Processes 2019, 7, 163
Figure 6. Frequency of occurrence of oscillatory dynamics in randomly-generated networks with a
1:1.5, number of species to number of reactions ratio, binned by number of floating species in the
network. (Top row) Frequency of oscillatory dynamics, including sustained and damped systems;
(Bottom row) Frequency of sustained oscillatory dynamics.
3.3. Components of Randomly-Generated Networks Responsible for Oscillation
Next, a population of 10,000 randomly-generated 10 species, 10 reaction networks were generated,
allowing for the existence of orphaned species, which are disconnected from the network. In addition,
42.6% of these networks had at least one orphaned species, and the frequency of oscillatory dynamics
was greatest in the subset of networks containing orphaned species. Figures 7 and 8 show two
randomly-generated networks which were pulled from the population to study the components of
the network topology, or species connectivity, responsible for oscillatory dynamics. Both networks
have nine total species that participate in reactions, indicating that one of the species was orphaned,
and therefore does not contribute to the network dynamics. The reduced forms of these networks,
which maintain oscillatory behavior, were generated by removing unnecessary and redundant nodes
or reactions, and compounding constant parameters within the rate law. These reduced forms show
the species and reactions responsible for oscillatory dynamics.
The reduced network in Figure 7 shows feedback loops that contribute to oscillatory behavior.
The first pathway of interest is the sequence of unimolecular–unimolecular reactions from species
S1 to S6, with two intermediate nodes, S3 and S5. This pathway shows that, as the concentration of
species S1 increases or decreases, there is an impact on species S6 in the same direction of change,
accompanied with a time delay in the signal due to the intermediate nodes. Continuing the cycle,
an increase in species S6 contributes to a rise in species S2 and subsequent production of S5, feeding
back positively into the pathway that produces species S6 and sustaining the cycle. However, negative
feedback of S6 on S1 is key for enabling oscillations. While species S6 increases, the bi-molecular
reaction between S6 and S1 which produces S2 simultaneously reduces the concentration of species S1
available for the uni-molecular reaction producing species S3. Together, the feedback and time delay
promote oscillation. This cycle repeats without dampening given appropriate global parameter values.
The reduced network in Figure 8 also contains feedback loops, involving the reversible reactions
between species S2, S9,and S4, as well as the reactions between species S2, S7 and S3. The optimized
model drives the conversions of S2 to S9 and S9 to S4 at a fast rate, while the conversion of S4 to S9
occurs much more slowly, as determined by the rate constants for these reactions. Since species S9
feeds back to S4 when generating S2, negative feedback arises. It is important to note that reactions
involving S8 effectively promote a time delay due to the involvement of an intermediate node in the
production of S2, which impacts the phase of the oscillations. However, removing species S8 allows for
a simplified reversible reaction between S4 and S9 in which the time delay created by the intermediate
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Figure 6. Frequency of occurrence of oscillatory dynamics in randomly-generated networks with a
1:1.5, number of species to number of reactions ratio, binned by number of floating species in the
network. (Top row) Frequency of oscillatory dynamics, including sustained and damped systems;
(Bottom row) Frequency of sustained oscillatory dynamics.
3.3. Components of Randomly-Generated Networks Responsible for Oscillation
Next, a population of 10,000 randomly-generated 10 species, 10 reaction networks were generated,
allowing for the existence of orphaned species, which are disconnected from the network. In addition,
42.6% of these networks had at least one orphaned species, and the frequency of oscillatory dynamics
was greatest in the subset of networks containing orphaned species. Figures 7 and 8 show two
randomly-generated networks which were pulled from the population to study the components of
the network topology, or species connectivity, responsible for oscillatory dynamics. Both networks
have nine total species that participate in reactions, indicating that one of the species was orphaned,
and therefore does not contribute to the network dynamics. The reduced forms of these networks,
which maintain oscillatory behavior, were generated by removing unnecessary and redundant nodes
or reactions, and compounding constant parameters within the rate law. These reduced forms show
the species and reactions responsible for oscillatory dynamics.
The reduced network in Figure 7 shows feedback loops that contribute to oscillatory behavior.
The first pathway of interest is the sequence of unimolecular–unimolecular reactions from species
S1 to S6, with two intermediate nodes, S3 and S5. This pathway shows that, as the concentration of
species S1 increases or decreases, there is an impact on species S6 in the same direction of change,
accompanied with a time delay in the signal due to the intermediate nodes. Continuing the cycle,
an increase in species S6 contributes to a rise in species S2 and subsequent production of S5, feeding
back positively into the pathway that produces species S6 and sustaining the cycle. However, negative
feedback of S6 on S1 is key for enabling oscillations. While species S6 increases, the bi-molecular
reaction between S6 and S1 which produces S2 simultaneously reduces the concentration of species S1
available for the uni-molecular reaction producing species S3. Together, the feedback and time delay
promote oscillation. This cycle repeats without dampening given appropriate global parameter values.
The reduced network in Figure 8 also contains feedback loops, involving the reversible reactions
between species S2, S9,and S4, as well as the reactions between species S2, S7 and S3. The optimized
model drives the conversions of S2 to S9 and S9 to S4 at a fast rate, while the conversion of S4 to S9
occurs much more slowly, as determined by the rate constants for these reactions. Since species S9
feeds back to S4 when generating S2, negative feedback arises. It is important to note that reactions
involving S8 effectively promote a time delay due to the involvement of an intermediate node in the
production of S2, which impacts the phase of the oscillations. However, removing species S8 allows for
a simplified reversible reaction between S4 and S9 in which the time delay created by the intermediate
14