Energy Minimization and Molecular Dynamics with Constraints

It may be that case that for some reason (usually an experimental observation) it is known that whatever conformation is taken by a molecule, a particular internuclear distance or bond angle is present. A common way to force a molecule to take up a particular conformation during computer modeling is to introduce an "artificial" term into the force field used during conformational energy minimization. This term, or constraint, assigns a high energy penalty if some distance or angle does not have a desired value. During energy minimization the algorithm will direct the calculation so that the desired feature is retained.

There are many forms for these penalty functions. One approach is define at function that adds zero to the potential (conformational) energy as long as an internuclear distance (or an angle) is within a certain range of values, but then rapidly increases the energy if the distance is less than the lowest value of the range or higher than the highest value of the range. Thus, if the smallest desired value for a distance d is dshort and the longest value is dlong, the force field would contain the term

Econstraint = 0 when dshort < d < dlong

but

Econstraint = k(d-dshort)n when d < dshort

or

Econstraint = k(d-dlong)n when dlong < d

The parameter k defines how rapidly the energy increases when the distance is not in the range between dshort and dlong; the power n is usually set to 2.

Such additional, contraining terms can be added to the force field during energy minimization or during molecular dynamics.

Exercise

Open SYBYL and click on Build/Edit, then Get Fragment. Load the chair form of cyclohexane and position the model so that you can clearly see the hydrogens attached to carbons 1 and 4. Calculate the conformational energy of chair cyclohexane. We will attempt to convert this model to the boat conformation by using a distance constraint. In the boat conformation the distance between axial hydrogens attached to carbons 1 and 4 is 2.351 angstroms while in the chair conformation the corresponding distance is over 4 angstroms.

To include a constraint in the force field within SYBYL, click on Build/Edit, then Constraints. Click on the box Distance and then select Range from the pop-up menu. Click on Define and then on the two atoms between which the distance constraint is to be developed, in this case the two hydrogens that will become axial in the boat form. In the Range boxes enter the lower and upper distance limits for the constraint; 2.2 and 2.4, respectively are suggested for starting values. Calculate the energy again. There should be a huge increase because now the 1-4 distance relation added to the force field is in violation. Now minimize the energy--the structure will quickly flip into a conformation close to the boat conformation.

If a dynamics simulation is run with the contraining term in the force field, the cyclohexane model quickly assumes and then fluctuates about the boat conformation.