the formula: the structure
factors
F(hkl) (amplitudes and phases!). We
will
see later on that the amplitudes of the structure factors are
directly obtained from the diffraction experiment.
As visual and
practical exercises on Fourier transform we recommend visiting:
 or, even better, the applet kindly provided by Nicholas
Schöni and Gervais Chapui (Ecole
Polytechnique Federal de Lausanne,
Switzerland). This applet calculates the Fourier transform of a two
dimensional density function ρ(x) yielding the complex
magnitude
G(S), the reciprocal space. The applet is also able to
calculate the
inverse Fourier transform of G(S). The density function can be either
periodic or nonperiodic. Numerous tools including drawing tools can be
applied in order to understand the role of amplitudes and phases which
are of particular importance in diffraction phenomena. As an
illustration, the Patterson function of a periodic structure can be
simulated.
The analytic expression of the structure
factors, F(hkl), is simple and involves a new
magnitude (ƒ_{j }) called atomic scattering factor (defined
in a previous chapter) which takes into account the
different scattering powers with which the electrons of the j atoms scatter the Xrays:
Formula
2. Structure factor for each diffracted beam. This
equation is the Fourier transform of the electron density (Formula
1).
It takes into account the scattering ƒ of all j atoms contained in the crystal unit cell.

The structure factors F(hkl) are waves and therefore can be
represented as vectors by their amplitudes, [F(hkl)], and phases Φ(hkl) measured on a common origin of phases. 
The structure factors F(hkl) are waves and therefore can be
represented by their amplitudes, [F(hkl)], and phases Φ(hkl), measured on a common origin of phases.
From the experimental point of view, it
is
relatively simple to measure the amplitudes [F(hkl)]
of all diffracted waves produced by a crystal. We just need an Xray
source, a single crystal of the material to be studied and an appropriate
detector. With these conditions fulfilled we can then measure
the intensities, I(hkl), of the diffracted beams in terms of:
Formula
3. Relationship
between the amplitude of a structure factor F(hkl) and its intensity I(hkl)
K is a factor that puts
the experimental structure factors, (F_{rel}) , measured on a relative scale (which
depends on the power of the Xray source, crystal size, etc.) into an absolute scale, which is to say,
the
scale of the calculated (theoretical) structure
factors (if we could know them from the real structure, Formula 2
above). As the structure is unknown at this stage, this factor can be
roughly evaluated using the experimental data by means of the socalled
Wilson plot.

Wilson
plot. I
_{rel}
represents the average intensity (in a relative scale) collected in a
given interval of θ
(the Bragg angle); f_{j} are the atomic scattering factors in
that angular range, and λ
is the Xray wavelength.
By plotting the magnitudes shown in the left figure (green
dots),
a straight line is obtained from which the folllowing information can
be derived:
 The value of
the yaxis intercept is the Neperian logarithm of C,
a magnitude related to the scale factor K
(= 1 / √C),
described above.
 The slope is
equivalent to 2B,
where B is the isotropic overall atomic thermal
vibration factor.

A is
an absorption factor, which also can be estimated from the dimensions
and composition of the crystal.
L is known as the Lorentz
factor, responsible
for correcting the different angular velocities with which the
reciprocal
points cross the surface of Ewald's sphere. For fourcircle
goniometers this factor can be calculated as 1/sin
2θ, where θ
is the Bragg angle of the reflections.
p is
the polarization factor which corrects the polarization effect of the of the incident beam, and
is given by the expression (1+cos^{2}2θ)/2, where θ also represents the Bragg angle of the
reflections (the reciprocal points).
THE
PHASE PROBLEM
However, in order to calculate the electron
density (ρ(xyz) in Formula 1, above), and
therefore to know the atomic positions inside the unit cell,
we also need to know the
phases of the different diffracted beams (Φ(hkl) in Formula 1 above). But, unfortunately,
this
valuable information is lost during the diffraction experiment
(there is no experimental technique available to
measure the phases!) Thus, we must face the socalled phase
problem if we want to solve Formula 1.
The phase problem can be very easily understood if we compare
the
diffraction experiment (as a procedure to see the internal structure of
crystals) with a conventional optical microscope...

Illustration on
the phase problem.
Comparison between an optical microscope and the "impossible" Xray
microscope. There are no optical lenses able to combine diffracted
Xrays to produce a zoomed image of the crystal contents (atoms and
molecules).

