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

5.6 Our Compartment Model 29
migrate to the cytokine compartment, i.e. become cytokines. This is naturally not the
case, so this is another issue we must address in the next section, where we present our
own version of compartment model.
5.6 Our Compartment Model
Now that we have touched upon what we perceive as inconsistencies in the original
model (figure 5.2), we will devise our own compartment model. But before we present
our version, we will specify how we use the fully drawn and stippled arrows, since
ambiguous use of the arrows was one of the major problems in the original model.
• Fully Drawn Arrow: is to be used only when there is an actual flow of molecules.
This means that if a fully drawn arrow points into a compartment, then the
differential equation that describes the rate of change in concentration of the
molecule in that compartment will contain a term that constitutes an inflow; e.g.
a in equation 5.6 or f1MBa in equation 5.7. If a fully drawn arrow points out of
a compartment, then the corresponding differential equation must contain a term
that signifies this outflow; e.g. −dBa in equation 5.8.
• Stippled Arrow: if a stippled arrow points into a compartment, it yields a positive
contribution in the corresponding differential equation, but without causing an
identical loss in the differential equation associated with the compartment the
arrow came from; e.g. αMaBn in equation 5.10 and AmaxC/(kc+C) in equation 5.8.
Now that these definitions are in place let us look at our compartment models. Yes,
we wrote models in plural, because we have found it necessary to divide the model
into two figures. We have split the model into a part that contains M, Ma and Ba
compartments which we call the non-necrotic part, and a part that contains M, Ma,
Bn and C compartments, which we call the necrotic or Bn-part. Had we not done so
arrows would be crossing each other, which does not facilitate the transparency of the
model. In figure 5.4 we have the non-necrotic part of our modified compartment model.
While figure 5.5 is the Bn-part of the compartment model. The compartment diagrams
are made in agreement with our definitions of full-drawn, and stippled arrows.
One feature of figure 5.4 that may seem a bit confusing is the stippled arrow that
leaves the Ma compartment to coincide with the fully drawn arrow leaving the Ba
compartment. This is done so because the apoptotic β-cells are phagocytized by the
active macrophages, which implies that the β-cells leave their compartment, hence
a full-drawn arrow. The active macrophages on the other hand do not leave their
compartment during phagocytosis (they remain activated), and they do not go to the
Ba compartment either, hence the stippled arrow from the Ma compartment points to
the fully drawn arrow.
Another feature that needs to be commented on is the stippled arrow that coincides
with the apoptosis arrow to the left of the Ba compartment. This arrow denotes the
cytokine induced apoptosis which appears as gain in equation 5.8. In figure 5.5 the same
arrow leaves the cytokine compartment. Another arrow that transcends the figures is
the arrow that represents the non-specific decay of β-cells – those that become necrotic.
This arrow enters the Bn compartment in figure 5.5, and thus entails a loss in equation
5.8 and a gain in equation 5.9.