Use of Phase Transition Curves for Testing Models of the pH

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The Use of Phase Transition Curves for Testing Models of the pH-oscillatory HPTS Reaction Tomas Veber, Lenka Schreiberova, and Igor Schreiber J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b00441 • Publication Date (Web): 22 Feb 2016 Downloaded from http://pubs.acs.org on February 22, 2016

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The Journal of Physical Chemistry

The Use of Phase Transition Curves for Testing Models of the pH-oscillatory HPTS Reaction Tomáš Veber, Lenka Schreiberová and Igor Schreiber* Department of Chemical Engineering, University of Chemistry and Technology, Prague, Technická 5, 166 28 Prague 6, Czech Republic KEYWORDS: phase transition curve, response dynamics, pH-oscillations, classification of chemical oscillators.

ACS Paragon Plus Environment

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ABSTRACT: The reaction system hydrogen peroxide-thiosulfate-sulfite in diluted sulfuric acid (HPTS) displays strongly nonlinear dynamics when operated in a continuous-flow stirred tank reactor. Due to a crucial role of hydrogen ion during the reaction this system is a prime example of an inorganic pH-oscillator. Under specific external conditions the system exhibits multiple steady states, periodic oscillations and chaotic behavior. We focus on evaluating alternative kinetic models by exploring phase resetting of the periodic oscillatory regime caused by a singlepulse perturbation with various reacting species. Phase transition curve (PTC) – the plot of phase after the resetting against the phase of perturbation – is a convenient characteristic of the oscillatory dynamics adopted as a major tool in this work. Experimental results for hydrogen ions, hydroxide ions, thiosulfate ions, sulfite ions and hydrogen sulfite ions used as perturbants are systematically compared with calculations under corresponding conditions using two available reaction mechanisms. In addition, we use the stoichiometric network analysis to identify possible core oscillatory subnetworks in the models and choose the one that corresponds best to the measured PTCs.


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I.

INTRODUCTION

Description of the dynamics of forced systems using the concept of a phase response curve (PRC) and the closely related phase transition curve (PTC) is widely accepted and used in many fields where intrinsic rhythmic phenomena influenced by natural or artificial external stimuli occur. How the oscillatory system reacts to external perturbations and what is the relationship between perturbed and unperturbed dynamics is comprehensively treated by Winfree.1 He collected, studied and discussed many examples of theoretical and experimental research focused on phase resetting experiments, particularly in biological oscillators, and brought new theoretical insights into these issues. Winfree has concluded that two different characteristic topological patterns can be distinguished depending on the phase and magnitude of the pulse and it is applicable to any periodic oscillator. Many other researchers have concentrated on studying response dynamics in biological oscillators that range from circadian rhythm2 through neural oscillators3 to cardiac oscillators.4,5 Phase resetting studies of cardiac oscillators are mostly focused on investigation and characterization of response dynamics of the sinoatrial node (SAN), a group of cells responsible for the normal beating of the heart. Guevara and co-workers6–8 investigated spontaneously beating chicken heart cells both by experiment and modelling. They periodically stimulated active heart cells by brief electrical current pulses and constructed PTCs and PRCs. Tsalikakis et al.9 performed numerical simulations using the Zhang model representing a cardiac model. They obtained PTCs for short external electrical pulses of varying amplitude on the electrical activity of peripheral and central SAN cells. Anumonwo et al.10 carried out phase resetting experiments with an enzymatically dissociated rabbit sinus nodal cell and numerical simulations using a single-cell model of the rabbit sinus nodal cell. Their results were compared and evaluated in terms of PRCs and PTCs. The reports on stimulation and


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resetting of neural oscillators are focused on synchronization in neuronal networks11, a system of electrically excitable nerve cells mediating information transfer. Schultheiss et al.12 modeled dendritic and perisomatic excitatory synaptic current inputs on a globus pallidus (GP) and obtained phase response curves. Extensive research on biological oscillators has inspired numerous studies of chemical oscillatory systems in an effort to identify analogies between the two types of systems. Phase resetting experiments in chemical systems can also provide information about mechanistic features of the examined system. In our previous work13 we aimed at classification and identification of the roles of essential species14 in the HPTS15 reaction based on the comparison of PTCs measured in a flow-through stirred reactor with those calculated for prototypes of Category 1 oscillators. In this work the concept of PTCs applied to the HPTS system is developed in two directions. First, the classification of the reaction based on experimental data is compared with the classification of proposed alternative reaction mechanisms; second, the experimental PTCs are directly compared to those obtained from models. The paper is organized as follows: Section II provides a brief report of materials, solutions, conditions and methods of perturbation experiments as well as the calculation background consisting in simulation of perturbation effect on the periodic regime in close agreement with the experimental procedure. A brief outline of the theory of phase resetting and stoichiometric network analysis (SNA) is also included in this section. In section III, we present a reaction mechanism proposed by Rábai and Hanazaki15 and an extended mechanism16 and analyze possible oscillatory subnetworks available in both models. In section IV our experimental results are presented. Calculated PTCs corresponding to the experiments are described in section V. All results are compared and discussed in section VI.


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II.

EXPERIMENTAL AND THEORETICAL METHODS

A. Experimental section. All measurements were carried out in a flow-through stirred cylindrical Plexiglas reaction cell. The volume of liquid in the cell was 17.6 mL. A Tefloncovered magnetic stirrer (1.3 cm long) placed on a metal bottom of the cell was used to ensure uniform mixing. The metal bottom of the cell allowed for a rapid heat exchange between reaction mixture and circulating water from RM6 Lauda E103 thermostat. The temperature in the cell was maintained at 26 ± 0.2°C. Inlet and outlet tubing and electrode probes were inserted through the reactor cap. The reactor was fed through inlet tubes with two reactant solutions by means of the Ismatec IPC peristaltic pump. A separate inlet was used for pulsed perturbations of a chosen perturbant using a syringe pump. Excess liquid from the reactor was continuously removed through an outlet tube. Two voltage probes were placed in the reactor. A glass semi-micro combined pH-electrode (Theta ´90, type HC 139) with the Orion 525A pH-meter for monitoring of pH and the RTD-860 probe connected to an A/D converter for monitoring of temperature. The signals were collected using a personal computer (Octek, Intel Pentium 200MHz) through a measuring card (LabVIEW). The computer also controlled timing of perturbations and registering of reference events. Two reactant solutions were prepared daily from fresh demineralized water and commercially available chemicals. The first solution contained diluted hydrogen peroxide (30% aqueous solution, Penta, Chrudim) and the second contained sodium thiosulfate (Sigma-Aldrich), sodium sulfite (Penta, Chrudim) and sulfuric acid (96%, Lachema a.s., Neratovice). The storage reagents were of p.a. grade. The inlet concentrations of the reactants were: [H2O2]0=0.0135 mol/L, [Na2S2O3]0=0.005 mol/L, [Na2SO3]0=0.0025 mol/L, [H2SO4]0=5.10-4 mol/L. These water solutions were bubbled with nitrogen for at least twelve hours before the experiments to avoid


