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pyqcprot.py
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__doc__ = """
QCP rotation calculation
This is an RMSD and optimal rotation calculator, written in pure Python. The
goal of this is to allow the code to be run in a JIT compiler such as PyPy,
Jython that cannot interface with extern C modules, such as numpy.
The algorithm was originally developed by Douglas Theobald as a C module,
[qcp][qcp], which solves the eigenvalue decomposition in quaternion space, and
thus avoids the expensive SVD decomposition of 3D rotational matrices. The
current code is based on a Cython adaption of qcp, [pyqcprot][pyqcprot],
written by Joshua Adelman.
[pyqcprot]: https://github.com/synapticarbors/pyqcprot
[qcp]: http://theobald.brandeis.edu/qcp/
References:
Douglas L. Theobald. (2005) "Rapid calculation of RMSD using a quaternion-
based characteristic polynomial." Acta Crystallographica A. 61(4):478-480
Pu Liu, Dmitris K. Agrafiotis and Douglas L. Theobald. (2010) "Fast
determination of the optimal rotational matrix for macromolecular
superpositions."J. Comput. Chem. 31, 1561-1563
Joshua L. Adelman (2011) "Pyqcprot"
https://github.com/synapticarbors/pyqcprot
# BSD License
-----------------------------------------------------------------------------
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright notice, this
list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright notice,
this list of conditions and the following disclaimer in the documentation
and/or other materials provided with the distribution.
* Neither the name of the <ORGANIZATION> nor the names of its contributors may
be used to endorse or promote products derived from this software without
specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
-----------------------------------------------------------------------------
"""
import math
def make_correlation_matrix(coords1, coords2):
"""
Returns E0, and A, a 3x3 matrix reprsented as a list of 9 values, which
represents the correlation matrix between the coords. E0 is the static
component of the RMSD, which is half the sum of the squared lengths of the
coordinate vectors.
Parameters:
- coords1, coords2: a list of 3 floats, a list of N coordinates
"""
N = len(coords1)
assert N == len(coords2)
G1 = 0.0
G2 = 0.0
A = [0.0 for i in range(9)]
for i in xrange(N):
x1 = coords1[i][0]
y1 = coords1[i][1]
z1 = coords1[i][2]
G1 += (x1*x1 + y1*y1 + z1*z1)
x2 = coords2[i][0]
y2 = coords2[i][1]
z2 = coords2[i][2]
G2 += (x2*x2 + y2*y2 + z2*z2)
A[0] += (x1 * x2)
A[1] += (x1 * y2)
A[2] += (x1 * z2)
A[3] += (y1 * x2)
A[4] += (y1 * y2)
A[5] += (y1 * z2)
A[6] += (z1 * x2)
A[7] += (z1 * y2)
A[8] += (z1 * z2)
E0 = (G1 + G2) * 0.5
return E0, A
def calc_rms_rot(coords1, coords2):
"""
Returns rms and a list of 9 values that represents a rotation
matrix.
Args:
coords1, coords2: a list of 3 floats, representing an Nx3 matrix,
or a list of N set of coordinate vectors.
"""
E0, A = make_correlation_matrix(coords1, coords2)
N = len(coords1)
oldg = 0.0
evecprec = 1e-6
evalprec = 1e-14
Sxx = A[0]
Sxy = A[1]
Sxz = A[2]
Syx = A[3]
Syy = A[4]
Syz = A[5]
Szx = A[6]
Szy = A[7]
Szz = A[8]
Sxx2 = Sxx * Sxx
Syy2 = Syy * Syy
Szz2 = Szz * Szz
Sxy2 = Sxy * Sxy
Syz2 = Syz * Syz
Sxz2 = Sxz * Sxz
Syx2 = Syx * Syx
Szy2 = Szy * Szy
Szx2 = Szx * Szx
SyzSzymSyySzz2 = 2.0*(Syz*Szy - Syy*Szz)
Sxx2Syy2Szz2Syz2Szy2 = Syy2 + Szz2 - Sxx2 + Syz2 + Szy2
C = [0.0 for i in range(3)]
C[2] = -2.0 * (Sxx2 + Syy2 + Szz2 + Sxy2 + Syx2 + Sxz2 + Szx2 + Syz2 + Szy2)
C[1] = 8.0 * (Sxx*Syz*Szy + Syy*Szx*Sxz + Szz*Sxy*Syx - Sxx*Syy*Szz - Syz*Szx*Sxy - Szy*Syx*Sxz)
SxzpSzx = Sxz + Szx
SyzpSzy = Syz + Szy
SxypSyx = Sxy + Syx
SyzmSzy = Syz - Szy
SxzmSzx = Sxz - Szx
SxymSyx = Sxy - Syx
