Processes 2019, 7, 163
2.5. Random Network Generation
To determine the frequency of oscillators in random networks, a network generator was used
which permitted four types of reactions and variable numbers of floating and boundary species.
Floating species are state variables, and therefore the concentrations of these species are variable
in time during the course of a simulation [19]. Boundary species are fixed and independent of the
model state, and are therefore either constant sources to the system or sinks, constant outputs [19].
The random networks were assigned simple mass-action kinetic rate laws and included the reactions
summarized in Figure 1. Mass-action kinetic rate laws are proportional to the concentration of the
reactant species in the biochemical reaction. Figure 1 therefore defines the mass-action rate laws, ν,
used in the random network generator as the product of a rate constant, k, and the concentration
of the reactants involved. Species concentrations are represented by placing brackets around the
species name. The generator excludes reactions that violate moiety conservation, and requires that
at least one species is a boundary species. For the purpose of analysis of networks with a specified
number of species and reactions, which are discussed in the frequency analysis, only networks which
did not have orphaned species and which had at least three floating species were passed to the final
populations. Three floating species was selected as the minimum cutoff because the smallest system
exhibiting a Hopf bifurcation contained three floating species [20]. For parameter value assignment,
the random network generator assigned concentrations and rate constants with arbitrary units (a.u).
Initial concentrations ranged from 1 to 10 a.u., and rate constants ranged from 1 × 10 −3 to 2 a.u.
The random network generator was used to create populations of random networks that could be
sent to the bifurcation–evolution software to evolve oscillatory dynamics. The default settings in the
tool were used for optimization, and, as a result, optimized species initial concentrations ranged from
0.1 to 10.0 a.u., and rate constant value assignments ranged from 1 × 10 −4 to 10.0 a.u. Models that
could not reach steady state, or which contained negative concentrations, were omitted from analysis.
Models that obtained a sufficiently low fitness value after optimization were reset and underwent
two additional rounds of optimization to increase the probability of achieving sustained oscillations
given an appropriate network architecture, accounting for stochasticity in the algorithm. Following
parameter optimization of all networks, populations of a minimum of 1100 randomly-generated
networks for each network size were manually assessed for oscillatory dynamics by simulating the
model with optimized parameters and inspecting the time-course of all floating species concentrations.
a.
c.
d.
b.
v = k [A]
v = k [A]
v = k [A][B]
v = k [A][B]
AB
AB + C
A + B
C
A + B
C + D
Figure 1. Types of reactions permitted in randomly-generated networks, governed by laws of
mass-action. Reactions are depicted visually and written with standard biochemical reaction
formatting. Mass-action kinetic rate laws, ν, for each reaction are defined by the product of rate
constant, k, and the concentrations of reactant species. (a) unimolecular–unimolecular reaction;
(b) unimolecular–bimolecular reaction; (c) bimolecular–unimolecular reaction; (d) bimolecular–
bimolecular reaction.
9
2.5. Random Network Generation
To determine the frequency of oscillators in random networks, a network generator was used
which permitted four types of reactions and variable numbers of floating and boundary species.
Floating species are state variables, and therefore the concentrations of these species are variable
in time during the course of a simulation [19]. Boundary species are fixed and independent of the
model state, and are therefore either constant sources to the system or sinks, constant outputs [19].
The random networks were assigned simple mass-action kinetic rate laws and included the reactions
summarized in Figure 1. Mass-action kinetic rate laws are proportional to the concentration of the
reactant species in the biochemical reaction. Figure 1 therefore defines the mass-action rate laws, ν,
used in the random network generator as the product of a rate constant, k, and the concentration
of the reactants involved. Species concentrations are represented by placing brackets around the
species name. The generator excludes reactions that violate moiety conservation, and requires that
at least one species is a boundary species. For the purpose of analysis of networks with a specified
number of species and reactions, which are discussed in the frequency analysis, only networks which
did not have orphaned species and which had at least three floating species were passed to the final
populations. Three floating species was selected as the minimum cutoff because the smallest system
exhibiting a Hopf bifurcation contained three floating species [20]. For parameter value assignment,
the random network generator assigned concentrations and rate constants with arbitrary units (a.u).
Initial concentrations ranged from 1 to 10 a.u., and rate constants ranged from 1 × 10 −3 to 2 a.u.
The random network generator was used to create populations of random networks that could be
sent to the bifurcation–evolution software to evolve oscillatory dynamics. The default settings in the
tool were used for optimization, and, as a result, optimized species initial concentrations ranged from
0.1 to 10.0 a.u., and rate constant value assignments ranged from 1 × 10 −4 to 10.0 a.u. Models that
could not reach steady state, or which contained negative concentrations, were omitted from analysis.
Models that obtained a sufficiently low fitness value after optimization were reset and underwent
two additional rounds of optimization to increase the probability of achieving sustained oscillations
given an appropriate network architecture, accounting for stochasticity in the algorithm. Following
parameter optimization of all networks, populations of a minimum of 1100 randomly-generated
networks for each network size were manually assessed for oscillatory dynamics by simulating the
model with optimized parameters and inspecting the time-course of all floating species concentrations.
a.
c.
d.
b.
v = k [A]
v = k [A]
v = k [A][B]
v = k [A][B]
AB
AB + C
A + B
C
A + B
C + D
Figure 1. Types of reactions permitted in randomly-generated networks, governed by laws of
mass-action. Reactions are depicted visually and written with standard biochemical reaction
formatting. Mass-action kinetic rate laws, ν, for each reaction are defined by the product of rate
constant, k, and the concentrations of reactant species. (a) unimolecular–unimolecular reaction;
(b) unimolecular–bimolecular reaction; (c) bimolecular–unimolecular reaction; (d) bimolecular–
bimolecular reaction.
9