In the context of this chapter, you will also be invited to visit these sections...
The
analysis and interpretation of the electron density function, ie the
resolution of a
crystal structure (molecular or nonmolecular) leads to an initial
distribution of atomic positions within the unit cell which can be
represented by points or small spheres:



Initial
model of the 3dimensional structure of a molecule. Atoms, also
labelled, are represented by small spheres. 
Initial
model of the 3dimensional structure of a molecule and an
ion.
Despite
the increased complexity or beauty of the model, the level of
information is the same as in the model on the left. 
Initial
model of the 3dimensional structure of a molecule, including its
crystal structure, that is, the molecular packing within the unit cell. 
Once
the structural model is completed, having stereochemical sense
and including its crystal packing, it is necessary to make use
of
all the information we can extract from the experimental data, since
the diffraction pattern generally contains much more data (intensities)
than needed to locate the atoms at their 3dimensional coordinates. For
instance, for a medium sized structure, with 50 independent
atoms in the asymmetric unit (in the structural unit which is
repeated
by the symmetry operations), the diffraction pattern usually
contains around 2500 structure factors, which implies approximately 50
observations per atom (each atom needs 3 coordinates). However, for
more complex structures, as in the case of macromolecules, the amount
of experimental data available normally does not reach these limits.
REFINING
THE FINAL MODEL
The basic parameters associated with a threedimensional structure are,
obviously, the three positional coordinates (x,
y,
z) for
each atom, given in terms of unit cell
fractions. But, in general, given the experimental overdetermination
mentioned above, the atomic model can become more complex. For
instance,
associating each atom with an additional parameter
reflecting
its thermal vibrational state, in a first approach as an isotropic
(spherical) thermal vibration
around its position of equilibrium. This new parameter is normally
shown in
terms of different radius of the sphere representing the atom. Thus an
isotropic structural model would be represented by 4 variables per
atom: 3 positional + 1 thermal.
However, for small and mediumsized structures (up to several hundred
of atoms), the diffraction experiment usually contains enough data
to complete the thermal vibration model, associating a tensor
(6
variables) to each atom which expresses the state of vibration in an
anisotropic manner, ie distinguishing between different directions of
vibration in the form of an ellipsoid (which resembles the shape of a
baseball). Therefore, a crystallographic anisotropic model
will
require 9 variables per atom (3 positional + 6 vibrational).


Three bonded atoms represented with the
isotropic thermal vibration model

The same three atoms shown on the left,
but represented using the anisotropic thermal vibration model 


Anisotropic
model of the 3dimensional structure of a molecule, showing some atoms
from neighboring molecules. 
Anisotropic
model of the 3dimensional structure of a molecule showing its crystal
packing. 
Regardless of the model type, isotropic or anisotropic, the
abovementioned overabundance of experimental data permits
description of
the structural model in terms of very precise parameters (positional
and vibrational) which lead to very precise geometrical
parameters
of the structure (interatomic distances, bond angles, etc.).
This refined model is obtained by the analytical method of
leastsquares. Using this technique, atoms are allowed to
"move"
slightly from their previous positions and thermal factors are
applied to each atom so that the diffraction pattern
calculated
with this model is essentially the same as the experimental one
(observed), ie minimizing the differences between the calculated and
observed structure factors. This process is carried out by
minimizing the function:
Σ
w 
F_{o}  F_{c}
^{2}
→ 0 
Leastsquares function used to refine the
final model of a crystal structure 
where w represents
a "weight" factor assigned to each observation (intensity),
weighting the effects of the lessprecise observations vs. the
more accurate ones and avoiding possible systematic errors in the
experimental observations which could bias the model.
Although usually the mentioned experimental overdetermination ensures
the success of this analytical process of refinement, it must always be
controlled through the stereochemical aspects, ie, ensuring that
the positional movements of the atoms are reasonable and which
therefore generate distances within the expected values. Similarly, the
thermal vibration factors (isotropic or anisotropic) associated with
the atoms must always show reasonable values.
In addition to the aforementioned control of the model
changes during the refinement process, it seems obvious that
(if
everything goes well), additionally the diffraction pattern calculated (F_{c}) with
the refined model (coordinates + thermal vibration factors) will show
increasing similarity to the observed pattern (F_{o}). The
comparison between both patterns (observed vs. calculated) is done via
the socalled R
parameter, which defines the "disagreement" factor between the
two patterns:
R =
Σ [
 F_{o}  F_{c}
 ] / F_{c} 
Disagreement
factor of a structural model calculated in terms of differences between
calculated and observed diffraction patterns 
The value of the disagreement
factor (R)
is estimated as a percentage (%), ie, multiplied by 100, so that "well"
solved structures, with an appropriate degree of
precision, will show an R
factor below 0.10 (10% ), which implies that the calculated pattern
differs from the observed one (experimental) less than 10%.
The diffraction patterns of macromolecules (enzymes, proteins,
etc.) usually do not show such large overdetermination of experimental
data and therefore it is difficult to reach an anisotropic final model.
Moreover, in these cases the values of the R
factor are greater than those for small and mediumsized molecules, so
that values around or below 20% are usually acceptable. In addition, as
a result
of this relative scarcity of experimental data, the analytical
procedure of refinement (leastsquares) must be combined with an
interactive stereochemical
modeling process and by imposing
certain "soft restraints" to the molecular geometry.
MODEL VALIDATION
The reliability of a structural model has to
be assessed in terms of several tests, a procedure known as
model
validation.
Thus, the structural model should be continuously checked and
validated using consistent stereochemical criteria (for example, bond
lengths and bond angles must be acceptable). For instance
a CO distance of 0.8 Angstrom would not be acceptable for
a carbonyl
group (C = O). Similarly, the bond angles must also be
consistent with an acceptable geometry. These criteria are very
restrictive for small or mediumsized structures, but even in
the
structures of macromolecules they must meet some minimum criteria.
Maximum dispersion values accepted for
interatomic distances and bond angles in the structural model of a
macromolecule. 
In the case of proteins, the peptide bond
(the bond between two consecutive amino acids) must
also satisfy some geometrical restrictions. The torsional angles of
this
bond should not deviate much from the acceptable values of the usual
conformations shown by the amino acid chains, as is shown in the socalled
Ramachandran plot:


Schematic representation of
the peptide bond, showing the two torsional angles (Ψ
y Φ) defining it.

Ramachandran plot showing the different
allowed
(acceptable) areas for the torsional angles of the peptide
bonds in a macromolecule. The different areas depend on the
different structural arrangements (αhelices,
βsheets,
etc.) 
Similarly, the values of the thermal factors associated with each atom
should show physically acceptable values. These parameters account for
the thermal vibrational mobility of the different structural parts.
Thus, in the structure of a macromolecule, these values should be
consistent with the internal or external location of the chain,
being
generally lower for the internal parts, and higher for external parts
near the solvent.

Graphic
representation of the thermal vibration factors in the main chain of a
macromolecule. The "cool" colors (blue) denote areas of low
mobility. "Hot" colors (red and green) denote areas with higher
mobility. 
DEGREE
OF RELIABILITY OF THE MODEL
A model that has been
"validated" according to the criteria described above, that is, which
demonstrates:
 a reasonable agreement between
observed and calculated structure factors,
 bond distances, bond angles
and torsional angles that meet stereochemical criteria, and
 physically reasonable thermal
vibration factors,
is a reliable model. However, the concept of reliability is not a
quantitative parameter which can be written in terms of a single
number.
Therefore, to interpret a structural model up to its logical
consequences
one has to bear in mind that it is just a simplified
representation, extracted from an electron density function:
on which the atoms have been positioned and which is being affected by
some conditions
described in another
section, which we invite you to read.
But,
in any case, welldone crystallographic work always provides
atomic parameters (positional and vibrational) along with their
associated
precision estimates. This means that any direct crystallographic
parameter (atomic coordinates and vibration factors) or derived
(distances, angles, etc.) is usually expressed by a number followed by
its standard deviation (in parentheses) affecting the last
figure.
For example, an interatomic distance expressed as 1.541 (2)
Angstroms means a distance of 1.541 and a standard deviation of 0.002.
THE ABSOLUTE CONFIGURATION (OR ABSOLUTE STEREOCHEMISTRY)
As stated in a
previous chapter, all molecules or structures in
which neither mirror planes nor centres of symmetry are present,
have an absolute
configuration, that is, that they are different from their
mirror images (they cannot be superimposed).

Structural models showing two
enantiomers of a compound (the molecules are mirror images) 
These particular structural differences, very important as far as the
molecular properties are concerned, can be unambiguously
determined through the diffraction experiment (without using any
external standard). This can be carried out using the socalled
anomalous scattering effect which atoms show when appropriate
Xray
wavelengths are used. This feature is also very succesfully
used
as a method
to solve the phase problem for macromolecular crystals.
It doesn't seem difficult to understand that the molecular
enantiomers have different properties, as in the end they are
different molecules, but regarding their biological activity (if
any) the situation is particularly striking.

Enantiomeric
molecules that are represented in the left figure were introduced in
the market by a pharmaceutical company and, obviously, they showed
different properties.
The properties of DARVON (Dextropropoxyphene Napsylate) are
available through this link, while production of NOVRAD
(Levopropoxyphene Napsylate) was discontinued.