SxxpSyy = Sxx + Syy
SxxmSyy = Sxx - Syy
Sxy2Sxz2Syx2Szx2 = Sxy2 + Sxz2 - Syx2 - Szx2
C[0] = (Sxy2Sxz2Syx2Szx2 * Sxy2Sxz2Syx2Szx2
+ (Sxx2Syy2Szz2Syz2Szy2 + SyzSzymSyySzz2) * (Sxx2Syy2Szz2Syz2Szy2 - SyzSzymSyySzz2)
+ (-(SxzpSzx)*(SyzmSzy)+(SxymSyx)*(SxxmSyy-Szz)) * (-(SxzmSzx)*(SyzpSzy)+(SxymSyx)*(SxxmSyy+Szz))
+ (-(SxzpSzx)*(SyzpSzy)-(SxypSyx)*(SxxpSyy-Szz)) * (-(SxzmSzx)*(SyzmSzy)-(SxypSyx)*(SxxpSyy+Szz))
+ (+(SxypSyx)*(SyzpSzy)+(SxzpSzx)*(SxxmSyy+Szz)) * (-(SxymSyx)*(SyzmSzy)+(SxzpSzx)*(SxxpSyy+Szz))
+ (+(SxypSyx)*(SyzmSzy)+(SxzmSzx)*(SxxmSyy-Szz)) * (-(SxymSyx)*(SyzpSzy)+(SxzmSzx)*(SxxpSyy-Szz)))
mxEigenV = E0
n_iter = 50
for i in range(n_iter):
oldg = mxEigenV
x2 = mxEigenV*mxEigenV
b = (x2 + C[2])*mxEigenV
a = b + C[1]
delta = ((a*mxEigenV + C[0])/(2.0*x2*mxEigenV + b + a))
mxEigenV -= delta
if (abs(mxEigenV - oldg) < abs((evalprec)*mxEigenV)):
break
else:
raise Exception("More iterations needed to find eigenvalue")
val = 2.0 * (E0 - mxEigenV)/float(N)
if abs(val) < evecprec:
rms = 0.0
else:
rms = math.sqrt(val)
rot = [0.0 for i in range(9)]
a11 = SxxpSyy + Szz-mxEigenV
a12 = SyzmSzy
a13 = - SxzmSzx
a14 = SxymSyx
a21 = SyzmSzy
a22 = SxxmSyy - Szz-mxEigenV
a23 = SxypSyx
a24= SxzpSzx
a31 = a13
a32 = a23
a33 = Syy-Sxx-Szz - mxEigenV
a34 = SyzpSzy
a41 = a14
a42 = a24
a43 = a34
a44 = Szz - SxxpSyy - mxEigenV
a3344_4334 = a33 * a44 - a43 * a34
a3244_4234 = a32 * a44-a42*a34
a3243_4233 = a32 * a43 - a42 * a33
a3143_4133 = a31 * a43-a41*a33
a3144_4134 = a31 * a44 - a41 * a34
a3142_4132 = a31 * a42-a41*a32
q1 = a22*a3344_4334-a23*a3244_4234+a24*a3243_4233
q2 = -a21*a3344_4334+a23*a3144_4134-a24*a3143_4133
q3 = a21*a3244_4234-a22*a3144_4134+a24*a3142_4132
q4 = -a21*a3243_4233+a22*a3143_4133-a23*a3142_4132
qsqr = q1 * q1 + q2 * q2 + q3 * q3 + q4 * q4
# The following code tries to calculate another column in the adjoint
# matrix when the norm of the current column is too small. Usually
# this commented block will never be activated. To be absolutely safe
# this should be uncommented, but it is most likely unnecessary.
if (qsqr < evecprec):
q1 = a12*a3344_4334 - a13*a3244_4234 + a14*a3243_4233
q2 = -a11*a3344_4334 + a13*a3144_4134 - a14*a3143_4133
q3 = a11*a3244_4234 - a12*a3144_4134 + a14*a3142_4132
q4 = -a11*a3243_4233 + a12*a3143_4133 - a13*a3142_4132
qsqr = q1*q1 + q2 *q2 + q3*q3+q4*q4
if (qsqr < evecprec):
a1324_1423 = a13 * a24 - a14 * a23
a1224_1422 = a12 * a24 - a14 * a22
a1223_1322 = a12 * a23 - a13 * a22
a1124_1421 = a11 * a24 - a14 * a21
a1123_1321 = a11 * a23 - a13 * a21
a1122_1221 = a11 * a22 - a12 * a21
q1 = a42 * a1324_1423 - a43 * a1224_1422 + a44 * a1223_1322
q2 = -a41 * a1324_1423 + a43 * a1124_1421 - a44 * a1123_1321
q3 = a41 * a1224_1422 - a42 * a1124_1421 + a44 * a1122_1221
q4 = -a41 * a1223_1322 + a42 * a1123_1321 - a43 * a1122_1221
qsqr = q1*q1 + q2 *q2 + q3*q3+q4*q4
if (qsqr < evecprec):
q1 = a32 * a1324_1423 - a33 * a1224_1422 + a34 * a1223_1322
q2 = -a31 * a1324_1423 + a33 * a1124_1421 - a34 * a1123_1321
q3 = a31 * a1224_1422 - a32 * a1124_1421 + a34 * a1122_1221
q4 = -a31 * a1223_1322 + a32 * a1123_1321 - a33 * a1122_1221
qsqr = q1*q1 + q2 *q2 + q3*q3 + q4*q4
if (qsqr < evecprec):
# if qsqr is still too small, return the identity matrix. #
rot[0] = rot[4] = rot[8] = 1.0
rot[1] = rot[2] = rot[3] = rot[5] = rot[6] = rot[7] = 0.0
return rms, rot
normq = math.sqrt(qsqr)
q1 /= normq
q2 /= normq
q3 /= normq
q4 /= normq
a2 = q1 * q1
x2 = q2 * q2
y2 = q3 * q3
z2 = q4 * q4
xy = q2 * q3
az = q1 * q4
zx = q4 * q2
ay = q1 * q3
yz = q3 * q4
ax = q1 * q2
rot[0] = a2 + x2 - y2 - z2
rot[1] = 2 * (xy + az)
rot[2] = 2 * (zx - ay)
rot[3] = 2 * (xy - az)
rot[4] = a2 - x2 + y2 - z2
rot[5] = 2 * (yz + ax)
rot[6] = 2 * (zx + ay)
rot[7] = 2 * (yz - ax)
rot[8] = a2 - x2 - y2 + z2
return rms, rot