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any effect of carbon dioxide. Both reactant solutions were stored at room temperature. The hydrogen peroxide stock solution was standardized with permanganate and kept in cool conditions to retard its decomposition. Sodium sulfite and sodium thiosulfate solutions were titrated with a standardized iodine solution. The concentration of sulfuric acid was checked using sodium hydrogen carbonate. B. Experimental techniques. An experimental procedure used in this study follows the same principle and conditions as described in our previous work.13 After filling the reaction cell with both feed solutions the other external constraints were set to bring the reaction system to a regular oscillatory regime. These external constraints were the temperature of the reaction mixture kept at 26 ± 0.2°C and the continuous-flow through the reaction cell with the rate (reciprocal residence time) set at k0 = 0.00252 s-1. We applied pulse perturbations of 0.0547 mL delivering a chosen perturbant within a short time interval (one or two seconds) to the ongoing oscillations using a computer-controlled syringe pump. A few free-running cycles were allowed between two consecutive perturbations to restore the original periodic cycle. Each perturbation is specified by two parameters: the concentration and the volume of added species solution. In our experiments we kept constant volume and varied concentration. To obtain a quantitative measure of the perturbation we introduce an amplitude of the perturbation ∆c as the moles of the added species divided by the volume of the liquid in the reactor. Aqueous solutions of hydrogen ions, hydroxide ions, thiosulfate ions, sulfite ions and hydrogen sulfite ions were used as perturbants. The stock solutions were kept carbon dioxide free also during the experiments by bubbling with nitrogen. The PTCs were measured as described in our previous work.13 Namely, a pulse addition to sustained pH-oscillations at phase φ is applied by injecting a chosen species. The pulse corresponds to an immediate concentration increase ∆c which we call the perturbation


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amplitude. Following the perturbation, a sequence of times corresponding to reference events chosen as crossings (in downward direction) of the reference value of pH = 6 was measured and used to determine both the phase shift ∆φ and the reestablishment of periodic oscillations. By adding the value of the phase shift to the phase of perturbation a point of the PTC is obtained. Experimental data sets are fitted and simultaneously plotted by using a nonlinear regression tool of the GNUPLOT17 program. A suitable regression function18 for the type 1 PTC is

Θ = a0 +

a1 sin (2π ϕ + a3 ) + a5 +ϕ a 2 cos(2π ϕ + a 4 ) + a 6

(mod1) ,

(1)

and the type 0 PTC is fitted by

Θ = a0 +

a1 sin (2π ϕ + a 3 ) + a5 a 2 cos(2π ϕ + a 4 ) + a 6

(mod1) ,

(2)

where φ is the phase prior to perturbation ("old" phase) and Θ is the asymptotic phase after the periodic oscillations are reestablished ("new" phase). The regression curves are used below as a convenient guide to distinguish primary phase advance (φ < Θ < φ + 0.5 < 1), primary phase delay (0 < φ − 0.5 < Θ < φ), secondary phase advance (0 < Θ < φ − 0.5) and secondary phase delay (φ + 0.5 < Θ < 1). Secondary phase delay occurs when primary phase advance exceeds half-cycle whereas secondary phase advance occurs when primary phase delay exceeds halfcycle. Also, convex/concave pattern of the curve is taken into account as well as location of the cross-point(s) of the regression curve with the primary diagonal (Θ = φ) and/or secondary diagonal (Θ = φ + 0.5 (mod 1)). C. Numerical techniques. Calculations were performed using the software package CONT19 which, in addition to numerical continuations, allows for direct dynamical simulations and Poincare intersections, which in turn can be conveniently used to determine the PTCs. In the


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calculations the inlet concentrations of reactants are set to correspond to experiments, but the models have their limitations and thus the value of flow rate needs to be adjusted to obtain regular oscillations matching the experiments as much as possible. As an appropriate value of the flow rate for the basic model we use k0 = 0.00245 s-1, while for the extended model the adjusted value of flow rate is k0 = 0.004 s-1. To keep consistence with the experiments, the chosen reference pH value for determining the phase shifts is identical with that in our experiments. The perturbation was simulated as an (Dirac) impulse with amplitude equal to ∆c used in the experiments. The resulting phase transition curves were calculated as follows. First, at chosen values of inlet concentrations and flow rate a numerically accurate periodic orbit is calculated and represented by a chosen number of points covering densely the limit cycle (typically a few hundred). The zero phase of oscillations is arbitrarily chosen to correspond to the reference event (i.e., pH = 6) which marks the fastest rate of autocatalysis. For each point on the cycle the perturbation is applied and the sequence of reference events is indicated as Poincare intersections until periodic regime is reestablished to an accuracy of 0.001 in the interspike time intervals, which typically required only a few oscillations. The phase shift and subsequently the point on the PTC are recovered from the cumulative interspike time interval. In this way PTCs for hydrogen ions, hydroxide ions, thiosulfate ions, sulfite ions and hydrogen sulfite ions are obtained. D. Stoichiometric network analysis and classification of oscillatory mechanisms. A chemical network is a set of n chemical species and m chemical reactions with known kinetics. The time evolution in a spatially homogeneous system at constant temperature is based on mass balance equations that may be written in a compact form as

dc = N r(c) , dt

(3)


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where c is the n-vector of concentrations of the species, r(c) is the m-vector of the rate laws and N is the stoichiometric matrix. Eq 3 may be readily extended to represent a flow system provided that N and r involve pseudoreactions corresponding to inflows and outflows (zero and first order terms, respectively). According to the stoichiometric network analysis (SNA), stability of steady states of eq 3 can be conveniently analyzed by determining the structure of the null space of N which corresponds to an hierarchy of subnetworks, and examining stability of the subnetworks that contribute to the analyzed steady state.20 Of particular interest are subnetworks that induce oscillatory instabilities via Hopf bifurcation. Furthermore, the subnetworks accounting for oscillations can be used to classify the chemical mechanism according to the arrangement of positive and negative feedback loops and to determine the role of species in the oscillations.14,21 There are three major types of species playing an essential role in the oscillations: type X is the autocatalytic species that appears in the autocatalytic loop; type Z provides negative feedback controlling the oscillations and type Y removes the type X species from the autocatalytic cycle. Based on different topologies of interaction between these species, two categories (called 1 and 2) of chemical oscillators are recognized, each having two subcategories (called B and C). Here 1 stands for first-order autocatalysis (most experimental systems fall in this group), B stands for batch system with internally generated negative feedback and C for continuous system with flow-assisted negative feedback.14,21 At the Hopf bifurcation or in the oscillatory regime, various methods can be applied to determine the role and category, for example, by calculating mutual phase shifts of the essential species. In our previous work we examined the possibility of using the PTCs for that purpose13 and below we use those results to characterize alternative mechanisms for the HPTS reaction.


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III.

