Biomedical Engineering Reference
In-Depth Information
The probabilities are related to the kinetic parameters
p
L?PL
~k
f
½
P
eq
p
PL?L
~k
b
ð
10
:
7
Þ
The
probabilities
obey
'detailed
balance',
i.e.,
if
[L] 5 [PL],
than
p
L?PL
~p
PL?L
, even when k
f
and k
b
are very different.
The two-site chemical exchange differential equations are more compactly
cast with the definition M
+
5 M
x
+ iM
y
as
"
#
M
L
M
PL
~
M
L
M
PL
iv
L
{R
2
{p
L?PL
d
dt
p
PL?L
ð
10
:
8
Þ
iv
PL
{R
P
2
{p
PL?L
p
L?PL
Cast for a resonance on the protein, these equations are:
"
#
M
P
M
PL
~
M
P
M
PL
iv
P
{R
2
{p
P?PL
p
PL?P
d
dt
ð
10
:
9
Þ
iv
PL
{R
P
2
{p
PL?P
p
P?PL
with
p
P?PL
~k
f
½
L
eq
p
PL?L
~k
b
ð
10
:
10
Þ
For two-site exchange between sites A and B, there is an analytical solution
valid for all exchange regimes, cast in the frequency domain:
3,5
M
z
(v)~M
tot
f
A
(v{v
A
{iR
2
)zf
B
(v{v
B
{iR
2
)zi(p
AB
zp
BA
)
(v{v
A
zip
AB
)(v{v
B
zip
BA
)zp
AB
p
BA
ð
10
:
11
Þ
where f
A
and f
B
are the fractions of species A and B, respectively.
The observable signal is given by the imaginary part (i.e., M
y
(v)) of this
equation, given by:
M
y
(v)~M
tot
(BC{AD)
=
(C
2
zD
2
)
ð
10
:
12
Þ
where
A~f
A
(v{v
B
)zf
B
(v{v
A
)
B~f
A
R
2
zf
B
R
2
zp
AB
zp
BA
C~(v{v
A
)(v{v
B
){R
2
R
2
{R
2
p
BA
{R
2
p
AB
D~v(R
2
zR
2
zp
AB
zp
BA
){v
A
(R
2
zp
BA
){v
B
(R
2
zp
AB
)
ð
10
:
13
Þ
