About DC: ServerBased Tools for Dipolar Coupling Calculations
The DC computation servers provide facilities for analyzing dipolar couplings
and their relation to molecular structure.
The mathematical form for dipolar couplings can be expressed in terms of an
alignment tensor, as described in more detail below.
The most basic DC analysis involves fitting a collection of measured
dipolar couplings to a known structure. This will result in a set of
bestfit alignment tensor parameters. It will also provide a collection of
calculated dipolar couplings which can be compared to the measured values.
It is also possible to use previouslyderived alignment parameters to calculate
dipolar couplings for different parts of the same molecular system. For example,
given a set of dipolar couplings and a proposed structure for a twodomain molecule,
we can fit the couplings for the first domain to determine alignment tensor parameters,
and then use these parameters to calculate dipolar couplings for the second domain.
An alignment tensor is also associated with one or more molecular
orientations. So, given alignment tensor parameters for a proposed structure,
we can also rotate that structure into the possible orientations consistent
with the alignment tensor.
Accordingly, the input for a typical DC Server calculation will include:
 A molecular structure in PDB format
 A table of dipolar couplings in NMRPipe table format, all measured in the same alignment medium.
 Optional specification of one or more alignment tensor parameters or limits.
 Selection of which parts of the molecular system to use in the calculation, for example as a range of residue numbers.
The typical output will include:
 A report of the tensor parameter values.
 A table of measured couplings with their corresponding calculated couplings.
 Statistics which describe the agreement between the measured and calculated couplings.
 Graphs of the measured and observed couplings in PDF format.
 Versions of the input PDB structure, rotated into orientations described by the alignment tensor.
The DC Server modules include two methods for fitting dipolar couplings to a given structure.
While both of these methods compute couplings, they perform the computation and
fitting differently:

In the Order Matrix Fitting method,
no prior assumption is made about the
alignment tensor parameters, and the fit is performed by linear
least squares analysis using equations 3, 4, and 8 below. Since
Singular Value Decomposition is used, this is also called the SVD method.
In the DC modules, the SVD fitting is weighted by uncertainty
values
DD from the input.

In the NonLinear Fitting method,
specific values or limits on one or more tensor parameters can be specified, and
the fit is performed according to equation 2 and 7 below using repeated
trials of nonlinear least squares optimization. The optimization attempts to minimize
the weighted RMS between the observed and calculated couplings.
In the DC modules, the nonlinear fitting is weighted by
W values from the input.
In addition to fitting dipolar couplings to a given structure, the DC modules also
include two related options for calculating dipolar couplings for a given structure
when the alignment tensor is specified completely:

In the Known Order Matrix
option, dipolar couplings for a given PDB structure are calculated according to five
order matrix values. These five values are usually determined by previous
Order Matrix Fitting using part of the same PDB structure.

In the Known Tensor Parameters
option, dipolar couplings are calculated according to given values for the
five parameters which specify the alignment tensor.
Calculating Dipolar Couplings from Alignment Tensor Parameters and Molecular Coordinates
The alignment tensor can be described according to five parameters:
 The probability of alignment, also called the alignment magnitude,
Da , which varies from 0.0
for perfectly isotropic systems to 1.0 for complete static alignment.
 The axial symmetry in the alignment, also called the rhombicity,
Rh , which varies from 0.0
for perfect axial symmetry to 2/3.

Three rotational angles, sometimes also called euler angles, which specify the orientation of
alignment with respect to the magnetic field. DC modules always specify rotations according to
an XYZ rotation convention, discussed in more detail below. The three XYZconvention rotational
angles are sometimes called
RX , RY , and RZ in DC reports.
For a structure rotated into a given orientation,
the dipolar coupling between atoms I and J at coordinates (xI,yI,zI) and
(xJ,yJ,zJ) separated by distance r^{IJ} is expressed as follows;
the direction cosines are given as:
X^{IJ} = (xIxJ) / r^{IJ}
Y^{IJ} = (yIyJ) / r^{IJ}
Z^{IJ} = (zIzJ) / r^{IJ}

[Eqn 1ac]

Then, given tensor values Da and Dr such that rhombicity Rh = Dr / Da ,
the dipolar coupling D is calculated as:
D = DI[Da(3Z^{IJ}Z^{IJ}  1) + 1.5 Dr(X^{IJ}X^{IJ}  Y^{IJ}Y^{IJ})]

