JHU 580.429 SB3
HW14: Diffusion and spatial patterning.
1. Diffusion.
(a) The diffusion constant of water in water is
D
=
10

5
cm
2
/
sec. We have interpreted
D
as
(
1
/
2
)
Δ
x
2
/
Δ
t
, where
Δ
x
is diffusive step and
Δ
t
is the typical time between steps.
For water, a typical step will be the diameter of a water molecule, about 0.3 nm. What
is the corresponding
Δ
t
in picoseconds?
(b) Suppose a hormone has to diffuse through a cell to activate a gene in the nucleus.
A typical diffusion constant for small peptides or drugs is
D
=
10

6
cm
2
/
sec. If the
diffusion distance is 5
μ
m, approximately how much time does the hormone require to
diffuse to the nucleus?
(c) Suppose a morphogen has a diffusion constant of
D
=
10

6
cm
2
/
sec and has a lifetime
of 1 hr. The distance over which patterning can occur is about equal to its rootmean
square diffusion length over its lifetime. What is this distance? At what developmental
stage does a human embryo have this diameter, and how many cells does it have at this
point?
2. Morphogen gradient patterning, transient solution. Suppose a morphogen diffuses in one
dimension according to the partial differential equation
˙
ρ
(
x
,
t
) =
D
(
d
/
dx
)
2
ρ
(
x
,
t
)

αρ
(
x
,
t
)
.
At time 0, the density is concentrated at the origin,
ρ
(
x
,
t
=
0
) =
n
0
δ
(
x
)
.
(a) Convert the PDE for
ρ
(
x
,
t
)
into an ODE for ˆ
ρ
(
k
,
t
)
.
(b) Evaluate ˆ
ρ
(
k
,
t
=
0
)
in terms of model parameters.
(c) Solve the ODE to obtain ˆ
ρ
(
k
,
t
)
in terms of model parameters.
(d) Use an inverse Fourier transform to obtain
ρ
(
x
,
t
)
.
(e) For no decay,
α
=
0, ﬁnd the time
t
?
(
x
)
when
ρ
(
x
,
t
)
has its maximum value.
(f) Whenever you have a diffusion problem,
R
2
=
2
Dt
is a good guess for relating time,
distance, and diffusion constant. For the previous problem, a reasonable guess would
therefore be
t
?
(
x
) =
x
2
/
2
D
. How does this guess compare to to the analytical answer?
(g) Evaluate the maximum morphogen density at position
x
, equal to
ρ
(
x
,
t
?
(
x
))
.
(h) For threshold
K
, ﬁnd the patterning length
x
?
deﬁned as
ρ
(
x
?
,
t
?
(
x
?
)) =
K
.
3. Morphogen gradient patterning, method of images for an absorbing barrier. Suppose a mor
phogen diffuses in one dimension according to the partial differential equation
˙
ρ
(
x
,
t
) =
D
(
d
/
)
2
ρ
(
x
,
t
)
.
At time 0, the density is concentrated at the origin,
ρ
(
x
,
t
=
0
) =
n
0
δ
(
x
)
. The cell membrane
is at
x
=
L
. Suppose that morphogens are absorbed and degraded at the cell membrane so that
Version: 2014/12/02 09:51:56
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