Processes 2019, 7, 163
oscillatory dynamics. The frequency of sustained oscillators increases for larger networks in both
the 1:1 and 1:1.5 network sizes. However, networks with a 1:1.5 ratio have a higher frequency of
oscillatory dynamics than networks with a 1:1 ratio when comparing networks with the same number
of species, therefore increasing the ratio of reactions to species increases the frequency of oscillators in
randomly-generated populations.
Table 1. Test cases from the BioModels Database. Table model names and descriptions adapted from
Chickarmane et al. with average runtime values generated by optimizing model parameter values for
all test cases using the Pythonic bifurcation–evolution software [6].
Turning Point Model
Description
Runtime (s)
Tyson et al., 2003 (Figure 1e) [24]
Irreversible bistable switch
9.10
Tyson et al., 2003 (Figure 1f) [24]
Reversible bistable switch
8.88
Edelstein, 1970 [25]
Autocatalytic bistable switch
17.70
Hervagault and Canu, 1987 [22]
Bistable switch
6.72
Angeli et al., 2004 [26]
Bistable switch cdc2/wee1
8.16
Oscillatory Model
Description
Runtime (s)
Tyson et al., 2003 (Figure 2a) [24]
Negative feedback oscillator
1.24
Tyson et al., 2003 (Figure 2b) [24]
Activator-inhibitor oscillator
1.07
Tyson et al., 2003 (Figure 2c) [24]
Substrate-depletion oscillator
0.99
Nicolis and Prigogine, 1977 [27]
Autocatalytic oscillator
0.16
Heinrich et al., 1977 [28]
Positive feedback oscillator
0.16
Seno et al., 1978 [23]
Modified Edelstein relaxation oscillator
1.54
Kholodenko, 2000 [29]
Mitogen-activated protein kinase feedback oscillator
42.57
Goldbeter, 1991 [30]
Mitotic oscillator
1.20
Francois et al., 2005 [31]
Mixed feedback loop oscillator
6.18
Lavrentovich and Hemkin, 2008 [32]
Spontaneous Ca 2+ oscillator
1.70
Figure 2. Optimized dynamic concentration changes in a bistable over time. The dashed black
trace is the result randomizing all parameter values in the bistable switch model from Hervagault
and Canu, 1987, using Tellurium and libRoadRunner functionalities by selecting from a uniform
distribution between 0.001 and 15.0 and performing a parameter sweep of parameter J1_k from 0.0 to
2.0. The green trace is the optimized bistable output for both floating species in the model in response
to an identical parameter sweep after the randomized parameters were evolved with default settings
in the bifurcation–evolution software.
11
oscillatory dynamics. The frequency of sustained oscillators increases for larger networks in both
the 1:1 and 1:1.5 network sizes. However, networks with a 1:1.5 ratio have a higher frequency of
oscillatory dynamics than networks with a 1:1 ratio when comparing networks with the same number
of species, therefore increasing the ratio of reactions to species increases the frequency of oscillators in
randomly-generated populations.
Table 1. Test cases from the BioModels Database. Table model names and descriptions adapted from
Chickarmane et al. with average runtime values generated by optimizing model parameter values for
all test cases using the Pythonic bifurcation–evolution software [6].
Turning Point Model
Description
Runtime (s)
Tyson et al., 2003 (Figure 1e) [24]
Irreversible bistable switch
9.10
Tyson et al., 2003 (Figure 1f) [24]
Reversible bistable switch
8.88
Edelstein, 1970 [25]
Autocatalytic bistable switch
17.70
Hervagault and Canu, 1987 [22]
Bistable switch
6.72
Angeli et al., 2004 [26]
Bistable switch cdc2/wee1
8.16
Oscillatory Model
Description
Runtime (s)
Tyson et al., 2003 (Figure 2a) [24]
Negative feedback oscillator
1.24
Tyson et al., 2003 (Figure 2b) [24]
Activator-inhibitor oscillator
1.07
Tyson et al., 2003 (Figure 2c) [24]
Substrate-depletion oscillator
0.99
Nicolis and Prigogine, 1977 [27]
Autocatalytic oscillator
0.16
Heinrich et al., 1977 [28]
Positive feedback oscillator
0.16
Seno et al., 1978 [23]
Modified Edelstein relaxation oscillator
1.54
Kholodenko, 2000 [29]
Mitogen-activated protein kinase feedback oscillator
42.57
Goldbeter, 1991 [30]
Mitotic oscillator
1.20
Francois et al., 2005 [31]
Mixed feedback loop oscillator
6.18
Lavrentovich and Hemkin, 2008 [32]
Spontaneous Ca 2+ oscillator
1.70
Figure 2. Optimized dynamic concentration changes in a bistable over time. The dashed black
trace is the result randomizing all parameter values in the bistable switch model from Hervagault
and Canu, 1987, using Tellurium and libRoadRunner functionalities by selecting from a uniform
distribution between 0.001 and 15.0 and performing a parameter sweep of parameter J1_k from 0.0 to
2.0. The green trace is the optimized bistable output for both floating species in the model in response
to an identical parameter sweep after the randomized parameters were evolved with default settings
in the bifurcation–evolution software.
11