Deciphering Fate Decision in Normal and Cancer Stem Cells: Mathematical Models and Their Experimental Verification - Mathematical Methods and Models in Biomedicine - page 212

and limits of the sigmoid) are denoted by subscripts ( f F , f S , etc.). Increasing and
decreasing functions are denoted by f ↑ and f ↓ , respectively.
In Eq. ( 7 ), W t is the total expression level of Wnt proteins in the environment
of the considered cell, calculated as W t =

1
2 · (

W 6

,where W is Wnt produced
by the particular cell, and W n is the sum of Wnt produced by all of the cells in the
adjacent environment of that cell (maximum of six neighbors, due to the CA grid
structure). D t , representing the total expression level of Dkk1 in the environment of
the cell, is calculated in a similar way, while D is the Dkk1 produced by the cell
itself.
The parameters

W + )

are the constant degradation rates for each different protein,
respectively, denoted by subscript (

μ

μ L ,etc.).
The number of bound E-cadherins in the cell, E b (Eq. ( 9 )), is the sum of the num-
ber of E-cadherins bound to any adjacent cell: E b (

μ D ,

6
i

t )= ∑

1 E b , i (

t ) ,where E b , i ( t ) is

=

the number of E-cadherins bound to the neighboring cell in the i th direction. E b , i ( )

is dependent (Eq. ( 11 )) on the level of E-cadherins in the considered cell E ,inthe
considered neighboring cell E i , and on the E-cadherin binding coefficient k b .
p N (Eq. ( 13 )) is the Notch receptor synthesis rate.
N r (Eq. ( 14 )) is the level of Notch receptor ready to be activated, which is
dependent on the number of Notch receptors in the cell ( N ) and also on the
level of DSL in the microenvironment, in the following way: N r =

t

min ( N , N l ) ,

1
6 · ∑

6
i

N l =

1 N l , i ,where N l , i is the DSL level of the neighboring cell being in i th
direction from the considered cell and N l is the total level of DSL directed to the cell.

=

6.2 Hybrid Cellular Automata Multi-scale Model

The new tissue model [ 5 ], formed by implementation of this ODE model into the CA
model described above, is considered a hybrid cellular automata (HCA) model since
it contains both continuous protein activities and discrete cellular developmental and
spatial states. This multi-scale model can be used to study consequences of specific
intracellular changes on the structure of the tissue.
In order to analyze the molecular proliferation-differentiation regulation
mechanism, the ODE system describing the signaling pathways was slightly
simplified, and stability analysis was conducted, concluding that the system has
a unique equilibrium point (i.e., stable values for all variables, in particular P and
M ), which is locally asymptotically stable [ 44 ]. Simulations showed that the system
converges to an equilibrium point under a wide range of biologically relevant values
of parameters and initial conditions.
The authors studied how the steady-state values for P and M , which reflect the
SC's tendency to proliferate or differentiate, are dependent on the microenviron-
mental conditions [ 44 ]. This was performed by examining the system's response to
changes in the various external signals, e.g., Dkk1 level, Wnt level, and the level
of DSL receptors on the neighboring cells. Results of this analysis showed that
the steady-state values for P and M depend on the level of local cell density. Under