A previously overlooked version of the so-called Olsen model of the peroxidase–oxidase reaction has been studied numerically using 2D isospike stability and maximum Lyapunov exponent diagrams and reveals a rich variety of dynamic behaviors not observed before. The model has a complex bifurcation structure involving mixed-mode and bursting oscillations as well as quasiperiodic and chaotic dynamics. In addition, multiple periodic and non-periodic attractors coexist for the same parameters. For some parameter values, the model also reveals formation of mosaic patterns of complex dynamic states. The complex dynamic behaviors exhibited by this model are compared to those of another version of the same model, which has been studied in more detail. The two models show similarities, but also notable differences between them, e.g., the organization of mixed-mode oscillations in parameter space and the relative abundance of quasiperiodic and chaotic oscillations. In both models, domains with chaotic dynamics contain apparently disorganized subdomains of periodic attractors with dinoflagellate-like structures, while the domains with mainly quasiperiodic behavior contain subdomains with periodic attractors organized as regular filamentous structures. These periodic attractors seem to be organized according to Stern–Brocot arithmetics. Finally, it appears that toroidal (quasiperiodic) attractors develop into first wrinkled and then fractal tori before they break down to chaotic attractors.

1.
L.
Glass
, “
Synchronization and rhythmic processes in physiology
,”
Nature
410
,
277
284
(
2001
).
2.
A.
Goldbeter
,
C.
Gerard
,
D.
Gonze
,
J. C.
Leloup
, and
G.
Dupont
, “
Systems biology of cellular rhythms
,”
FEBS Lett.
586
,
2955
2965
(
2012
).
3.
P. L.
Lakin-Thomas
, “
Circadian rhythms—New functions for old clock genes?
,”
Trends Genet.
16
,
135
142
(
2000
).
4.
A. N.
Dodd
,
N.
Salathia
,
A.
Hall
,
E.
Kévei
,
R.
Tóth
,
F.
Nagy
,
J. M.
Hibberd
,
A. J.
Millar
, and
A. A. R.
Webb
, “
Plant circadian clocks increase photosynthesis, growth, survival, and competitive advantage
,”
Science
309
,
630
633
(
2005
).
5.
K.
Aihara
and
I.
Tokuda
, “
Possible neural coding with interevent intervals of synchronous firing
,”
Phys. Rev. E
66
,
026212
(
2002
).
6.
J.
Bing
,
G.
H.
,
L.
Li
, and
X.
Zhao
, “
Dynamics of period-doubling bifurcation to chaos in the spontaneous neural firing patterns
,”
Cogn. Neurodyn.
6
,
89
106
(
2012
).
7.
A. Z.
Larsen
,
L. F.
Olsen
, and
U.
Kummer
, “
On the encoding and decoding of calcium signals in hepatocytes
,”
Biophys. Chem.
107
,
83
99
(
2004
).
8.
S.
Khan
, “
Conformational spread drives the evolution of the calcium-calmodulin protein kinase II
,”
Sci. Rep.
12
,
8499
(
2022
).
9.
H.
Degn
, “
Bistability caused by substrate inhibition of peroxidase in an open reaction system
,”
Nature
217
,
1047
1050
(
1968
).
10.
H.
Degn
, “
Compound-3 kinetics and chemiluminescence in oscillatory oxidation reactions catalyzed by horseradish peroxidase
,”
Biochim. Biophys. Acta
180
,
271
290
(
1969
).
11.
S.
Nakamura
,
K.
Yokota
, and
I.
Yamazaki
, “
Sustained oscillations in a lactoperoxidase NADPH and O2 system
,”
Nature
222
,
794
795
(
1969
).
12.
L. F.
Olsen
and
H.
Degn
, “
Chaos in an enzyme reaction
,”
Nature
267
,
177
178
(
1977
).
13.
M. S.
Samples
,
Y.-F.
Hung
, and
J.
Ross
, “
Further experimental studies on the horseradish-peroxidase oxidase reaction
,”
J. Phys. Chem.
96
,
7338
7342
(
1992
).
14.
T.
Hauck
and
F. W.
Schneider
, “
Mixed-mode and quasi-periodic oscillations in the peroxidase oxidase reaction
,”
J. Phys. Chem.
97
,
391
397
(
1993
).
15.
A.
Scheeline
,
D. L.
Olson
,
E. P.
Williksen
,
G. A.
Horras
,
M. L.
Klein
, and
R.
Larter
, “
The peroxidase-oxidase oscillator and its constituent chemistries
,”
Chem. Rev.
97
,
739
756
(
1997
).
16.
L. F.
Olsen
and
H.
Degn
, “
Oscillatory kinetics of peroxidase-oxidase reaction in an open system—Experimental and theoretical studies
,”
Biochim. Biophys. Acta
523
,
321
334
(
1978
).