In a conventional
optical microscope
the visible light illuminates the sample and the scattered
beams
can be recombined (with intensity and phase) using a system
of
lenses, leading to an enlarged image of the sample under observation.
In what we might call the impossible
Xray microscope
(the process of viewing inside the crystals to
locate
the atomic positions), the visible light is replaced by Xrays (with
wavelengths close to 1 Angstrom)
and
the sample (the crystal) also scatters this "light" (the Xrays).
However, we do not have any system of lenses that could play the role
of the optical lenses, to recombine the diffracted waves providing us
with a direct "picture" of the internal structure of the crystal. The
Xray diffraction experiment just gives us a picture
of the reciprocal lattice of the crystal on a photographic plate
or detector. The only thing we can do at this stage
is to measure the positions and intensities of the spots collected on the detector.
These intensities are proportional
to the structure factor amplitudes, [F(hkl)].
But regarding the phases, Φ(hkl),
nothing can be concluded for the moment, preventing us from
obtaining a
direct solution of the electron density function
(Formula 1
above).
We therefore need some alternatives in order to retrieve the phase
values, lost during the diffraction experiment...
STRUCTURAL RESOLUTION
Once the phase
problem is known and understood, let's now see the general
steps (see the scheme below) that a crystallographer must face in order
to solve
the structure of a crystal and therefore locate the
positions of atoms, ions or molecules contained in the unit cell...

General
diagram illustrating the process of resolution of molecular and crystal
structures by Xray diffraction The process consists of different steps
that have been treated previously or are described below:
 Getting a crystal suitable for the experiment, with
adequate quality and size. Something related will be seen in another
section.
 Obtaining the diffraction pattern with the
appropriate wavelength. This has been described in another
chapter.
 Solving the electron density function, obtaining any
information about the phases of the diffracted beams. This is a key
point for the structural resolution that will be discussed below.
 Building an initial structural model to explain the
values
of the electron density function and completing the model locating the
remaining atomic positions. This will be seen below.
 Refining the model, adjusting all atomic positions to
get
the calculated diffraction pattern as similar as possible to the
experimental diffraction pattern, and finally validate and show the
total structural model obtained. This will be seen in another
chapter.

 The compound under study must be pure to be crystallized
(if not already, as in the case of natural minerals).
 Crystals can be
obtained
using different techniques, from the most simple evaporation
or
slow cooling method up to the more complex: vapor (or solvent)
diffusion, sublimation, convection, etc. There is enough literature
available.. See, for example, the pages of the LEC, Laboratory of
Crystallographic Studies,
for additional information on specific crystallization techniques. For
proteins, the procedure most extensively used is based on
vapour
diffusion experiments, usually with the "hanging drop" technique, described elsewhere
in these pages. In this sense it is very relevant to note
the recent advances introduced in
the field of femtosecond Xray protein nanocrystallography, which will
mean a giant step to practically eliminate most difficulties in the
crystallization process, and in particular for proteins (see
the small paragraph dedicated to the Xray free electron laser).
 If appropriate crystals are obtained, they are exposed to Xrays and their diffraction
intensities measured using the methods and equipment described
in a previous chapter. A careful data evaluation will provide
us with the dimensions
of the unit cell, the symmetry
and, directly
from the intensities,
the amplitudes of the structure
factors [F(hkl)] . Of all these
subjects at this stage, the most difficult one concerns the
determination of the crystal symmetry,
a key question for the successful resolution of the structure. To
obtain crystal symmetry, a visual study of the crystal would make
no sense and therefore it must be deduced from the symmetry of the
diffraction pattern, as indicated in a specific section
of these pages.
 At this stage,
the question about the unknown phases, Φ(hkl), arises, so that they must be somehow
evaluated, as we will see below...
 If the evaluated
phases are correct, the electron density function ρ(xyz)
will show a distribution of maxima (atomic positions) consistent
and meaningful from the stereochemical point of view. Once an
initial structure is known, some additional steps (construction of the
detailed model, mathematical refinement
and validation) must
be carried out. This will lead us to the
socalled final
model of the structure.
But let us come back to the most
important issue: how
do we solve the phase problem?
THE PATTERSON FUNCTION

The very
first solution to the phase problem was introduced
by Arthur
Lindo Patterson.
Basing his work on the inability to directly solve the
electron
density function (Formula 1 above or below), and after his training
(under the U.S. mathematician Norbert
Wiener) on Fourier transforms
convolution, Patterson introduced a new
function P(uvw)
(Formula 4, below) in 1934. This
formula, which
defines a new space (the
Patterson space), can be considered as the most important
single development in crystalstructure analysis since the discovery
of Xrays by Röntgen in 1895 or Xray
diffraction by Laue in 1914.
His elegant formula, known as the Patterson
function
(Formula 4, below), introduces a simplification of the information
contained in the electron density function. The Patterson function
removes the term containing the phases, and the amplitudes of
the
structure factors are replaced by their squares. It is thus a function
that can be calculated immediately from the available experimental data (intensities,
which are related to the amplitudes of the structure factors). Formally,
from the mathematical point of view, the Patterson function is
equivalent to the convolution
of the electron density (Formula 1,
below) with its inverse: ρ(x,y,z)
* ρ(x,y,z). 