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MODEL FORMULATION AND NETWORK ANALYSIS

A. Characteristics of the HPTS reaction. The reaction of hydrogen peroxide with thiosulfate and sulfite belongs to the extensive family of pH-regulated oscillators because the concentration of hydrogen ions or concentrations of hydroxide ions are the key governing species responsible for the dynamics. The HPTS system displays a wide spectrum of dynamic behavior as one of the control parameters is varied. Complex dynamics of this reaction is very rich, ranging from multiple stable steady states to regular periodic oscillations to chaotic behavior, which occur over a wide range of operating conditions in the continuous-flow stirred tank reactor. Dynamical regimes depend on external constraints including temperature, flow rate, presence of carbon dioxide in the reaction mixture and feed concentrations. The reaction is quite temperature sensitive; oscillations arise in a limited range of temperature which may also influence the amplitude and period of the oscillations.22 However, the significance of the HPTS reaction consists in the first experimental observation of the temperature-compensation in a chemical system22 akin to temperature-compensated biological systems such as circadian clock. Rábai and Hanazaki22 performed detailed numerical simulations and experimental measurements building on their earlier studies and observed that temperature-compensation of oscillatory period exists under specific conditions. Recently Rábai et al.23 discovered that temperatureinduced period-doubling route from simple low-frequency pH-oscillations to chemical chaos and a reverse transition from chaos to high-frequency oscillations with rather good reproducibility occur in this system. B. Basic mechanism. A basic mechanism of the HPTS reaction due to Rábai and Hanazaki15 is derived from a previously proposed mechanism for a complex reaction between hydrogen peroxide and thiosulfate ion catalyzed by a small amount of cupric ions (HPTCu) proposed by


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Kurin-Csörgei et al.24 Rábai and Hanazaki eliminated direct catalytic effect of cupric ion on the oxidation of thiosulfate by hydrogen peroxide from the original HPTCu reaction mechanism and dissociation equilibrium of hydrogen peroxide to the H + and HO −2 was also neglected. They found that the resulting eight step catalyst free hydrogen peroxide-thiosulfate reaction mechanism consists of two core oscillatory parts. However, the two core mechanism leads to a different type of pH oscillations than found experimentally in the HPTS system. These observations lead them to redesign the mechanism, where they also took into account that oscillations were observed only when sulfite was introduced in the feed. They proposed the following basic mechanism consisting of seven steps with the corresponding kinetics as shown in Table 1: H 2 O 2 + S 2 O 32 − → HOS 2 O 3− + OH −

(R1)

H 2 O 2 + HOS 2 O 3− →2HSO 3− + H +

(R2)

S 2 O 3 + S 2 O 32− → S 4 O 62−

(R3)

H 2 O ⇔ H + + OH −

(R4)

H+

H 2 O 2 + HSO 3− → SO 24− + H 2 O + H +

(R5)

HSO 3− ⇔ H + + SO 32 −

(R6)

HOS 2 O 3− + H + ⇔ S 2 O 3 + H 2 O

(R7)


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Table 1. Rate laws and rate constants for the basic mechanism.15 Reaction

Rate law

Rate constant

v1 = k1 [H 2 O 2 ][S 2 O 32− ]

k1 =1.9 × 10 −2 dm 3 mol −1 s −1

(R1)

v 2 = k 2 [H 2 O 2 ][HOS 2 O 3− ]

k 2 =1.0 × 10 −2 dm 3 mol −1 s −1

(R2)

v3 = k 3 [S 2 O 3 ] [S 2 O 32− ]

k 3 = 5.0 dm 3 mol −1 s −1

(R3)

v 4 = k 4 [H 2 O] − k −4 [H + ] [OH − ]

k 4 [H 2 O]=1.0 × 10 −3 mol1 dm −3 s −1 , k −4 =1.0 ×1011 dm 3 mol −1 s −1

v5 = (k 5 + k 5* [H + ]) [H 2 O 2 ][HSO 3− ]

k 5 = 4.0 dm 3 mol −1 s −1 , k 5* =1.0 × 10 7 dm 3 mol −1 s −1

v 6 = k 6 [HSO 3− ]− k −6 [H + ] [SO 32− ]

(R4)

(R5)

k 6 = 3 .0 × 10 3 s −1 , k −6 = 5 .0 × 1010 dm 3 mol −1 s −1

v7 = k 7 [HOS 2 O 3− ] [H + ] − k −7 [S 2 O 3 ]

(R6)

k 7 =1.0 ×10 3 dm 3 mol −1 s −1 , k −7 = 2.0 ×10 −2 s −1

(R7)

C. Extended mechanism. All steps except R2 are used from the previous mechanism but, in addition, some reactions from the mechanism proposed for the hydrogen peroxide-thiosulfate reaction25 combined with reactions involving various polythionates, disulfite and disulfurtrioxide considered. The extended mechanism is as follows (for the corresponding rate laws see Table 2):

H 2 O 2 + S 2 O 32− → HOS 2 O 3− + OH −

(R1)

S 2 O 3 + S 2 O 32 − → S 4 O 62 −

(R3)

H 2 O ⇔ H + + OH −

(R4)


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H+

H 2 O 2 + HSO 3− → H + + SO 24− + H 2 O

(R5)

HSO 3− ⇔ H + + SO 32 −

(R6)

HOS 2 O 3− + H + ⇔ S 2 O 3 + H 2 O

(R7)

H 2 O 2 + HOS 2 O 3− → S 2 O 52− + H + + H 2 O

(R8)

S 2 O 32− + HOS 2 O 3− → S 4 O 62− + OH −

(R9)

H 2 O 2 +S 2 O 52 − → HSO 3− + H + + SO 24 −

(R10)

H 2 O 2 ⇔ H + + HO 2−

(R11)

H 2 O 2 + SO 32 − → SO 24 − + H 2 O

(R12)

S 2 O 52− + H 2 O → 2HSO 3−

(R13)

HO 2− + HOS 2 O 3− → S 2 O 52− + H 2 O

(R14)

S 4 O 62 − + HO −2 → S 2 O 32 − +S 2 O 52− + H +

(R15)

S 4 O 62− + SO 32− → S3 O 62− +S 2 O 32−

(R16)

S 2 O 52 − + HOS 2 O 3− → S 3 O 62− + HSO 3−

(R17)

HSO 3− + HOS 2 O 3− → S3 O 62− + H 2 O

(R18)

SO 23− + HOS 2 O 3− → S 3O 62− + OH −

(R19)

S 2 O 3 + H 2 O 2 → S 2 O 52− + 2H +

(R20)

S 2 O 3 + HSO 3− → S3 O 62− + H +

(R21)

H + + CO 23 − ⇔ HCO 3−

(R22)

H + + HCO 3− ⇔ H 2 CO 3

(R23)

H 2 CO 3 ⇔ CO 2 + H 2 O

(R24)


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Table 2. Rate laws and rate constants for the extended mechanism.16, 25 Reaction

Rate laws

Rate constants

v1 = k1 [H 2 O 2 ] [S 2 O 32− ]

k1 = 2.3 ×10 −2 dm 3 mol −1 s −1

(R1)

v3 = k 3 [S 2 O 3 ] [S 2 O 32− ]

k 3 = 7.0 ×10 2 dm 3 mol −1 s −1

(R3)

v 4 = k 4 [H 2 O] − k −4 [H + ] [OH − ]

k 4 [H 2 O]=1.0× 10 −3 mol1 dm −3 s −1 , k − 4 =1.0 ×1011 dm 3 mol −1 s −1

v5 = (k 5 + k 5* [ H + ]) [H 2 O 2 ] [HSO 3− ]

k 5 = 7.0 dm 3 mol −1 s −1 , k 5* =1.5 × 10 7 (dm 3 mol −1 ) 2 s −1

v 6 = k 6 [HSO 3− ]− k −6 [H + ] [SO 32− ]

