In this chapter, the build-up curve processing command set available
in Gifa is exposed. This set permits to calculate the distance from the
results of a multi-exponential analysis (RELAX, DIST), and to evaluate
the quality of this analysis (RCRYST). The processing of build-up curves
is closely related to the Linear Prediction Package available in Gifa,
and the manual related to the latter should be read before going into
this one.
Theory
The multi-exponential expression of the NOE
intensities
One can express the nOe intensities obtained from a NOESY
experiment, as a series of symmetric matrices
,
where
holds the intensity of the cross peak
between atoms i and j, for a NOESY mixing time of
. The relaxation parameters for each pair of spins
can be presented as the relaxation matrix
.
is bound to the nOe intensity matrix
by the relation :
(1)
where Io is a diagonal matrix with elements equal to the equilibrium magnetizations of each respective spin.
If the complete experimental determination of the matrix can be performed at a given
, then the determination of
is straightforward. However this is seldom the
case, due to overlaps in the 2D NOESY spectra.
The problem is then one of incompleteness of and several procedures have been proposed to
interpret quantitative NOESY information in terms of the relaxation
matrix
. We will show here, that this
incompleteness can be avoided if one takes a different approach to the
problem.
The relaxation matrix is symmetric, provided we use the construction
proposed by Olejniczak (Olejniczak, E. T. (1989) J. Magn. Res.
81, 392-394). It can be diagonalized :(2)
where is a diagonal matrix, and I is thus
computed by :
(3)
From Eq. (2) we can rewrite each element of matrix in the following way :
(4)
so that each element of matrix I becomes :
(5)
where the is the (i,k) element of the matrix
L, and
is the
element of the diagonal matrix
. We can rewrite Eqs. (4) and (5) as follows :
(6)
and :
(7)
where :
(8)
The form of Eq. (6) proves that NOESY build-up curves are sums of exponentials. Eqs. (6) and (7) show that we can deduce the value of the relaxation matrix elements from the analytical form of the build-up curve.
The process we propose for computing the relaxation matrix elements from
the NOESY experimental intensities is then, given an experimental nOe
build-up curve measured between atoms i and j,
to extract the parameters
and
of the multi-exponential decay.
From the parameters and
it is then easy to obtain the values of the
relaxation parameter
. Subsequently, it is
possible to extract distance information, by assuming a dynamic model
for the molecule (for instance rigid spherical tumbling).
In Gifa, the LP-SVD method is used for multiexponential analysis of the curves, because it permits the separation of the signal parameters from additional parameters arising from the noise.
A Froissard doublet is characterized by two related roots, located within the unit-circle, with opposite frequencies, opposite imaginary phases and the same amplitudes and damping factors:
(A1)
time dependence generated by such a pattern is:
(A2)
if
the rotation leads to:
(A3)
which gives:
(A4)
if
and
The developments
of the two expressions are very similar and equivalent for small values
of bt. The operation permitting the replacement of Eq. (A2) by (A4) is
equivalent to pivoting the root doublet by about the center of the doublet, bringing the roots
back onto the real axis.
This operation does not modify the first order of the build-up curve
reconstruction and can be safely used when the imaginary part of the
complex root is small compared to the real part. This condition can be
translated into geometric terms, by enforcing that the complex roots
must lie within a cone of angular extent .
In some cases however, particularly when the signal to noise ratio is very low, the LP-SVD analysis produces complex root doublets corresponding to higher frequencies for which the equations A2) and (A4) are no longer equivalent. In this case the fast frequencies detected in the signal are probably associated to the noise. We thus chose to ignore the related roots altogether for the amplitude reconstruction step. This decision was supported by the fact that the corresponding amplitudes, when computed, often appear to be at least one order of magnitude smaller than the other amplitudes.
(10)
Here is the value of the calculated build-up
curve reconstructed from the multi-exponential parameters, and
is the "experimental" data, for k running on the
different values of
. We found the R factor to
be more discriminant when computed only for small values of
.
(11)
Practising the build-up curves processing
A set of command as been implemented into Gifa in order to permit
the processing as described above. The set-up is such that these
commands should be used in conjonction with the Lineara Prediction
Module of Gifa. The processing should thus be performed on a build-up
curve, held into the regular 1D area of Gifa. This build-up curve should
be such as obtained with a set of NOESY spectra, obtained for various
value of the mixing time , regularly sampled
from 0 to Tmax. It
should be noted that the value for
is assumed
to be present. However it is rarely usefull to measure this value, and
it is preferable to insert a 0 value as the
point of the build-up curve.
RELAXRATE Permits to get the relaxation rate from the amplitude and the damping factors, which are obtained by the multi-exponential analysis of the build-up curve.
METH p Determine the hydrogen pair type. p is either 1, 2 or 3 for hydrogen-hydrogen, hydrogen-methyl or methyl-methyl pairs.
DIST Performs the distance calculation from the relaxation rate, using a reference distance, and the type of hydrogen pair observed.
CALIBDI dist_ref rate_ref Permits to define a reference rate, which corresponds to a reference distance, a the motion model considered.
RCRYST n Performs the computation of a "crystallographic factor", between the data obtained from the multi-exponential analysis and the initial data. This factor is calculated from the first n curve points.
SLOPE n Performs a least-square fit on the n first curve points, to determine his initial slope.
RTPIV Performs the processing of the
pairs of complexe
conjugate roots, which can be obtained when solving the PE polynomial.
The roots found inside the cone of extent
, are pivoted, those
outside are removed:
= cotan(
).
Works only with
forward roots.
The following internal variables are available :
$RELAX $RCRYST $CALIBDI[] $DIST
print"Calibration relaxation rate?" set ratecal = $_
calibdi $dstcal $ratecal
Processing a build-up curve
print "Name of the build-up curve file?"
set bldp = $_
print "Number of points on which the R-factor is computed?"
set nb_pts_R = $_
print "Number of points on which initial-slope is computed?"
set nb_pts_slop = $_
print "Order of the linear prediction analysis?"
set ord = $_
read $bldp ; read data file
reverse chsize +1 reverse ; add the first (null) point
slope $nb_pts_slop dist ; determine the distance
; by initial slope method
chsize *2 swa ; makes the data complex by iterleaving zeros
; (the Gifa linear prediction package
; processes only complexe data).
order $ord ; defines the order of the linear ; prediction analysis
dt->svd % svd->ar 2 ; calculates the AR coefficients
rtclean 2 rtinv 2 rtfreq % 1 ; PE root processing
rt->pk % 1 ; calculate the damping factors and the ; amplitudes
relaxrate ; calculate the relaxation rate
dist ; calculate the distance
rcryst $nb_pts_R ; calculate the crystallographic factor
References :
T.-E. Malliavin, M.-A. Delsuc and J.-Y. Lallemand, "Computation of
Redfield matrix elements for incomplete NOESY data-sets",J. Biomolec.
NMR 2, 349-360 (1992)
Reisdorf, C., Malliavin, T.-E. et Delsuc, M.-A. "Accurate estimation of inter-atomic distances in large proteins by NMR", Biochimie 74, 809-813 (1992)