Formula
1. The electron density function calculated at
the point (x,y,z). 


Formula
4.
The Patterson function calculated at the point (u, v, w). This is a
simplification of Formula 1; the summation is done on F^{2}(hkl)
and all phases are assumed to be zero. 
It seems obvious that after omitting the crucial information contained in the phases [Φ(hkl) in Formula
1],
the Patterson function will no longer show the direct positions of the
atoms in the unit cell, as the electron density function would do.In
fact, the Patterson function only provides a map
of interatomic vectors
(relative atomic positions), the height of its maxima being
proportional to the number of electrons of the atoms implied.
We
will see that this feature means an advantage in detecting the
positions
of "heavy" atoms (with many electrons) in structures where the
remaining atoms have lower atomic numbers. Once the Patterson map is
calculated, it has to be correctly interpreted (at least partially)
to get the absolute positions (x,y,z)
of the heavy atoms within the unit cell. These atomic positions can now
be used to obtain the phases Φ(hkl) of the diffracted beams by inverting
Formula 1 and therefore this will allow the calculation of the electron
density function ρ(xyz), but this will be the object of another
section of these pages.
THE
DIRECT METHODS
The phase problem for crystals formed by small
and medium size molecules was solved
satisfactorily by several authors throughout the twentieth
century with special mention to Jerome
Karle (19182013) and Herbert
A. Hauptmann (1917), who shared the Nobel Prize in
Chemistry in 1985 (without forgetting the role of Isabella Karle, 1921). The methodology
introduced by these authors, known as the
direct methods, generally exploit constraints or
statistical correlations between the phases of different Fourier
components.
The atomicity of molecules, and the fact that the electron density should
be zero or positive, at any point of the unit cell, creates certain
limitations in the distribution of phases associated
with the structure factors. In this context, the direct
methods establish systems of equations
that use the intensities of diffracted beams to describe these
limitations. The resolution of these systems of equations provides direct
information on the distribution of phases. However, since
the validity of each of these equations is established in terms of
probability, it is necessary to have a large number of equations to
overdetermine the phase values of the unknowns (phases Φ(hkl)).

The direct methods
use equations that relate the phase of a
reflection (hkl)
with the phases of other neighbor reflections (h',k,'l' y hh',kk',ll'),
assuming that these relationships are "probably
true" (P)
... 

