Processes 2019, 7, 163
exclude global bifurcations from the evolving population. While a system with a zero real component
of one member of a complex conjugate pair would be a true Hopf bifurcation that produces sustained
oscillations, the bifurcation–evolution software only approximates this behavior. Additionally, since the
bifurcation–evolution software introduces stochasticity and depends on appropriate parameter ranges
for convergence, it is not guaranteed to produce the bifurcation behavior selected on every run.
This may require that the function is called multiple times before a satisfactory solution is reached.
4.2. Oscillator Frequency in Randomly-Generated Network Populations
The frequency data presented shows the intuitive result that larger random networks are more
likely to contain components that permit oscillatory dynamics. All randomly-generated networks
which exhibited sustained oscillations contained at least four floating species, although a system
with three floating species could conceivably achieve a Hopf bifurcation with the appropriate
network topology [20]. This suggests that oscillatory networks with three floating species are rare in
randomly-generated populations and sensitive to the parameter regime, such that oscillatory networks
occupy only a small region of parameter space. Oscillatory dynamics arise in networks that have
negative feedback and a time delay, as described for the networks in Figures 7 and 8. These are more
likely to arise in larger networks because there are more nodes to participate in complex loops and
engage in multiple reactions, increasing the likelihood of randomly generating a motif that provides
the feedback or time delay architecture. Figure 8 shows that it is advantageous to have a large number
of floating species to produce complex dynamics, as the reduced network has two three-species loops
that incorporate negative feedback, but the individual loops could not sustain oscillatory dynamics
when the second loop was removed. However, the data also suggests that the number of reactions
available to floating species in the network exerts greater control over the frequency of oscillatory
dynamics than does the total number of species.
Intentionally increasing the ratio of reactions to species greatly increases the frequency of
oscillatory dynamics following parameter optimization, as shown by the oscillator enrichment in the
1:1.5 networks shown in Figure 5 compared to the 1:1 networks shown in Figure 4. In the case where
the number of reactions and species are equal, as in Figure 4, oscillatory dynamics only appear in
networks with fewer than n − 2 floating species, where n is the total number of boundary and floating
species in the network. The networks which produce oscillators tend to have the reaction density
shifted onto the floating species, with minimal reaction density between boundary species. This shift
creates a local enrichment of reaction density on the floating species, permitting greater control over
the dynamic interactions between these species by providing flexibility to the network. This is not
necessary in the networks that have a 1:1.5 ratio, in which networks with the maximal number of
floating species can give rise to oscillatory dynamics because sufficient flexibility is conferred by the
intentional enrichment in the number of reactions. As a result, networks with a 1:1 ratio of total species
to reactions can approximate the behavior seen in networks with a 1:1.5 ratio by decreasing the number
of floating species in the network and minimizing the number of trivial reactions between boundary
species to locally enrich the reaction density between species that contribute to network dynamics.
The importance of reaction enrichment is further confirmed by the study of the 10 species, 10 reactions
randomly-generated network populations in which orphaned species were permitted. The majority
of oscillatory networks in this population included at least one orphaned species, which enriches the
reaction density between the remaining species in the network and provides additional flexibility.
Together, these data demonstrate the importance of providing a mass-action network
with sufficient connectivity to enable dynamic bifurcation behaviors to arise. True biological
systems, which may incorporate complex rate laws involving cooperativity and enzyme kinetics,
could circumvent such limitations in network size and reaction enrichment. In addition to the
studies presented describing the frequency of oscillatory dynamics, further studies should explore
the sizes of the parameter landscapes which permit desired bifurcation behaviors. This could better
explain the sensitivity of some optimized networks to variation in the parameter regime, and could
17
exclude global bifurcations from the evolving population. While a system with a zero real component
of one member of a complex conjugate pair would be a true Hopf bifurcation that produces sustained
oscillations, the bifurcation–evolution software only approximates this behavior. Additionally, since the
bifurcation–evolution software introduces stochasticity and depends on appropriate parameter ranges
for convergence, it is not guaranteed to produce the bifurcation behavior selected on every run.
This may require that the function is called multiple times before a satisfactory solution is reached.
4.2. Oscillator Frequency in Randomly-Generated Network Populations
The frequency data presented shows the intuitive result that larger random networks are more
likely to contain components that permit oscillatory dynamics. All randomly-generated networks
which exhibited sustained oscillations contained at least four floating species, although a system
with three floating species could conceivably achieve a Hopf bifurcation with the appropriate
network topology [20]. This suggests that oscillatory networks with three floating species are rare in
randomly-generated populations and sensitive to the parameter regime, such that oscillatory networks
occupy only a small region of parameter space. Oscillatory dynamics arise in networks that have
negative feedback and a time delay, as described for the networks in Figures 7 and 8. These are more
likely to arise in larger networks because there are more nodes to participate in complex loops and
engage in multiple reactions, increasing the likelihood of randomly generating a motif that provides
the feedback or time delay architecture. Figure 8 shows that it is advantageous to have a large number
of floating species to produce complex dynamics, as the reduced network has two three-species loops
that incorporate negative feedback, but the individual loops could not sustain oscillatory dynamics
when the second loop was removed. However, the data also suggests that the number of reactions
available to floating species in the network exerts greater control over the frequency of oscillatory
dynamics than does the total number of species.
Intentionally increasing the ratio of reactions to species greatly increases the frequency of
oscillatory dynamics following parameter optimization, as shown by the oscillator enrichment in the
1:1.5 networks shown in Figure 5 compared to the 1:1 networks shown in Figure 4. In the case where
the number of reactions and species are equal, as in Figure 4, oscillatory dynamics only appear in
networks with fewer than n − 2 floating species, where n is the total number of boundary and floating
species in the network. The networks which produce oscillators tend to have the reaction density
shifted onto the floating species, with minimal reaction density between boundary species. This shift
creates a local enrichment of reaction density on the floating species, permitting greater control over
the dynamic interactions between these species by providing flexibility to the network. This is not
necessary in the networks that have a 1:1.5 ratio, in which networks with the maximal number of
floating species can give rise to oscillatory dynamics because sufficient flexibility is conferred by the
intentional enrichment in the number of reactions. As a result, networks with a 1:1 ratio of total species
to reactions can approximate the behavior seen in networks with a 1:1.5 ratio by decreasing the number
of floating species in the network and minimizing the number of trivial reactions between boundary
species to locally enrich the reaction density between species that contribute to network dynamics.
The importance of reaction enrichment is further confirmed by the study of the 10 species, 10 reactions
randomly-generated network populations in which orphaned species were permitted. The majority
of oscillatory networks in this population included at least one orphaned species, which enriches the
reaction density between the remaining species in the network and provides additional flexibility.
Together, these data demonstrate the importance of providing a mass-action network
with sufficient connectivity to enable dynamic bifurcation behaviors to arise. True biological
systems, which may incorporate complex rate laws involving cooperativity and enzyme kinetics,
could circumvent such limitations in network size and reaction enrichment. In addition to the
studies presented describing the frequency of oscillatory dynamics, further studies should explore
the sizes of the parameter landscapes which permit desired bifurcation behaviors. This could better
explain the sensitivity of some optimized networks to variation in the parameter regime, and could
17