[Eqn 2]


This is the functional form used to calculate dipolar couplings in the DC server restrained fit modes.
The dipolar interaction value, DI , is the dipolar coupling that would be
observed for the static (completely aligned) molecule.
It is about 21,585 Hz for an HNN amide coupling, as described in more detail below.
Calculating Dipolar Couplings from Order Matrix Values and Molecular Coordinates
The alignment tensor can also be described in terms of a symmetric 3x3 matrix called the order matrix, or sometimes called the
saupe matrix. The relationship between the order matrix and the tensor parameters is described
in more detail elsewhere.
Since it is symmetric, the 3x3 order matrix can be described in terms of five unique values.
In the order matrix fitting scheme, the dipolar coupling D is expressed
as the linear combination of five terms:
DD^{1}
D = c_{1}*q_{1} + c_{2}*q_{2} + c_{3}*q_{3} + c_{4}*q_{4} + c_{5}*q_{5}

[Eqn 3]


where DD is the estimated uncertainty
in the measured coupling, and the five terms in the basis set are:
q_{1} =
^{1}/_{2}
DD^{1}
DI (3Z^{IJ}Z^{IJ}  1)
q_{2} =
^{1}/_{2}
DD^{1}
DI (X^{IJ}X^{IJ}  Y^{IJ}Y^{IJ})

q_{3} =
2 DD^{1}
DI X^{IJ}Y^{IJ}
q_{4} =
2 DD^{1}
DI X^{IJ}Z^{IJ}
q_{5} =
2 DD^{1}
DI Y^{IJ}Z^{IJ}


[Eqn 4ae]


Given the five coefficients c_{1} ... c_{5} , the elements of the
order matrix S are:
S_{xx} = ^{1}/_{2}(c_{1}  c_{2})

S_{xy} = S_{yx} = c_{3}

S_{yy} = ^{1}/_{2}(c_{1} + c_{2})

S_{xz} = S_{zx} = c_{4}

S_{zz} = c_{1}

S_{yz} = S_{zy} = c_{5}


[Eqn 5af]


Given five or more measured couplings, it is possible to use linear least squares fitting
to estimate values for the coefficients c1 ... c5 using a series of equations
built from the q1 ... q5 terms above. In practice, the linear least squares
analysis is performed via Singular Value Decomposition (SVD), so this method of tensor
estimation is often called the SVD method.
Once an order matrix is determined, it can be used to compute the five tensor parameters of
magnitude, rhombicity, and rotational angles. Likewise, given these five tensor
parameters, it is possible to calculate the corresponding order matrix.
So, as a convenience, all DC fitting modes report both order matrix
information and tensor parameters. The five coefficients
c1 ... c5 are reported in the DATA SAUPE line of
a DC output table.
DC Calculations for More than Two Atoms: Methyl and Methylene Couplings and Ensembles of Structures
In some cases, a measured dipolar coupling can correspond to more than one pair of atoms,
in particular HC couplings for methyl groups, such as ALA HBCB, and methylene groups, such as GLY HACA.
In these cases, the DC modules will compute couplings based on the sum of individual couplings in
the group. For example, a GLY HACA coupling corresponds to the sum of two terms, one for
the HA2CA atom pair, and another for the HA3CA atom pair.
Likewise, methyl HC dipolar couplings can be expressed as the sum of three terms,
such as for ALA HB1CB, HB2CB, and HB3CB. However, methyl HC dipolar couplings are a special case,
because of the rapid rotation of the methyl protons
about the CC axis. In this case, the HC couplings are scaled by a contant which depends on the
tetrahedral geometry of the methyl group. Details are given
in
Ottiger and Bax, J. Magn. Reson. 134, 365369 (1998).
Dipolar coupling calculations using a single alignment tensor for an ensemble of structures
can be treated in a similar way. In this case, one measured coupling is expressed as a sum of terms,
one for each structure in the ensemble.
Along these lines, we can modify the equations above to accommodate the cases of multiple
atom pairs and ensembles of structures. In the general case, we can consider each
dipolar coupling according to a group of one or more atom pairs I and J .
For example, an amide HN coupling would be a group with a single atom pair I = HN and J = N,
while a GLY HACA coupling would be a group consisting of two atom pairs,
one with I = HA2 and J = CA, and the other with I = HA3 and J = CA.
Then, given an ensemble of structures K = 1 to N_PDB
rotated into a given orienation, the overall dipolar coupling for the group of atom pairs I and J
over the ensemble is expressed as follows:
If the coordinates of atom I and atom J in the K th structure of the ensemble
are (x_{K}I,y_{K}I,z_{K}I) and
(x_{K}J,y_{K}J,z_{K}J) separated by distance r_{K}^{IJ}
the direction cosines are given as:
X_{K}^{IJ} = (x_{K}Ix_{K}J) / r_{K}^{IJ}
Y_{K}^{IJ} = (y_{K}Iy_{K}J) / r_{K}^{IJ}
Z_{K}^{IJ} = (z_{K}Iz_{K}J) / r_{K}^{IJ}