(R5)

k 6 = 3.0 ×10 3 s −1 , k −6 = 5.0 ×1010 dm 3 mol −1 s −1

v7 = k 7 [HOS 2 O 3− ] [H + ] − k −7 [S 2 O 3 ]

(R4)

(R6)

k 7 = 4.0 ×10 7 dm 3 mol −1 s −1 , k −7 = 4.0 s −1

(R7)

v8 = k 8 [HOS 2 O 3− ] [H 2 O 2 ]

k 8 ≈ 50 dm 3 mol −1 s −1

(R8)

v9 = k 9 [HOS 2 O 3− ] [S 2 O 32− ]

k 9 ≈ 50dm 3 mol −1 s −1

(R9)

v10 = k10 [S 2 O 52− ][H 2 O 2 ]

k10 ≈ 5.0 dm 3 mol −1 s −1

(R10)

v11 = k11 [H 2 O 2 ]− k −11 [H + ] [HO 2− ]

k11 = 2.5 ×10 −2 s −1 , k −11 =1.0 ×1010 dm 3 mol −1 s −1

(R11)

v12 = k12 [SO 32− ] [H 2 O 2 ]

k12 = 0. 2 dm 3 mol −1 s −1

(R12)

v13 = k13 [S 2 O 52− ]

k13 = 5.0 ×10 −3 s −1

(R13)

v14 = k14 [HOS 2 O 32− ] [HO −2 ]

k14 =1.0 ×10 5 dm 3 mol −1 s −1

(R14)

v15 = k15 [S 4 O 62− ] [HO −2 ]

k15 = 0.60 dm 3 mol −1 s −1

(R15)

v16 = k16 [S 4 O 62− ] [SO 32− ]

k16 = 0.27 dm 3 mol −1 s −1

(R16)

v17 = k17 [HOS 2 O 3− ] [S 2 O 52− ]

k17 ≈ 4.0 ×10 4 dm 3 mol −1 s −1

(R17)

v18 = k18 [HOS 2 O 3− ] [HSO 3− ]

k18 = 5.0 dm 3 mol −1 s −1

(R18)

v19 = k19 [HOS 2 O 3− ] [SO 32− ]

k19 = 5.0 dm 3 mol −1 s −1

(R19)


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v 20 = k 20 [S 2 O 3 ] [H 2 O 2 ]

k 20 = 4.0 dm 3 mol −1 s −1

(R20)

v 21 = k 21 [S 2 O 3 ] [HSO 3− ]

k 21 ≤ 5.0 × 10 4 dm 3 mol −1 s −1

(R21)

v 22 = k 22 [H + ] [CO 32− ]− k −22 [HCO 3− ]

k 22 =1.0 ×1011 dm 3 mol −1 s −1 , k − 22 = 4.8 s −1

v 23 = k 23 [H + ] [HCO 3− ]− k −23 [H 2 CO 3 ]

k 23 = 5 .0 ×1010 dm 3 mol −1 s −1 , k − 23 = 8.6 ×10 6 s −1

v 24 = k 24 [H 2 CO 3 ]− k − 24 [CO 2 ]

(R22)

(R23)

k 24 =16.5 s −1 , k − 24 = 4.3 ×10 −2 s −1

(R24)

Remark: Values of the rate coefficients of shared reactions (R1), (R3), (R5) and (R7) differ in the two mechanisms.

D. Model formulation. Since our experimental system is an isothermal continuous-flow well-stirred tank reactor with a constant reaction volume, the basic mass balance equations for n reacting species taking part in r reactions constitute a system of ordinary differential equations: r d xi = vi (x1 ,L, xn ) = ∑ υi j r j + k 0 (x0i − xi ) , i =1,L ,n , dt j =1

(4)

where x i is the concentration of species i, v i is the net rate of change due to reactions involving species i and inflow/outflow, k 0 is the flow rate and x 0i is the inflow concentration of species i. The reaction rates r j and stoichiometric coefficients υi j are given in section III for each mechanism. The basic mechanism includes 8 reacting species and 7 reactions, while the extended mechanism has 11 species (inert products are not independent dynamical species, water is assumed in excess and of constant concentration) taking part in 23 reactions of which the last three are omitted, because experiments were done in the absence of carbon dioxide.


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E. Identification of oscillatory subnetworks. For both mechanisms we reformulate eq 4 in the format of eq 3 by formally treating inflows as zero order reactions and outflows as first order reactions and apply the stoichiometric network analysis. This enables us to determine interplay of positive and negative feedbacks leading to oscillations and identify subnetworks shown below that may underlie the experimentally observed oscillations. Relevance to experimental observations is discussed in section IV. For convenience, Figure 1a,b shows graphical representations of the two mechanisms – the so called networks diagrams, which display each reaction as a multiple-tail multiple-head arrow connecting reactants and products, where the number of barbs at any head is the stoichiometric coefficient of the corresponding product and the number of feathers at any tail is the stoichiometric coefficient of the corresponding reactant (feather is omitted when the coefficient is equal to 1).


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Figure 1. Network diagrams of the two HPTS mechanisms: (a) basic mechanism (reaction 5 is split into noncatalyzed (R5) and catalyzed (R5*) parts, (b) extended mechanism (reactions R5, R11, R14, R15 and R22-24 are omitted due to their marginal role in instabilities). Inflows and outflows are included, inert products are omitted, square box around H2O indicates fixed concentration.

Basic mechanism. The SNA predicts that the system possesses several candidate oscillatory subnetworks. As already discussed by Rábai and Hanazaki15 the most important feature of the reaction mechanism is the autocatalytic production of hydrogen ions H + provided via oxidation of hydrogen sulfite ions HSO 3− by hydrogen peroxide H 2 O 2 (step R5). However, this reaction alone does not generate instability/positive feedback. The SNA indicates that an unstable


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subnetwork must involve removal of H + via a bimolecular step (exit reaction), which in this case may be provided by either of the steps R4 or R6. Such an instability may only be a bistable system, an oscillatory instability requires a specific type of negative feedback.26 In ref 15, the role of R3 together with R1 and R7 in negative feedback is emphasized. However, the SNA predicts that negative feedback is an integral part of the autocatalytic step R5 where HSO 3− controls autocatalysis by its limited availability (tangent negative feedback). Rather than being provided directly by inflow, HSO 3− is produced via two different pathways, either through reaction R2 or through protonation of sulfite in R6. Accordingly, there are two types of subnetworks generating oscillatory behavior. The first one consists of the positive feedback part involving R5 and R4 and the negative feedback part involving R2. Namely, the subnetwork combining R1, R2, R4 and R5 is oscillatory. The role of species in the oscillations is as follows: H + is of type X, HSO 3− of type Z and OH − is of type Y, the rest of species are nonessential for oscillations (i.e., can be buffered without losing the oscillations). Notice that this subnetwork does not require inflow of sulfite. The second type involves R5 and R6 only, because the negative feedback species HSO 3− is directly produced by the exit reaction R6. The type X and Z species are the same as before but the type Y species is SO 32− . This oscillatory subnetwork does not require feed of thiosulfate. Apart from the two outlined oscillatory subnetworks, there are others that should be considered. Interestingly, the instability in these additional subnetworks is not associated with an autocatalytic cycle. Rather, they are built around a topological feature called competitive autocatalysis, which is distinguished by possessing two non-cyclic species acting like type X species competing for a type Y species via two exit-like reactions. In addition, the two type X


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species must be linked in a consecutive chain of reactions. An example of this situation is a subnetwork consisting of steps R1, R2, R7 and R3. Here H 2O 2 and S2O 3 compete for S2 O 32− via R1 and R3, and they are linked through R2 and R7. The type X species are H 2O 2 , H + and S2O 3 , the type Y species is S2 O 32− and the type Z species is HOS2 O 3− . An interesting feature is that the essential species involve the disulfurtrioxide S2O 3 . Which of the three subnetworks (or a combination thereof) is a realistic core oscillator will be discussed after evaluation of experimentally obtained PTCs.