where E_{hkl}, E_{h´k´l´}
and E_{hh',kk',ll' } are
the so called normalized
structure factors,
that is, structure factors corrected for thermal motion, brought to an
absolute scale and assuming that structures are made of point atoms. In
other words, structure factor normalization converts measured F
values into "point atoms at rest" coefficients known as E
values. 
At present, direct methods are the preferred method for phasing
structure factors produced by small or medium sized molecules having up
to 1,000 atoms in the asymmetric unit. However, they are generally not
feasible by themselves for larger molecules such as proteins. The
interested reader should look into an excellent
introduction to direct methods through this link.
METHODS OF STRUCTURAL RESOLUTION FOR MACROMOLECULES
For crystals composed of large
molecules, such as proteins and enzymes, the phase
problem can be solved succesfully with three main methods,
depending of the case:
(i)
introducing atoms in the structure with high scattering
power. This
methodology, known as MIR (Multiple
Isomorphous
Replacement) is therefore based on the Patterson
method.
(ii)
introducing atoms that scatter Xrays anomalously, also known
as MAD
(Multiwavelength
Anomalous
Diffraction),
and
(iii)
by means of the method known as MR (Molecular
Replacement),
which uses the previously known structure of a similar protein.
MIR (Multiple
Isomorphous
Replacement)
This technique, based on the Patterson method, was
introduced by David
Harker, but was successfully applied for the
first time by Max
F. Perutz and John
C. Kendrew who received the Nobel Prize in
Chemistry in 1962, for solving the very first structure of a protein,
hemoglobin.
The MIR method is applied after introducing "heavy" atoms
(large scatterers) in the crystal structure. However, the
difficulty of this methodology lies in the fact that the
heavy atoms should not affect the crystal formation or unit cell
dimensions in comparison to its native form, hence, they should be
isomorphic
This method is conducted by soaking the crystal of the sample to be
analyzed with a heavy atom solution or by cocrystallization with the
heavy atom, in the hope that the heavy atoms go through the channels of
the crystal structure and remain linked to amino acid side chains with
the
ability to coordinate metal atoms (eg SH groups of cysteine).
In the case of metalloproteins, one can replace their
endogenous metals by heavier ones (eg Zn by Hg, Ca by Sm,
etc.).
Heavy atoms (with a large number of electrons) show a higher
scattering power than the normal atoms of a protein (C, H, N,
O and S), and therefore they appreciably change the intensities of the diffraction
pattern when compared with the native protein. These differences in
intensity between the two spectra (heavy and native
structures) are used to calculate a map
of interatomic vectors between the heavy atom positions
(Patterson
map), from which it is relatively easy to determine their
coordinates within the unit cell.
Scheme of a Patterson function
derived from a crystal containing three atoms in the unit cell. To
obtain this function graphically from a known crystal structure
(left figure) all possible interatomic vectors are
plotted (center figure). These vectors are then
moved parallel to themselves to the origin of the
Patterson unit cell (right figure). The calculated
function will show maximum values at the end of these vectors, whose
heights are proportional to the product of the atomic numbers of the
involved atoms. The positions at these maxima (with
coordinates u,
v, w) represent the differences
between the coordinates of each pair of atoms in the crystal,
ie u=x_{1}x_{2}, v=y_{1}y_{2}, w=z_{1}z_{2}.
With the known positions of the heavy atoms, the structure factors are
now calculated using Formula 2 (see also the diagram below), that is
their amplitudes F_{c}(hkl) and phases Φ_{c}(hkl), where the c subscript means "calculated". By using
Formula 1, an electron density map,
ρ(xyz),
is now
calculated using the amplitudes of the structure
factors
observed in the experiment, F_{o}(hkl) (containing the contribution of the
whole structure) combined with the calculated phases Φ_{c}(hkl).
If these phases are good enough, the
calculated electron density map will show not only the known heavy
atoms, but will also yield additional information
on further atomic positions (see diagram below).
In summary, the MIR methodology steps are:
 Prepare one or several heavy atom
derivatives that must be isomorphic with the native protein. A first
test of isomorphism is done in terms of the unit cell parameters.
 Collect diffraction data from
both native and heavy atom derivative(s).
 Apply the Patterson method to get
the heavy atom positions.
 Refine these atomic positions and
calculate the phases for all diffracted beams.
 Obtain an electron density map
with those calculated phases.
MAD (Multiwavelength
Anomalous
Diffraction)
The changes in the intensity of the diffraction
data produced by introducing heavy atoms in the
protein crystals can be regarded as a chemical modification of the
diffraction experiment. Similarly, we can cause changes in the
intensity of diffraction by modifying the physical properties of atoms.
Thus, if the incident Xray radiation has a frequency close to the
natural vibration frequency of the electrons in a given atom, the atom
behaves as an "anomalous scatterer". This produces some changes in the
atomic scattering factor, ƒ_{j } (see Formula 2), so that its expression
is modified by two terms, ƒ'_{} and ƒ''' which account for its real and
imaginary components, respectively. For atoms which behave
anomalously, its scattering factor is given by the expression shown
below (Formula 5).
Formula
5. In the presence of anomalous scattering, the atomic
scattering factor, ƒ_{0}
,
has to be modified adding two new terms, a real and an imaginary
part.
The advanced reader
should also read
the section about the phenomenon of anomalous dispersion.
The ƒ' and ƒ''_{} corrections vs. Xray energy
(see
below for the case of Cu Kα) can be calculated taking into
account some
theoretical considerations...
Real and imaginary components of the Selenium scattering factor vs. the
energy of the incident Xrays. The vertical line indicates
the
wavelength for CuKα. 
For Xray energy values where
resonance exists, ƒ' increases
dramatically, while the value of ƒ'' decreases.
This has practical importance considering that
many heavy atoms used in crystallography
show absorption
peaks at energies (wavelengths) which can be easily obtained with
synchrotron radiation.
Diffraction data collected in these conditions will show a
normal component, mainly due to the light atoms (nitrogen, carbon and
hydrogen), and an anomalous part produced by the heavy atoms,
which will produce a global change in the phase of each reflection. All
this leads to an intensity change between those reflections known
as Friedel pairs (pairs of reflections which under normal conditions
should have the same amplitudes and identical phases, but with opposite
signs). The detectable change in intensity between these reflection
pairs (Friedel pairs) is what we call anomalous diffraction.

