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

5.4 The DuCa Model – Compartment Model and Equation System 23
for the change in the concentration of A, B, C (Othmer (2006)):
da
dt = k1 − k2a (5.5)
db
dt = k2a − k3b
dc
dt = k3b − k4c
dp
dt
= k4c
where we have assumed that we have an inexhaustible source and sink, and lower-case
letters denotes concentrations.
Using a compartment description of a system has the benefit that it provides an easy
overview of the molecular flow in the system, and thereby it facilitates a visual understanding
of how the differential equations interact.
Now that we have presented some fundamentals regarding compartment models we
present a slightly modified version of Marée et al. (2006)’s compartment model; see figure
5.2. In the original compartment diagram in Marée et al. (2006), the arrow pointing
from the compartment of macrophages toward the compartment of active macrophages
was associated with a rate constant called g. But in the Marée et al. (2006) article
they write that g = f1 several times so we have decided only to use f1 to avoid confusion
– this is our modification. Besides this figure 5.2 is a replication of the original
compartment diagram as presented on page 1277 in Marée et al. (2006).
5.4 The DuCa Model – Compartment Model and Equation System
As we have just learned the fully drawn arrows should signify an actual flow of molecules
from one compartment to another; e.g. resting macrophages become activated,
and leave the macrophage compartment, only to enter the compartment of activated
macrophages (at rate f1); figure 5.2. But as we will soon learn from equations 5.6 to
5.10 the fully drawn arrow from the Ma compartment to the cytokine compartment
does not obey this rule. The interpretation of the stippled arrows is also a little abstruse.
On one hand the stippled arrows seem to indicate that a stimuli is mediated to
the appropriate receptors/cells thus facilitating an increase/decrease in the molecular
concentration at hand; e.g. the activated macrophages do not themselves flow into
the surrounding tissue and reenter the macrophage compartment – rather they recruit
macrophages by signaling, signified by b. On the other hand the secretion of cytokines,
at rate α, is evidently not a signaling process, but Marée et al. (2006) also identify it
by a stippled arrow that meets the fully drawn arrow from the Ma-compartment which
we mentioned just before. We will get back to these sources of confusion in section
5.5, where we will present a modified compartment model that we have constructed
ourselves.
Just as there was a correspondence between figure 5.1 and equations 5.5 so Marée et al.