[Eqn 6ac]

Then, we can generalize Equation 2 such that the overall dipolar coupling D for
the group of atom pairs I and J over the ensemble is calculated as:
D = 
Σ 
Σ 
Σ 
S_{rot}DI[Da(3Z_{K}^{IJ}Z_{K}^{IJ}  1)
+ 1.5 Dr(X_{K}^{IJ}X_{K}^{IJ}  Y_{K}^{IJ}Y_{K}^{IJ})] / N_PDB

[Eqn 7] 
 K  I  J   
where S_{rot} is an empirical scaling constant which is 3*0.309107 for HC methyl groups,
and 1.0 otherwise.
In the case of order matrix calculation, the terms in the basis set of Equation 4ae are generalized
accordingly for a group of atom pairs I and J over an ensemble of structures:
q_{1} = 
Σ 
Σ 
Σ 
^{1}/_{2} S_{rot} DD^{1} DI (3Z_{K}^{IJ}Z_{K}^{IJ}  1) / N_PDB

 K  I  J  
q_{2} = 
Σ 
Σ 
Σ 
^{1}/_{2} S_{rot} DD^{1} DI (X_{K}^{IJ}X_{K}^{IJ}  Y_{K}^{IJ}Y_{K}^{IJ}) / N_PDB 
 K  I  J  
q_{3} = 
Σ 
Σ 
Σ 
2 S_{rot} DD^{1} DI X_{K}^{IJ}Y_{K}^{IJ} / N_PDB 
 K  I  J  
q_{4} = 
Σ 
Σ 
Σ 
2 S_{rot} DD^{1} DI X_{K}^{IJ}Z_{K}^{IJ} / N_PDB 
 K  I  J  
q_{5} = 
Σ 
Σ 
Σ 
2 S_{rot} DD^{1} DI Y_{K}^{IJ}Z_{K}^{IJ} / N_PDB 
 K  I  J  

[Eqn 8ae]

Dipolar Interaction (DI) Values and Standardized Bond Distances
As noted above, the dipolar interaction value, DI, is the dipolar coupling that would be
observed for the static (completely aligned) molecule, and is given as:
DI = (u0*hbar)*gammaI*gammaJ/(4*PI*PI*rIJ^3)

[Eqn 9]