Extended mechanism. Upon noticing that the sum of the steps R8 and R13 yields R2, oscillatory subnetworks identified by the SNA are in part analogous to those described above with the minor modification of using R8 and R13 in place of R2. Thus all three types of core oscillators as in the basic model are present. However, an additional type of oscillatory subnetworks exists owing to the extended set of reactions. It is based on an autocatalytic cycle mediated by reactions R7 and R20 which in combination with the exit reaction R4 or R6 produces a positive feedback. Negative feedback necessary for oscillations is due to limited availability of HOS 2 O 3− . Thus the autocatalytic species (type X) in such an oscillator are H + and S2O 3 , the exit species (type Y) is either OH − or SO 32− and the negative feedback species (type Z) is HOS2 O 3− . In the following we will use the PTCs as a comparative tool to discriminate which of the mechanisms gives better agreement with the experiments and which subnetwork is at the core of observed oscillations.

IV.

PTCs OBTAINED FROM EXPERIMENTS

As mentioned in section II, results presented here extend our previous work.13


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Figures 2 – 6 shows representative experimental results of sequential single pulse perturbations applied to sustained pH-oscillations at various initial phases. By using this approach we determined the PTCs for pulses of a given amplitude ∆c. The data were fitted to eq 1or 2 and the resulting PTCs include data from at least two measurements with varying old phase through the entire range.

(a)

(b)

Figure 2. Phase transition curves obtained from experiments with single pulse additions of H+. Amplitude of perturbations: (a) ∆c(H+) = 2.1756 × 10-6 mol/L; (b) ∆c(H+) = 1.8648 × 10-5 mol/L. Blue squares (■) correspond to experimental data and red curve (─) displays the fit of experimental data. Conditions: [H2O2]0 = 0.0135 mol/L, [Na2SO3]0 = 0.0025 mol/L, [Na2S2O3]0=0.005 mol/L, [H2SO4]0 = 5 × 10-4 mol/L, k0 = 0.00252 s-1, t = 26 ± 0.2°C.

Perturbations by H+. In the first series of experiments hydrogen ion was chosen as the perturbant. In Figure 2a the PTC of type 1 is found. Although the experimental data near the


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reference point are quite dispersed the regression curve consistently indicates initial phase delays followed by pronounced phase advances in the second half of the cycle. Figure 2b provides a well developed PTC of type 0 with a maximum followed by a minimum. In other words, the corresponding PTC has the concave/convex order, with secondary delay in the first half of the cycle turning into primary advance and primary delay in the second half-cycle. The PTC at phases near φ = 0.4 exhibits two points of apparent discontinuities where the secondary delay becomes primary delay and vice versa as indicated in Figure 2b. By comparison with the corresponding type 0 PTC from our previous work13 an increase of the concentration of injected solution leads to a shift of the data points downwards.

(a)

(b)

Figure 3. Phase transition curves obtained from experiments with single pulse additions of OH−. Amplitude of perturbations: (a) ∆c(OH−) = 1.2431 × 10-5 mol/L; (b) ∆c(OH−) = 2.4864 × 10-5 mol/L. Blue squares (■) correspond to experimental data and orange curve (─) displays the fit of experimental data. Conditions: [H2O2]0 = 0.0135 mol/L, [Na2SO3]0 = 0.0025 mol/L, [Na2S2O3]0=0.005 mol/L, [H2SO4]0 = 5 × 10-4 mol/L, k0 = 0.00252 s-1, t = 26 ± 0.2°C.


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Perturbations by OH–. The next series of experiments were focused on perturbation by solutions of sodium hydroxide (Figure 3). In contrast to the previous case, the PTC of type 1 (Figure 3a) for the perturbation by base is characterized by phase advance followed by phase delay and occurrence of pronounced maximum and minimum of Θ. Figure 3b corresponds to the PTC of type 0 again displaying apparent discontinuities. Perturbations in a middle range of the cycle (around φ = 0.4-0.6) imply a transition from secondary to primary delay at the left boundary and another transition from primary to secondary advance at the right boundary of the range thus providing two apparent discontinuities.

(a)

(b)

Figure 4. Phase transition curves obtained from experiments with single pulse additions of HSO3−. Amplitude of perturbations: (a) ∆c(HSO3−) = 3.1079 × 10-6 mol/L; (b) ∆c(HSO3−) = 2.1755 × 10-5 mol/L. Blue squares (■) correspond to experimental data and light blue curve (─) displays the fit of experimental data. Conditions: [H2O2]0 = 0.0135 mol/L, [Na2SO3]0 = 0.0025 mol/L, [Na2S2O3]0=0.005 mol/L, [H2SO4]0 = 5 × 10-4 mol/L, k0 = 0.00252 s-1, t = 26 ± 0.2°C.


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Perturbations by HSO3–. In our experiments with perturbations by the solution of hydrogen sulfite (Figure 4) both topological types of the PTC were found again. The type 1 PTC in Figure 4a is initially convex with phase delay and at about φ=0.4 passes through a cross-point and displays phase advance. When compared with the cross-point of type 1 PTC for hydrogen ion (Figure 2a) the location of cross-point is shifted to the left. This feature is consistent with our previous experimental results13 and leads to conclusion that with increasing amplitude of perturbation the cross-point is systematically shifting to the left. When a relatively large amplitude of perturbation is applied, a well-developed type 0 PTC with maximum and minimum is found (Figure 4b). A somewhat different shape of the type 0 PTC for hydrogen ion corroborates our previous conclusion13 about different roles of both species in the reaction mechanism (see Discussion).

(a)

(b)

Figure 5. Phase transition curves obtained from experiments with single pulse additions of SO32−. Amplitude of perturbations: (a) ∆c(SO32−) = 3.1080 × 10-5 mol/L; (b) ∆c(SO32−) = 1.5540


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× 10-4 mol/L. Blue squares (■) correspond to experimental data and violet curve (─) displays the fit of experimental data. Conditions: [H2O2]0 = 0.0135 mol/L, [Na2SO3]0 = 0.0025 mol/L, [Na2S2O3]0=0.005 mol/L, [H2SO4]0 = 5 × 10-4 mol/L, k0 = 0.00252 s-1, t = 26 ± 0.2°C.

Perturbations by SO32–. In the following series of experiments, the system was forced by adding sulfite ion as displayed in Figure 5. There is a monotonous type 1 PTC (Figure 5a) with no maximum and minimum unlike the corresponding type 1 PTC for a higher amplitude of perturbation presented in our previous work.13 The present curve shows initially phase advances and subsequently phase delays. Figure 5b displays the PTC of type 0.

