Supplementary Methods
1.
PDE Model
Denote the normal and altered stromal cells as N and A. We assume that they are produced from
point sources in a certain locations   {( x i , y i ) i  1,2,L 20} and A  {( xi, yi) i  1,2,L 20} .
Also, let us M1 and M2 be morphogen I and II and m1 and m2 be their concentrations (M). Because M1
and M2 are produced by the altered and normal stromal cells, the production rate of morphogen I and
II (dm1=dt and dm2=dt) will be proportional to the concentrations of altered cells (A(x, y)) and
normal cells (N(x; y)). Also, 
since M1 and M2 diffuse freely in the
inter-ductal space, the kinetics of m1
and m2 can be described as uncoupled reaction diffusion type model:
20
m1
 k1  (xi, yi)  k d1m1  D1 2 m1
t
i1
(1)
20
m 2
 k 2  (x i, y i)  k d 2 m2  D2 2 m2
t
i1

where k1 and k2 are production rates, kd1 and kd2 are decay rates, and D1 and D2 are diffusion
coefficients of morphogen I and II. The Dirac delta function (xi;yi) is defined as

1 if (x, y)  (xi, yi)
otherwise
0
(xi,yi) (x, y)  
The initial conditions for Eq. 1 are given by
m1(0) = 0
m2(0) = 0,

and the boundary condition on the ducts and the outer rectangle are given by
m1
0
n
m 2
0
n
Now, let us call normal, altered and cancerous epithelial cells as N, A, and C around the ducts
and their concentrations as n, a, and c (M). Let us assume that the changes of epithelial cell types
are much slower than the diffusionof morphogen I and II (Time scale of n, a, and c is much slower
than the time scale of m1 and m2). At the time scales of n, a, and c, we may regard that m1 and m2 are




in their quasi-steady states, m1 and m 2 . For m1 and m 2 , we solve the Poisson equation type steady
state equations for 1,

20
D1 m kd1m  k1 (xi,yi)
2

1

1
i1

(2)
20
D2 2 m2  kd 2 m2  k 2 (x i,y i)
i1
Next, the proposed underlying mechanism can be written as

Non
k

NM
A


kNoff

*
1
Aon
k

A M
C


kAoff
*
2

k2

(3)
VII 
k1 = 500, k2 = 1
IV
k1 = 100, k2 = 1
I
k1 = 1, k2 = 1
VIII
k1 = 500, k2 = 100
IV
k1 = 100, k2 = 100
I
k1 = 1, k2 = 100
IX
k1 = 500, k2 = 500
IV
k1 = 100, k2 = 500
I
k1 = 1, k2 = 500
k1
Table 1: Parameter Regimes used in the simulation
*
*
where M1 and M 2 are morphogen
 I and II the ductal boundaries from quasi-steady states of Eq. 1.
Assuming N, A, and C are immobile around the ducts, the kinetics of n, a, and c can be written
as

dn
 k Non m1  n  kNoff a
dt
da
 kAon m2  a  kAof fc  kNon m1  n  kNoff a
dt
dc
 kAon m2  a  kAoff c
dt
*
(4)
*
where kNon and kAon are binding rate constants of M1 and M 2 to N and A. Because the total
concentration of cells is constant (n+a+c = C0), we may drop one of n, a, and c equations. Here, we

drop equation
for c.
2

Steady State Analysis
Let us consider following regimes of k1 and k2 (the production rates of morphogen I, II) as described
in the following table:
In these regimes, the steady states of Eq. 1 given by Eq. 2 were numerically computed by the
finite element methods and shown in Figure 1 1.


Given the steady state solutions, m1 (x, ) and m 2 (x, y) where (x, y) is at the circumference
of ducts, the densities of normal, altered and cancerous epithelial cells are dertermined by the steady

states of Eq. 4 as
 k

C0  Noff  
k Non m1 
n 
k m    k

1  Aon 2   Noff  
 k Aoff  k Non m1 
C0
a 
k m    k

1  Aon 2   Noff  
 k Aoff  k Non m1 
k m  
C0  Aon 2 
 k Aoff 
c 
k m    k Noff 
1  Aon 2  
 
 k Aoff  k Non m1 
Let kNon=kNoff = kN = 1, and kAon=kAoff = kA = 1 for simplisty then


n 
C0
1 m  m1 m2
a 
C0 m1
1 m1  m1 m2
c 
C0 m1 m2
1 m1  m1 m2

1
(5)
Figure 1: Steady state of morphogen I, II
For simulation, the locations of 20 normal and 20 altered stromal cells are arbituary chosens. I–IX
correspond to the parameter values shown in the Table 1.
Figure 2: Steady state distribution of normal, altered and cancerous epithelial cells in the
parametric regime I, II and III


From the m1 (x, y) and m2 (x, y) determined from the numerical solution to Eq. 2, the
steady state distribution of normal, altered and cancerous epithelial cells are dertermined by Eq. 5
and numerical solutions are shown in Figure 2~4.
If we assume that cancer cells are invasive if their local density is over 10% of total cell
density,putting Figure 2~4 together, we have following phase diagram
k 2*
Normal
+ Cancer
Normal
+ Cancer
Normal

Normal
+ Altered + Cancer
Normal
+ Altered + Cancer
Normal
+ Altered
Normal
+ Altered + Cancer
Normal
+ Altered + Cancer
Normal
+ Altered
k1*
3.
Numerical Scheme

First we solve Eq. 2 by finite element method. If we let the inter-ductal area as , then weak form of
Eq. 2 are
Figure 3: Steady state distribution of normal, altered and cancerous epithelial cells in the parametric
regime
Figure 4: Steady state distribution of normal, altered and cancerous epithelial cells in the parametric
regime

20

  (D1m1 ) j  kd1m1 dx  k1  (xi ,yi ) j dx

i1
   (D m
1


2
20
) j  kd1m dx  k1  (x i ,y i ) j dx

2
i1
for some basis functions (tent function) associated with the triangulation of . From the choice of the
basis functions, we have



 j dx   j (x i, y i )  (x ,y )
 (x i ,y i )
i
i
By integrations by parts


(D1m1 )   j  kd1m1 dx 
 (D m

2

2
)   j  k d 2 m dx 
where n is outer normal vectors on .


2
20

n  (m1 ) j ds  k1 (x i ,y i )

n  (m ) j ds  k2 (x i ,y i )


i1

2
20
i1
Considering the representation of the solution by a linear combination of the basis functions
(linear interpolation of the solutions) as
n
m1  U k  k
k1
n
m  Vk  k

2
k1
then we obtain an Eq 2–equivalent linear system in (Uk; Vk) as
n
 (D  )    k

k
n
1
k
j

k
 (D2k )   j  kd 2k dx 
With the solutions



n  ( k ) j ds U k  k1 (x i ,y i )

i1
20
 n  (m2 ) j ds Vk  k2 (xi ,yi )
i1
m1   U kk and m2   Vkk on the boundaries of ducts, we can find
n
n
k1
k1
steady state solution for Eq. 3 by 4.


 dx 
d1 k
20