such that your answer has 5% accuracy in the region of interest, namely, the
FRAPed region, and tell us how many terms you used.
Here is some stuﬀ that will come in handy when thinking about this
problem. First, you should try to obtain this equation
d
2
ρ
dz
2
+
1
z
dρ
dz
+
ρ
= 0
.
(1)
The only solution to this equation that does not diverge for
z
→
0 is the
zero order Bessel function
J
0
(
z
). Next, the boundary conditions at the edge
of the cell will lead to a condition of the form
J
0
0
(
kR
) = 0
.
(2)
Interestingly, the roots of
J
0
0
are just the roots of
J
1
because of the identity
J
0
0
(
z
) =

J
1
. The full solution you are looking for will emerge as (make sure
you demonstrate this clearly and convincingly)
c
(
r,t
) =
a
0
+
∞
X
i
=1
a
i
e

DK
2
i
t
J
0
(
K
i
r
)
.
(3)
We can determine the coeﬃcients
a
i
using the initial condition
c
(
r,
0). An
other identity that will prove useful when doing the calculation of the coef
ﬁcients is:
R
zJ
0
(
z
)
dz
=
zJ
1
(
z
).
2. Average Occupancy of Receptors.
In class, I did the problem of ligandreceptor binding in three diﬀerent
ways. Redo the derivation that I did using chemical potentials (i.e. the grand
canonical ensemble) and show that the average number of ligands bound is
given by
h
N
i
=
1
β
∂
∂μ
ln
Z
.
(4)
2
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View Full DocumentYou will ﬁnd it helpful to note that the average particle number can be
written as
h
N
i
=
1
Z
X
i
N
i
e

β
(
E
i

N
i
μ
)
,
(5)
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 Winter '09
 Nicotinic acetylcholine receptor, PBOC, photobleached region, ﬁnal concentration c∞

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