init(self,
params=None,
time_scale=1.0,
d_scale=1.0,
angle_units='deg',
param_types=0,
spheroid_type=None,
fixed=True)
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Function for initialising the diffusion tensor.
Keyword Arguments
~~~~~~~~~~~~~~~~~
params: The diffusion tensor data.
time_scale: The correlation time scaling value.
d_scale: The diffusion tensor eigenvalue scaling value.
angle_units: The units for the angle parameters.
param_types: A flag to select different parameter combinations.
spheroid_type: A string which, if supplied together with spheroid parameters, will restrict
the tensor to either being 'oblate' or 'prolate'.
fixed: A flag specifying whether the diffusion tensor is fixed or can be optimised.
The sphere (isotropic diffusion)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
When the molecule diffuses as a sphere, all three eigenvalues of the diffusion tensor are
equal, Dx = Dy = Dz. In this case, the orientation of the XH bond vector within the
diffusion frame is inconsequential to relaxation, hence, the spherical or Euler angles are
undefined. Therefore solely a single geometric parameter, either tm or Diso, can fully and
sufficiently parameterise the diffusion tensor. The correlation function for the global
rotational diffusion is
-----
1 - tau / tm
C(tau) = - e ,
5
-----
To select isotropic diffusion, the parameters argument should be a single floating point
number. The number is the value of the isotropic global correlation time, tm, in seconds.
To specify the time in nanoseconds, set the 'time_scale' argument to 1e-9. Alternative
parameters can be used by changing the 'param_types' flag to the following integers
0: {tm} (Default),
1: {Diso},
where
1 / tm = 6Diso.
The spheroid (axially symmetric diffusion)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
When two of the three eigenvalues of the diffusion tensor are equal, the molecule diffuses
as a spheroid. Four pieces of information are required to specify this tensor, the two
geometric parameters, Diso and Da, and the two orientational parameters, the polar angle
theta and the azimuthal angle phi describing the orientation of the axis of symmetry. The
correlation function of the global diffusion is
-----
_1_
1 \ - tau / tau_i
C(tau) = - > ci . e ,
5 /__
i=-1
-----
where
c-1 = 1/4 (3 dz^2 - 1)^2,
c0 = 3 dz^2 (1 - dz^2),
c1 = 3/4 (dz^2 - 1)^2,
and
1 / tau -1 = 6Diso - 2Da,
1 / tau 0 = 6Diso - Da,
1 / tau 1 = 6Diso + 2Da.
The direction cosine dz is defined as the cosine of the angle alpha between the XH bond
vector and the unique axis of the diffusion tensor.
To select axially symmetric anisotropic diffusion, the parameters argument should be a tuple
of floating point numbers of length four. A tuple is a type of data structure enclosed in
round brackets, the elements of which are separated by commas. Alternative sets of
parameters, 'param_types', are
0: {tm, Da, theta, phi} (Default),
1: {Diso, Da, theta, phi},
2: {tm, Dratio, theta, phi},
3: {Dpar, Dper, theta, phi},
4: {Diso, Dratio, theta, phi},
where
tm = 1 / 6Diso,
Diso = 1/3 (Dpar + 2Dper),
Da = Dpar - Dper,
Dratio = Dpar / Dper.
The spherical angles {theta, phi} orienting the unique axis of the diffusion tensor within
the PDB frame are defined between
0 <= theta <= pi,
0 <= phi <= 2pi,
while the angle alpha which is the angle between this axis and the given XH bond vector is
defined between
0 <= alpha <= 2pi.
The 'spheroid_type' argument should be 'oblate', 'prolate', or None. The argument will be
ignored if the diffusion tensor is not axially symmetric. If 'oblate' is given, then the
constraint Da <= 0 is used while if 'prolate' is given, then the constraint Da >= 0 is
used. If nothing is supplied, then Da will be allowed to have any values. To prevent
minimisation of diffusion tensor parameters in a space with two minima, it is recommended
to specify which tensor is to be minimised, thereby partitioning the two minima into the two
subspaces along the boundry Da = 0.
The ellipsoid (rhombic diffusion)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
When all three eigenvalues of the diffusion tensor are different, the molecule diffuses as
an ellipsoid. This diffusion is also known as fully anisotropic, asymmetric, or rhombic.
The full tensor is specified by six pieces of information, the three geometric parameters
Diso, Da, and Dr representing the isotropic, anisotropic, and rhombic components of the
tensor, and the three Euler angles alpha, beta, and gamma orienting the tensor within the
PDB frame. The correlation function is
-----
_2_
1 \ - tau / tau_i
C(tau) = - > ci . e ,
5 /__
i=-2
-----
where the weights on the exponentials are
c-2 = 1/4 (d + e),
c-1 = 3 dy^2 dz^2,
c0 = 3 dx^2 dz^2,
c1 = 3 dx^2 dy^2,
c2 = 1/4 (d + e).
