On suitable observables for the chaotic Peroxidase-Oxidase autocatalytic reaction

Proceedings, 2nd IFAC Conference on Proceedings, 2nd on Proceedings, 2nd IFAC IFAC Conference Conference Modelling, Identification and Controlon of Nonlinear Systems Proceedings, 2nd IFAC Conference on Modelling, Identification and Control of Nonlinear Systems Available online at www.sciencedirect.com Modelling, Identification and Control of Nonlinear Systems Guadalajara, Mexico, June 20-22, 2018 Proceedings, 2nd IFAC Conference on Modelling, Identification and Control of Nonlinear Systems Guadalajara, Mexico, Mexico, June June 20-22, 20-22, 2018 2018 Guadalajara, Modelling, Identification and Control of Nonlinear Systems Guadalajara, Mexico, June 20-22, 2018 Guadalajara, Mexico, June 20-22, 2018

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IFAC PapersOnLine 51-13 (2018) 109–114

On On suitable suitable observables observables for for the the chaotic chaotic On suitable observables for the chaotic Peroxidase-Oxidase autocatalytic reaction On suitable observables for the chaotic Peroxidase-Oxidase autocatalytic reaction Peroxidase-Oxidase autocatalytic reaction Peroxidase-Oxidase autocatalytic reaction J. M. M´ endez-Gonz´ alez ∗∗ R. Femat ∗∗ ∗∗

J. M. M´ e ndez-Gonz´ a lez ∗∗ R. Femat ∗∗ J. e a J. M. M. M´ M´ endez-Gonz´ ndez-Gonz´ alez lez ∗ R. R. Femat Femat ∗∗ ∗∗ ∗ J. M. M´ e ndez-Gonz´ a lez R. Femat Divisi´ o n de Matem´ a ticas Aplicadas, IPICYT. Camino a la Presa San ∗ ∗ Divisi´ o n de Matem´ a ticas Aplicadas, IPICYT. Camino a la Presa San Divisi´ o n de Matem´ a ticas Aplicadas, IPICYT. Camino a la Presa San ∗Jos´ e 2055, Col. Lomas 4a. secci´ o n C.P. 78216. San Luis Potos´ ı, S. L. Divisi´ o n de Matem´ a ticas Aplicadas, IPICYT. Camino a la Presa San Jos´ e 2055, Col. Lomas 4a. secci´ o n C.P. 78216. San Luis Potos´ ı, S. ∗Jos´ e 2055, Col. Lomas 4a. secci´ on C.P. 78216. San Luis Potos´ı, S. L. L. Divisi´ on de Matem´ aticas IPICYT. la Presa P., M´ eeLomas xico. (e-mail: [email protected]). Jos´ e 2055, Col. 4a.Aplicadas, secci´ on C.P. 78216. Camino San Luisa Potos´ ı, S.San L. M´ xico. (e-mail: [email protected]). ∗∗e 2055,P., P., M´ eeLomas xico. (e-mail: [email protected]). Jos´ Col. 4a. secci´ on C.P. IPICYT. 78216. San Luis Potos´ ı, S. L. Divisi´ o n de Matem´ a ticas Aplicadas, Camino a la Presa ∗∗ P., M´ xico. (e-mail: [email protected]). ∗∗ Divisi´ on n de de Matem´ Matem´ aticas ticas Aplicadas, Aplicadas, IPICYT. IPICYT. Camino Camino a a la la Presa Presa a ∗∗ Divisi´ eCol. xico.Lomas (e-mail: [email protected]). San Jos´ee o 2055, secci´ o n C.P. 78216. San Luis Divisi´ oP., n deM´ Matem´ aticas 4a. Aplicadas, IPICYT. Camino a la Potos´ Presaı, San 2055, Col. Lomas 4a. secci´ o n C.P. 78216. San Luis Potos´ ı, ∗∗ Jos´ San Jos´ e 2055, Col. Lomas 4a. secci´ o n C.P. 78216. San Luis Potos´ n de Matem´ axico. ticas(e-mail: Aplicadas, IPICYT. Camino a la Potos´ Presaı, S. L. P., M´ e [email protected]) SanDivisi´ Jos´e o2055, Col. Lomas 4a. secci´ o n C.P. 78216. San Luis ı, S. L. P., M´ e xico. (e-mail: [email protected]) S. eexico. [email protected]) San Jos´e 2055, Col.M´ Lomas 4a. secci´ on C.P. 78216. San Luis Potos´ı, S. L. L. P., P., M´ xico. (e-mail: (e-mail: [email protected]) S. L. P., M´exico. (e-mail: [email protected])

