Biomedical Engineering Reference
In-Depth Information
12.5
Discussion
The degradation rate
(
M
of the exogenous depolymerization model is the ratio
of the total weight of
M
-molecules degraded per unit time. It also represents the
ratio of the number of
M
-molecules that undergo exogenous depolymerization
processes. It might be assumed to be independent of the molecular size
M
, for
the exogenous depolymerization processes take place only at the terminals of the
molecules. In practice, however, this is the case for molecules of moderate sizes.
As metabolic enzymes are located in cell membranes, they take effect in peri-
plasms, and at least one end of a molecule must penetrate through its outer
membrane to become subject to an exogenous depolymerization process [7].
In the presence of suffi cient number of microorganism, the number ratio of
molecules in contact with microorganisms in a fi xed period of time can be assumed
to be independent of the molecular weight. When a molecule makes contact with
a microorganism, a part of a fi xed length should be taken into a cell in a fi xed period.
If that part happens to contain one of the terminals, the enzyme takes effect and an
exogenous depolymerization process starts. The possibility for a part of a fi xed
length to contain a terminal becomes less when the molecule becomes large. This
is the reason why the oxidation rate is a decreasing function of the molecular
weight. The result shown in Figure 12.5. clearly indicates the dependence of the
rate of membrane transport with respect to the molecular weight, and our mathe-
matical analysis has revealed the role of membrane transport in exogenous depo-
lymerization processes, which is unpredictable from experimental results alone.
The exogenous depolymerization model originally developed for PE biodegrada-
tion has been applied successfully to PEG biodegradation. The numerical results
show how molecules are incorporated into cells in exogenous depolymerization
processes. The validity of the result concerning the oxidation rate has been con-
fi rmed by the numerical simulation (Figures 12.7 and 12.8). This is a typical
microbial depolymerization process of exogenous type, where monomer units are
split from the terminals of molecules.
The only factor assumed in construction of the endogenous depolymerization
model was random separation of molecules. There are other factors of weight
changes in processes through the metabolic pathways of PVA as was described,
but those changes should be negligible when compared to the weight shift due to
random separation of molecules due to enzymatic degradation. As a mathematical
model for enzymatic degradation of PVA, a linear second-order hyperbolic partial
differential has been derived from the original model. Given a prescribed function
that represents the degradation rate and an initial condition, it forms an initial
value problem of a linear partial differential equation. On the other hand, given
the initial condition and an additional fi nal condition, it forms an inverse problem
to determine the degradation rate for which the solution of the initial value
problem also satisfi es the fi nal condition.
It is shown that the inverse problem can be reduced to a nonlinear ordinary
differential equation whose unknown variable represents the weight fl ux into the
λ