(a)

(b)

Figure 6. Phase transition curves obtained from experiments with single pulse additions of S2O32−. Amplitude of perturbations: (a) ∆c(S2O32−) = 6.8375 × 10-4 mol/L; (b) ∆c(S2O32−) = 7.7699 × 10-4 mol/L. Blue squares (■) correspond to experimental data and green curve (─)


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displays the fit of experimental data. Conditions: [H2O2]0 = 0.0135 mol/L, [Na2SO3]0 = 0.0025 mol/L, [Na2S2O3]0=0.005 mol/L, [H2SO4]0 = 5 × 10-4 mol/L, k0 = 0.00252 s-1, t = 26 ± 0.2°C.

Perturbations by S2O32–. Finally, we also perturbed the system by thiosulfate and obtained rather peculiar results. The curve displayed in Figure 6a corresponds to the PTC of type 1. It is monotonous and displays a range where the phase follows the secondary diagonal and after an apparent point of discontinuity near the midpoint of the cycle, the phase is within the primary delay belt. The regression curve has no cross-points with the primary diagonal, which is quite unusual. This feature would imply a highly skewed primary resonance wedge (Arnold tongue) in a corresponding periodically perturbed system. However, as shown in our previous work13 the expected pair of cross-points exists at a lower perturbation amplitude. Figure 6b apparently represents a PTC of type 0, even though there is an ambiguity of new phases above φ > 0.5 reflecting different outcomes from our two measurements. We chose to draw the regression curve through the set of experimental data corresponding to the type 0 PTC. Despite this ambiguity the conclusions about the role of sulfite can be drawn clearly, see section VI.

V.

PTCs CALCULATED FROM MODELS

Our simulations using eq 4 with the basic15 and extended mechanisms16 show PTCs (Figures 7-11) for the corresponding perturbants and the external constraints set as in the experiments except for the flow rate, which is for each model adjusted so as to fall within the model’s oscillatory range. Each figure includes four representative PTCs chosen within a proper range of the concentration amplitudes. The chosen concentration range used in simulations takes into account the experimental results presented here and reported in ref 13. The color coding of the


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curves in each graph is arranged according to the amplitude of the perturbation, from dark blue through light blue to green to red. The main purpose of the graphs is to obtain a representative set of PTCs to be compared with the experimental counterparts. (a)

(b)

Figure 7. Calculated phase transition curves for perturbation by H+: (a) PTCs for the basic mechanism; (b) PTCs for the extended mechanism. Parameters: [H2O2]0 = 0.0135 mol/L, [Na2SO3]0 = 0.0025 mol/L, [Na2S2O3]0=0.005 mol/L, [H2SO4]0 = 5 × 10-4 mol/L; the flow rate chosen within the oscillatory region: (a) k0 = 0.00245 s-1; (b) k0 = 0.004 s-1. First we constructed PTCs for stimulations by hydrogen ions (Figure 7). The curves of type 1 for both models yield the same convex/concave order. Within the region of phase delays the curves are convex while they are concave in the region of phase advance. As a convenient distinguishing feature between the PTC of type 1 for hydrogen ion stimulations the position of cross-points on the primary diagonal line can be used. For the basic mechanism (Figure 7a) the


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two cross-points are located at low phase (0 < φ < 0.1) and high phase (φ ≈ 0.9) creating a broad range of phase advance. For the extended mechanism the cross-points are found at medium to high values of φ (0.65 < φ < 0.9) forming a much narrower range of phase advance. In addition, there is also a corresponding difference between the type 0 PTCs in both mechanisms. The same value of amplitude of the perturbation (red curve in Figure 7a and green curve in 7b) implies higher values of the new phase and a negative slope at the intersection with the secondary diagonal in the extended model.

(a)

(b)

Figure 8. Calculated phase transition curves for perturbation by OH−: (a) PTCs for the basic mechanism; (b) PTCs for the extended mechanism. Parameters: [H2O2]0 = 0.0135 mol/L, [Na2SO3]0 = 0.0025 mol/L, [Na2S2O3]0=0.005 mol/L, [H2SO4]0 = 5 × 10-4 mol/L; the flow rate chosen within the oscillatory region: (a) k0 = 0.00245 s-1; (b) k0 = 0.004 s-1.


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Figure 8 shows PTCs for stimulation by hydroxide ions. The shape of calculated PTCs indicates an expected opposite effect of OH − when compared to that caused by H + and likewise reveals considerable differences in both models. The curves of type 1 go through the concave/convex/concave sequence. The type 1 PTCs for the basic mechanism (Figure 8a) show a distinct spike at values of φ immediately above zero followed by a much broader trough. At significantly larger amplitudes of perturbation the type 0 PTC (red curve) loses the spike. Instead, an asymmetric maximum/minimum pair occurs, connected by a steeply decreasing branch of the PTC. In contrast, the extended model (Figure 8b) yields type 1 PTC with gradual maximum/minimum shifted to the right. As the perturbation amplitude grows, the maximum shifts to the left while the minimum is virtually stagnant until the type 0 PTC results with a fairly symmetric pair of gradual extremes.

(a)

(b)

Figure 9. Calculated phase transition curves for perturbation by HSO3−: (a) PTCs for the basic mechanism; (b) PTCs for the extended mechanism. Parameters: [H2O2]0 = 0.0135 mol/L,


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[Na2SO3]0 = 0.0025 mol/L, [Na2S2O3]0=0.005 mol/L, [H2SO4]0 = 5 × 10-4 mol/L; the flow rate chosen within the oscillatory region: (a) k0 = 0.00245 s-1; (b) k0 = 0.004 s-1.

Phase transition curves calculated for hydrogen sulfite stimulations (Figure 9) at different amplitudes of perturbation are very similar to those shown in Figure 7 in both mechanisms. Only a slightly different feature can be seen in the extended mechanism, see green curves in Figure 7b and in Figure 9b. Here the same amplitude results in type 0 PTC for H + but type 1 PTC for HSO 3− . Such differences are expected close to the transition between type 1 and type 0 PTCs since the curve is extremely sensitive to amplitude variations in that range.

(a)

(b)

Figure 10. Calculated phase transition curves for perturbation by SO32−: (a) PTCs for the basic mechanism; (b) PTCs for the extended mechanism. Parameters: [H2O2]0 = 0.0135 mol/L,


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For stimulations by sulfite ion the resulting PTCs (Figure 10) are reminiscent of those for hydroxide ion (Figure 8). For the basic mechanism (Figures 8a and 10a), the type 1 PTCs have similar shape including the spike at low values of φ, but the type 0 PTC is shifted upwards. Likewise, there is considerable similarity between responses to SO 32− and OH − perturbations when using the extended model (Figures 8b and 10b). As the amplitude is increasing, the inconspicuous maximum of the type 1 PTC shifts to the left in both cases until type 0 PTC occurs, which is again shifted upwards in the present case.

(a)

(b)

Figure 11. Calculated phase transition curves for perturbation by S2O32−: (a) PTCs for the basic mechanism; (b) PTCs for the extended mechanism. Parameters: [H2O2]0 = 0.0135 mol/L,


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Finally, we examined the effect of stimulation by thiosulfate ion. In both mechanisms we found that no cross-point with the primary diagonal occurs in the range of our studied parameters. When the stimulation amplitude is increased, the basic mechanism displays initially only phase delayed PTCs, which eventually include also phase advance, whereas the extended mechanism displays initially phase advanced PTCs, which eventually include also phase delays.