Abstract: Abstract: Abstract: Neurodegenerative diseases such as Alzheimer’s and Parkinson present oxidative damage of Abstract: Neurodegenerative diseases diseases such such as as Alzheimer’s Alzheimer’s and and Parkinson Parkinson present present oxidative oxidative damage damage of of Neurodegenerative Abstract: neurocytes generally caused by reactive oxygen species (ROS). Heme peroxidases consume Neurodegenerative diseases such as Alzheimer’s and Parkinson present oxidative damage of neurocytes generally generally caused caused by by reactive reactive oxygen oxygen species species (ROS). (ROS). Heme Heme peroxidases peroxidases consume consume neurocytes Neurodegenerative diseases such as suppresors Alzheimer’s and Parkinson present oxidative damage of and produce ROS thus serving as promoters ROS-related pathology. A neurocytes caused by reactive oxygenand species (ROS).of peroxidases consume and produce producegenerally ROS thus thus serving as suppresors suppresors and promoters ofHeme ROS-related pathology. A and ROS serving as and promoters of ROS-related pathology. A neurocytes generally caused by reactive oxygen species (ROS). Heme peroxidases consume core reaction network of production and consumption of hydrogen peroxide and superoxide and produce ROS thus serving as suppresors and promoters of ROS-related pathology. A core reaction network of production and consumption of hydrogen peroxide and superoxide core reaction network production and consumption of peroxide superoxide and produce ROS thusof as suppresors andamong promoters of ROS-related pathology. A is the Peroxidase-Oxidase (PO). PO reaction was the first biochemical oscillators to core network of serving production consumption of hydrogen hydrogen peroxide and and superoxide is the thereaction Peroxidase-Oxidase (PO). PO PO and reaction was among among the first first biochemical biochemical oscillators to is Peroxidase-Oxidase (PO). reaction was the oscillators to core reaction network of production and consumption of hydrogen peroxide and superoxide be experimentally and numerically studied, with dynamical scenarios such as multiple steady is Peroxidase-Oxidase (PO). PO reaction wasdynamical among the first biochemical oscillators to be the experimentally and numerically studied, with scenarios such as multiple multiple steady be experimentally and studied, with scenarios such steady is the Peroxidase-Oxidase (PO). PO reaction wasdynamical among the firstperoxidase biochemical oscillators to states, complex oscillations and chaos. In particular, the oxyferrous (compound III) be experimentally and numerically numerically studied, with dynamical scenarios such as as multiple steady states, complex oscillations and chaos. In particular, the oxyferrous peroxidase (compound III) states, complex oscillations and chaos. In particular, the oxyferrous peroxidase (compound III) be experimentally and numerically studied, with dynamical scenarios such as multiple steady and O have been used as observable variables in experiments to record the chaotic dynamics. 2 states, complex oscillations and chaos. In particular, the oxyferrous peroxidase (compound III) and O O2 have been used as observable variables in experiments to the chaotic dynamics. and have been used observable in experiments to record record the chaotic dynamics. states, oscillations and chaos.variables In particular, the oxyferrous peroxidase (compound III) Moreover, PO reaction has been in nanoparticles in to serve as sensor and O22complex have used as as variables experiments record chaotic Moreover, PObeen reaction hasobservable been embedded embedded in in nanoparticles intoorder order tothe serve as aaa dynamics. sensor for for Moreover, PO reaction has been embedded in nanoparticles in order to serve as sensor for and O have been used as observable variables in experiments to record the chaotic dynamics. ROS. In this contribution we quantified, using observability coefficients, which chemical species 2 Moreover, POcontribution reaction has been embedded inobservability nanoparticlescoefficients, in order towhich servechemical as a sensor for ROS. In this we quantified, using species ROS. In this contribution we quantified, using observability coefficients, which chemical species Moreover, POcontribution reaction has embedded nanoparticles in complex order towhich servechemical as a of sensor for is more to observe and reconstruct characteristic dynamics the PO ROS. In suitable this webeen quantified, usinginthe observability coefficients, species is more suitable to observe and reconstruct the characteristic complex dynamics of the PO is more to observe reconstruct characteristic complex dynamics of the PO ROS. In suitable this contribution we and quantified, usingthe observability coefficients, which chemical species reaction. is more suitable to observe and reconstruct the characteristic complex dynamics of the PO reaction. reaction. is more suitable to observe and reconstruct the characteristic complex dynamics of the PO reaction. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. reaction. Keywords: Chaotic behaviour, Observability indices, Reactive oxygen species, Keywords: Chaotic behaviour, Observability indices, Reactive oxygen species, Keywords: Chaotic Chaotic diseases, behaviour, Observability indices, indices, Reactive Reactive oxygen oxygen species, species, Neurodegenerative Autocatalysis. Keywords: behaviour, Observability Neurodegenerative diseases, Autocatalysis. Neurodegenerative diseases, Autocatalysis. Keywords: Chaotic diseases, behaviour, Observability indices, Reactive oxygen species, Neurodegenerative Autocatalysis. Neurodegenerative diseases, Autocatalysis. 1. INTRODUCTION perimentally and numerically studied (Aguda and Clarke, 1. INTRODUCTION perimentally and numerically studied (Aguda and Clarke, 1. perimentally and numerically studied (Aguda and Clarke, 1987; Larter et al., 1987, 1988; Aguda et al., 1989; Stein1. INTRODUCTION INTRODUCTION perimentally and numerically studied (Aguda and Clarke, 1987; Larter et al., 1987, 1988; Aguda et al., 1989; Stein1987; Larter et al., 1987, 1988; Aguda et al., 1989; Stein1. INTRODUCTION perimentally andal., numerically studied (Aguda and Clarke, metz and Larter, 1991; Bronnikova et al., 1996; Sensse Peroxidases are enzymes which catalyze the oxidation of a 1987; Larter et 1987, 1988; Aguda et al., 1989; Steinand Larter, 1991; Bronnikova et al., 1996; Sensse Peroxidases are are enzymes enzymes which which catalyze catalyze the the oxidation oxidation of of aa metz metz and Larter, 1991; Bronnikova et al., 1996; Sensse Peroxidases 1987; Larter et al.,1991; 1987, 1988; Aguda etfirst al.,1996; 1989; Sensse Steinet al., 2006; Olsen, 1983). One of the proposed relarge class of substrates using H O . Because peroxidases Peroxidases are enzymes which catalyze the oxidation of a metz and Larter, Bronnikova et al., 2 2 et al., 2006; Olsen, 1983). One of the first proposed relarge class class of of substrates substrates using using H H22 O . Because peroxidases 2 et al., 2006; Olsen, 1983). One of the proposed reO peroxidases large 2 .. Because and Larter, 1991; Bronnikova et first al., 1996; Sensse Peroxidases are enzymes whichHcatalyze the oxidation of a metz action network, the so-called BFSO network (Bronnikova are able to use O as oxidizing substrate they are also large class of substrates using et al., 2006; Olsen, 1983). One of the first proposed re2 O Because peroxidases 2 2 action network, the so-called BFSO network (Bronnikova are able to use O as oxidizing substrate they are also 2 action network, the so-called BFSO network (Bronnikova are able to use O as oxidizing substrate they are also 2 et al., 2006; Olsen, 1983). One of the first proposed relarge class of substrates using H O . Because peroxidases al., 1996) comprised all the dynamical features observed act as oxidases. The catalytic reduction of H O to water 2 substrate 2 action network, the so-called BFSO network (Bronnikova 2 2 are able to use O as oxidizing they are also 2 et al., 1996) comprised all the dynamical features observed act as oxidases. The catalytic reduction of H O22 to water 2 et al., 1996) comprised all the dynamical features observed act as oxidases. catalytic reduction of H to water 2O action network, the so-called BFSO network (Bronnikova are able to use The O oxidizing substrate they are also in experiments. Later, Sensse et al. (2006) derived a new has an important physiological function because it protects et al., 1996) comprised all the dynamical features observed 2 as act as oxidases. The catalytic reduction of H O to water 2 2 in experiments. Later, Sensse et al. (2006) derived aa new has an important physiological function because it protects in experiments. Later, Sensse et al. derived new has an important physiological function because it protects et al., 1996) comprised all thesix dynamical features observed act as The catalytic reduction ofH2HO2 O to water reaction network considering species and the cell from the toxic oxidative power of bein experiments. Later, Sensse etchemical al. (2006) (2006) derived a thirnew 2 ,, 2 has an oxidases. important physiological function because itwhich protects reaction network considering six chemical species and thirthe cell from the toxic oxidative power of H O which be2 2 reaction network considering six chemical species and thirthe cell from the toxic oxidative power of H O , which be2 2 in experiments. Later, Sensse et al. (2006) derived a new has an important physiological function because it protects teen reactions; further reduction lead to a core network longs to the so-called chemically Reactive Oxygen Species reaction network considering six chemical species and thirthe cell from the toxic oxidative power of H O , which be2 2 teen reactions; reactions; further further reduction reduction lead lead to to aa core core network network longs to the so-called chemically Reactive Oxygen Species – teen longs to the so-called chemically Reactive Oxygen Species reaction network considering six chemical species and thirthe cell from the toxic oxidative power of H O , which beof three chemical species: NAD · radical, O , and com(ROS) among others such as superoxide ion radical (O ) 2 2 2 , and teen reactions; further reduction to aOcore network 2– longs toamong the so-called chemically Reactiveion Oxygen Species of three chemical species: NAD ·· lead radical, com(ROS) others such as superoxide radical (O 2 +6 2–) three chemical species: NAD radical, com(ROS) others such as superoxide ion radical (O 2 ,, and –– Per teen reactions; further reduction to core network longs toamong the so-called chemically Reactive Oxygen Species pound III (coIII ). The addition of aa O chemiand molecular oxygen (O are chemically produced +6 (ROS) among others such asROS superoxide ion radical (O22 –– )) of of three chemical species: NAD · lead radical, Ofourth and com2 ). 