where u0 is the magnetic permittivity of vacuum, hbar is Planck's
constant over 2*PI , gammaI and gammaJ
are the magnetogyric ratios for spins I and J, and rIJ is the distance between nuclei I and J.
Some common dipolar interaction values for protein dipolar couplings are given in the following table.
Since dipolar couplings depend on the cube of the distance between
atoms, they are very sensitive to structural variations. In practice,
the minor uncertainties in atomic coordinates may give rise
to substantial errors in calculated dipolar couplings. For this
reason, the DC modules will attempt to use standardized distances
when possible. The DC software uses the following standardized
values for couplings, according to atom names (HN N C HA CA etc.)
in the input:
Atom Pair I  J 
rIJ Angtroms 
Dipolar Interaction Value DI 
HN  N  1.041  21,585 Hz 
HN  C'  2.085  6,666 Hz 
C'  N  1.329  2,609 Hz 
HA  CA  1.107  44,539 Hz 
H  CH3  1.117  43,353 Hz 
C'  CA  1.525  4,285 Hz 
N  CA  1.458  1,976 Hz 
The DC server modules include the option to use existing Dipolar Interaction (DI) values
from the input table. This is intended to allow use of standardized DI values
for molecules other than proteins if desired.
If DI values are not specified in the input, they will be calculated
according to information in the PDB file.
In the case of common fixedgeometry peptide couplings
(HNN, HNC ', NC ', NCA, CA  C ', H  CH3 and HACA)
standardized interatomic distances in the table above will be used to calculate DI. Otherwise,
atomic distances from the PDB file will be used. If a DI value of 0.0 is specified
in the input file, it will always be replaced by a calculated DI value.
Computing Dipolar Couplings from Measured Isotropic and Aligned Values: Sign Issues
A coupling observed in an aligned sample is the sum of a dipolar coupling D and a scalar coupling J.
So, experimental dipolar coupling values are generally computed as a coupling from an aligned sample minus
the corresponding Jcoupling from an isotropic sample, in other words D = aligned  isotropic .
Proper handling of the signs of couplings can be a confusing issue, since the
intrinsic signs of couplings are often ignored, or at least not measured directly.
For example, when an HNN protein amide coupling is measured from peak
separation in an HNN doublet, the sign of the coupling isn't determined directly.
Since a 1H atom has a positive gamma value and a 15N atom has a negative gamma
value, we expect that the HNN isotropic Jcoupling should also be negative,
usually around 95 Hz. In practice however, HNN Jcoupling values are commonly
recorded as positive numbers, since the sign is not directly apparent from typical coupling measurements.
Correspondingly, the DC modules compute dipolar couplings independently of the
sign of gamma values, using the absolute value of the dipolar interaction value, DI .
So, when preparing dipolar couplings for use with DC modules, the signs of isotropic and aligned
couplings should not be adjusted according to the sign of gamma values. For example,
if an HNN doublet has a 94 Hz separation in an isotropic sample, and a 82 Hz separation in
an aligned sample, then the dipolar couping D is simply 82 Hz minus 94 Hz = 12 Hz.
The physical implication of this is that DCmodule dipolar couplings for vectors which point in the same direction
will have the same sign, regardless of the atom types. So for example, if an HACA bond vector
points in the same direction as an HNN bond vector, the HACA and HNN dipolar couplings will
both have the same sign.
Additional examples of dipolar couplings computed from isotropic and aligned coupling values
are shown in the following table:
RESID 
RESNAME 
HACA Isotropic 
HACA Aligned 
HACA DC 
HNN Isotropic 
HNN Aligned 
HNN DC 
CCA Isotropic 
CCA Aligned 
CCA DC 
CN Isotropic 
CN Aligned 
CN DC 
1  MET  .  .  .  .  .  .  .  .  .  .  .  . 
2  GLN  143.16  130.66  12.50  .  .  .  .  .  .  .  .  . 
3  TYR  145.31  132.64  12.67  93.98  84.72  9.27  53.07  53.24  0.17  4.71  10.18  5.47 
4  LYS  140.12  124.60  15.52  93.58  87.57  6.01  51.85  53.22  1.37  4.82  3.76  8.58 
5  LEU  142.47  130.10  12.37  92.69  86.67  6.01  53.32  56.34  3.02  4.74  4.76  0.02 
6  VAL  140.17  129.42  10.75  93.37  87.90  5.47  52.18  50.71  1.47  4.54  1.02  5.56 
7  ILE  139.92  126.07  13.84  93.17  89.30  3.87  53.16  56.77  3.61  4.74  1.78  2.96 
8  ASN  140.14  125.14  15.00  94.60  86.04  8.56  53.55  52.20  1.34  5.18  1.47  3.71 
9  GLY  .  .  .  94.58  94.18  0.40  53.86  57.46  3.60  5.37  0.30  5.67 
10  LYS  147.12  123.31  23.81  93.28  82.07  11.21  50.88  51.36  0.49  4.63  8.41  3.78 
[ Show Complete Table ]
DC Uncertainty Values (DD ) and Weighting Factors (W )
A given DC input table might contain more than one type of dipolar coupling.
For example, DC input for a protein might include both HNN amide
and HACA dipolar couplings. As listed in the previous section, each type of coupling has
a characteristic Dipolar Interaction (DI ) value. So, for example
since the DI value for an HACA coupling is about two times larger than
the DI value for an HNN coupling, the HACA dipolar couplings will be about two
times larger than HNN dipolar couplings in the same sample. In order to
compensate for the overall size differences between different types of
couplings, each dipolar coupling value D in the input table can
have two associated values:

The
DD value is the uncertainty in the coupling, given in Hz.
As shown in equations 3 and 4 above, the Order Matrix Fitting method
weights each dipolar coupling D by the inverse of its corresponding uncertainty DD , so that
couplings with large DD uncertainty values have low importance in the
Order Matrix SVD Fitting calculation.
Since the DD value determines the importance that a coupling has in calculating order matrix coefficients,
it can affect the tensor parameters determined by order order matrix fitting.

The
W value is a statistical weighting factor. The W value
does not affect the Order Matrix SVD calculation directly. Instead, it determines how a coupling contributes
to the RMS value and other reported statistics. Couplings with small W values have
low importance in the statistics calculations. For example, in a DC output,
the reported RMS between N observed couplings D_OBS and N coresponding
calculated couplings D is computed as:
RMS = [ Σ ( W*D_OBS  W*D )^{2} / N ] ^{1/2}
Since the DC NonLinear Fitting method
works by minimizing this RMS statistic, W values can affect the tensor parameters
found by nonlinear fitting. By contrast, while the W values will change reported statistics
like RMS, they will not affect the tensor parameters determined by order matrix fitting.
The DD and W values are optional for input tables used
with the DC server modules. If these columns are missing from the input table,
default values will be used. These default values will be proportional to
the DI value for the given coupling, and relative to the
standard DI value for a protein HNN coupling with a 1.0 Hz uncertainty,
so that the default DD = DI/21,585 and the
default W = 21,585/DI
Atom Pair I  J 
Default DD 
Default W 
HN  N  1.000  1.000 
HN  C'  0.309  3.238 
C'  N  0.121  8.273 
HA  CA  2.063  0.485 
H  CH3  2.008  0.498 
C'  CA  0.199  5.037 
N  CA  0.092  10.923 
Dipolar Coupling Input Table Format
The DC modules use dipolar coupling input tables in the NMRPipe
table format. A DC table groups together all couplings associated with a given aligned sample.
So, a DC input table might contain more than one type of coupling measurement, for example
both HNN couplings and HACA couplings.
In its simplest form, entries in a DC input table identify a pair of atoms I and J,
and give the coupling in Hz. The atoms are each identified by a residue ID, a residue name, and an atom name,
as shown in this example of the simplest DC input format. As shown in the example
every dipolar coupling input table for the DC servers must contain columns for RESID_I RESNAME_I ATOMNAME_I,
RESID_J RESNAME_J ATOMNAME_J, and the dipolar coupling value in Hz, column D .
Atom names in the DC input table must correspond to atom names used in the PDB file.
For example, if an atom name for a given residue is given as HN in the DC input, then the
PDB file should also have an amide proton called HN for that residue,
rather than one called H . Methylene and Methyl protons in the DC input
can be listed using hashmarkstyle naming, for example GLY HA# CA, ALA HB# CB,
VAL HG1# CG1 etc. In these cases of couplings for more than one atom pair,
the coupling should be given as the sum of its components. For example,
a GLY HA#CA coupling is given as the sum of HA2CA and HA3CA couplings, while
an ALA HB#CB coupling is given as the sum of of the three (identical) onebond HC couplings.
In the case of dipolar coupling tables for proteins, the input table can also
include amino acid sequence information in
the form of a DATA FIRST_RESID line and one or more DATA SEQUENCE LINES as
in this example. In current versions of the DC modules, this sequence
data is used to label graphs of dipolar couplings with respect to residue.
The DC input table can also contain columns for the estimated uncertainties in the couplings DD ,
and the statistical weighting factors W , as shown in this example.
If the PDB file has multiple chain IDs, the DC input table will require
specification of CHAINNAME_I and CHAINNAME_J .
Likewise if the PDB file has multiple segment IDs, the DC input table will require
specification of SEGNAME_I and SEGNAME_J .
Note: in a PDB ATOM line, which spans characters 1 to 80,
the chain ID is given in character 22,
and the segment ID is given in characters 73 to 76.
Column Name 
Usage 
Description 
RESID_I 
required 
Residue ID of Atom I. 
RESNAME_I 
required 
Residue name of Atom I. 
ATOMNAME_I 
required 
Atom name of Atom I, which must match the corresponding PDB atom name. 
SEGNAME_I 
optional 
PDB Segment ID name of Atom I (four characters max) 
CHAINNAME_I 
