def superpositionFit(confs):
"""
:param confs: the weight, reference position, and alternate
position for each atom
:type confs: sequence of (float, Vector, Vector)
:returns: the quaternion representing the rotation,
the center of mass in the reference configuration,
the center of mass in the alternate configuraton,
and the RMS distance after the optimal superposition
"""
w_sum = 0.
wr_sum = N.zeros((3,), N.Float)
for w, r_ref, r in confs:
w_sum += w
wr_sum += w*r_ref.array
ref_cms = wr_sum/w_sum
pos = N.zeros((3,), N.Float)
possq = 0.
cross = N.zeros((3, 3), N.Float)
for w, r_ref, r in confs:
w = w/w_sum
r_ref = r_ref.array-ref_cms
r = r.array
pos = pos + w*r
possq = possq + w*N.add.reduce(r*r) \
+ w*N.add.reduce(r_ref*r_ref)
cross = cross + w*r[:, N.NewAxis]*r_ref[N.NewAxis, :]
k = N.zeros((4, 4), N.Float)
k[0, 0] = -cross[0, 0]-cross[1, 1]-cross[2, 2]
k[0, 1] = cross[1, 2]-cross[2, 1]
k[0, 2] = cross[2, 0]-cross[0, 2]
k[0, 3] = cross[0, 1]-cross[1, 0]
k[1, 1] = -cross[0, 0]+cross[1, 1]+cross[2, 2]
k[1, 2] = -cross[0, 1]-cross[1, 0]
k[1, 3] = -cross[0, 2]-cross[2, 0]
k[2, 2] = cross[0, 0]-cross[1, 1]+cross[2, 2]
k[2, 3] = -cross[1, 2]-cross[2, 1]
k[3, 3] = cross[0, 0]+cross[1, 1]-cross[2, 2]
for i in range(1, 4):
for j in range(i):
k[i, j] = k[j, i]
k = 2.*k
for i in range(4):
k[i, i] = k[i, i] + possq - N.add.reduce(pos*pos)
from Scientific import LA
e, v = LA.eigenvectors(k)
i = N.argmin(e)
v = v[i]
if v[0] < 0: v = -v
if e[i] <= 0.:
rms = 0.
else:
rms = N.sqrt(e[i])
from Scientific.Geometry import Quaternion
return Quaternion.Quaternion(v), Vector(ref_cms), \
Vector(pos), rms
def superpositionFit(confs):
"""
:param confs: the weight, reference position, and alternate
position for each atom
:type confs: sequence of (float, Vector, Vector)
:returns: the quaternion representing the rotation,
the center of mass in the reference configuration,
the center of mass in the alternate configuraton,
and the RMS distance after the optimal superposition
"""
w_sum = 0.
wr_sum = N.zeros((3, ), N.Float)
for w, r_ref, r in confs:
w_sum += w
wr_sum += w * r_ref.array
ref_cms = wr_sum / w_sum
pos = N.zeros((3, ), N.Float)
possq = 0.
cross = N.zeros((3, 3), N.Float)
for w, r_ref, r in confs:
w = w / w_sum
r_ref = r_ref.array - ref_cms
r = r.array
pos = pos + w * r
possq = possq + w*N.add.reduce(r*r) \
+ w*N.add.reduce(r_ref*r_ref)
cross = cross + w * r[:, N.NewAxis] * r_ref[N.NewAxis, :]
k = N.zeros((4, 4), N.Float)
k[0, 0] = -cross[0, 0] - cross[1, 1] - cross[2, 2]
k[0, 1] = cross[1, 2] - cross[2, 1]
k[0, 2] = cross[2, 0] - cross[0, 2]
k[0, 3] = cross[0, 1] - cross[1, 0]
k[1, 1] = -cross[0, 0] + cross[1, 1] + cross[2, 2]
k[1, 2] = -cross[0, 1] - cross[1, 0]
k[1, 3] = -cross[0, 2] - cross[2, 0]
k[2, 2] = cross[0, 0] - cross[1, 1] + cross[2, 2]
k[2, 3] = -cross[1, 2] - cross[2, 1]
k[3, 3] = cross[0, 0] + cross[1, 1] - cross[2, 2]
for i in range(1, 4):
for j in range(i):
k[i, j] = k[j, i]
k = 2. * k
for i in range(4):
k[i, i] = k[i, i] + possq - N.add.reduce(pos * pos)
from Scientific import LA
e, v = LA.eigenvectors(k)
i = N.argmin(e)
v = v[i]
if v[0] < 0: v = -v
if e[i] <= 0.:
rms = 0.
else:
rms = N.sqrt(e[i])
from Scientific.Geometry import Quaternion
return Quaternion.Quaternion(v), Vector(ref_cms), \
Vector(pos), rms
def findTransformationAsQuaternion(self, conf1, conf2=None):
universe = self.universe()
if conf1.universe != universe:
raise ValueError("conformation is for a different universe")
if conf2 is None:
conf1, conf2 = conf2, conf1
else:
if conf2.universe != universe:
raise ValueError("conformation is for a different universe")
ref = conf1
conf = conf2
weights = universe.masses()
weights = weights / self.mass()
ref_cms = self.centerOfMass(ref).array
pos = Numeric.zeros((3, ), Numeric.Float)
possq = 0.
