Biophysical studies of membrane proteins/peptides. Interaction with ...

2
2w
J ( t)
=
'2
R − R
+
w /
1
∫
d
2 2
w + Rp
2
p
⎛
⎜
⎜
⎝
w /
w /
⎡⎛
t
exp⎢
⎜−
⎢⎣
⎝ τ
D
2 2
w + Rp
−3
∫
α
2 ' 2
w + Rd
⎞⎛
R0
⎟⎜
⎠⎝
w
dα −
6
⎞
⎟
⎠
w /
w /
2 2
w + Rp
∫
2 ' 2
w + Rd
⎤
6
α ⎥ f ( α)
α
⎥⎦
⎪⎧
⎡⎛
t
⎨1
− exp⎢
⎜−
⎪⎩ ⎢⎣
⎝ τ
−3
⎞
⎟
dα⎟
⎠
D
⎞⎛
R0
⎟⎜
⎠⎝
w
6
⎞
⎟
⎠
⎤ ⎪⎫
6
α ⎥ f ( α)
⎬dα +
⎥⎦
⎪⎭
(A7)
and denote the integrals on the right hand side of Eq. A7 by J 1 , J 2 (t) and J 3 (t), according
to the order in which they appear in this equation. J 1 is easily calculated, leading to
2
2w
J ( t)
= 1−
[ J
2
( t)
− J
3(
t)
]
(A8)
'2 2
R − R
d
p
Being the average concentration of acceptors in the bilayer given by,
n
2
= N
'2 2
π(
R d
− R p
)
(A9)
it follows that
2
2πn2w
J ( t)
= 1−
[ J
2
( t)
− J
3(
t)
]
(A10)
N
Inserting this expression for J(t) in Eq. A1, and taking the limit (N → ∞, R’ d →
∞), one obtains the macroscopic decay law. The result is,
2
{ − 2πn
w [ J ( t)
− J ( )]}
ρ ( t)
= exp
2 2 3
t
(A11)
which is equivalent to each of the i items in the product of the right hand side of Eq. 6.
Acknowledgements:
Financial support for this work was provided by “Fundação para a Ciência e
Tecnologia” (FCT, Lisboa) through projects POCI/SAU-FCF/56003/2004 and
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