Biomedical Engineering Reference
In-Depth Information
receptor-covered plane. The statistical properties of the time and distance to the
first binding event depend on the forward binding rate constant,
k
on
, the cell sur-
face density of receptors,
R
cell
, the ligand diffusion coefficient in the extracellular
matrix,
D
, and the height of the extracellular medium,
h
. For the relevant ranges
of these parameters, the distribution of time to capture is given by an exponen-
tially distributed random variable: The probability that a ligand is bound for the
first time after time
t
(
P
first
) is given by
exp
¬
-
kR
Pt
first
()
=
on
cell
t
.
[1]
-
-
-
h
®
This expression leads to the probability density function for the lateral distance
traveled until the first binding event (Figure 6C): The probability that a ligand is
bound in the ring between radii
r
and
r
+
dr
is given by
¬
¬
kR
kR
-
-
-
g rdr
()
=
on
cell
K r
on
cell
rdr
,
[2]
-
-
-
-
-
0
Dh
®
Dh
-
®
where
K
0
is the modified Bessel function (41). We see that for reasonable ranges
of receptor densities and binding rate constants, a majority of the secreted
ligands is bound for the first time after traveling a very short distance. As an
immediate consequence, the ligand-producing cell can recapture a significant
fraction of the secreted ligands (Figure 6D).
Once secreted, a ligand undergoes several cycles of binding, dissociation,
and extracellular diffusion before it is removed from the extracellular medium
by receptor-mediated endocytosis (Figure 6B). The rate constants for dissocia-
tion and endocytosis of ligand-receptor complexes,
k
off
and
k
e
, determine the
number of binding events until the first endocytosis event. In the simplest
model, the number of binding events is a geometrically distributed random vari-
able with the mean equal to 1 + (
k
off
/
k
e
) (25). Based on the measurements for the
human EGFR-TGF
interactions (
k
off
x k
e
0.1 min
-1
) (40), the ligand will be
internalized after ~2 binding events. Thus, this simple model predicts that inhibi-
tion of receptor-mediated endocytosis can extend the range of secreted ligands.
This prediction can be tested in mutants with defects in the genes mediating re-
ceptor-mediated endocytosis (42,43). (In other systems, where ligand is effi-
ciently recycled to the cell surface, the opposite can be true. In fact, endocytosis
might actually be the main mechanism for the spatial propagation of secreted
signals (44).)
According to this simple model, the relative rates between binding interac-
tions, endocytosis, and diffusion influence the spatial range of secreted ligands.
This might explain the apparent differences in the spatial ranges of Gurken and
Spitz. An experimental test of this explanation requires measurement of the
relevant rate constants of Gurken and Spitz.
