Subsections

Restraints and their derivatives

The chain rule is used to find the partial derivatives of the feature pdf with respect to the atomic coordinates. Thus, only the derivatives of the pdf with respect to the features are listed here.

Single Gaussian restraint

The pdf for a geometric feature (e.g., distance, angle, dihedral angle) is

 (A.62)

A corresponding restraint in the sum that defines the objective function is

 (A.63)

(Note that since the second term is constant for a given restraint, it is ignored. is also scaled by in kcal/mol with to allow these scores to be summed with CHARMM energies.)

The first derivatives with respect to feature are:

 (A.64)

The relative heavy violation with respect to is given as:

 (A.65)

Multiple Gaussian restraint

The polymodal pdf for a geometric feature (e.g., distance, angle, dihedral angle) is

 (A.66)

A corresponding restraint in the sum that defines the objective function is (as before, this is scaled by ):

 (A.67)

The first derivatives with respect to feature are:

 (A.68)

When any of the normalized deviations is large, there are numerical instabilities in calculating the derivatives because are arguments to the exp function. Robustness is ensured as follows. The ‘effective’ normalized deviation is used in all the equations above when the magnitude of normalized violation is larger than cutoff rgauss1 (10 for double precision). This scheme works up to rgauss2 (200 for double precision); violations larger than that are ignored. This trick is equivalent to increasing the standard deviation . A slight disadvantage is that there is a discontinuity in the first derivatives at rgauss1. However, if continuity were imposed, the range would not be extended (this is equivalent to linearizing the Gaussian, but since it is already linear for large deviations, a linearization with derivatives smoothness would not introduce much change at all).

 (A.69) (A.70) (A.71) (A.72) (A.73) (A.74)

Now, Eqs. A.66-A.68 are used with instead of . For single precision, , rgauss1 = 4, rgauss2 = 100.

The relative heavy violation with respect to is given as:

 (A.75)

Multiple binormal restraint

The polymodal pdf for a geometric feature (e.g., a pair of dihedral angles) is

 (A.76)

where . is the correlation coefficient between and . MODELLER actually uses the following series expansion to calculate :

 (A.77)

A corresponding restraint in the sum that defines the objective function is (as before, this is scaled by ):

 (A.78)

The first derivatives with respect to features and are:

 (A.79) (A.80)

The relative heavy violation with respect to is given as:

 (A.81)

Lower bound

This is like the left half of a single Gaussian restraint:

 (A.82)

where is a lower bound and is given in Eq. A.62. A similar equation relying on the first derivatives of a Gaussian holds for the first derivatives of a lower bound.

Upper bound

This is like the right half of a single Gaussian restraint:

 (A.83)

where is an upper bound and is given in Eq. A.62. A similar equation relying on the first derivatives of a Gaussian holds for the first derivatives of an upper bound.

Cosine restraint

This is usually used for dihedral angles :

 (A.84)

where is CHARMM force constant, is phase shift (tested for 0 and 180°), and is periodicity (tested for 1, 2, 3, 4, 5, and 6). The CHARMM phase value from the CHARMM parameter library corresponds to °. The force constant can be negative, in effect offsetting the phase for 180° compared to the same but positive force constant.

 (A.85)

Coulomb restraint

 (A.86) (A.87)

where and are the atomic charges of atoms and , obtained from the CHARMM topology file, that are at a distance . is the relative dielectric, controlled by the energy_data.relative_dielectric variable. Function is a switching function that smoothes the potential down to zero in the interval from to ( ). The total Coulomb energy of a molecule is a sum over all pairs of atoms that are not in the same bonds or bond angles. 1-4 energy for the 1-4 atom pairs in the same dihedral angle corresponds to the ELEC14 MODELLER term; the remaining longer-range contribution corresponds to the ELEC term.

The first derivatives are:

 (A.88) (A.89)

The violations of this restraint are always reported as zero.

Lennard-Jones restraint

Usually used for non-bonded distances:

 (A.90)

The parameters and of the switching function can be different from those in Eq. A.87. The parameters and are obtained from the CHARMM parameter file (NONBOND section) where they are given as and such that in kcal/mole for in angstroms and ; the minimum of is at , and its zero is at . The total Lennard-Jones energy should be evaluated over all pairs of atoms that are not in the same bonds or bond angles. The parameters and for 1-4 pairs in dihedral angles can be different from those for the other pairs; they are obtained from the second set of and in the CHARMM parameter file, if it exists. 1-4 energy corresponds to the LJ14 MODELLER term; the remaining longer-range contribution corresponds to the LJ term.

The first derivatives are:

 (A.91) (A.92)

As tends toward zero, the repulsive part of the energy dominates, and approaches infinity. Near-infinite forces result in unstable trajectories during optimization. This is particularly a problem in the first few steps of optimization starting from randomized, interpolated, or otherwise non-physical atomic coordinates. To avoid this, the potential is simply artificially truncated: if exceeds 6, is treated as being equal to .

The violations of this restraint are always reported as zero.

Spline restraint

Any restraint form can be represented by a cubic spline [Press et al., 1992]:

 (A.93) (A.94) (A.95) (A.96) (A.97)

where .

The first derivatives are:

 (A.98)

The values of and beyond and are obtained by linear interpolation from the termini. A violation of the restraint is calculated by finding the global minimum. A relative violation is estimated by using a standard deviation (e.g., force constant) obtained by fitting a parabola to the global minimum.

Variable spacing of spline points could be used to save on memory. However, this would increase the execution time, so it is not used.

To calculate the relative heavy violation, the feature value that results in the smallest value of the restraint is obtained by interpolation, and a Gaussian function is fitted locally around this value to obtain the standard deviation . These are then used in Eq. A.65.

Symmetry restraint

The asymmetry penalty added to the objective function is defined as

 (A.99)

where the sum runs over all pairs of equivalent atoms , is an atom weight for atom , is an intra-molecular distance between atoms in the first segment, and is the equivalent distance in the second segment.

For each , the first derivatives are:

 (A.100) (A.101)

Thus, the total first derivatives are obtained by summing the two expressions above for all and distances.

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