17.
L. F.
Olsen
, “
Studies of the chaotic behavior in the peroxidase-oxidase reaction
,”
Z. Naturforsch. A
34
,
1544
1546
(
1979
).
18.
L. F.
Olsen
, “
An enzyme reaction with a strange attractor
,”
Phys. Lett. A
94
,
454
457
(
1983
).
19.
S.
Alexandre
and
H. B.
Dunford
, “
A new model for oscillations in the peroxidase oxidase reaction
,”
Biophys. Chem.
40
,
189
195
(
1991
).
20.
B. D.
Aguda
and
R.
Larter
, “
Sustained oscillations and bistability in a detailed mechanism of the peroxidase oxidase reaction
,”
J. Am. Chem. Soc.
112
,
2167
2174
(
1990
).
21.
B. D.
Aguda
and
R.
Larter
, “
Periodic chaotic sequences in a detailed mechanism of the peroxidase oxidase reaction
,”
J. Am. Chem. Soc.
113
,
7913
7916
(
1991
).
22.
D. L..
Olson
,
E. P.
Williksen
, and
A.
Scheeline
, “
An experimentally based model of the peroxidase-NADH biochemical oscillator—An enzyme-mediated chemical switch
,”
J. Am. Chem. Soc.
117
,
2
15
(
1995
).
23.
T. V.
Bronnikova
,
V. R.
Fed’kina
,
W. M.
Schaffer
, and
L. F.
Olsen
, “
Period-doubling bifurcations and chaos in a detailed model of the peroxidase-oxidase reaction
,”
J. Phys. Chem.
99
,
9309
9312
(
1995
).
24.
A.
Sensse
,
M. J. B.
Hauser
, and
M.
Eiswirth
, “
Feedback loops for Shilnikov chaos: The peroxidase-oxidase reaction
,”
J. Chem. Phys.
125
,
014901
(
2006
).
25.
J. A. C.
Gallas
and
L. F.
Olsen
, “
Complexity in subnetworks of a peroxidase–oxidase reaction model
,”
Chaos
32
,
063122
(
2022
).
26.
B. D.
Aguda
,
R.
Larter
, and
B. L.
Clarke
, “
Dynamic elements of mixed-mode oscillations and chaos in a peroxidase-oxidase model network
,”
J. Chem. Phys.
90
,
4168
4175
(
1989
).
27.
C. G.
Steinmetz
,
T.
Geest
, and
R.
Larter
, “
Universality in the peroxidase oxidase reaction: Period doublings, chaos, period-3, and unstable limit-cycles
,”
J. Phys. Chem.
97
,
5649
5653
(
1993
).
28.
D. R.
Thompson
and
R.
Larter
, “
Multiple time-scale analysis of 2 models for the peroxidase-oxidase reaction
,”
Chaos
5
,
448
457
(
1995
).
29.
M.
Desroches
,
B.
Krauskopf
, and
H. M.
Osinga
, “
The geometry of mixed-mode oscillations in the Olsen model for the peroxidase-oxidase reaction
,”
Discrete Contin. Dyn. Syst. - S
2
,
807
827
(
2009
).
30.
M. R.
Gallas
and
J. A. C.
Gallas
, “
Nested arithmetic progressions of oscillatory phases in Olsen’s enzyme reaction model
,”
Chaos
25
,
064603
(
2015
).
31.
C.
Kuehn
and
P.
Szmolyan
, “
Multiscale geometry of the Olsen model and non-classical relaxation oscillations
,”
J. Nonlinear Sci.
25
,
583
629
(
2015
).
32.
E.
Musoke
,
B.
Krauskopf
, and
H. M.
Osinga
, “
A surface of heteroclinic connections between two saddle slow manifolds in the Olsen model
,”
Int. J. Bifurcation Chaos
30
,
2030048
(
2020
).
33.
C.
Kuehn
,
N.
Berglund
,
C.
Bick
,
M.
Engel
,
T.
Hurth
,
A.
Iuorio
, and
C.
Soresina
, “
A general view on double limits in differential equations
,”
Physica D
431
,
133105
(
2022
).
34.
S.
Hoops
,
S.
Sahle
,
R.
Gauges
,
C.
Lee
,
J.
Pahle
,
N.
Simus
,
M.
Singhal
,
L.
Xu
,
P.
Mendes
, and
U.
Kummer
, “
COPASI—A COmplex PAthway SImulator
,”
Bioinformatics
22
,
3067
3074
(
2006
).
35.
P.
Frederickson
,
J. L.
Kaplan
,
E. D.
Yorke
, and
J. A.
Yorke
, “
The Lyapunov dimension of strange attractors
,”
J. Differ. Equ.
49
,
185
207
(
1983
).