2 +6 pound III (coIII ). The addition of fourth chemiPer and molecular oxygen (O ). ROS are chemically produced 2 – pound III (coIII ). The addition of a fourth chemiPer +2 +3 – and molecular oxygen (O ). ROS are chemically produced +6 2 three chemical species: NAD ·) leads radical, Ofourth andchemicom(ROS) among others such asof superoxide ion radical (O2 ) of within the cells by means multiple reaction networks 2 , emergence – , Per or Per to the cal species (O pound III (coIII ). The addition of a Per +2 +3 – and molecular oxygen (O ). ROS are chemically produced 2 2 +3 ) leads to the emergence – , Per+2 within the cells by means of multiple reaction networks +6 or Per cal species (O 2 within the cells by means of multiple reaction networks or Per ))these leads to emergence cal species (O +2 ). +3 pound III (coIII The addition of athe fourth chemiPer and molecular oxygen (O ROS are chemically produced 2 – ,,–Per depending on the cell and tissue types thus playing 2 ). of of Shilnikov chaos provided that species enter in aa within the cells by means multiple reaction networks Per or Per leads to the emergence cal species (O 2 and depending on the cell and tissue types thus playing of Shilnikov chaos provided that species enter in +2 +3 these – and depending on cell and tissue types thus playing of Shilnikov chaos provided that these species enter in aa within the cellsrole bythe means of multiple reaction networks an important in cell signaling (Sensse, 2005). The , Per or Per ) leads to the emergence cal species (O supplementary feedback-loop (Sensse and Eiswirth, 2005; and depending on the cell and tissue types thus playing 2 of Shilnikov chaos provided that these species enter in an important important role role in in cell cell signaling signaling (Sensse, (Sensse, 2005). 2005). The The supplementary feedback-loop (Sensse and Eiswirth, 2005; an supplementary feedback-loop (Sensse and Eiswirth, 2005; and dependingof onROS the cellan and tissue(Sensse, types thus playing concentration in organism may dramatically of Shilnikov chaos provided that these species enter in a Sensse et al., 2006; Sensse, 2005). Thus, we have three an important role in cell signaling 2005). The supplementary feedback-loop (SensseThus, and Eiswirth, 2005; concentration of ROS in an organism may dramatically Sensse et al., 2006; Sensse, 2005). we have three concentration of ROS in an organism may dramatically Sensse et al., 2006; Sensse, 2005). Thus, we have three an important role in cell signaling (Sensse, 2005). The increase under environmental stress hence damaging cell supplementary feedback-loop (Sensse and Eiswirth, 2005; subnetworks that share three chemical species and the concentration of ROS in an organism may dramatically Sensse et al., 2006; Sensse, 2005). Thus, we have three increase under environmental stress hence damaging cell subnetworks that share three chemical species and the increase under stress hence cell that share three chemical and the concentration ofenvironmental ROS in an organism maydamaging dramatically structures. Neural tissue nerve fibrils are Sensse et al., 2006; Sensse, 2005). Thus,species we have three same route to chaos despite the distinct chemical nature increase under environmental stress cell subnetworks subnetworks that share three chemical species and the structures. Neural tissue such such as as nervehence fibrilsdamaging are susceptisusceptisame route to chaos despite the distinct chemical nature structures. Neural tissue such as nerve fibrils are susceptisame route to chaos despite the distinct chemical nature increase under environmental stress hence damaging cell ble to the increase concentration of ROS that might lead to subnetworks that share three chemical species and the of the fourth chemical species. structures. Neural tissue such as nerve fibrils are susceptisame route to chaos despite the distinct chemical nature ble to to the the increase increase concentration concentration of of ROS ROS that that might might lead lead to to of of the the fourth fourth chemical chemical species. species. ble structures. Neural tissue such as nerve fibrils are susceptiaxonopathy, an early event in different neurodegenerative same route to chaos despite the distinct chemical nature ble to the increase concentration of ROS that might lead to of the fourth chemical species. that there are kinetic rate axonopathy, an early event in different neurodegenerative it has been reported axonopathy, an early event neurodegenerative ble to thesuch increase concentration of ROS that might lead to Recently, diseases Alzheimer’s and Parkinson’s (Belenky of the fourth chemical species. that Recently, it has been reported there are kinetic rate axonopathy, anas early event in in different different neurodegenerative Recently, it has been reported that there kinetic rate diseases such as Alzheimer’s and Parkinson’s (Belenky constants least one Action Kinetic diseases such as Alzheimer’s and Parkinson’s (Belenky Recently, itsuch has that been at reported that Mass there are are kinetic rate axonopathy, an early event in different neurodegenerative et al., 2006). Peroxidase-Oxidase (PO) reaction is linked constants such that at least one Mass Action Kinetic diseases such as Alzheimer’s and Parkinson’s (Belenky constants such that at least one Mass Action Kinetic et al., 2006). Peroxidase-Oxidase (PO) reaction is linked Recently, it has been reported that there are kinetic rate (MAK) ODE induced by one of the three PO subnetworks et al., 2006). Peroxidase-Oxidase (PO) reaction is linked constants such that at least one Mass Action Kinetic diseases such as Alzheimer’s and Parkinson’s (Belenky to the appearance of neurodegenerative diseases because (MAK) ODE induced by one of the three PO subnetworks et al., 2006). Peroxidase-Oxidase (PO) reaction isbecause linked (MAK) ODE induced by one of the three PO subnetworks to the appearance of neurodegenerative diseases constants such that at least one Mass Action Kinetic can support the same pair of steady states of a second to the appearance of neurodegenerative diseases because (MAK) ODE induced by one of the three PO subnetworks et its al., 2006). Peroxidase-Oxidase (PO) reaction linked of role suppressor and promoter ROScan support the same pair of steady states of a second to theregulating appearance ofas diseases isof because can support same of steady states of of its regulating role asneurodegenerative suppressor and promoter of ROS(MAK) ODE the induced oneof ofthe the chemical three POspecies subnetworks PO subnetwork irrespective being of its role suppressor and promoter of ROScan support the samebypair pair steady states of aa second second to theregulating appearance ofas neurodegenerative diseases because related pathology (Atamna and Boyle, 2006). This twofold PO subnetwork irrespective of the chemical species being of its regulating role as suppressor and promoter of ROSPO subnetwork irrespective of the chemical species being related pathology (Atamna and Boyle, 2006). This twofold can support the same pair ofand steady states of aThat second measured (M´ e ndez-Gonz´ a lez Femat, 2016). is, related pathology (Atamna and Boyle, 2006). This twofold PO subnetwork irrespective of the chemical species being of its regulating role as suppressor and promoter of ROSrole has been exploited to sensor ROS embedding the PO measured (M´ e ndez-Gonz´ a lez and Femat, 2016). That is, related pathology (Atamna and Boyle, 2006). This twofold measured (M´ e ndez-Gonz´ a lez and Femat, 2016). That is, role has has been been exploited exploited to to sensor sensor ROS ROS embedding embedding the the PO PO PO subnetwork irrespective of the chemical species being unless we measure the fourth chemical species we cannot role measured (M´ e ndez-Gonz´ a lez and Femat, 2016). That is, related (Atamna and Boyle, 2006). This twofold reaction in nanoparticles et al., 2007). we measure the fourth chemical species we cannot role haspathology been exploited to(Poulsen sensor ROS embedding the PO unless unless we measure the fourth chemical species we cannot reaction in nanoparticles (Poulsen et al., 2007). measured (M´ e ndez-Gonz´ a lez and Femat, 2016). That is, distinguish which PO subnetwork is operative or domireaction in nanoparticles (Poulsen et al., 2007). unless we measure the fourth chemical species we cannot role has been exploited to(Poulsen sensor ROS embedding the PO distinguish which PO subnetwork is operative or domireaction in70’s, nanoparticles et al., 2007). distinguish which PO subnetwork is operative or domiSince the PO reaction was among the first biochemunless we measure the fourth chemical species we cannot nant. The biological implications of this redundancy or distinguish which PO subnetwork is operative or domireaction in70’s, nanoparticles (Poulsen et al.,the 2007). Since the PO reaction was among first biochemnant. The biological implications of this redundancy or Since the PO was the biochemnant. The biological redundancy or ical reactions the capacity to exhibit chaotic oscildistinguish which POimplications subnetwork of is this operative or domiSince the 70’s, 70’s,with PO reaction reaction was among among the first first biochemnant. The biological implications of this redundancy or ical reactions with the capacity to exhibit chaotic oscilical reactions with the capacity to exhibit chaotic oscilSince the 70’s,with PO reaction was among the first biochemlations (among other dynamical scenarios) that were exnant. The biological implications of this redundancy or ical reactions the capacity to exhibit chaotic oscillations (among other dynamical scenarios) that were exlations (among other were exical reactions thedynamical capacity scenarios) to exhibit that chaotic oscillations (amongwith other dynamical scenarios) that were exlations (among otherConference dynamical scenarios)of that wereControl) ex-109 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2018, IFAC (International Automatic Proceedings, 2nd IFAC onFederation Proceedings, 2nd Conference on 109 Proceedings, 2nd IFAC IFAC Conference on 109 Control. Peer reviewIdentification under responsibility of International Federation of Automatic Modelling, and Control of Nonlinear Proceedings, 2nd IFAC Conference 109 Modelling, Identification and Control Controlon of Nonlinear 10.1016/j.ifacol.2018.07.263 Modelling, Identification and of Nonlinear Systems Proceedings, 2nd IFAC Conference on 109 Modelling, Identification and Control of Nonlinear Systems Systems Guadalajara, Mexico, June 20-22, 2018 Modelling, Identification and Control of Nonlinear