VI.

DISCUSSION

The comparison of the specific shape of the PTCs predicted by the models and experiments using the systematic approach developed in ref 13 gives us the opportunity not merely to choose from the two mechanisms but to distinguish between specific features of the mechanisms. A distinctive feature of the calculated transition curves shared by both mechanisms is a considerable similarity between the effect of hydrogen ion and hydrogen sulfite ion (Figure 7 and 9). Both mechanisms also display similarity between hydroxide ion and sulfite ion (Figure 8 and 10), even though both models consistently show higher values of the new phase in the type 0 PTC for sulfite ion. This suggests that in both models the indicated species play the same mechanistic role. The models differ in the effect caused by perturbing with thiosulfate. For sufficiently small amplitude of perturbation the basic model displays phase delays only while the extended model predicts phase advance only. Generally, the above features are in line with experiments when considering the results from our previous study13 in combination with the results presented here. However, to uncover more


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subtle relations between experimental and calculated PTCs a direct focus on each species separately is in order. Starting with hydrogen ion, calculated PTCs for the extended mechanism (Figure 7) are corresponding more tightly with those obtained from experiments (Figures 2a,b and Figures 6a,b in ref 13). This claim is supported especially by the presence of a pair of crosspoints in the type 1 PTC locating the range of phase advance to higher values of φ and by a gradual waveform of the type 0 PTC shifted to higher values of Θ. The same conclusion holds for the hydrogen sulfite PTCs. Again, both type 1 and type 0 experimentally determined PTCs (Figures 4a,b, see also Figures 8a,b in ref 13) correspond better with the curves based on the extended mechanism (Figure 9b). The experimental PTCs for hydroxide ions (Figures 3a,b, see also Figures 7a,b in ref 13) follow opposite trend to hydrogen ion PTCs which was also found in the simulations (Figure 8). The basic mechanism (Figure 8a) corresponds better to experiments resulting in the type 1 PTC. The type 0 PTCs obtained for either of the mechanisms differ significantly from the experimental response curves, particularly in the average value of Θ. Experimental PTCs for sulfite ion (Figure 5a,b, see also Figures 9a,b in ref 13) are partly captured in both mechanisms. The presence of a narrow range of phase advances just above the origin followed by a wide range of phase delays in the type 1 PTC points to the basic mechanism (Figure 10a). On the other hand, the experimental PTC of type 0 corresponds well with the extended mechanism (Figure 10b). As already noted, the PTCs of type 1 for perturbations by thiosulfate – both experimental (Figure 6a) and calculated (Figure 11a,b) indicate no intersection with the primary diagonal and consequently zero phase shift The PTC in Figure 6a shows a broad range of phase delays and is consistent with the basic mechanism. Interestingly, even for extremely high amplitudes of stimulation there is no type 0 PTC generated by either model, implying that the shift in the phase space caused by the stimulation is roughly parallel with the


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stable manifold of the steady state from which the limit cycle originates. By contrast, our experiments at large stimulation amplitudes do seem to provide the type 0 PTC (see Figure 6b). To summarize: i) response dynamics to stimulations by hydrogen ion and hydrogen sulfite ion indicate better consistency between experiments and the extended model in both types of the PTC, ii) stimulations by hydroxide ion point to the basic model for the type 1 PTC while both models fail to reproduce type 0 PTC, iii) stimulations by sulfite correspond a bit better to the basic model for the type 1 PTC but they are convincingly consistent with the extended model for the type 0 PTC, iv) stimulations by thiosulfate point to the basic model, even though both models successfully reproduce the absence of the cross-points with primary diagonal. Another way of comparing the experiments with the two mechanisms is based on the stoichiometric network analysis and the implied role of the species in the mechanism. In ref 13 we identified the role of species by comparing the experimental PTCs against those of mechanistic prototypes representing various categories of chemical oscillators. We concluded that the category is 1CX (flow-system with no X-like species located outside the autocatalytic cycle) where H + is the autocatalytic (type X) species, SO 32− is the exit (type Y) species, HSO 3− is the negative feedback (type Z) species and OH − and S 2 O 32− are nonessential for oscillations. When the same approach is applied to the models we obtain somewhat different results in both cases. More specifically, in both models H + is the autocatalytic species and HSO 3− is the negative feedback species. However, SO 32− is the exit species for the extended model while for the basic model the PTCs of type 0 do not correspond well to the PTCs of the prototype by having the average value of the new phase close to zero rather than close to the expected middle of the range; thus its role of type Y species is dubious. On the other hand, OH − is another exit


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species according to PTCs for both mechanisms, whereas it is indicated as nonessential from experiments. Finally, S 2 O 32− is a nonessential species according to both models. As outlined in section III.E the classification is readily applied to both mechanisms based on the earlier established methods of determining the role of species.14,20 We identified three candidate oscillatory subnetworks for the basic model and an additional one for the extended model. Based on the above discussion we are finally in the position of picking among them the one most closely corresponding to experiments. The first choice it the oscillatory subnetwork involving reactions R1, R2, R4 and R5 corresponding to OH − being the exit species. The basic model is consistent with such a choice. The second choice is the subnetwork based on reactions R5 and R6, which corresponds to SO 32− being the exit species. The extended model is consistent with a combination of both subnetworks. The third choice is the competitive catalysis subnetwork involving R1, R2, R3 and R7. However, such an oscillator would require S 2 O 32− as essential type Y species, which is ruled out by both models. The additional possible core oscillatory subnetwork occurs only in the extended model and involves R7, R20 and R4 or R6. Here H + and S 2 O 32− would be the autocatalytic species and either OH − or SO 32− or both would be the exit species, which is consistent with the experiments, and HOS2 O 3− would be the negative feedback species, for which no experimental data are available. However, given the slow rate of R20, such a subnetwork is unlikely to be the core of oscillations in the real system. Moreover, HSO 3− would be nonessential, which is not supported by the experimental data. When all the evidence is considered, the second subnetwork is the best candidate to explain the core oscillator in the experiments and thus the extended model provides better description. We note that the basic model does provide a more consistent explanation of certain features, such as


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the thiosulfate response dynamics. However, being nonessential thiosulfate should not be considered as a species of primary importance.

VII.

CONCLUSIONS

We investigated response dynamics of the pH-oscillatory hydrogen peroxide-thiosulfate-sulfite reaction in terms of phase transition curves. The reaction system was studied in a continuousstirred tank reactor at 26 ± 0.2°C minimizing the influence of air impurities in the reaction mixture by purging all solutions with nitrogen. Phase transition curves were experimentally observed as a response to single-pulse addition of a small volume of a dissolved reaction species including hydrogen ions, hydroxide ions, thiosulfate ions, sulfite ions and hydrogen sulfite ions. Phase transition curves are found to be a suitable tool for determination of how the original unperturbed periodic oscillatory dynamics relates to the response dynamics to brief pulsing by various reaction species. Our aim was using the phase transition curves to obtain a deeper insight into alternative reaction mechanisms. We carried out numerical simulations using a mechanism proposed by Rábai and Hanazaki15, which we call a basic mechanism, and simulations based on an extended mechanism16 for this reaction leading to the construction of phase transition curves for five reaction species and compared them with the results obtained from experiments. Although it is not surprising that the calculated PTCs somewhat differ in many aspects from the measured, we systematically compared the shapes of the phase transition curves by using quantitative as well as qualitative measures introduced in our previous work13 and searched for common features. According to this analysis, we found that the basic mechanism better reflects the role of thiosulfate ion and some aspects of hydroxide ion and sulfite ion, while the extended


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mechanism better reflects the role of hydrogen ion and hydrogen sulfite ion, and some aspects of sulfite ion. We also analyzed core subnetworks available in both mechanisms by using the stoichiometric network analysis and assigned one of them as consistent with our experimental findings. None of studied reaction mechanisms were found to accurately capture the oscillatory core of the experimental system, but the extended model has a slight edge. The results of the current study demonstrate that the methodology of phase transition curves can serve as a useful tool to refine the proposed reaction mechanism.

