Biomedical Engineering Reference
In-Depth Information
Now the space available to the second species is reduced to 1 -
P
1
-
P
2
, i.e.,
those patches occupied by
S
- 1 cannot be invaded. This simple model predicts
that coexistence will occur provided that some critical tradeoffs are satisfied.
Specifically, it can be shown that coexistence will take place if
c
1
>
m
1
,
[16]
c
2
> [
c
1
(
c
1
-
m
2
-
m
1
)]/
m
1
.
[17]
Essentially what these inequalities define is a set of requirements that the two
competitors have to verify in order to not exclude each other. The inferior colo-
nizer, for example, will be able to persist if it is able to exploit local resources
more efficiently. Since energy resources have to be distributed in different sur-
vival strategies, it is not difficult to understand that such tradeoffs will be ex-
pected to be common.
Of course, the previous models can seem too simplistic to say anything
relevant. But we have learned from years of modeling and data analysis that
simple models often capture the underlying causes responsible for the observed
patterns. A step beyond the previous metapopulation models is to consider space
explicitly (13,20). This was done within the context of cancer heterogeneity for
the spatial dynamics of the two genes mentioned above (TGFBR-2 and BAX). A
three-dimensional space was considered in terms of a cubic lattice (18). Each
lattice point was occupied by a single cell, whose internal state was defined in
terms of two copied of the two genes. The reproduction and death rates of each
cell were determined by the state of their two key genes and their mutations.
Mutations in TGFBR-2 triggered phenotypic responses in terms of increased
replication, whereas mutations in BAX led to decreased levels of apoptosis. Mu-
tations within this framework are independent random events not coupled with
environmental factors.
Available experimental data provided well-defined frequencies of each mu-
tation for each tumor analyzed. The simulation model, starting from a small
group of non-mutated cells, could produce a whole spectrum of possible out-
comes through its growth dynamics. In order to determine the most likely com-
bination of growth and death rates compatible with the available information, a
search algorithm was used in order to find the optimal parameter set compatible
with experimental data. In this way, we can go beyond the limitations imposed
by a high-dimensional parameter space and simply leave the algorithm to search
for candidate solutions.
The model revealed a very good parameter combination able to reproduce
the observed frequencies of mutants and their spatial distribution (18). In Figure
4 we show an example of the model outcome for the optimal parameter combi-
nation. As expected from the competition models described in the previous sec-
tions, spatial heterogeneity was achieved because of a combination of tradeoffs