optional 
PDB Chain ID name of Atom I (single character) 
RESID_J 
required 
Residue ID of Atom J. 
RESNAME_J 
required 
Residue name of Atom J. 
ATOMNAME_J 
required 
Atom name of Atom J, which must match the corresponding PDB atom name. 
CHAINNAME_J 
optional 
PDB Chain ID name of Atom J (single character) 
SEGNAME_J 
optional 
PDB Segment ID name of Atom J (four characters max) 
D 
required 
Observed dipolar coupling, Hz. 
DI 
optional 
Dipolar Interaction value, Hz. 
DD 
optional 
Estimated uncertainty in the observed coupling, Hz. 
W 
optional 
Weighting factor for computing RMS etc. 
Selection of Residues for a DC Calculation
The many DC calculation servers provide an option to select which residues of the
molecule will be included in the DC calculation. The selection is specified as a list of
residue IDs or ranges separated by spaces. The keywords First and Last
are used for the lowest and highest residue IDs in the molecule, respectively. The keyword to is used
to specify a range of residues. So, for example, the following specification includes all couplings, i.e.,
all couplings from the lowest residue ID to the highest residue ID in the molecule (this is usually the
default selection in the DC modules):
first to last
Likewise, a specific range of residues can be specified, for example:
22 to 34 .
Multiple ranges can be specified, as well as individual residues. For example, the
following specification selects residues 10 and 13 as well as the range of
residues 22 to 34 and the range of residues 40 to 50:
10 13 22 to 34 40 to 50
The specifications can include commas for clarity, but the commas are
ignored. So, the following two specification lines select the same residues:
10, 13, 22 to 34, 40 to 50
10 13 22 to 34 40 to 50
The keywords chain and seg are used
to specify residues in systems with multiple chains or segments,
for example:
chain A 30 to 40 .
After a given chain or segment is specified, that selection applies to all
residues that follow, until a new chain or segment is specified. For example,
the following specification selects residues 30 to 40 from chain A,
ranges 50 to 60 and 80 to 90 from chain B, and residues 10 to 20 from chain C:
chain A 30 to 40 chain B 50 to 60 80 to 90 chain C 10 to 20 .
Note that if the PDB file has multiple chain IDs, the DC input table will require
specification of CHAINNAME_I and CHAINNAME_J . Likewise
if the PDB file has multiple segment IDs, the DC input table will require
specification of SEGNAME_I and SEGNAME_J .
Rotational Angles and Tensor Orientation
There are several conventions for specifying rotational angles. In the case of the DC
server modules, orientations are always specified as "XYZ" rotations: a rotation about
the XAxis, followed by a rotation about the YAxis, followed by a rotation about the ZAxis.
An orientation can also be described by its 3x3 rotation matrix, which does not
depend on any particular convention for specifying rotational angles.
As a general point, the relationship between rotational angles and orientation is not unique.
In practice, there will usually be two different sets of rotational angles which both result in
the same orientation, and give the same rotation matrix. The results of a DC computation
will report orientations as one or more sets of XYZconvention rotational angles,
given as EULER_ANGLES in
a DC output table.
The DC results will also report the values
from the corresponding 3x3 rotation matrix, given as three ROTATION_MATRIX entries
in the output.
In the specific case of dipolar couplings, the tensor parameters describe the equivalent of a 3D elipsoid.
As a result, the tensor parameters are fourfold degenerate, because as with an elipsoid, there is no way
to distinguish "top" from "bottom", "left" from "right", or "front" from "back". So, for a given choice
of Da and Rh, and rotation angles RX RY RZ, there are at least four different alignment orientations which
will give rise to the same couplings:
 The orientation resulting from rotation by RX RY RZ.
 The orientation resulting from rotation by RX RY RZ, followed by an additional 180degree rotation about the XAxis.
 The orientation resulting from rotation by RX RY RZ, followed by an additional 180degree rotation about the YAxis.
 The orientation resulting from rotation by RX RY RZ, followed by an additional 180degree rotation about the ZAxis.
Dipolar Coupling Output Table
An example dipolar coupling calculation output is given here.
Some details follow:

In the DC output table table,
N is the number of couplings,
and N_PDB is the number of molecular structures.