36.
J. A. C.
Gallas
, “
Dissecting shrimps—Results for some one-dimensional physical models
,”
Physica A
202
,
196
223
(
1994
).
37.
T.
Gedeon
,
A. R.
Humphries
,
M. C.
Mackey
,
H. O.
Walther
, and
Z.
Wang
, “
Operon dynamics with state dependent transcription and/or translation delays
,”
J. Math. Biol.
84
,
2
(
2022
).
38.
M.
Kalia
,
Y. A.
Kuznetsov
, and
H. G. E.
Meijer
, “
Homoclinic saddle to saddle-focus transitions in 4D systems
,”
Nonlinearity
32
,
2024
2054
(
2019
).
39.
C.
Letellier
,
V.
Messager
, and
R.
Gilmore
, “
From quasiperiodicity to toroidal chaos: Analogy between the Curry-Yorke map and the van der Pol system
,”
Phys. Rev. E
77
,
046203
(
2008
).
40.
J. A. C.
Gallas
, “
Structure of the parameter space of the Henon map
,”
Phys. Rev. Lett.
70
,
2714
2717
(
1993
).
41.
J. R. B.
M. Araujo
and
J. A. C.
Gallas
, “
Nested sequences of period-adding stability phases in a CO2 laser map proxy
,”
Chaos, Solitons Fractals
150
,
111180
(
2021
).
42.
M.
Niqui
, “
Exact arithmetic on the Stern–Brocot tree
,”
J. Discrete Algorithms
5
,
356
379
(
2007
).
43.
C.
Grebogi
,
E.
Ott
,
S.
Pelikan
, and
J. A.
Yorke
, “
Strange attractors that are not chaotic
,”
Physica D
13
,
261
268
(
1984
).
44.
K.
Suresh
,
A.
Prasad
, and
K.
Thamilmaran
, “
Birth of strange nonchaotic attractors through formation and merging of bubbles in a quasiperiodically forced Chua’s oscillator
,”
Phys. Lett. A
377
,
612
621
(
2013
).
45.
S.
Kumarasamy
,
A.
Srinivasan
,
M.
Ramasamy
, and
K.
Rajagopal
, “
Strange nonchaotic dynamics in a discrete Fitzhugh–Nagumo neuron model with sigmoidal recovery variable
,”
Chaos
32
,
073106
(
2022
).
46.
C. G.
Steinmetz
and
R.
Larter
, “
The quasi-periodic route to chaos in a model of the peroxidase oxidase reaction
,”
J. Chem. Phys.
94
,
1388
1396
(
1991
).
47.
T. V.
Bronnikova
,
W. M.
Schaffer
, and
L. F.
Olsen
, “
Quasiperiodicity in a detailed model of the peroxidase-oxidase reaction
,”
J. Chem. Phys.
105
,
10849
10859
(
1996
).
48.
T. V.
Bronnikova
,
W. M.
Schaffer
,
M. J. B.
Hauser
, and
L. F.
Olsen
, “
Routes to chaos in the peroxidase-oxidase reaction. 2. The fat torus scenario
,”
J. Phys. Chem. B
102
,
632
640
(
1998
).
49.
L. F.
Olsen
and
A.
Lunding
, “
Chaos in the peroxidase-oxidase oscillator
,”
Chaos
31
,
013119
(
2021
).
50.
H.
Degn
,
L. F.
Olsen
, and
J. W.
Perram
, “
Bistability, oscillation and chaos in an enzyme reaction
,”
Ann. N.Y. Acad. Sci.
316
,
623
637
(
1979
).
51.
J. A. C.
Gallas
,
M. J. B.
Hauser
, and
L. F.
Olsen
, “
Complexity of a peroxidase-oxidase reaction model
,”
Phys. Chem. Chem. Phys.
23
,
1943
1955
(
2021
).
52.
L.
Kocic
and
L.
Stefanovska
, “A property of Farey tree,” in Numerical Analysis and Its Applications: 3rd International Conference on Numerical Analysis and Its Applications, Univ. Rousse, Bousse, Bulgaria, 29 Jun–3 Jul 2004, Lecture Notes in Computer Science Vol. 3401, edited by Z. Li, L. Vulkov, and J. Wasniewski (Department of Mathematics, North Carolina State University, 2005), pp. 345–351.
53.
J. G.
Freire
and
J. A. C.
Gallas
, “
Stern-Brocot trees in cascades of mixed-mode oscillations and canards in the extended Bonhoeffer-van der Pol and the Fitzhugh-Nagumo models of excitable systems
,”
Phys. Lett. A
375
,
1097
1103
(
2011
).
54.
M. J. B.
Hauser
and
L. F.
Olsen
, “
Mixed-mode oscillations and homoclinic chaos in an enzyme reaction
,”
J. Chem. Soc. Faraday Trans.
92
,
2857
2863
(
1996
).