2018 IFAC MICNON 110 Guadalajara, Mexico, June 20-22, 2018J.M. Méndez-González et al. / IFAC PapersOnLine 51-13 (2018) 109–114

capacity to activate different subnetworks to reach the same steady state values are unknown. Experimentally, PO chaotic dynamics have been recorded using detectable chemical species such as the oxyferrous peroxidase (coIII – Per+6 ) and O2 through their absorbances (Hung and Ross, 1995; Møller et al., 1998). However, it is known that absorbances measurements might hide the contribution of reaction intermediaries to dynamics (Maraun et al., 2004). Furthermore, a frequent scenario in the study of biochemical reaction systems is the presence of fragmentary or unreliable data, thus it is of paramount importance to select properly the chemical species to be measured (the observable) to reconstruct the chaotic dynamics of PO subnetworks. Fortunately, it is assured by Takens’ theorem (Takens, 1981) that a single time series contains sufficient information to reconstruct the dynamics of the phase space. A procedure to unfold the dynamic is the proper construction of an equivalent space using the recursive Lie derivatives of an observable variable (i.e. a chemical species concentration) (Gilmore and Lefranc, 2002; Letellier et al., 2006). However, this procedure does not guarantee that the chosen observable will be the best among all possible choices: there are better observables than others. The quantification of the degree of observability depending on the variable used can be done it trough the computation of observability coefficients (Letellier et al., 1998; Letellier and Aguirre, 2002; Letellier et al., 2006). These observability coefficients are a local quantity averaged over the (chaotic) attractor, derived from the observability matrix obtained using recursive Lie derivatives of the observable variable as well (Letellier et al., 2006). Comparing the observability coefficients computed for every chemical species it is possible to rank the variables in descending degree of observability (Letellier and Aguirre, 2002; Letellier et al., 2006). In other words, we can quantify which chemical species may influence the quality of a dynamical analysis for the PO reaction under a chaotic regime, for example. In this contribution we quantified, using observability coefficients, which chemical species is more suitable to observe in order to reconstruct the complex dynamics of the PO reaction. This paper is organised as follows: section 2 presents the MAK ODEs that model all three PO extended reaction networks that exhibit Shilnikov chaos. The construction of the nonlinear observability matrix using Lie derivatives and further derivation of observability coefficients is presented in section 3. Ranking of chemical species depending on their degree of observability is given in section 4. Conclusions are given in section 5.

