nr. 477 - 2011 - Institut for Natur, Systemer og Modeller (NSM)

100 Discussion of the Bifurcation Analysis
The bifurcation diagrams compiled from Balb/c parameters (figures 8.11 and 8.15)
reveal that, for the estimated Balb/c-values of f1 and f2, there is only one fixed point,
namely the HRS. In essence the macrophages of the Balb/c-mice should be able to
overcome any apoptotic wave! This may seem like ludicrous since we could just imagine
a wave that would wipe out all of the β-cells thus inducing a very severe case of T1D.
However we must remember that such a wave would (probably) never occur, and we
should always judge a model based on what it is intended to model, hence, not based
on some extreme event that is science-fiction rather than science.
If we are to learn anything from the Balb/c-diagrams, it must be how low the two
phagocytosis rates should be (from birth) if chronic inflammation were to appear in
Balb/c-mice. Again the phagocytosis rates of the resting macrophages is the one that
needs to be modified the least to induce this detrimental change. The difference between
chronic inflammation or not appears at f1 ≈ 1.2 × 10 −5 when f2 is kept at 5 × 10 −5 . 3
An interesting question to raise is: suppose we were mad scientists, and we had
found a way of engineering the phagocytosis rates to be higher or lower at birth as we
saw fit. Would it then be easier to induce NOD-like behavior in a Balb/c-mouse or
could we sooner save an NOD-mouse from the dire future of it’s unengineered peers?
The bifurcation diagram that reveals the smallest margin between NOD and Balb/c
behavior is the one compiled using the Balb/c-value of f2 with f1 as the bifurcation
parameter (figure 8.11). Here the Hopf bifurcation takes place at f1 = 1.21 × 10 −5
which is 0.79 × 10 −5 from the estimated value of f1 = 2 × 10 −5 . If we take f1-values
from other articles into account we find that we are close to loosing our job as mad
scientists. In the 2007 article of Marée et al. (Marée et al. (2007)) they give estimates
f1 = 1.31 ± 0.05 × 10 −5 and f1 = 1.29 ± 0.04 × 10 −5 for their reversible and irreversible
models respectively; cf. table 5.3 and the subsection titled “Reversible or irreversible
activation?” in section 5.8. Even with these intervals of deviation we are still a little
above the bifurcation value, but we must remember that the deviations of ±0.05 and
±0.04 are based on several measurements, so it is not entirely impossible that some of
the Balb/c-macrophages had a phagocytosis rate that was ≤ 1.21. 4 Furthermore there
is an uncertainty associated with the step size of f1 or f2 when locating the bifurcation
value which could shift the result either way.
All in all the DuCa model corresponds well its underlying biological system and it
fully works on the premises that were made with it, and it also provides an answer to
the hypothesis that is was constructed to investigate. An answer that does not change
if the phagocytosis rates are changed within a sensible range. Naturally you can always
criticize the choice of assumptions and/or simplifications in a mathematical model, but,
without repeating our thoughts from chapter 4 we will state that, fact of the matter is
that these are necessary means if you want to be able to model anything.
3 Interestingly enough this is very close to the value at which chronic inflammation occurs upon
tweaking f2.
4 This is also in agreement with the fact that some Balb/c mice develop diabetes.