Deciphering Fate Decision in Normal and Cancer Stem Cells: Mathematical Models and Their Experimental Verification - Mathematical Methods and Models in Biomedicine - page 213

high local density, the high E-cadherin signal is dominant and causes differentiation.
Under lower cell density, Wnt and Dkk1 signal intensities are dominant, and SCs
proliferate at a rate that is dependent on the Dkk1 signal intensity. As will be
explained later, the Dkk1 signal reflects the feedback regulation of the SC proportion
in the population. Low cell density is generally characterized by a high proliferation
rate; however, under extremely low cell density, low Notch signal leads to SC
differentiation.
In addition, numerical simulations of the HCA model dynamics have been carried
out. The CA honeycomb grid was initially seeded with randomly placed cells.
To provide a stable model, parameter values for normal SCs were chosen in a range
that promises tissue survival to confluence. These parameters were estimated to fit
real characteristics of mammary SCs, based on relevant literature (for details see
[ 5 ]). Then, for every time step, intracellular dynamics for all the cells were simulated
by calculation of per-cell expression levels of all modeled proteins. Accordingly,
cell fate was determined for each of the automata cells. This way, the effects of
changes in specific protein concentrations, e.g., Dkk1, on the tissue dynamics, could
be explored.
Simulations were also used to examine possible effects of defects in signaling
pathways on SC proliferation-differentiation balance. Parameters were changed
such that Notch receptor synthesis, Wnt ligand expression, or E-cadherin concentra-
tion required for LEF/TCF activity inhibition would increase by 5-20 %. The model
was re-simulated, and results were compared to the control simulation result with
“normal” parameter values.

6.3 Dkk1 as a Key Regulating Factor for Fate Decision

Regulation

Mathematical analysis of the model [ 44 ] showed that Dkk1 is a key, biologically
plausible factor in fate decision regulation. The protein Dkk1 is secreted by SCs
into the microenvironment and hence may serve as a potential QS modulator, as it
can indicate the number of SCs in the close neighborhood. The model predicts that
above a specific level, Dkk1 reduces proliferation, thus increasing differentiation.
The numerical simulation results suggest that the Dkk1 effect is biphasic. Below
a critical concentration Dkk1 will not affect, and may even somewhat increase,
the proportion of SCs in the population. Above this threshold, increasing Dkk1
concentration leads to a significant decrease in the numbers of both proliferating
and quiescent SCs, as a result of differentiation.
Simulating dose effects of Dkk1 with changed model parameters, representing
increasingly activated pathways due to mutations, did not change the qualitative
dependence of SC proportion on Dkk1. However, the critical Dkk1 concentration
under which proliferating SCs switch to differentiation depends on the pathway
activity, as affected by the specific mutation. This implies that application of