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pyqcprot.pyx
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# -----------------------------------------------------------------------------
# Author(s) of Original Implementation:
# Douglas L. Theobald
# Department of Biochemistry
# MS 009
# Brandeis University
# 415 South St
# Waltham, MA 02453
# USA
#
#
# Pu Liu
# Johnson & Johnson Pharmaceutical Research and Development, L.L.C.
# 665 Stockton Drive
# Exton, PA 19341
# USA
#
#
# For the original code written in C see:
# http://theobald.brandeis.edu/qcp/
#
#
# Author of Python Port:
# Joshua L. Adelman
# Department of Biological Sciences
# University of Pittsburgh
# Pittsburgh, PA 15260
#
#
#
# If you use this QCP rotation calculation method in a publication, please
# reference:
#
# 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.
#
#
# Copyright (c) 2009-2010, Pu Liu and Douglas L. Theobald
# Copyright (c) 2011 Joshua L. Adelman
# All rights reserved.
#
# 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 numpy as np
cimport numpy as np
import cython
cdef extern from "math.h":
double sqrt(double x)
double fabs(double x)
@cython.boundscheck(False)
@cython.wraparound(False)
def InnerProduct(np.ndarray[np.float64_t,ndim=1] A,
np.ndarray[np.float64_t,ndim=2] coords1,
np.ndarray[np.float64_t,ndim=2] coords2,
int N,
np.ndarray[np.float64_t,ndim=1] weight):
"""
Calculate the inner product of two structures.
If weight array is not None, calculate the weighted inner product.
Input:
A[9] -- inner product array (modified in place)
coords1 -- reference structure
coords2 -- candidate structure
N -- the size of the system
weight -- the weight array of size N: set to None if not needed
Output:
A[9] -- the inner product matrix
Return:
(G1 + G2) * 0.5; used as E0 in function 'FastCalcRMSDAndRotation'
Warning:
1. You MUST center the structures, coords1 and coords2, before calling this function.
2. Please note how the structure coordinates are stored as 3xN arrays,
not Nx3 arrays as is also commonly used. The difference is
something like this for storage of a structure with 8 atoms:
Nx3: xyzxyzxyzxyzxyzxyzxyzxyz
3xN: xxxxxxxxyyyyyyyyzzzzzzzz
"""
cdef double x1, x2, y1, y2, z1, z2
cdef unsigned int i
cdef double G1, G2
G1 = 0.0
G2 = 0.0
A[0] = A[1] = A[2] = A[3] = A[4] = A[5] = A[6] = A[7] = A[8] = 0.0
if (weight != None):
for i in xrange(N):
x1 = weight[i] * coords1[0,i]
y1 = weight[i] * coords1[1,i]
z1 = weight[i] * coords1[2,i]
G1 += x1*coords1[0,i] + y1*coords1[1,i] + z1*coords1[2,i]
x2 = coords2[0,i]
y2 = coords2[1,i]
z2 = coords2[2,i]
G2 += weight[i] * (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)
else:
for i in xrange(N):
x1 = coords1[0,i]
y1 = coords1[1,i]
z1 = coords1[2,i]
G1 += (x1*x1 + y1*y1 + z1*z1)
x2 = coords2[0,i]
y2 = coords2[1,i]
z2 = coords2[2,i]
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)
return (G1 + G2) * 0.5
@cython.boundscheck(False)
@cython.wraparound(False)
def FastCalcRMSDAndRotation(np.ndarray[np.float64_t,ndim=1] rot, np.ndarray[np.float64_t,ndim=1] A, double E0, int N):
"""
Calculate the RMSD, and/or the optimal rotation matrix.
Input:
rot[9] -- rotation matrix (modified in place)
A[9] -- the inner product of two structures
E0 -- (G1 + G2) * 0.5
N -- the size of the system
Output:
rot[9] -- the rotation matrix in the order of xx, xy, xz, yx, yy, yz, zx, zy, zz
rmsd -- the RMSD value
Return:
only the rmsd was calculated if rot is None
both the RMSD & rotational matrix calculated if rot is not None
"""
cdef double rmsd
cdef double Sxx, Sxy, Sxz, Syx, Syy, Syz, Szx, Szy, Szz
cdef double Szz2, Syy2, Sxx2, Sxy2, Syz2, Sxz2, Syx2, Szy2, Szx2,
cdef double SyzSzymSyySzz2, Sxx2Syy2Szz2Syz2Szy2, Sxy2Sxz2Syx2Szx2,
cdef double SxzpSzx, SyzpSzy, SxypSyx, SyzmSzy,
cdef double SxzmSzx, SxymSyx, SxxpSyy, SxxmSyy
cdef np.ndarray[np.float64_t,ndim=1] C = np.zeros(4,)
cdef unsigned int i
cdef double mxEigenV
cdef double oldg = 0.0
cdef double b, a, delta, rms, qsqr
cdef double q1, q2, q3, q4, normq
cdef double a11, a12, a13, a14, a21, a22, a23, a24
cdef double a31, a32, a33, a34, a41, a42, a43, a44
cdef double a2, x2, y2, z2
cdef double xy, az, zx, ay, yz, ax
cdef double a3344_4334, a3244_4234, a3243_4233, a3143_4133,a3144_4134, a3142_4132
cdef double evecprec = 1e-6
cdef double evalprec = 1e-14
cdef double a1324_1423, a1224_1422, a1223_1322, a1124_1421, a1123_1321, a1122_1221
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[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
for i in xrange(50):
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 (fabs(mxEigenV - oldg) < fabs((evalprec)*mxEigenV)):
break
#if (i == 50):
# print "\nMore than %d iterations needed!\n" % (i)
# the fabs() is to guard against extremely small, but *negative* numbers due to floating point error
rms = sqrt(fabs(2.0 * (E0 - mxEigenV)/N))
if (rot == None):
return rms # Don't bother with rotation.
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
normq = 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
def CalcRMSDRotationalMatrix(np.ndarray[np.float64_t,ndim=2] ref,
np.ndarray[np.float64_t,ndim=2] conf,
int N,
np.ndarray[np.float64_t,ndim=1] rot,
np.ndarray[np.float64_t,ndim=1] weights):
"""
Calculate the RMSD & rotational matrix.
Input:
ref -- reference structure
conf -- candidate structure
N -- the size of the system
rot[9] -- array to store rotation matrix; set to None if only calculating rmsd (modified in place)
weight -- the weight array of size len; set to None if not needed
Output:
rot[9] -- rotation matrix
Return:
RMSD value
"""
cdef double E0, rmsd
cdef np.ndarray[np.float64_t,ndim=1] A = np.zeros(9,)
E0 = InnerProduct(A,conf,ref,N,weights)
rmsd = FastCalcRMSDAndRotation(rot,A,E0,N)
return rmsd