Chemistry Reference
In-Depth Information
4.4 Bloch-McConnell Equation and Related Equations
Parameter optimization of CPMG
R
2
dispersion data is carried out by minimizing
chi squared as given by
!
X
X
R
i;exp
2
a
R
i;cal
2
a
x
2
¼
(5)
s
i;err
m
i
Here, R
2a
i,exp
and R
2a
i,cal
are experimental and calculated R
2
values of
i
th
s
CP
value, respectively, and is ex-perimental uncertainty of the
i
th' R
2
value. R
2a
i,exp
is
calculated as described below. The number, m, indicates the number of residues to
be analyzed. When fitting each residue,
m
= 1.R
2a
i,cal
is calculated by solving the
Bloch-McConnell equation including the effects of 180
pulses iteratively or by
using its analytical solutions [
94
,
95
,
98
,
103
]. Typical analytical equations applied
for CPMG R
2
dispersion are (1) Luz-Meiboom equation that is suitable to analyze
fast exchange and easy to incorporate in optimization programs because R
ex
(
t
CP
)is
expressed by a single equation [
95
] or (2) Carver-Richards equation that is also
suitable to analyze intermediate and fast exchange and when there are differences
in R
2
0
in two sites [
98
,
103
]. Violation of these equations in the slow limit has been
well described in the literature [
116
]. In contrast to these analytical solutions,
the Bloch-McConnell equation is applicable to any exchange regime and any
relaxation rates [
94
]. However, since inten-sity is calculated step by step for each
t
CP
, a relatively longer computation time is required.
When optimizing the parameters for on/off-resonance
R
1
r
experiments, the same
principles apply in the minimization of
w
2
in (5). Since
R
1
r
is applied to investigate
dynamics on the time scale faster than that of CPMG
R
2
, the fast exchange equation
is often applied [
39
,
106
]. Recently, the equation that is applicable for the slow time
scale has become available, and has been applied to proteins undergoing slow
conformational exchange [
129
-
131
].
4.5 Practical Aspects Parameter Optimization
Two steps are involved in optimizing the fitted exchange parameters: first,
parameters are optimized for each residue (
m
¼
1 in (5)) and, second, parameters
are optimized for a group of residues (
m
1 in (5)). Overall flow is described as
follows. First, the data of individual residues are fit to get verification that the
R
2
dispersion data is in reasonable agreement with theory and to estimate the time
scale of motions. This step may also select only the sites that exhibit significant
dispersion profiles. Next, once this is done, one fits the data of a group of residues to
determine global exchange parameters. When only one
R
2
dispersion is fit, the
maximum number of unknown parameters in the two-site exchange model is four:
>
