Deciphering Fate Decision in Normal and Cancer Stem Cells: Mathematical Models and Their Experimental Verification - Mathematical Methods and Models in Biomedicine - page 200

5 General Description of Stem Cell Dynamics in Tissue:

A Discrete Model

5.1 A General Cellular Automaton Tissue Model

The first model by Agur et al. was a general model describing tissues with
hierarchical (SC-based) structures [ 4 ]. This model formed the basis for all SC
models that followed, and its aim was to describe the simplest possible system cap-
turing the essential properties of developing tissues which is capable of retrieving
homeostasis in living systems.
The model is a simple, discrete dynamical system that can represent any tissue
containing SCs. As the replication-differentiation balance in SCs is essential
for maintenance of tissue homeostasis, the model assumes that replication and
differentiation decisions are regulated by feedback regarding the condition of the
tissue as a whole. Specifically, an SC's fate is assumed to be determined by
feedback it receives from neighboring cell populations (referred to as quorum
sensing , QS). The SC “reads” and responds to signals from other SCs in its
local microenvironment. Thus, QS is the fate decision mechanism controlling the
SC replication-differentiation balance. The QS mechanism exists among Gram-
negative bacteria, e.g., Vibrio harveyi and Vibrio cholera [ 10 , 54 ]. In these bacteria,
gene expression is regulated through the monitoring of population density, using
diffusible molecules for communication.
To be able to take cell-cell feedback interactions into account, without assuming
spatial homogeneity of the environmental signals, Agur et al. [ 4 ]useda cellular
automata (CA) model, in which the behavior of each individual cell is tracked.
In CA models, cells are discrete sites on a lattice. Time is also discretized, and at
every time step, the state of each cell is defined by fixed rules. The rules can be
deterministic or include stochasticity and probability distributions, but they must be
determined by local conditions at the site of the specific cell.
The basic conceptual model includes the minimum of details necessary to
represent a normally functioning tissue, as can be seen in the scheme in Fig. 2 .
Tissue cells are represented by three types of automata cells: stem (S), differentiated
(D), and null (N) cells, the latter representing vacant space in the tissue. An SC
can either replicate, generating new SCs, or differentiate and become a DC. A DC
is assumed to live in the system for a certain maturation time, and then die or
migrate from the tissue, leaving an unoccupied space (N cell). This N cell may
eventually become occupied by a new SC, created via a proliferation process (i.e.,
when a neighboring SC replicates). A DC in the model represents an entire cell
line of progenitors and differentiated cells before they die or migrate from the
tissue, generalized in the model through the DC life span. An SC's “decision”
to differentiate or proliferate depends on the number of SCs and N cells in its
neighborhood, respectively. This dependency represents the effects of a variety of
secreted cytokines in the cell's microenvironment, enabling the cell to sense which
types of cells are in its proximity.