Restraints and Constraints

A systematic analysis of a potential energy surface often requires minimization of the strain energy with one or more internal coordinates fixed. Successive variation (stepping) of fixed internal coordinates can be used in the analysis of activation barriers and energetic minima along a reaction coordinate. Applications include  

Some of these applications are discussed in detail in various chapters of Part II. 

A crystal structure is described by a collection of parameters that give the arrangement of the atoms, their motions and the probability that each atom occupies a given location. These parameters are the atomic fractional coordinates, atomic displacement or thermal parameters, and occupancy factors. A scale factor then relates the calculated structure factors to the observed values. This is the suite of parameters usually encountered in a single crystal structure refinement. In the case of a Rietveld refinement an additional set of parameters describes the powder diffraction profile via lattice parameters, profile parameters and background coefficients. Occasionally other parameters are used these describe preferred orientation or texture, absorption and other effects. These parameters may be directly related to other parameters via space group symmetry or by relations that are presumed to hold by the experimenter. These relations can be described in the refinement as constraints and as they relate the shifts, Ap,-, in the parameters, they can be represented by 

The ky are then the elements of a sparse rectangular matrix that relates the suite of parameters of interest to a smaller set that are actually refined. This reduces the number of partial derivatives via  

In their simplest form, these constraints can be used for atomic sites on special positions (e.g. x,x,x positions in cubic space groups) where there are special relationships between the individual atomic coordinates and also among the individual anisotropic thermal motion parameters (e.g. Ui i = U22 = U33 and Ui2 = Ui3 = U23 for cubic x,x,x sites). 

Least-squares methods are usually used for fitting a model to experimental data. They may be used for functions consisting of square sums of nonlinear functions. The well-known Gauss-Newton method often leads to instabilities in the minimization process since the steps are too large. The Marquardt algorithm [9 1 is better in this respect but it is computationally expensive. 

2 Another technique used for the computation of metal ion selectivities, where plots of strain energy vs. metal ligand equilibrium distance r0 are produced and interpreted, is discussed in Chapter 8[°21. 

A systematic analysis of a potential energy surface requires often minimization of the strain energy with one or more internal coordinates fixed. Successive 

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In many cases, it is also helpful to have the path repel itself so that the transition pathway is self-avoiding. An acmal dynamic trajectory may oscillate about a minimum energy configuration prior to an activated transition. In the computed restrained, selfavoiding path, there will be no clusters of intermediates isolated in potential energy minima and no loops or redundant segments. The self-avoidance restraint reduces the wasted effort in the search for a characteristic reaction pathway. The constraints and restraints are essential components of the computational protocol. 

The use of constraints and restraints arise from unrealistic bond lengths and angles. It is a reasonable use of statistical analysis to include prior information in a refinement procedure, and geometry is one such restraint/constraint. Poor geometries are often a consequence of missing data down one axis, invariably the result of the missing cone or lack of an epitaxially grown crystal. This lack of data has a profound effect on the refinement process... 

Once the arrangement of atoms in the macromolecule has been determined, one can proceed to analyze the results. At this stage it is essential to take into consideration the resolution of the structure determination, the extent to which refinement of the structure has been possible, the constraints and restraints used in the refinement, and the overall R value. At 6 A resolution a protein molecule looks like a folded solid tube, and no atomic detail can be found. At 3.5 A resolution more information is obtained and bulky side chains might even be discerned. Many protein structure determinations can only be carried out to 2.2-2.5 Aresolution and the overall atomic arrangement can be deduced by use... 

So, how do we drive these torsion angles to the values we want and keep them there There are two methods, constraints and restraints. Constraints are applied by... 

As their name suggests, free variables can be used to refine a multitude of different parameters and facilitate the formulation of constraints and restraints. The first free variable is always the overall scale factor (osf), which is used to bring the reflections in the dataset to an absolute scale. The example in Section 4.4.3 shows the effects of incorrect scaling on the refinement. Additional free variables can be linked to the site occupancy factors (sof) of groups of disordered atoms (for details see Chapter 5), but can also be related to other atomic parameters (x, y, z, sof, U, etc.) and even interatomic distances, chiral volumes, and other parameters. 

If an atom is NPD because of incorrectly assigned element type (carbon instead of sulfur or so) or due to an unresolved disorder, it usually suffices to correct the error to rectify the situation. In other, more difficult situations, constraints and restraints can help. 

If it can not be avoided, disorder may be reduced or its effects ameliorated by low-temperature data collection, which is almost always to be recommended. Modern crystallographic software provides many effective and powerful methods for modeling disorder, even in quite complex manifestations, and the judicious application of refinement techniques such as sensible geometrical constraints and restraints, and an appropriate treatment of atomic displacement parameters, can often salvage what would otherwise be a structural mess. 

Bansal, M. Sasiekharan, V. Moleculctr Model-Building of DNA Constraints and Restraints, Bansal, M. Sasiekharan, V., Ed. Elsevier New York, 1986 pp 127-214. 

The optimal values of the parameters and their associated standard uncertainties are meaningful only in relation to the model employed in the final least-squares cycle. All constraints and restraints used should be reported. Note that the positions and displacement parameters of hydrogen atoms are often derived from the parameters of the heavier atoms and subsequently suitably restrained or constrained corresponding bond lengths, such as C-H, are then features of the model and not experimental evidence. 

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