2.1 PO subnetwork No. 1 The set of MAK ODEs for the first PO subnetwork as reported by Sensse et al. (2006) is: x˙ 1 = k1 x1 x3 − k2 x1 x2 − 2k3 x21 + k4

x˙ 2 = −k2 x1 x2 − k5 x2 + k6 + k13 x ˆ24 x˙ 3 = −k1 x1 x3 − k7 x3 + k8 ˆ2 x ˆ˙ 4 = k2 x1 x2 − 2k13 x

(1) +6 – where x1 =NAD · , x2 =O2 , x3 =coIII Per , and the fourth species x ˆ4 =O2 – . We note, however, that in Sensse et al. (2006) paper the chemical species O2 – is denoted as x6 . The chaotic attractor obtained from numerical integration of MAK-ODEs (1) is depicted in Fig. 1. 4

Fig. 1. Chaotic attractor for PO subnetwork No 1. Chemical species: x1 =NAD · , x2 =O2 , x3 =coIII – Per+6 and x ˆ4 =O2 – . Kinetic rate constants for Shilnikov chaos (Sensse et al., 2006): k1 = 0.0062, k2 = 0.415, k3 = 0.007, k4 = 0.59, k5 = 0.14, k6 = 0.71, k7 = 0.0001, k8 = 0.353, k13 = 0.0006. 2.2 PO subnetwork No. 2 The fourth chemical species added to the core PO network is x ˜4 =Per+2 (denoted x5 by Sensse et al. (2006)), which leads to the following MAK ODEs: x˙ 1 = k1 x1 x3 − k2 x1 x2 − 2k3 x21 + k4 − k11 x1

2. PEROXIDASE-OXIDASE REACTION SUBNETWORKS

x˙ 2 = −k2 x1 x2 − k5 x2 + k6 − k12 x2 x ˜4

According to Sensse et al. (2006), the interplay among the core PO reaction network chemical species, i.e., NAD · , O2 and Per+6 , with an additional fourth chemical species, O2 – , Per+2 or Per+3 , gives rise to three four dimensional MAK ODEs. We shall denote with capital letter Xi , i = 1 . . . 4 the chemical species whereas lower case letters xi , i = 1 . . . 4 stand for chemical species concentrations. Next, we present the associated MAK-ODEs induced by the PO subnetworks and their chaotic behaviour. 110

x˙ 3 = −k1 x1 x3 − k7 x3 + k8 + k12 x2 x ˜4 ˜4 x ˜˙ 4 = k11 x1 − k12 x2 x

(2) again with x1 =NAD · , x2 =O2 , x3 =coIII – Per . Fig. 2 shows the chaotic attractor associated to MAK ODEs (2). +6

2.3 PO subnetwork No. 3 The addition of x4 =Per+3 to the core PO network gives the following MAK ODEs:



J.M. Méndez-González et al. / IFAC PapersOnLine 51-13 (2018) 109–114

111

Fig. 2. Chaotic attractor for PO subnetwork No. 2. Chemical species: x1 =NAD · , x2 =O2 , x3 =coIII – Per+6 and x4 =Per+2 . Kinetic rate constants for Shilnikov chaos (Sensse et al., 2006): k1 = 0.032, k2 = 0.15, k3 = 0.008, k4 = 0.0009, k5 = 0.132, k6 = 1, k7 = 0.01, k8 = 0.59, k11 = 0.23, k12 = 0.01642.

