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12.1.1 Overview of the Backbone and Chain Closure

To generate the positions of the backbone atoms, an extension (Macromolecules (1985) 18, 2767-2773) of the local chain deformation and chain closure procedure of Go and Scheraga (Macromolecules (1970) 3, 178-187) is used. Given fixed, oriented endpoints and a chain of bonded atoms containing six freely rotatable torsions, their procedure determines a set of values for the torsion angles that permit the chain to bridge the endpoints. The free torsions are the phi and psi angles so three is the minimum number of residues for which a search over an internal polypeptide segment can be performed (it is presumed that the omega peptide torsion angle is planar and normally trans, although cis peptides are considered as described below).

It turned out that the original Go & Scheraga procedure was overly restrictive, particularly for bridging regular structures like alpha-helicies. To avoid this problem, we have modified the method to permit limited alterations in the bond angles and the procedure is named CLSCHN. The main option controlling bond angle variations is MAXDT which gives the maximum variation from standard bond angles. Its default value is 5 degrees which improves its performance significantly while incurring a bond angle energy penalty of at most 1 kT per angle. The number of solutions obtained from the chain closure method is always even and has not exceeded eight in our experience with peptides.

The ring in proline creates special problems. The proline ring constrains the phi torsion to be close to -65 degrees; any deviation from -65 degrees distorts the ring. Prior to running CONGEN, we determine the minimum energy configuration of the proline ring (specifically, 1,2 dimethyl pyrrolidine) for a range of phi angles (+/- 90 degrees) about -65 degrees using energy minimization with a constraint on phi, and we construct a file (PRO.CNS) which contains these energies and the construction parameters necessary to calculate the position of CB, CG and CD of the proline. All of these energies are adjusted relative to a minimum ring energy equal to zero. After a chain closure is performed, we discard any conformations which have a proline phi angle whose energy exceeds the minimum energy by more than the parameter, ERINGPRO. Generally, we use a large value for ERINGPRO, 50 kcal/mole, so CLSCHN does not overly restrict proline closures. We handle cis-trans peptide isomerization by trying all possible combinations of cis and trans configurations. The user has complete control over which residues can be built in the cis isomer. Since there are only three residues involved in the chain closure, this results in no more than eight (2^3) attempts at chain closure.

The backbone search of an N residue segment begins by using backbone degrees of freedom to sample the free torsions of N-3 residues and then using the chain closure degree of freedom to close the chain. As the free torsions are sampled, we can discard any segment if the end of the constructed chain is too far from the other framework end for closure to be possible. See Backbone Degree of Freedom, for a description of the CLSA and CLSD options which control this process. The determination is made by calculating the distance between the last atom constructed and the other fixed endpoint and comparing that to the distance spanned by m peptides with all torsions being trans and all bond angles increased by MAXDT, where m is the number of peptides still to be constructed.

The direction of backbone construction is arbitrary, although the endpoints of the search are conserved regardless of the order. The N-terminus of the internal segment is anchored on the peptide nitrogen; the C-terminus is anchored on the alpha carbon. When the construction direction is from the N terminus to the C terminus, the first torsion to be sampled in a residue is the omega angle (which normally is sampled just at 180 degrees, and sampled at 0 degrees and 180 degrees for prolines). It determines the alpha-carbon and the peptide hydrogen positions. The phi angle determines the position of the carbonyl carbon and the beta carbon of the sidechain; and finally, the psi angle determines the carbonyl oxygen and peptide nitrogen of the next residue. When the construction is in the reverse direction; the psi angle determines the peptide nitrogen; the phi angle determines the carbonyl carbon of the preceding residue, the peptide hydrogen, and the beta carbon; and the omega angle determines the position of the preceding residue's alpha carbon and carbonyl oxygen.

Rather than treating each of the three torsion angles in a amino acid residue as three separate degrees of freedom, we combine them into a single degree of freedom. This permits the use of Ramachandran type plots to limit the range of phi, psi values to those that are energetically acceptable and found in known structures.

To determine the allowed phi,psi angles, the CONGEN command uses energy maps. These energy maps are stored as files, see Backbone Maps, and they are expressed in tabular form with entries composed of omega, phi, and psi angle values along with the energy for that angle combination. Thus, any arbitrary criterion may be used in place of the energy in these maps. There are three different types of maps, one for glycine, one for proline, and one for the other amino acids which are modeled by alanine. Typically, the glycine and alanine maps are computed by modeling a dipeptide and using the van der Waals energy, whereas the proline maps is computed using a dipeptide, but the energy in the map is the sum of the van der Waals energy plus the ring energy. The actual values for phi and psi in these tabulations are usually multiples of 15 degrees or 30 degrees with all possible combinations of angles present. Thus, these maps determine the sampling grid. For the alanine and glycine map, phi and psi both range over -180 degrees to 180 degrees, and the omega angle can be either 0 or 180 degrees. Normally, the omega angle of 0 degrees is not used. In the proline map, the range of phi angles is -150 degrees to 30 degrees; psi goes from 0 degrees to 360 degrees; and omega has values of 0 degrees and 180 degrees.

Each backbone degree of freedom can specify its own map. However, in most applications, each backbone residue will use the correct map for its type (proline, glycine, or alanine), and the grid spacing in all the maps will be the same. Therefore, default maps are typically specified for each type of amino acid, and the user can override these maps for individual residues. Fortran unit numbers for default maps for glycine, proline, and other amino acids are specified with the global variables; GLYMAP, PROMAP, and ALAMAP; respectively.

The particular values of torsion angles used for generating conformations is determined by these maps and the so-called EMAX options. The maps specify all the possible angles. The EMAX options restrict these sets as they specify the maximum allowed energy relative to the minimum energy value found in each map. The global options; GLYEMAX, ALAEMAX, and PROEMAX; specify the selection of the default backbone maps, see Global Options for Conformational Search. GLYEMAX specifies that for the glycine map; PROEMAX for the proline map; and ALAEMAX for the alanine map. The backbone degree of freedom option, EMAX, specifies the allowed energy for an individual backbone degree of freedom. For example, when using the alanine map with a sequence of alanines and a value of ALAEMAX of 2.0 kcal/mole, all the conformations generated will have phi, psi angles corresponding to only right-handed alpha-helices or beta-sheets. For a value around 5 kcal/mole, phi, psi angles for left-handed alpha-helices will also be selected. If values for the EMAX options are set to very large values then the entire phi, psi space will be sampled.

D amino acids are indicated by the presence of the word, “D”, in the residue attributes. These residues are handled by inverting all torsion angles for the backbone maps and for the proline constructor files.

For more details on the commands which implement these degrees of freedom, see Backbone Degree of Freedom, and Chain Closure.