Biomedical Engineering Reference
In-Depth Information
previous section to analyze the operation of the Rhomboid/Spitz feedback in a
multicellular system such as an epithelial layer. The resulting description is use-
ful in analyzing the effects of exogenous signals presented to the epithelial layer.
For example, in the case of oogenesis, it is important to characterize the domain
affected by Gurken and Gurken-induced EGFR ligands (4,11,45).
To understand the operation of the Rhomboid/Spitz circuit we started to
develop models of autocrine signaling in epithelial layers (25). In addition to
ligand transport, binding, and internalization, these models account for Rhom-
boid induction and Rhomboid-mediated Spitz release. Rhomboid induction was
modeled as a threshold-like function, T, of the total number of ligand-receptor
complexes on the cell surface. The balance for the level of Rhomboid in the cell
(
i
,
j
),
P
i,j
, takes the following form:
dP
P
(
)
ij
,
ij
,
tot
,
=
+
T
CC
,
[3]
ij T
dt
U
where
tot
i
C
is the total number of occupied EGF receptors in the cell (
i
,
j
) and
C
T
is the threshold-value for Rhomboid induction. The time-scale for Rhomboid
degradation, U ~ 20 minutes, can be estimated from the experiments in the em-
bryo (49).
Receptor occupancy on any given cell within the epithelial layer depends on
the pattern of ligand release, and hence the pattern of Rhomboid expression in
the entire layer. Our analysis suggests that ligand binding and transport rapidly
adjust to the much slower dynamics of Rhomboid expression. In other words,
the equations for ligand binding and transport reach the steady state dictated by
the pattern of Rhomboid across the epithelial layer. In the ligand-limited regime,
receptor occupancy for a given cell is computed from the linear superposition of
ligand fields due to individual cells (Figure 7B). As a result, the dynamics of
cells coupled by secreted signals can be described entirely in terms of the intra-
cellular variables:
¬
-
dP
P
-
ij
,
ij
,
=
+
T
I PC
-
,
[4]
-
imjn
,
mn
,
T
dt
U
-
-
-
®
mn
,
where
I
|i-m|,|j-n|
is the cell-cell coupling coefficient that quantifies the strength and
the spatial range of autocrine and paracrine signals. Importantly, these coeffi-
cients were derived as a function of the biophysical parameters of the problem,
such as the diffusion and binding rates, as well as the rates and levels of ligand
release by single cells within the layer. We found that the coupling coefficients
decay rapidly as a function of cell-cell distance. This suggests that only a small
number of cell-cell interactions must be taken into account in calculating recep-