The
experimental diffraction signal that allows this structural
differentiation is
a consequence of the fact that the atomic scattering
factor does not behave as a real number when the frequency
of Xrays is similar to the natural frequency of the atomic absorption.
See also the chapter
dedicated to anomalous dispersion.
Under these conditions, Friedel's Law is
no longer fulfilled and therefore structure factors such
as F_{h,k,l}  and F_{h,k,l}  will
be slightly different. These differences are evaluated in terms of the
socalled Bijvoet
estimators,
which compare the
ratios for observed structure factors for such reflection pairs
with the corresponding ratios for the calculated structure
factors
using the two possible absolute models. Only one of these two
comparisons will maintain the same type of bias:
Thus, if the quotient between the observed structure factors is
<1,
the same quotient for the calculated structure factors should
also
be <1. Or, on the contrary, both quotients should be
>1. If
this is true for a large number of reflection pairs it will indicate
that the absolute model is the right one. If it is not so, the
structural model has to be inverted.
The interested reader should also
have a look
into the web pages on anomalous scattering, prepared
by Ethan A.
Merritt.
THE FINAL RESULT
The information describing a final crystallographic model is
composed of:
 Data
from the diffraction experiment: wavelength and diffraction
pattern (the intensity of thousands or even hundreds of thousands of
diffracted waves with their hkl
indices),
 Unit cell dimensions as derived from the
diffraction pattern (from the reciprocal cell),
 The symmetry
present in the crystal, derived from the reciprocal lattice (the
diffraction pattern), and
 Atomic
positions (coordinates and thermal vibration factors) and, if needed,
the socalled population factor, as indicated in the table below.
The
atomic positions are usually given as fractional coordinates (fractions
of the unit cell axes), but sometimes, especially for macromolecules
where the information usually refers to the isolated molecule, they are
given as absolute coordinates, ie, expressed in Angstrom and referred
to a system of orthogonal axes independent of the crystallographic ones
(see below).

Information about several atoms of a
protein structure using the socalled PDB format (Protein Data Bank),
ie atomic coordinates in Angstrom on a system of orthogonal
axes,
different from the crystallographic ones. For clarity, the
estimated standard deviations have been omitted. 
The population factor is the fraction of
atom located in a
specific position, although this factor is usually 1. The
meaning
of this parameter requires an explanation for the beginner,
since
it could be understood that atoms could be divided in parts,
which
obviously has no physical meaning. Due to
atomic
vibrations, and to the fact that the diffraction experiment has a
duration in time, it is possible that in some of the unit cells atoms
are missing. Thus, instead of a complete occupancy
(population
factor = 1), the corresponding site, in an average unit cell, will
contain
only a fraction of the atom. In these cases it is said that the crystal
lattice has defects and population factors smaller than 1
reflect
a fraction of unit cells where a specific atomic position is occupied.
Obviously, a fraction of unit cells where the same position is empty
complements the population factor to unity. Therefore, the
crystallographic model reflects the average structure of all unit cells
during the experiment time.
The atomic coordinates and in general all information collected from a
crystallographic study, is stored in accessible databases. There are
different databases, depending of the type of compound or molecule, but
this will be
discussed in another
chapter of these pages.
GRAPHICAL REPRESENTATIONS OF THE MODEL

The
final structural model (atomic coordinates, thermal factors and,
possibly, population factors) directly
provide additional
information which leads to a detailed knowledge of the structure
itself,
including bond lengths, bond angles, torsional angles,
molecular planes, dipole momentum, etc., and any other structural
detail
that might be useful for understanding the functionality and/or
properties of the material under study.
In the case of
complex biological
molecules, the use of highquality graphic processors and relatively
simple models, greatly facilitates the understanding of the
relationship between structure and function, as shown in the figure on
the left. 
At present the available computational and graphic
techniques allow us to obtain beautiful and very descriptive
models which help to visualize and understand structures, as is shown
in the examples below:


Model of balls and sticks to represent
the structure of a simple inorganic compound 
Representation of an inorganic compound,
in which a partial polyhedral representation has been added 




Animated model of sitcks to
represent the packing and molecular structure of a simple
organic compound 
Given the complexity of
biological
molecules, the models which represent them are
usually simple, showing the overall folding and the different
structural motifs (αhelices,
βstrands,
loops, etc.) shown with the ribbon model. The example also
shows
a stick representation of a cofactor linked to the enzyme. 




Combined
model of ribbons and sticks to represent the dimmeric structure of a
protein which also shows a sulfate ion in the middlerepresented with
balls 
Representation
of the surface of a biological molecule where the colours represent
different properties of hydrophobia. The arrow represents the
dipolar momentum of the molecule. 


Finally, using
additional information
from other techniques (such as highresolution electron microscopy), or
combining two different crystal conformations of a molecule, other
models are available as shown below: 




Combined
model of the molecular structure of a protein and an envelope (as
obtained by highresolution electron
microscopy) showing a
pore formed by the association of four protein molecules 
Simplified animated
model showing the backbone folding of an enzyme and the structural
changes between two molecular states: active (open)
and inactive
(closed). The structures of both states were determined by
crystallography 