Let
R = sqrt(1 + 3Dr),
then
d = 3 (dx^4 + dy^4 + dz^4) - 1,
e = - 1 / R ((1 + 3Dr)(dx^4 + 2dy^2 dz^2) + (1 - 3Dr)(dy^4 + 2dx^2 dz^2) - 2(dz^4 + 2dx^2 dy^2)).
The correlation times are
1 / tau -2 = 6Diso - 2Da . R,
1 / tau -1 = 6Diso - Da (1 + 3Dr),
1 / tau 0 = 6Diso - Da (1 - 3Dr),
1 / tau 1 = 6Diso + 2Da,
1 / tau 1 = 6Diso + 2Da . R.
The three direction cosines dx, dy, and dz are the coordinates of a unit vector parallel to
the XH bond vector. Hence the unit vector is [dx, dy, dz].
To select fully anisotropic diffusion, the parameters argument should be a tuple of length
six. A tuple is a type of data structure enclosed in round brackets, the elements of which
are separated by commas. Alternative sets of parameters, 'param_types', are
0: {tm, Da, Dr, alpha, beta, gamma} (Default),
1: {Diso, Da, Dr, alpha, beta, gamma},
2: {Dx, Dy, Dz, alpha, beta, gamma},
3: {Dxx, Dyy, Dzz, Dxy, Dxz, Dyz},
where
tm = 1 / 6Diso,
Diso = 1/3 (Dx + Dy + Dz),
Da = Dz - (Dx + Dy)/2,
Dr = (Dy - Dx)/2Da.
The angles alpha, beta, and gamma are the Euler angles describing the diffusion tensor
within the PDB frame. These angles are defined using the z-y-z axis rotation notation where
alpha is the initial rotation angle around the z-axis, beta is the rotation angle around the
y-axis, and gamma is the final rotation around the z-axis again. The angles are defined
between
0 <= alpha <= 2pi,
0 <= beta <= pi,
0 <= gamma <= 2pi.
Within the PDB frame, the XH bond vector is described using the spherical angles theta and
phi where theta is the polar angle and phi is the azimuthal angle defined between
0 <= theta <= pi,
0 <= phi <= 2pi.
When param_types is set to 3, then the elements of the diffusion tensor matrix defined
within the PDB frame can be supplied.
Units
~~~~~
The 'time_scale' argument should be a floating point number. The only parameter affected by
this value is tm.
The 'd_scale' argument should also be a floating point number. Parameters affected by this
value are Diso, Dpar, Dper, Da, Dx, Dy, and Dz. Significantly, Dr is not affected.
The 'angle_units' argument should either be the string 'deg' or 'rad'. Parameters affected
are theta, phi, alpha, beta, and gamma.
Examples
~~~~~~~~
To set an isotropic diffusion tensor with a correlation time of 10 ns, type:
relax> diffusion_tensor.init(10e-9)
relax> diffusion_tensor.init(params=10e-9)
relax> diffusion_tensor.init(10.0, 1e-9)
relax> diffusion_tensor.init(params=10.0, time_scale=1e-9, fixed=True)
To select axially symmetric diffusion with a tm value of 8.5 ns, Dratio of 1.1, theta value
of 20 degrees, and phi value of 20 degrees, type:
relax> diffusion_tensor.init((8.5e-9, 1.1, 20.0, 20.0), param_types=2)
To select a spheroid diffusion tensor with a Dpar value of 1.698e7, Dper value of 1.417e7,
theta value of 67.174 degrees, and phi value of -83.718 degrees, type one of:
relax> diffusion_tensor.init((1.698e7, 1.417e7, 67.174, -83.718), param_types=3)
relax> diffusion_tensor.init(params=(1.698e7, 1.417e7, 67.174, -83.718), param_types=3)
relax> diffusion_tensor.init((1.698e-1, 1.417e-1, 67.174, -83.718), param_types=3,
d_scale=1e8)
relax> diffusion_tensor.init(params=(1.698e-1, 1.417e-1, 67.174, -83.718), param_types=3,
d_scale=1e8)
relax> diffusion_tensor.init((1.698e-1, 1.417e-1, 1.1724, -1.4612), param_types=3,
d_scale=1e8, angle_units='rad')
relax> diffusion_tensor.init(params=(1.698e-1, 1.417e-1, 1.1724, -1.4612), param_types=3,
d_scale=1e8, angle_units='rad', fixed=True)
To select ellipsoidal diffusion, type:
relax> diffusion_tensor.init((1.340e7, 1.516e7, 1.691e7, -82.027, -80.573, 65.568),
param_types=2)
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