In the DC output table columns,
D_OBS is the observed
dipolar coupling value as taken from the input, and D
is the calculated dipolar coupling.

Qfactor, RMS, and Correlation factor are each computed from
couplings multiplied by W values from the table. For example,
for a collection of
N couplings, the RMS and
QFactor are computed as:
RMS = [ Σ( W*D_{observed}  W*D_{calculated} )^{2}/ N ]^{1/2}
Q_FACTOR = ( Σ( W*D_{observed}  W*D_{calculated} )^{2}/[N*(Da^{2}(4 + 3Rh^{2})/5)] )^{1/2}

In fitting via SVD (saupe order matrix method) coupling values
are divided by uncertainty
DD to determine the order matrix values.
So, in this case, the DD values will influence the tensor parameters
but the W values will only influence the reported statistics.

In nonlinear fitting mode, the fit attempts to minimize the RMS,
which is weighted by the
W values. So in this case, the DD values
will not influence the tensor parameters, but the W values will.

Methylene CH2 couplings, such as
GLY HA#/CA , are given as the
sum of both HC couplings. Likewise, methyl HC couplings such
as ALA HB#/CB are given as the sum of three onebond HC values,
as in the full separation between the two outer peaks of a
methyl CH3 quartet.

All rotational angles are given as an XYZ Rotation which rotates the
PDB file coordinates onto the alignment tensor frame.

For nonlinear fitting,
SAUPE (order matrix) values are
backcalculated from the fitted Da, Dr, and rotations.
The five values reported in the SAUPE line of the table
correspond to the five coefficients c1 ... c5 which define
the 3x3 order matrix S, as given in Equation 5af above.

The three values reported in
S_XYZ are the diagonal
elements of the diagonalized order matrix S, sorted so that
S_{zz} > S_{yy} > S_{xx}.

In the case of nonlinear fit modes, values reported for RX=PSI RY=THETA RZ=PHI
are the actual XYZ rotations applied by the fit, also given together as ROT_XYZ.
Values for
EULER_ANGLES are the XYZ rotations backcalculated from
the order matrix values.
DC Error Analysis Modes
The DC modules include several error analysis options. In each case, a given dipolar
coupling calculation will be repeated for many trials, with the input data
adjusted in some way at each trial. At each trial, the fitted alignment tensor
parameters are recorded, along with statistics about the quality of fit.

Error Analysis by Randomized Couplings: this analysis is intended
to show how random uncertainty in the measured couplings
influences the fitted alignment tensor parameters.
Each trial uses couplings
D _{trial} formed by adding noise
to the original couplings D , as
D _{trial} = D + c*R(DD) ,
where DD is the estimated uncertainty in the coupling as specified
in the input table, R(DD) is a Gaussianrandom noise function with standard
devition DD , and c is an adjustable scaling
parameter for the error analysis, typically around 1.0.

Error Analysis by Randomized Atomic Coordinates: this analysis is
intended to show how uncertainties in the structure influence the
the fitted alignment tensor parameters and the quality of the dipolar
coupling fit. In some cases, this might also indicate the influence
of molecular motions on the fitted parameters.
Each trial of this error analysis uses a PDB structure computed by adding Gaussianrandom noise
of a given standard deviation to the atomic coordinates of
the original PDB input.

CrossValidation by Systematic Deletion: this analysis is intended
to show how well a given collection of couplings defines the alignment
tensor parameters, particularly when a only a small number of couplings
is available. In each trial of this analysis, one of the couplings
is deleted, and the fit is performed on the remainder. The
alignment tensor parameters from the fit of the remainder
are then used to calculate a value for the deleted coupling,
which can be compared to the value originally calculated
by fitting the complete data set.
The trials are repeated so that each coupling is systematically deleted in turn.

CrossValidation by Random Deletion: this analysis is similar to the
systematic crossvalidation above, but in this analysis, each trial deletes a
randomly selected fraction of the original data. In each trial,
the fit is performed on the remainder, and the
alignment tensor parameters from the fit of the remainder
are then used to calculate values for the deleted couplings.
