Quick Guide to Equations and Assumptions
Mouse Prostate Cross-Section
Our computational model couples a discrete, stochastic description of cells at the
macroscopic scale with a continuous, deterministic description of morphogen
diffusion at the microscopic scale. The diffusion region and cell positions were
defined using a 250 m x 400 m experimental image (Fig. 3A) of a crosssection of the mouse prostate.
Major Assumptions
Although the experimental system is 3-dimensional (3-D), our computational
model approximates a section of mouse prostate as a finite, rectangular 2-D
region. Cell division, cell apoptosis, and cell movement were not modeled. In
particular, the model is not explicitly time resolved as we consider the response
of cells to steady state morphogen concentrations.
Morphogen Production, Diffusion and Decay
The (x,y) positions of 44 stromal cells are point sources for diffusive signals M1
and M2. Tgfbr2-KO (altered) stromal cells produce M1 and Tgfbr2-WT (normal)
stromal cells produce M2. The kinetics of the morphogen concentrations are
described as uncoupled reaction diffusion:
Equation 1:
ns
m1
 k1  xi , yi   kd1m1  D12m1
t
i1
a
s
m2
 k2 xi ,yi   kd 2 m2  D2 2 m2
t
i 1
where SN xi , yi  i 1,2,, ns  and SA xi , yi  i 1,2,,as are the locations
of normal and altered stromal cells ( n s  a s  44 ), m1 and m2 are the
concentrations
(M), k1 and k2 are the production rates (s-1), kd1 and kd2 are the
decay rates (s-1), and D1 and D2 are the diffusion coefficients (m2/s) of
morphogen M1 and M2. The Dirac delta function (xi,yi) is defined as
 if x, y  xi , yi 
The initial conditions are given by m1(x,y,0) = 0
 xi , yi   
0
o
th
erwise

.
and m2(x,y,0) = 0. The boundary condition on the ducts and the outer rectangle
n1
n 2
 0 and
 0.
are given by
n
n
Major Assumptions
Stromal cells are point sources for morphogens and the production rates are
proportional to the concentration of Tgfbr2-KO cells and Tgfbr2-WT cells.
Diffusion occurs in a diffusive region between and surrounding the prostatic
ducts. Boundary conditions for the diffusive lattice area are reflective, so that
morphogen cannot diffuse beyond the duct boundary.
Morphogen Response
Epithelial cells respond to steady-state levels of morphogens M1 and M2 so we
consider m1 and m2 in their quasi-steady states, m1 and m2 . From Equation 1, we
solve:
Equation 2:
as
D1 m  kd1 m  k1  xi , yi 
2

1

1
i1
ns
D22m2  kd2 m2  k2 xi , yi 
i1
Denote normal, proliferative and cancerous (invasive) epithelial cells around the
ducts as EN, EA, and EC and their concentrations as en, ea, and ec (M). The
proposed underlying mechanism for epithelial transformation can be written:
Equation 3:
EN  M1* EA
(3a)
EA M2* EC
for quasi-steady morphogen levels M 1 and M 2 that are above threshold levels
TM1 and TM2 . On the other hand, for M 1 and M 2 below threshold levels:
EN  M1 EN
(3b)
EA M2 EA
Assuming EN, EA, and EC are immobile around the ducts, the kinetics of en, ea,
and ec can be written:
Equation 4:
den
 k N  H (m1  m1* )  m1  en
dt
dea
 k A  H (m2  m2* )  m2  ea  k N  H (m1  m1* )  m1  en
dt
dec
 k A  H (m2  m2* )  m2  ea
dt
where kN and kA are transformation rate constants of M1 and M2 to EN and EA.
0 x  0
The Heaviside function H (x) is defined as: H ( x)  
.
1 x  0
Major Assumptions
Epithelial cells respond to morphogen by transforming if they are co-local with
threshold steady state levels. This assumes that the changes of epithelial cell
types are much slower than the diffusion of morphogen M1 and M2 (i.e., the time
scale of en, ea, and ec dynamics are much slower than the time scale of m1 and
m2 dynamics). This is reasonable, as steady state morphogen concentrations
establish at time scales that are fast compared to cell responses (19, 20).
Correspondence of model and physical parameters. The computational
model generates steady state morphogen levels as a function of the diffusion
rates D1, D2, production rates k1, k2 and decay rates kd1, kd2 of each morphogen.
We equivalently describe these parameters in more physically useful terms as
the diffusion rates D1, D2, diffusion lengths L1, L2 and abundance A1, A2. The
diffusion length L and the morphogen abundance A are more useful than the
decay rate and the production rate they replace because they do not depend
explicitly on time.
L 
A 
4D
kd
The diffusion length L is defined from the diffusion rate D (m2/s)
and the decay rate kd (s-1). Physically, it corresponds to the
average distance diffused by a morphogen during its average
lifetime.
k
kd
The total abundance of each morphogen produced per source (per
cell) is defined from the production rate k (s-1) and the decay rate
kd (s-1). Physically, it corresponds to the average steady state
amount of morphogen contributed by a single morphogenproducing cell.
Parameters
In all simulations, the diffusion rate was fixed at D = 0.19 m2/s and in all
simulations where the production was not explicitly varied, we fixed the
abundance for both morphogens at A=10,000 morphogen units by varying the
production P as a function of the decay rate (A=P/kd). The diffusion length L for
each morphogen was a free parameter in our model.