Fig. 3. Chaotic attractor for PO subnetwork No. 3. Chemical species: x1 =NAD · , x2 =O2 , x3 =coIII – Per+6 and x4 =Per+3 . Kinetic constants for Shilnikov chaos (Sensse et al., 2006): k1 = 0.05796, k2 = 0.3, k3 = 0.008, k4 = 0.00525, k5 = 0.285, k6 = 1.25, k7 = 0.001, k8 = 0.07, k9 = 0.052, k10 = 0.00396

x˙ 1 = k1 x1 x3 − k2 x1 x2 − 2k3 x21 + k4

observable function y = h(x) along the vector field, f (x, k), is defined as: s ∑ ∂h(x) Lf h(x) = fi (x, k) (5) ∂xi i=1

x˙ 2 = −k2 x1 x2 − k5 x2 + k6

x˙ 3 = −k1 x1 x3 − k7 x3 + k8 + k9 x4 x˙ 4 = k1 x1 x3 − k9 x4 − k10 x4

(3)

Chemical species x1 , x2 and x3 are the same as in both previous PO subnetworks. The chaotic attractor for PO subnetwork No. 3 is displayed in Fig. 3. 3. OBSERVABILITY COEFFICIENTS Let us recall that all three PO subnetworks exhibit chaotic dynamics of Shilnikov type (Sensse, 2005). However, it has been overlooked which chemical species provides a better observability for the dynamics. Thus, it is the purpose of this paper to point out the best observable among the chemical species that conform the chaotic PO subnetworks using the concept of observability coefficients. Briefly, the set of MAK ODEs induced by chemical reaction networks can be written as x˙ = f (x, k), where f is the vector field that maps points from some open set D ⊂ Rn to a tangent space, that is, a space spanned by derivatives. Further, consider the existence of a smooth output function y = h(x), where h : Rn → R. In the case of chemical reactions, any measurable chemical species concentration will be valid as an output function. A coordinate transformation (diffeomorphism) z = Φ(x) around x0 can be constructed through recursive Lie derivatives as follows (Isidori, 1995; Nijmeijer and van der Schaft, 1995; Slotine and Li, 1991): h(x))T z = Φ(x) = (h(x), Lf h(x), . . . , Ls−1 f

(4)

where s is the dimension of the chemical species concentration vector, x = (x1 , . . . , xs ). The Lie derivative of the 111

The jacobian matrix of Φ(x) defines a more general observability matrix, J(Φ(x)) = Q(x), which becomes the standard observability matrix (Letellier et al., 2006). Considering that J(Φ(x)) = Q(x), Letellier and Aguirre (2002) defined an observability coefficient to quantify a degree of observability as follows: |λmin [QT Q, x(t)]| (6) δ(x) = |λmax [QT Q, x(t)]| where the numerator (denominator) of (6) stands for the minimum (maximum) eigenvalue of matrix QT Q evaluated at point x(t). As a consequence, the observability coefficient is bounded between 0 ≤ δ(x) ≤ 1. The lower bound is attained when the system becomes unobservable, that is, when J(Φ(x)) is singular because of the local nature of the diffeomorphic map. In other words, as a parameter is varied the dynamical system may gradually become unobservable. Additionally, depending on the singularities of the determinant of the jacobian matrix J(Φ(x)) there might be regions in phase space more or less observables than others (Letellier et al., 2006; Letellier and Aguirre, 2010). Following ideas in (Letellier and Aguirre, 2002) we have the averaged value for δ(x): ∑τ 1 λmin [QT Q, x(t)] τ ¯ δ = 1 ∑τt=0 (7) T t=0 λmax [Q Q, x(t)] τ where τ is the final time considered. Combining the results obtained from Eqs. (6)-(7), the variables used as observables are ranked in descending degree of observability us-

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ing the order relation denoted by the symbol “” (Letellier and Aguirre, 2002). For example, x1  x2 means that x1 provides a better observability of the underlying dynamics than x2 . The above computation is implemented in the next section for all chemical species involved in PO subnetworks presented in section 2. 4. RESULTS AND DISCUSSION For the four dynamical variables of the PO subnetwork No. 1, observability coefficients (and their standard deviation) averaged using Eq. (7) over the chaotic dynamics are:  δ¯ = 3.70 × 10−9 ± 3.71 × 10−9    ¯x1 δ = 1.09 × 10−9 ± 1.20 × 10−9 PO subnetwork No. 1 ¯x2 δ = 1.03 × 10−5 ± 7.47 × 10−6    ¯x3 δx4 = 3.70 × 10−8 ± 1.83 × 10−8 From the above coefficient values, we can rank the variables in descending degree of observability according to: x3  x4  x1  x2 . The four coefficients have different orders of magnitude which suggest a highly sensitive dependency on the choice of the observable (Letellier and Aguirre, 2002). The chaotic dynamics from PO subnetwork No. 1 is most observable recording the x3 =coIII – Per+6 variable. The worst observable variable corresponds to x2 =O2 , which has been used by Bronnikova et al. (1995) and Møller et al. (1998) to record the chaotic behaviour in experiments; and by Hung and Ross (1995) for deduction and classification of oscillatory mechanisms.