cross = Numeric.zeros((3, 3), Numeric.Float)
for a in self.atomList():
r = a.position(conf).array
r_ref = a.position(ref).array - ref_cms
w = weights[a]
pos = pos + w * r
possq = possq + w*Numeric.add.reduce(r*r) \
+ w*Numeric.add.reduce(r_ref*r_ref)
cross = cross + w * r[:,
Numeric.NewAxis] * r_ref[Numeric.NewAxis, :]
k = Numeric.zeros((4, 4), Numeric.Float)
k[0, 0] = -cross[0, 0] - cross[1, 1] - cross[2, 2]
k[0, 1] = cross[1, 2] - cross[2, 1]
k[0, 2] = cross[2, 0] - cross[0, 2]
k[0, 3] = cross[0, 1] - cross[1, 0]
k[1, 1] = -cross[0, 0] + cross[1, 1] + cross[2, 2]
k[1, 2] = -cross[0, 1] - cross[1, 0]
k[1, 3] = -cross[0, 2] - cross[2, 0]
k[2, 2] = cross[0, 0] - cross[1, 1] + cross[2, 2]
k[2, 3] = -cross[1, 2] - cross[2, 1]
k[3, 3] = cross[0, 0] + cross[1, 1] - cross[2, 2]
for i in range(1, 4):
for j in range(i):
k[i, j] = k[j, i]
k = 2. * k
for i in range(4):
k[i, i] = k[i, i] + possq - Numeric.add.reduce(pos * pos)
from Scientific import LA
e, v = LA.eigenvectors(k)
i = Numeric.argmin(e)
v = v[i]
if v[0] < 0: v = -v
if e[i] <= 0.:
rms = 0.
else:
rms = Numeric.sqrt(e[i])
return Quaternion.Quaternion(v), Vector(ref_cms), \
Vector(pos), rms
def findTransformationAsQuaternion(self, conf1, conf2 = None):
universe = self.universe()
if conf1.universe != universe:
raise ValueError("conformation is for a different universe")
if conf2 is None:
conf1, conf2 = conf2, conf1
else:
if conf2.universe != universe:
raise ValueError("conformation is for a different universe")
ref = conf1
conf = conf2
weights = universe.masses()
weights = weights/self.mass()
ref_cms = self.centerOfMass(ref).array
pos = Numeric.zeros((3,), Numeric.Float)
possq = 0.
cross = Numeric.zeros((3, 3), Numeric.Float)
for a in self.atomList():
r = a.position(conf).array
r_ref = a.position(ref).array-ref_cms
w = weights[a]
pos = pos + w*r
possq = possq + w*Numeric.add.reduce(r*r) \
+ w*Numeric.add.reduce(r_ref*r_ref)
cross = cross + w*r[:, Numeric.NewAxis]*r_ref[Numeric.NewAxis, :]
k = Numeric.zeros((4, 4), Numeric.Float)
k[0, 0] = -cross[0, 0]-cross[1, 1]-cross[2, 2]
k[0, 1] = cross[1, 2]-cross[2, 1]
k[0, 2] = cross[2, 0]-cross[0, 2]
k[0, 3] = cross[0, 1]-cross[1, 0]
k[1, 1] = -cross[0, 0]+cross[1, 1]+cross[2, 2]
k[1, 2] = -cross[0, 1]-cross[1, 0]
k[1, 3] = -cross[0, 2]-cross[2, 0]
k[2, 2] = cross[0, 0]-cross[1, 1]+cross[2, 2]
k[2, 3] = -cross[1, 2]-cross[2, 1]
k[3, 3] = cross[0, 0]+cross[1, 1]-cross[2, 2]
for i in range(1, 4):
for j in range(i):
k[i, j] = k[j, i]
k = 2.*k
for i in range(4):
k[i, i] = k[i, i] + possq - Numeric.add.reduce(pos*pos)
from Scientific import LA
e, v = LA.eigenvectors(k)
i = Numeric.argmin(e)
v = v[i]
if v[0] < 0: v = -v
if e[i] <= 0.:
rms = 0.
else:
rms = Numeric.sqrt(e[i])
return Quaternion.Quaternion(v), Vector(ref_cms), \
Vector(pos), rms
def symmetricTensorBasis(cell, space_group):
from CDTK.Crystal import UnitCell
subspace = 1. * N.equal.outer(N.arange(6), N.arange(6))
for tr in cartesianCoordinateSymmetryTransformations(cell, space_group):
rot = symmetricTensorRotationMatrix(tr.tensor.array)
ev, axes = LA.eigenvectors(rot)
new_subspace = []
for i in range(6):
if abs(ev[i] - 1.) < 1.e-12:
p = N.dot(N.transpose(subspace), N.dot(subspace, axes[i].real))
new_subspace.append(p)
m, s, subspace = LA.singular_value_decomposition(N.array(new_subspace))
nb = N.sum(s / s[0] > 1.e-12)
subspace = subspace[:nb]
return [SymmetricTensor(a) for a in subspace]
from MMTK import *
from MMTK.Proteins import Protein
from Scientific import N, LA
# Import the graphics module. Substitute any other graphics
# module name to make the example use that module.
from Scientific.Visualization import VRML2
module = VRML2
# Create the protein and find its center of mass and tensor of inertia.
protein = Protein('insulin')
center, inertia = protein.centerAndMomentOfInertia()
# Diagonalize the inertia tensor and scale the axes to a suitable length.
mass = protein.mass()
diagonal, directions = LA.eigenvectors(inertia.array)
diagonal = N.sqrt(diagonal / mass)
# Generate the backbone graphics objects.
graphics = protein.graphicsObjects(graphics_module=module,
model='backbone',
color='red')
# Add an atomic wireframe representation of all valine sidechains
valines = protein.residuesOfType('val')
sidechains = valines.map(lambda r: r.sidechain)
graphics = graphics + sidechains.graphicsObjects(
graphics_module=module, model='wireframe', color='blue')
# Add three arrows corresponding to the principal axes of inertia.
for length, axis in map(None, diagonal, directions):