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Table 1
Functions assigned to continuous transitions in the simulation of
apoptosis induced by Fas ligand, where mA and mB represent con-
tents of the corresponding continuous places
Rate
Unimolecular reaction
Bimolecular reaction
Self-effacement
MA/200
Oligomer
mA/20
mA * mB/10000
Monomer
mA/10
mA * mB/5000
Enzyme binding
mA/5
mA * mB/2500
Enzyme reaction
MA * 10
steps where two different pathways from caspase 8 are assumed and many molecules including Fas
receptors, caspase family which includes aspartic acid-dependent cysteine proteases and produced from
their zymogens, Bcl-2 family which includes pro- and anti-apoptotic proteins, cytochrome c and DNA
fragmentation factor. The apoptosis starts from the Fas ligand binding to Fas receptors and ends in the
fragmentation of genomic DNA, which is used as a hallmark of apoptosis. Thus the amount of DNA
fragmentation can be assumed to be proportional to the cell death.
We have designed an HFPN by using the facts about the Fas-induced apoptosis pathways shown in
Fig. 9 and biochemical knowledge about reactions. Figure 10 shows the whole HFPN model that we have
described with GON. All places/transitions are continuous and parameters are roughly tuned by hand. For
Bid ( m 11) , procaspase-9 ( m 21) , procaspase-3 ( m 25) , DFF ( m 30) ,DNA( m 37) , the initial concentration
of each compound is assumed to be 100. On the other hand, for FADD ( m 4) , procaspase-8 ( m 5) , Apaf-1
( m 17) , dATP/ADP ( m 18) , when two compounds react together without the stimulation of apoptosis,
the initial concentrations and the rate are assumed to be 39.039 and m 1
m 2 /5000, respectively to keep
the stable state condition. Each compound is assumed to be produced by the rate of 0.5 (represented
by a transition without any incoming arc) and to degrade by the rate of its concentration divided by 200
(represented by a transition without any outgoing arc), which will keep its concentration at 100 under the
stable state condition. This degradation rate also applies to other compounds in the network. The rate
of other processes are determined roughly by following Table 1. Synthesis and catabolism processes are
added in the model for all proteins. Autocatalytic processes are also added in the model to all caspases
since they exist as proenzymes. The pathway from caspase 8 to caspase 3 is assumed when the caspase 8
concentration is over 30. Protease is often synthesized as a proenzyme (zymogen) and changed to active
form by other enzymes or by itself. So an autocatalytic process is added to every caspase reaction.
By using the apoptosis scheme modeled as an HFPN, we simulated the DNA fragmentation amount
by varying the Fas ligand concentration; Figure 11 shows the simulated relationship. It shows that under
very weak stimulation (very low amount of Fas ligand), DNA fragmentation does not occur since the
stimulation stops at the intermediate point because of the assumption of degradation processes. With the
increase of the stimulation, the reaction proceeds to the backward intermediates and DNA fragmentation
(cell death) occurs finally, which increases with the increase of the Fas ligand concentration.
There are two pathways from activated caspase 8 to caspase 3, one through several steps including the
cytochrome c release from mitochondria when the concentration of activated caspase 8 is low, and the
direct one to caspase 3 when the concentration of activated caspase 8 is high [20]. We assume arbitrarily
that the direct pathway starts when the concentration of activated caspase 8 is larger than 30. Reportedly
the removal of Bid by gene knockout method increases the resistance of liver cell apoptosis by Fas ligand,
while it does not affect the apoptosis of thymus and embryonic cells. If the second pathway is included
to the scheme, DNA fragmentation increases slightly, especially when the Fas ligand concentration is
high (Fig. 11). However the detailed mechanism of the selection of these two pathways from caspase 8
is still unclear and necessary to be studied in future in the laboratory.
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