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.
The pdf for a geometric feature (e.g., distance, angle, dihedral angle) is
The first derivatives with respect to feature are:
(A.64) |
The relative heavy violation with respect to is given as:
The polymodal pdf for a geometric feature (e.g., distance, angle, dihedral angle) is
(A.67) |
The first derivatives with respect to feature are:
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) |
The polymodal pdf for a geometric feature (e.g., a pair of dihedral angles) is
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:
The relative heavy violation with respect to is given as:
(A.81) |
This is like the left half of a single Gaussian restraint:
This is like the right half of a single Gaussian restraint:
This is usually used for dihedral angles (improper dihedrals generally use a Gaussian restraint instead):
(A.85) |
The first derivatives are:
(A.88) | |||
(A.89) |
The violations of this restraint are always reported as zero.
Usually used for non-bonded distances:
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.
Any restraint form can be represented by a cubic spline [Press et al., 1992]:
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.
The asymmetry penalty added to the objective function is defined as
For each , the first derivatives are:
(A.100) | |||
(A.101) |