AUTHOR INFORMATION

Corresponding Author * (I.S.) E-mail: [email protected] Tel.: +420220443167

ACKNOWLEDGEMENT This work was supported by the grant 15-17367S from the Czech Science Foundation. ABBREVIATIONS PTC, phase transition curve; PRC, phase response curve; CSTR, continuous-flow stirred tank reactor; HPTS, hydrogen peroxide-thiosulfate-sulfite reaction; HPTCu, hydrogen peroxidethiosulfate-cupric ion


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REFERENCES (1) Winfree, A. T. The Geometry of Biological Time; Springer-Verlag: New York Inc., 1980. (2) Moore, R. Y. Circadian Rhythms: Basic Neurobiology and Clinical Applications. Annu. Rev. Med. 1997, 48, 253-266. (3) Perkel, D. H.; Schulman, J. H.; Bullock, T. H.; Moore, G. P.; Segundo, J. P. Pacemaker Neurons: Effects of Regularly Spaced Synaptic Inputs. Science 1964, 145, 61-63. (4) Jalife, J.; Antzelevitch, C. Phase Resetting and Annihilation of Pacemaker Activity in Cardiac Tissue, Science 1976, 206, 695-697. (5) Sano, T.; Sawanobori, T.; Adaniya, H. Mechanism of Rhythm Determination Among Pacemaker Cell of the Mammalian Sinus Node. Am. J. Physiol. Heart Circ. Physiol. 1978, 235, 378-384. (6) Guevara, M. R.; Glass L.; Shrier, A. Phase Locking, Period-Doubling Bifurcations, and Irregular Dynamics in Periodically Stimulated Cardiac Cells. Science 1981, 214, 1350-1353. (7) Guevara, M. R.; Shrier, A.; Glass, L. Phase Resetting of Spontaneously Beating Embryonic Ventricular Heart Cell Aggregates, Am. J. Physiol. 1986, 251, 298-305. (8) Clay, J. R.; Guevara, M. R.; Shrier, A. Phase Resetting of the Rhythmic Activity of Embryonic Heart Cell Aggregates. Experiment and Theory. Biophys J. 1984, 45, 699-714. (9) Tsalikakis, D. G.; Zhang, H. G.; Fotiadis, D. I.; Kremmydas, G. P.; Michalis, Ł. K. Phase Response Characteristics of Sinoatrial Node Cells, Comput. Biol. Med. 2007, 37, 8-20.


37


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 38 of 40

(10) Anumonwo, J. M.; Delmar, M.; Vinet, A.; Michaels D. C.; Jalife, J. Phase Resetting and Entrainment of Pacemaker Activity in Single Sinus Nodal Cells, Circ. Res. 1991, 68, 1138-1153. (11) Smeal, R. M.; Ermentrout, G. B.; White, J. A. Phase-Response Curves and Synchronized Neural Network, Phil. Trans. R. Soc. B 2010, 365, 2407-2422. (12) Schulthiess, N. W.; Edgerton, J. R.; Jaeger, D. Phase Response Curve Analysis of a Full Morphological Globus Pallidus Neuron Model Reveals Distinct Perisomatic and Dendritic Modes of Synaptic Integration, J. Neurosci. 2010, 30, 2767-2782. (13) Veber, T.; Schreiberová, L.; Schreiber, I. Classification of the pH-Oscillatory Hydrogen Peroxide-Thiosulfate-Sulfite Reaction, J. Phys. Chem. A. 2013, 117, 12196-12207. (14) Eiswirth, M.; Freund, A.; Ross, J. Mechanistic Classification of Chemical Oscillators and the Role of Species. Adv. Chem. Phys. 1991, 80, 127-197. (15) Rábai, G.; Hanazaki, I. Chaotic pH Oscillations in the Hydrogen Peroxide-ThiosulfateSulfite Flow System. J. Phys. Chem. A 1999, 103, 7268-7273. (16) Voslař, M.; Matějka, P.; Bakeš, D.; Schreiberová, L., Schreiber, I. Complex Reaction Mechanism and Kinetics of Oxidation of Thiosulfate by Hydrogen Peroxide in the Presence of Sulfite, in preparation. (17) Janert, P. K. Gnuplot in Action: Understanding Data with Graphs; Manning Publications Co., 2010. (18) Zagora, J.; Voslař, M.; Schreiberová, L.; Schreiber, I. Excitability in Chemical and Biochemical pH-autocatalytic systems, Faraday Discuss. 2001, 120, 313-324.


38

Page 39 of 40

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(19) Kohout, M.; Schreiber, I.; Kubíček, M. A Computational Tool for Nonlinear Dynamical and Bifurcation Analysis of Chemical Engineering Problems. Comp. and Chem. Eng. 2002, 26, 517-527. (20) Clarke, B. L. Stability of Complex Reaction Networks. Adv. Chem. Phys. 1980, 43, 1– 278. (21) Ross, J.; Schreiber, I.; Vlad, M. O. Determination of Complex Reaction Mechanisms; Oxford University Press, Inc.: New York, 2006. (22) Rábai, G.; Hanazaki, I. Temperature Compensation in the Oscillatory Hydrogen Peroxide– Thiosulfate–Sulfite Flow System. Chem. Commun. 1999, 19, 1965 -1966. (23) Rábai, G.; Szántó, T. G.; Kovács K. Temperature-Induced Route to Chaos in the H2O2−HSO3-−S2O32- Flow Reaction System. J. Phys. Chem. A 2008, 112, 12007-12010. (24) Kurin-Csörgei, K.; Orbán, K.; Rábai, G.; Epstein, I. R. Model for Oscillatory Reaction Between Hydrogen Peroxide and Thiosulfate Catalysed by Copper(II) ions. J. Chem. Soc. Faraday Trans. 1996, 92, 2851-2855. (25) Voslař, M.; Matějka, P.; Schreiber, I. Oscillatory Reactions Involving Hydrogen Peroxide and Thiosulfate–Kinetics of the Oxidation of Tetrathionate by Hydrogen Peroxide. Inorg. Chem.

2006, 45, 2824-2834. (26) Eiswirth, M.; Bürger, J.; Strasser, P.; Ertl, G. Oscillating Langmuir-Hinshelwood Mechanisms. J. Phys. Chem. 1996, 100, 19118-19123.


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TOC Contents


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Use of Phase Transition Curves for Testing Models of the pH

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