With respect to PO subnetwork No. 2 the observable coefficients are:  δ¯ = 8.43 × 10−7 ± 5.88 × 10−6    ¯x1 δ = 1.75 × 10−6 ± 8.69 × 10−6 PO network No. 2 ¯x2 −5 −5   δx3 = 2.94 × 10−5 ± 7.28 × 10−4  ¯ δx4 = 4.44 × 10 ± 2.01 × 10 Thus, the rank of variables is x4  x3  x2  x1 . In this case, PO subnetwork No. 2 is more observable from the x4 =Per+2 variable meanwhile x1 =NAD · will lead to a poor observability of the chaotic regime. The projection of the differential embedding induced by x1 =NAD · shows a domain around the origin where the dynamics is squeezed, a signature of poor observability (Letellier et al., 2013); see Fig. 5. We point out that the differential embedding produced by x1 =NAD · is alike (up to a scaling factor) to the differential embedding induced by the variable z in the hyperchaotic R¨ossler attractor (cf. with Fig. 1 in (Letellier et al., 2005)), suggesting the possibility of a topological equivalence between both dynamical systems.

Fig. 4 shows the poorer observability obtained from the differential embedding induced by x2 =O2 : the dynamic is compressed near the domain of z2 = x˙ 2 = −0.02, making difficult to distinguish among trajectories. On the contrary, the differential embedding constructed from x3 =coIII – Per+6 leads to a better “unfolding” of the trajectories; regions of compressed dynamics are absent. Fig. 5. Differential embeddings produced by the four variables of PO subnetwork No. 2. Finally, for PO extended network No. 3 the observability coefficients are:   δ¯x1 = 2.15 × 10−6 ± 4.42 × 10−6  ¯ δ = 1.41 × 10−7 ± 4.29 × 10−7 PO network No. 3 ¯x2  δx3 = 4.49 × 10−5 ± 4.13 × 10−5  ¯ δx4 = 5.09 × 10−4 ± 8.70 × 10−4

The ranking of variables are: x4  x3  x1  x2 . As a consequence of the above averaged values we can conclude that variable x4 =Per+2 provides a better observability of the chaotic dynamics than x2 =O2 , which remains as the worst observable variable among all three PO extended networks. The differential embeddings are depicted in Fig. 6.

Fig. 4. Differential embeddings produced by the four variables of PO subnetwork No. 1. 112

Summing up, we have shown that chemical species x2 =O2 is the less appropriate observable to record the chaotic dynamics of the three PO subnetworks. On the contrary, x4 =Per+2 is ranked as the best observable for PO subnetwork No. 2 and 3, whereas x3 =coIII – Per+6 is best for PO

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113

5. CONCLUSIONS Using the concept of observability coefficients, we have shown that, in the presence of noise-free data, chemical species x2 =O2 is the less appropriate observable to record the chaotic dynamics of the three chaotic PO subnetworks. Chemical species x4 =Per+2 was ranked as the more convenient observable for PO subnetwork No. 2 and 3, whereas chemical species x3 =coIII – Per+6 is best for PO subnetwork No. 1. REFERENCES

Fig. 6. Differential embeddings produced by the four variables of PO subnetwork No. 3.

subnetwork No. 1. However, let us recall that the fourth chemical species is different in all PO subnetworks. Thus, in order to have a better unfolding of the chaotic dynamics for all three PO subnetworks, the choice of the second best observable chemical species is x3 =coIII – Per+6 which is present in all three PO subnetworks. This decision agrees with the experimental measurements of chaotic dynamics reported by Møller et al. (1998). Furthermore, the aforementioned experiments were performed at 28◦ C within a pH interval of 5.2 − 6.3. Hence, the present observability analysis suggest a degree of robustness against pH variations within the chaotic region. In real biochemical reactions, however, the time evolution of chemical species are always noisy. In theory, the selection of the greater observability coefficients (x4 =Per+2 for PO subnetwork No. 2 and 3; x3 =coIII – Per+6 for PO subnetwork No. 1) provides also a greatness robustness in the presence of noisy data which tends to diminish the degree of observability (Maquet et al., 2004). A further in-depth analysis need to be done for all three PO subnetworks for the ranked observables in order to quantify how the experimental noise affects the chaotic dynamics and its observability. For example, experimental evidence also suggests that oscillatory dynamics in PO reaction might be a seasonal event (Møller et al., 1998). Nevertheless, the observable chemical species used to argue seasonal oscillations was x2 =O2 (Møller et al., 1998), which the present observability analysis ranked it as the worst to analyse PO reaction dynamics. Further studies are required to determine if such seasonal event is the product of the interaction between experimental noise and the intrinsic oscillatory dynamics, or the selection of x2 =O2 as the observable to unfold the chaotic dynamics. 113

