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About DC: Server-Based Tools for Dipolar Coupling Calculations

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See Also:

Bax Group DC Calculation Servers

NMRPipe Table Format

Chemical Shifts, Dipolar Couplings, and Molecular Fragment Replacement

Contacts:
Yang Shen     shenyang@nih.gov
Ad Bax     bax@nih.gov

About DC: Server-Based 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 best-fit 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 previously-derived 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 two-domain 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:

  1. 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.

  2. In the Non-Linear 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 non-linear least squares optimization. The optimization attempts to minimize the weighted RMS between the observed and calculated couplings. In the DC modules, the non-linear 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:

  1. 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.

  2. 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 XYZ-convention 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 rIJ is expressed as follows; the direction cosines are given as:

        XIJ = (xI-xJ) / rIJ    YIJ = (yI-yJ) / rIJ    ZIJ = (zI-zJ) / rIJ    [Eqn 1a-c]

Then, given tensor values Da and Dr such that rhombicity Rh = Dr / Da, the dipolar coupling D is calculated as:

        D = |DI|[Da(3ZIJZIJ - 1) + 1.5 Dr(XIJXIJ - YIJYIJ)]       [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 HN-N 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 = c1*q1 + c2*q2 + c3*q3 + c4*q4 + c5*q5          [Eqn 3]

where DD is the estimated uncertainty in the measured coupling, and the five terms in the basis set are:

       q1 = 1/2 DD-1 |DI| (3ZIJZIJ - 1)
       q2 = 1/2 DD-1 |DI| (XIJXIJ - YIJYIJ) 		       q3 = 2 DD-1 |DI| XIJYIJ
       q4 = 2 DD-1 |DI| XIJZIJ
       q5 = 2 DD-1 |DI| YIJZIJ
         [Eqn 4a-e]

Given the five coefficients c1 ... c5, the elements of the order matrix S are:

       Sxx = -1/2(c1 - c2)        Sxy = Syx = c3
       Syy = -1/2(c1 + c2)        Sxz = Szx = c4
       Szz = c1        Syz = Szy = c5
             [Eqn 5a-f]

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 H-C couplings for methyl groups, such as ALA HB-CB, and methylene groups, such as GLY HA-CA. In these cases, the DC modules will compute couplings based on the sum of individual couplings in the group. For example, a GLY HA-CA coupling corresponds to the sum of two terms, one for the HA2-CA atom pair, and another for the HA3-CA atom pair.

Likewise, methyl H-C dipolar couplings can be expressed as the sum of three terms, such as for ALA HB1-CB, HB2-CB, and HB3-CB. However, methyl H-C dipolar couplings are a special case, because of the rapid rotation of the methyl protons about the C-C axis. In this case, the H-C 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, 365-369 (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 HA-CA 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 (xKI,yKI,zKI) and (xKJ,yKJ,zKJ) separated by distance rKIJ the direction cosines are given as:

     XKIJ = (xKI-xKJ) / rKIJ    YKIJ = (yKI-yKJ) / rKIJ    ZKIJ = (zKI-zKJ) / rKIJ    [Eqn 6a-c]

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 Srot|DI|[Da(3ZKIJZKIJ - 1) + 1.5 Dr(XKIJXKIJ - YKIJYKIJ)] / N_PDB      [Eqn 7]
  K I J    
where Srot is an empirical scaling constant which is 3*0.309107 for H-C methyl groups, and 1.0 otherwise.

In the case of order matrix calculation, the terms in the basis set of Equation 4a-e are generalized accordingly for a group of atom pairs I and J over an ensemble of structures:

     q1 = Σ Σ Σ 1/2 Srot DD-1 |DI| (3ZKIJZKIJ - 1) / N_PDB
  K I J  
     q2 = Σ Σ Σ 1/2 Srot DD-1 |DI| (XKIJXKIJ - YKIJYKIJ) / N_PDB
  K I J  
     q3 = Σ Σ Σ 2 Srot DD-1 |DI| XKIJYKIJ / N_PDB
  K I J  
     q4 = Σ Σ Σ 2 Srot DD-1 |DI| XKIJZKIJ / N_PDB
  K I J  
     q5 = Σ Σ Σ 2 Srot DD-1 |DI| YKIJZKIJ / N_PDB
  K I J  
        [Eqn 8a-e]

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 fixed-geometry peptide couplings (HN-N, HN-C ', N-C ', N-CA, CA - C ', H - CH3 and HA-CA) 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 J-coupling 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 HN-N protein amide coupling is measured from peak separation in an HN-N 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 HN-N isotropic J-coupling should also be negative, usually around -95 Hz. In practice however, HN-N J-coupling 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 HN-N 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 DC-module 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 HA-CA bond vector points in the same direction as an HN-N bond vector, the HA-CA and HN-N 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 HA-CA
Isotropic		 HA-CA
Aligned		 HA-CA
DC		 HN-N
Isotropic		 HN-N
Aligned		 HN-N
DC		 C-CA
Isotropic		 C-CA
Aligned		 C-CA
DC		 C-N
Isotropic		 C-N
Aligned		 C-N
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 HN-N amide and HA-CA 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 HA-CA coupling is about two times larger than the DI value for an HN-N coupling, the HA-CA dipolar couplings will be about two times larger than HN-N 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:

  1. 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.

  2. 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 Non-Linear Fitting method works by minimizing this RMS statistic, W values can affect the tensor parameters found by non-linear 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 HN-N 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 0.09210.923

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 HN-N couplings and HA-CA 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 hashmark-style 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 HA2-CA and HA3-CA couplings, while an ALA HB#-CB coupling is given as the sum of of the three (identical) one-bond H-C 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 X-Axis, followed by a rotation about the Y-Axis, followed by a rotation about the Z-Axis. 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 XYZ-convention 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 four-fold 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:

  1. The orientation resulting from rotation by RX RY RZ.
  2. The orientation resulting from rotation by RX RY RZ, followed by an additional 180-degree rotation about the X-Axis.
  3. The orientation resulting from rotation by RX RY RZ, followed by an additional 180-degree rotation about the Y-Axis.
  4. The orientation resulting from rotation by RX RY RZ, followed by an additional 180-degree rotation about the Z-Axis.

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.

  • Q-factor, 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 Q-Factor are computed as:

       RMS = [ Σ( W*Dobserved - W*Dcalculated )2/ N ]1/2
       Q_FACTOR = ( Σ( W*Dobserved - W*Dcalculated )2/[N*(Da2(4 + 3Rh2)/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 non-linear 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 H-C couplings. Likewise, methyl H-C couplings such as ALA HB#/CB are given as the sum of three one-bond H-C 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 non-linear fitting, SAUPE (order matrix) values are back-calculated 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 5a-f above.

  • The three values reported in S_XYZ are the diagonal elements of the diagonalized order matrix S, sorted so that |Szz| > |Syy| > |Sxx|.

  • 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 back-calculated 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 Dtrial formed by adding noise to the original couplings D, as Dtrial = D + c*R(DD), where DD is the estimated uncertainty in the coupling as specified in the input table, R(DD) is a Gaussian-random 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 Gaussian-random noise of a given standard deviation to the atomic coordinates of the original PDB input.

  • Cross-Validation 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.

  • Cross-Validation by Random Deletion: this analysis is similar to the systematic cross-validation 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.

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last update: Dec 6 2011