The
MAD method, developed by Hendrickson and Kahn, involves diffraction
data measurement of the protein crystal (containing a strong
anomalous scatterer) using Xray radiations with different energies
(wavelengths): one that maximizes ƒ''_{}, another which minimizes ƒ'_{}
and a third measurement at an energy value distinct from these two.
Combining these diffractions data sets, and specifically analyzing the
differences between them, it is possible to calculate the distribution
of amplitudes and phases generated by the anomalous scatterers. The
subsequent use of the phases generated by these anomalous scatterers,
as a first approximation, can be used to calculate an electron density
map for the whole protein.
In general, there is no current need to introduce individual atoms as
anomalous scatterers in protein crystals. It is relatively easy to
obtain recombinant proteins in which methionine residues are replaced
by seleniummethionine. Selenium (and even sulfur) atoms of methionine
(or cysteine), behave as suitable anomalous scatterers for
carrying out a MAD experiment.
The MAD method presents
some advantages vs.
the MIR technique:
 As
the MAD technique uses data collected from a single crystal, the
problems derived from lack of isomorphism, common in the MIR method, do
not apply.
 While in the absence of anomalous dispersion, the atomic scattering factor (ƒ_{0}) decreases dramatically with the angle
of dispersion, its anomalous component (ƒ'_{} + iƒ''_{} )
is independent of that angle, so that this relative signal increases at
a
higher resolution of the spectrum, which is to say, at high
Bragg angles.
Thus, the estimates of phases by MAD are generally better at
high
resolution. On the contrary, with the MIR method, the lack of
isomorphism is larger at high resolution angles and therefore the high
resolution intensities (> 3.5 Angstrom) are not suitable for
phasing.

Argand
diagram showing the scattering contribution from an anomalous scatterer
in a matrix of normal scatterers. This effect implies that
Friedel's law fails. Image taken from "Crystallography
101".
Fp represents the contribution from the normal scatterers
to the structure factor (of indices hkl).
Fa
and Fa'' represent
the real (ƒ_{0}
+ ƒ'_{} ) and imaginary (ƒ''_{} ) parts,
respectively, of the scattering factor from the anomalous scatterers.
Fp, Fa
and Fa"
represent the same as Fp,
Fa and Fa'', but for the
reflection with indices h,
k, l

The anomalous behavior of the atomic scattering factor only
produces small differences between the intensities (and
therefore
among the amplitudes of the structure factors) of the reflections that
are related by a centre of symmetry or a mirror plane (such as
for instance, I(h,k,l)
vs. I(h,k,l),
or I(h,k,l)
vs.
I(h,k,l).
Therefore, to estimate these small differences
between the
experimental intensities, additional precautions must be taken into
account. Thus, it is recommended that reflections expected to show
these differences are collected on the same diffraction
image, or
alternatively, after each collected image, rotate the crystal 180
degrees
and collect a new image. Moreover, since changes in ƒ' and ƒ''_{} occur by minimum Xray energy
variations, it is necessary to have good
control of the energy values (wavelengths). Therefore, it is
essential to use a synchrotron radiation facility, where wavelengths
can be tuned easily.
The advanced
reader should also have a look
into the web pages on anomalous scattering, prepared
by Ethan A.
Merritt, as well as the practical summary
prepared by Georg M. Sheldrick.
MR (Molecular
Replacement)
If
we know the structural model of a protein with a homologous amino acid
sequence, the phase problem can be solved by using the
methodology
known as molecular replacement (MR). The known structure of
the homologous protein
is regarded as the protein to be determined and serves as a first model
to be subsequently refined. This procedure is obviously based on the
observation that proteins with similar peptide sequences show a very
similar folding. The problem in this case is transferring the molecular
structure of the known protein from its own crystal structure
to a
new crystal packing of the protein with an unknown structure. The
positioning of the known molecule into the unit cell of the unknown
protein requires determining its correct orientation
and position within the unit cell. Both operations, rotation
and
translation, are calculated using the socalled rotation and
translation
functions (see below).