Aguda, B.D. and Clarke, B.L. (1987). Bistability in chemical reaction networks: Theory and application to the peroxidase-oxidase reaction. J. Chem. Phys., 87, 3461. doi:http://dx.doi.org/10.1063/1.452991. Aguda, B.D., Larter, R., and Clarke, B.L. (1989). Dynamic elements of mixed mode-oscillations and chaos in a peroxidase-oxidase model network. J. Chem. Phys., 90, 4168. doi:http://dx.doi.org/10.1063/1.455774. Atamna, H. and Boyle, K. (2006). Amyloid-β peptide binds with heme to form a peroxidase: Relationship to the cytopathologies of alzheimers disease. Proc. Natl. Acad. Sci. USA, 103(9), 3381–3386. Doi:10.1073/pnas.0600134103. Belenky, P., Bogan, K.L., and Brenner, C. (2006). NAD∗ metabolism in health and disease. Trends in biochemical sciences, 32(1), 12–19. doi:10.1016/j.tibs.2006.11.006. Bronnikova, T.V., Fed’kina, V.R., Schaffer, W.M., and Olsen, L.F. (1995). Period-doubling bifurcations and chaos in a detailed model of the peroxidase-oxidase reaction. J. Phys. Chem., 99(23), 9309–9312. doi: 10.1021/j100023a001. Bronnikova, T.V., Schaffer, W.M., and Olsen, L.F. (1996). Quasiperiodicity in a detailed model of the peroxidaseoxidase reaction. J. Chem. Phys., 105, 10849. doi: http://dx.doi.org/10.1063/1.472927. Gilmore, R. and Lefranc, M. (2002). The topology of chaos. Alice in stretch and squeezeland. John Wiley and Sons, Inc., 1st edition. Hung, Y.F. and Ross, J. (1995). New experimental methods toward the deduction of the mechanism of the oscillatory peroxidase-oxidase reaction. J. Phys. Chem., 99(7), 1974–1979. doi:10.1021/j100007a030. Isidori, A. (1995). Nonlinear control systems. Springer– Verlag, 3rd. edition. Larter, R., Bush, C.L., Lonis, T.R., and Aguda, B.D. (1987). Multiple steady states, complex oscillations, and the devil’s staircase in the peroxidaseoxidase reaction. J. Chem. Phys., 87, 5765. doi: http://dx.doi.org/10.1063/1.453550. Larter, R., Steinmetz, C.G., and Aguda, B.D. (1988). Fast-slow variable analysis of the transition to mixed-mode oscillations and chaos in theperoxidase reaction. J. Chem. Phys., 89, 6506. doi: http://dx.doi.org/10.1063/1.455370. Letellier, C., Denis, F., and Aguirre, L. (2013). What can be learned from a chaotic cancer model? J. Theor. Biol., 322, 7–16. doi: http://dx.doi.org/10.1016/j.jtbi.2013.01.003. Letellier, C., Maquet, J., Sceller, L.L., Gouesbet, G., and Aguirre, L.A. (1998). On the non-equivalence of observables in phase-space reconstructions from recorded

2018 IFAC MICNON 114 Guadalajara, Mexico, June 20-22, 2018J.M. Méndez-González et al. / IFAC PapersOnLine 51-13 (2018) 109–114

time series. J. Phys. A: Math. Gen., 31(39), 7913. doi: http://dx.doi.org/10.1088/0305-4470/31/39/008. Letellier, C., Aguirre, L., and Maquet, J. (2006). How the choice of the observable may influence the analysis of nonlinear dynamical systems. Commun. Nonlinear Sci. Numer. Simul., 11(5), 555–576. doi: http://dx.doi.org/10.1016/j.cnsns.2005.01.003. Letellier, C. and Aguirre, L.A. (2002). Investigating nonlinear dynamics from time series: The influence of symmetries and the choice of observables. Chaos, 12(3), 549–558. doi:http://dx.doi.org/10.1063/1.1487570. Letellier, C. and Aguirre, L.A. (2010). Interplay between synchronization, observability, and dynamics. Phys. Rev. E, 82, 016204. doi:10.1103/PhysRevE.82.016204. Letellier, C., Aguirre, L.A., and Maquet, J. (2005). Relation between observability and differential embeddings for nonlinear dynamics. Phys. Rev. E, 71, 066213. doi: 10.1103/PhysRevE.71.066213. Maquet, J., Letellier, C., and Aguirre, L.A. (2004). Scalar modeling and analysis of a 3d biochemical reaction model. J. Theor. Biol., 228(3), 421 – 430. doi: http://dx.doi.org/10.1016/j.jtbi.2004.02.004. Maraun, D., Horbelt, W., Rust, H., Timmer, J., Happersberger, H.P., and Drepper, F. (2004). Identification of rate constants and nonobservable absorption spectra in nonlinear biochemical reaction dynamics. Int. J. Bifurcation Chaos, 14(06), 2081–2092. doi: 10.1142/S0218127404010473. M´endez-Gonz´alez, J. and Femat, R. (2016). Steady state equivalence among autocatalytic peroxidase-oxidase reactions. J. Chem. Phys., 145, 225101. doi: 10.1063/1.4968554. Møller, A.C., Hauser, M.J., and Olsen, L.F. (1998). Oscillations in peroxidase-catalyzed reactions and their potential function in vivo. Biophys. Chem., 72(12), 63 – 72. doi:http://dx.doi.org/10.1016/S03014622(98)00123-9. Nijmeijer, H. and van der Schaft, A.J. (1995). Nonlinear dynamical control systems. Springer–Verlag, 3rd edition. Olsen, L.F. (1983). An enzyme reaction with a strange attractor. Phys. Lett. A, 94(9), 454–547. doi: 10.1016/0375-9601(83)90853-8. Poulsen, A.K., Scharff-Poulsen, A.M., and Olsen, L.F. (2007). Horseradish peroxidase embedded in polyacrylamide nanoparticles enables optical detection of reactive oxygen species. Analytical Biochemistry, 366, 29–36. doi:10.1016/j.ab.2007.04.004. Sensse, A. (2005). Convex and toric geometry to analyze complex dynamics in chemical reaction systems. Ph.D. thesis, Otto-von-Guericke-Universitat, Magdeburg. Sensse, A. and Eiswirth, M. (2005). Feedback-loops for chaos in activator-inhibitor systems. J. Chem. Phys., 122, 44516. doi:doi:10.1063/1.1840511. Sensse, A., Hauser, M.J.B., and Eiswirth, M. (2006). Feedback loops for shil’nikov chaos: The peroxidaseoxidase reaction. J. Chem. Phys., 125, 014901. doi: http://dx.doi.org/10.1063/1.2207140. Slotine, J.J.E. and Li, W. (1991). Applied nonlinear control. Prentice-Hall. Steinmetz, C.G. and Larter, R. (1991). The quasiperiodic route to chaos in a model of the peroxidaseoxidase reaction. J. Chem. Phys., 94, 1388. doi: http://dx.doi.org/10.1063/1.459996. 114

Takens, F. (1981). Detecting strange attractors in turbulence. In D. Rand and L.S. Young (eds.), Dynamical Systems and Turbulence, Warwick 1980, volume 898 of Lecture Notes in Mathematics, 366–381. Springer Berlin Heidelberg. doi:10.1007/BFb0091924.