55.
T.
Geest
,
C. G.
Steinmetz
,
R.
Larter
, and
L. F.
Olsen
, “
Period-doubling bifurcations and chaos in an enzyme reaction
,”
J. Phys. Chem.
96
,
5678
5680
(
1992
).
56.
A.
Lekebusch
and
F. W.
Schneider
, “
Two biochemical oscillators coupled by mass exchange
,”
J. Phys. Chem. B
101
,
9838
9843
(
1997
).
57.
L. F.
Olsen
,
T. V.
Bronnikova
, and
W. M.
Schaffer
, “
Secondary quasiperiodicity in the peroxidase-oxidase reaction
,”
Phys. Chem. Chem. Phys.
4
,
1292
1298
(
2002
).
58.
T.
Hauck
and
F. W.
Schneider
, “
Chaos in a Farey sequence through period-doubling in the peroxidase-oxidase reaction
,”
J. Phys. Chem.
98
,
2072
2077
(
1994
).
59.
L.
Aufinger
,
J.
Brenner
, and
F. C.
Simmel
, “
Complex dynamics in a synchronized cell-free genetic clock
,”
Nat. Commun.
13
,
2852
(
2022
).
60.
J. M.
Kembro
,
S.
Cortassa
,
D.
Lloyd
,
S. J.
Sollott
, and
M. A.
Aon
, “
Mitochondrial chaotic dynamics: Redox-energetic behavior at the edge of stability
,”
Sci. Rep.
8
,
15422
(
2018
).
61.
S.
Schuster
,
M.
Marhl
, and
T.
Höfer
, “
Modelling of simple and complex calcium oscillations—From single-cell responses to intercellular signalling
,”
Eur. J. Biochem.
269
,
1333
1355
(
2002
).
62.
R.
Malho
,
D.
Kaloriti
, and
E.
Sousa
, “
Calcium and rhythms in plant cells
,”
Biol. Rhythm Res.
37
,
297
314
(
2006
).
63.
E.
Smedler
and
P.
Uhlen
, “
Frequency decoding of calcium oscillations
,”
Biochim. Biophys. Acta: Gen. Subj.
1840
,
964
969
(
2014
).
64.
G.
Dupont
,
L.
Combettes
,
G. S.
Bird
, and
J. W.
Putney
, “
Calcium oscillations
,”
Cold Spring Harb. Perspect. Biol.
3
,
a004226
(
2011
).
65.
U.
Kummer
,
L. F.
Olsen
,
C. J.
Dixon
,
A. K.
Green
,
E.
Bornberg-Bauer
, and
G.
Baier
, “
Switching from simple to complex oscillations in calcium signaling
,”
Biophys. J.
79
,
1188
1195
(
2000
).
66.
N.
Manhas
,
J.
Sneyd
, and
K. R.
Pardasani
, “
Modelling the transition from simple to complex Ca2+ oscillations in pancreatic acinar cells
,”
J. Biosci.
39
,
463
484
(
2014
).
67.
C.
Allan
,
R.
Morris
, and
C.
Meisrimler
, “
Encoding, transmission, decoding, and specificity of calcium signals in plants
,”
J. Exp. Bot.
73
,
3372
3385
(
2022
).
68.
K.
Hashimoto
and
J.
Kudla
, “
Calcium decoding mechanisms in plants
,”
Biochimie
93
,
2054
2059
(
2011
).
69.
G.
Lenzoni
,
J.
Liu
, and
M. R.
Knight
, “
Predicting plant immunity gene expression by identifying the decoding mechanism of calcium signatures
,”
New Phytol.
217
,
1598
1609
(
2018
).
70.
H.
Gu
,
B.
Pan
, and
J.
Xu
, “
Experimental observation of spike, burst and chaos synchronization of calcium concentration oscillations
,”
Europhys. Lett.
106
,
50003
(
2014
).
71.
G.
Langlhofer
,
A.
Kogel
, and
M.
Schaefer
, “
Glucose-induced Ca2+ oscillations in betacells are composed of trains of spikes within a subplasmalemmal microdomain
,”
Cell Calcium
99
,
102469
(
2021
).
72.
R. J.
Field
,
J. G.
Freire
, and
J. A. C.
Gallas
, “
Quint points lattice in a driven Belousov-Zhabotinsky reaction model
,”
Chaos
31
,
053124
(
2021
).
73.
C. K.
Volos
and
J. A. C.
Gallas
, “
Experimental evidence of quint points and non-quantum chirality in a minimalist autonomous electronic oscillator
,”
Eur. Phys. J. Plus
137
,
154
(
2022
).

Supplementary Material

You do not currently have access to this content.