Scheme of the molecular replacement (MR) method.
The molecule with known structure (A)
is rotated through the [R]
operation and shifted through T to bring it over the position of the
unknown molecule (A’). 
The
rotation function. If we consider the case of two
identical molecules, oriented in a different way, then the Patterson
function
will contain three sets of vectors. The first one will contain the
Patterson vectors of one of the molecules, ie all interatomic
vectors within molecule one (also called
eigenvectors). The
second set will contain the same vectors but for the second
molecule, identical to the first one, but rotated due to their
different orientation. The third set of vectors will be the interatomic
cross vectors between the two molecules. While the
eigenvectors
are confined to the volume occupied by the molecule, the cross vectors
will extend beyond this limit. If both molecules (known and unknown)
are very similar in structure, the rotation function R(α,β,γ)
would try to bring the Patterson vectors of one of the molecules to be
coincident with those of the other, until they are in good agreement.
This methodology was first described by Rossman and Blow.
R(α,β,γ)
= ∫_{u}
P_{1}(u) x P_{2}(u_{r})
du
Formula 6. Rotation function
P_{1
}is the Patterson function and P_{2}
is the rotated Patterson function, where u
is the volume of the Patterson map, where interatomic vectors are
calculated.
The quality of the solutions of these functions is expressed by the
correlation coefficient between both Patterson functions: the
experimental one and the calculated one (with the known protein). A
high correlation coefficient between these functions is equivalent to a
good agreement between the experimental diffraction pattern and the
diffraction pattern calculated with the known protein structure. Once
the known protein structure is properly oriented and translated (within
the unit cell of the unknown protein), an electron density map is
calculated using these atomic positions and the experimental structure
factors.
COMPLETING THE STRUCTURE
All these methods (Patterson, direct methods, MIR, MAD, MR)
provide (directly or indirectly) knowledge about approximate phases
which must be upgraded. As indicated above, the calculated
initial phases, Φ_{c}(hkl), together with the observed experimental
amplitudes, F_{o}(hkl),
allow us to calculate an electron density map, also approximate, over
which we can build the structural model. The overall process is
summarized in the cyclic diagram shown below.
The initial phases, Φ_{c}(hkl), are combined with the
amplitudes of the experimental (observed) structure factors, F_{o}(hkl),
and an electron density map is calculated (shown at the bottom of the
scheme). Alternatively, if the initial known data are the coordinates (xyz) of
some atoms, they will provide the initial phases (shown at the top of
the scheme), and so on in a cyclic way until the process does not
produce any new information.
Scheme showing a cyclic process to
calculate electron density maps ρ(xyz) which produce further structural
information.
From several known atomic positions we can always calculate the
structure factors: amplitudes, F_{c}(hkl), and phases, Φ_{c}(hkl),as
shown at the top of the scheme. Obviously, the calculated
amplitudes can be rejected, because they are calculated from a partial
structure and the experimental ones represent the whole structure.
Therefore, the electron density map (shown at the bottom of the scheme)
is calculated with the experimental (or observed) amplitudes,
F_{o}(hkl), and the calculated phases, Φ_{c}(hkl). This
function is now evaluated in terms of possible new atomic
positions that are added to the previously known ones,
and the cycle repeated. Historically this process was known as
"succesive Fourier syntheses", because the electron density is
calculated in terms of a Fourier sum. The mentioned magnitudes are
shown in the scheme as F_{c}(hkl), F_{o}(hkl) and Φ_{cal}(hkl).
In
any case, from atomic positions or directly from phases, if
the
information is correct, the function of electron density will be
interpretable and will contain additional information (new coordinates)
that can be injected into the cyclic procedure shown
above until structure completion, which is to say
until
the calculated function ρ(xyz) shows no changes from the last
calculation.
The lighter atoms of the structure (those with lower atomic number, ie,
usually hydrogen atoms) are the most difficult ones to find on
an electron density map. Their scattering power is almost
obscured
by the scattering of the remaining atoms . For this reason, the
location of H atoms is normally done via a somewhat
modified electron density function (the difference
electron density), whose coefficients are the differences
between the observed and calculated structure factors of the model
known so far:

Formula 7.
Function of "difference" electron
density
In practice, if the structural model obtained is good enough, if the
experiment provided precise structure factors, and there are no
specific errors such as Xray absorption, the difference map Δρ
will contain enough signal (maxima) where H atoms can be
located. Additionally, to get an enhanced signal from the
light
atoms scattering, this function is usually calculated with the
structure factors appearing at lower diffraction angles only, usually
with those appearing at sin
θ / λ < 0.4,
that is, using the region where the scattering factors for hydrogens
are still "visible".