Biomedical Engineering Reference
In-Depth Information
dP
/
dt
=
CP
(1 -
P
) -
mP
,
[12]
where
c
is the colonization rate and
m
the extinction rate. Both are local rates
and describe the ability of the given species to occupy neighboring patches and
to get locally extinct. The model has two equilibrium points:
p
*
= 0 and
P
*
= 1 -
m
/
c
.
[13]
It can be seen from the last expression that the species will persist provided that
c
>
m
. In other words, there is a minimum requirement for colonization rates in
relation to extinction rates. The choice of strategy for a population will in part
depend on the environmental constraints and the contingencies that may influ-
ence these constraints.
In cancer both strategies can be adopted, leading to coexistence of diverse
cell populations (18). The Levins model enables us to explore what type of
tradeoffs between competing populations allow them to coexist. Let us consider
two competitors, which for simplicity we will assume to be ordered in a hierar-
chical way. The superior competitor will be more likely to colonize available
adjacent patches, and the second competitor will be less likely to die.
We can map these two strategies in the cancer context by choosing two
genes in epithelial cells: one coding for the receptor of tumor growth factor beta
(TGFBR-2), and the second encoding a pro-apoptotic protein named BAX.
These two genes will allow us to couple phenotypic traits associated with repli-
cation and senescence with the underlying genetic traits. Loss of function of
TGFBR-2 makes cells grow faster, while loss of BAX function increases cell
longevity. In terms of a model approach, this means that more mutations in
TGFBR-2 will map into increased proliferation rates, whereas mutations in
BAX will reduce cell mortality. Homozygous mutants will presumably have a
stronger phenotype, although the biological consequences of haploinsufficiency
in tumor suppressor genes are only emerging (18,24,25) and have not yet been
explored in much detail.
These constraints can be implemented as follows according to Levins dy-
namics:
dP
1
/
dt
=
c
1
P
1
(1 -
P
1
) -
m
1
P
1
.
[14]
From this equation we conclude that the first population senses as available
habitat all spaces not occupied by
S
- 1, but perhaps occupied by
S
- 2. The sec-
ond population is described by:
dP
2
/
dt
=
c
2
P
1
(1 -
P
1
-
P
2
) -
m
2
P
2
-
c
1
P
1
P
2
.
[15]
