Specific Binding of Pu to Ca-Binding Sites of Protein
J. Am. Chem. Soc., Vol. 119, No. 22, 1997 5123
resonance of the 4(240Pu )‚CaM signal (a correction of 2.5%
from the beam current monitor signal) and then fitted with a
straight line in the vicinity of the resonance. The theoretical
3+
240
Pu cross section must be convoluted with Doppler broadening
and with the instrument time-of-flight resolution, which has
terms due to the width of the recorded time slice (4.47 µs), the
proton pulse width (0.27 µs), and the pulse shape distribution
of the moderator. This last function has been calculated by a
detailed Monte Carlo simulation of the MLNSC target/moderator
2
9,30
system
by an Ikeda-Carpenter function, (t/λ) e
of the neutrons emitted directly and 30% convoluted with an
and is very well fitted near the resonance wavelength
31
2 -(0.257t/λ)
, with 70%
-
(t/6.8)
exponential decay, e
(for t in µs and λ in Å). Doppler
Figure 5. Weighted averages of resonant scattering length, including
effects of the asymmetric neutron pulse shape, resonant absorption,
and Doppler broadening for the first neutron-scattering experiment.
3
2
240
broadening results from thermal motion of the Pu nuclei.
The velocities of the CaM molecules are small compared to
the neutron velocity, but the 2 Pu atoms are vibrating within
their CaM binding sites. If we neglect Doppler broadening,
the resulting computed resolution is too narrow, and if we use
the value of the mean-square velocity appropriate for metallic
Pu as in ref 32, it is too broad. We have therefore taken the
thermal width to be a free parameter and have chosen 12 meV
as the best fit. Since the resolution is asymmetric and the
transmission is nonlinear, Monte Carlo is the best technique
for applying both the Doppler and the resolution corrections.
A program using library MCLIB33 determined the number of
counts per time slice, averaged over all wavelengths which may
40
blocked-beam measurement, and suppose the relative normal-
ization factors of the three measurements are N , N , and N .
S
B
K
Then we can subtract the blocked-beam component of the
signals before normalizing the transmitted beams as
(S) - (K)N /NK (B) - (K)N /NK
S
B
I )
-
(
T )
(T )
S
B
(
S)
(B) (K) N
S
N
B
)
-
-
-
(7)
2
40
(TS)
(T )
NK
[
(TS)
(T )
]
contribute to the slice, as a function of the density of Pu
atoms. The histogram line in Figure 4 is thus a theoretical curve,
with density and Doppler width as the two free parameters. The
B
B
The second form is preferred because it allows propagation of
errors without correlations, and also shows (K) as a correction
to the standard form. Since (TS)/NS and (TB)/NB are exactly the
functions which were fitted in the construction of Figure 4, the
coefficient of (K) can be computed precisely. To minimize the
statistical uncertainties in the transmission measurements the
measured (TB) was fitted over the relevant time range with a
second-order curve and (TS) was computed by multiplying by
the transmission curve corresponding to Figure 5. The blocked-
beam correction was a major fraction of the signal and
contributed to the statistical uncertainty of the data points; it
did not, however, affect the final result.
240
area density of Pu atoms was found by iteration, giving 2.37
1
8
2
×
10 atoms/cm , which is 6% lower than 4 times the estimated
CaM density; this is partly because the sample did not
completely cover the exposed area of the cell, but may also
represent some small deterioration of the sample.
The histogram in Figure 4 shows that the wavelength
calibration is excellent and the shape of the resonance is very
well understood. We can therefore use this measurement,
corrected for the different 240Pu density, in the reduction of the
first neutron data set. Also, the Monte Carlo procedure provides
independent averages of λ(t) and its standard deviation,
Re{b(t)}, Im{b(t)}, and |b(t)| appropriately weighted by resolu-
tion and transmission probability. These averages depend on
For each of 24 time slices around the resonance, the detector
(X, Y, t) histogram was binned in a uniform Q-scale. For each
2
40
value of Q the three orthogonal form factor terms of eq 5 were
then extracted by least squares. Correlations of the errors with
the small F00 and F0r terms prevent assignment of simple error
bars to Frr obtained in this extraction. Therefore simple
expressions for F00 and F0r as a function of Q were subtracted
from the data and Frr was extracted again. The error bars could
then be propagated from the counting statistics (see Figure 6).
In this process all of the errors are assigned to Frr. Systematic
bias may occur in the processes of dividing by the computed
transmission and normalizing the background and blocked-beam
subtractions. The dashed line on Figure 6 is a linear correction
computed in the model-fitting process. The resolution in Q is
dominated by the size and shape of the neutron beam spot, which
is a triangular distribution with a measured full width half
maximum of 28 mm. The resulting Q resolution, with full width
the Pu density; e.g., the peak value of |b| is reduced because
fewer neutrons are transmitted at the peak. The terms needed
for the decomposition of eq 5 are shown for the first neutron
sample density in Figure 5, which may be compared to Figure
1
A. The wavelength resolution (rms), including the width of
the recorded slices, is 0.63%.
The usual normalization and background subtraction for LQD
data is to divide the recorded histogram for the signal, (S), by
the transmitted beam, (TS), and subtract the buffer, (B), divided
by its transmitted beam, (TB). If, however, there is a significant
difference in the two transmission functions, and if there is also
a significant background that is sample independent, then an
additional correction term is necessary. Let (K) represent a
(29) Pitcher, E. J.; Russell, G. J.; Seeger, P. A.; Fergusson, P. D. ICANS-
XIII; Paul Scherrer Institut: Villigen, Switzerland, Oct 11-14, 1995; PSI
-
1
half maximum ) 0.15 Å , is illustrated in Figure 6.
Proceedings 95-02, pp 323-329.
(30) P. D. Fergusson provided Monte-Carlo calculations of the perfor-
If we assume the CaM molecules are non-interacting, then
the range of the double sum of eq 6 is just the four binding
sites within the molecule, a total of sixteen terms. We further
simplify by assuming the distances between the two sites within
each globular domain are equal, d, based on the knowledge that
the domains are structurally homologous (the crystal structure
mance of the as-built MLNSC target-moderator system.
(
44.
31) Ikeda, S.; Carpenter, J. M. Nucl. Instrum. Methods 1985, A239, 536-
5
(
32) Seeger, P. A.; Taylor, A. D.; Brugger, R. M. Nucl. Instrum. Methods
1
985, A240, 98-114.
(
33) Seeger, P. A. ICANS-XIII, Paul Scherrer Institut: Villigen, Swit-
zerland, Oct 11-14, 1995; PSI Proceedings 95-02, 323-329. The Monte
Carlo program referenced here and its description are available by
anonymous ftp from azoth.lansce.lanl.goV/pub/mclib.
2
+
2+
of 4Ca ‚CaM shows the separations of Ca ions within each
globular domain to be 11.85 and 11.49 Å which are equal within