  Rapidly dividing tissues, like intestinal crypts, are frequently chosen to
investigate the process of tumor initiation, because of their high rate of
mutations. To study the interplay between normal and mutant as well as immortal
cells in the human colon or intestinal crypt, we developed a 4-compartmental
stochastic model for cell dynamics based on current discoveries. Recent studies
of the intestinal crypt have revealed the existence of two stem cell groups.
Therefore, our model incorporates two stem cell groups (central stem cells
(CeSCs) and border stem cells (BSCs)), plus one compartment for transit
amplifying (TA) cells and one compartment of fully differentiated (FD) cells.
However, it can be easily modified to have only one stem cell group. We find
that the worst-case scenario occurs when CeSCs are mutated, or an immortal cell
arises in the TA or FD compartments. The probability that the progeny of a
single advantageous CeSC mutant will take over the entire crypt is more than
$0.2$, and one immortal cell always causes all FD cells to become
immortals.Moreover, when CeSCs are either mutants or wild-type (w.t.)
individuals, their progeny will take over the entire crypt in less than 100
days if there is no immortal cell. Unexpectedly, if the CeSCs are wild-type,
then non-immortal mutants with higher fitness are washed out faster than those
with lower fitness. Therefore, we suggest one potential treatment for colon
cancer might be replacing or altering the CeSCs with the normal stem cells.
