^{1,a)}, David G. Míguez

^{1,b)}and Irving R. Epstein

^{1,c)}

### Abstract

The reaction of glucose with ferricyanide catalyzed by glucose oxidase from *Aspergillus niger* gives rise to a wide range of bistability as the flow rate is varied in a continuous flow stirred tank reactor. Oscillations in can be obtained by introducing a negative feedback on the autocatalytic production of that drives the bistability. In our experiments, this feedback consists of an inflow of hydroxide ion at a rate that depends on in the reactor as . oscillations are found over a broad range of enzyme and ferricyanide concentrations, residence times , and feedback parameters. A simple mathematical model quantitatively accounts for the experimentally found oscillations.

This work was supported by National Science Foundation Grant Nos. CHE-0306262 and CHE-0615507.

I. INTRODUCTION

II. GLUCOSE OXIDASE

III. STRATEGY

IV. MATERIALS AND METHODS

V. EXPERIMENTAL RESULTS. AUTOCATALYSIS AND BISTABILITY

VI. EXPERIMENTAL RESULTS: FROM BISTABILITY TO OSCILLATIONS

VII. MODELING

VIII. LINEAR STABILITY ANALYSIS AND SIMULATIONS

IX. DISCUSSION AND CONCLUSION

### Key Topics

- Enzymes
- 31.0
- Oscillators
- 23.0
- Hydrogen reactions
- 21.0
- Enzyme kinetics
- 12.0
- Protons
- 7.0

## Figures

(a) Bistability in system (15) and (16) at , , , and . Dashes denote unstable steady state. (b) Bistability in a CSTR with influxes of GO, glucose, , and ferricyanide . The volume of the reactor, , is , , , , is varied from , and is varied from , where the subscript 0 denotes input concentrations in the CSTR without reaction.

(a) Bistability in system (15) and (16) at , , , and . Dashes denote unstable steady state. (b) Bistability in a CSTR with influxes of GO, glucose, , and ferricyanide . The volume of the reactor, , is , , , , is varied from , and is varied from , where the subscript 0 denotes input concentrations in the CSTR without reaction.

Experimental setup. -E, electrode; MS, magnetic stirrer; black bar inside the CSTR is a magnetic stirrer bar; PP1, PP2, and PP3 are peristaltic pumps. Dotted lines are electrical connections. A meter between -E and PC is not shown. In preliminary experiments we used five pumps (two additional syringe pumps) to vary input concentrations of some reactants.

Experimental setup. -E, electrode; MS, magnetic stirrer; black bar inside the CSTR is a magnetic stirrer bar; PP1, PP2, and PP3 are peristaltic pumps. Dotted lines are electrical connections. A meter between -E and PC is not shown. In preliminary experiments we used five pumps (two additional syringe pumps) to vary input concentrations of some reactants.

Reaction of glucose with catalyzed by GO in a batch reactor. (a) experiment with , , 10, (curve 2) 7.5, (curves 3 and 4) 5, (curve 5) 2.5, and [GO] 0.5 and (curve 4) 1. (b) Comparison between experiment (curve 1) and model (C1)–(C3), (C5)–(C11), (C17), and (C18) (curves 2 and 3). For curve 1: , , , and . For curves 2 and 3: (curve 2) and (curve 3), , , , , , , , , , , , , and .

Reaction of glucose with catalyzed by GO in a batch reactor. (a) experiment with , , 10, (curve 2) 7.5, (curves 3 and 4) 5, (curve 5) 2.5, and [GO] 0.5 and (curve 4) 1. (b) Comparison between experiment (curve 1) and model (C1)–(C3), (C5)–(C11), (C17), and (C18) (curves 2 and 3). For curve 1: , , , and . For curves 2 and 3: (curve 2) and (curve 3), , , , , , , , , , , , , and .

Experimentally observed oscillations in a CSTR for GO-catalyzed oxidation of glucose by ferricyanide supplemented by a negative feedback, . , , and . (a) , , , and . (b) , , , and . [(c) and (d)] , , , and 4.0 and (d) 3.8.

Experimentally observed oscillations in a CSTR for GO-catalyzed oxidation of glucose by ferricyanide supplemented by a negative feedback, . , , and . (a) , , , and . (b) , , , and . [(c) and (d)] , , , and 4.0 and (d) 3.8.

Experimental and theoretical dynamic phase diagrams. (a) . Experiment (symbols) in a CSTR. Black rhombs denote oscillations, “+” are high steady state, and “×” are low steady state. Concentrations: , , , , and . Curves 1 and 2 are result of linear stability analysis of Eqs. (32) and (33) with parameters , , , , , , , , , and . Theoretical oscillatory region is between curves 1 and 2; curve 2 is supercritical Hopf bifurcation and curve 1 is subcritical Hopf or saddle bifurcation. (b) . Symbols and parameters as in (a), except , , and . (c) . Symbols and parameters as in (a), except (experiment); 0.0025 and (curves 3 and 4) 0.006 (calculations); curves 2 and 4 show supercritical Hopf bifurcation; curves 1 and 3 show subcritical Hopf or saddle. Oscillatory region lies between curves 1 (3) and 2 (4). (d) Bistability region for experimental system GO-glucose-ferricyanide in a CSTR, Symbols: full circles, bistability between and ; +, low ; , high . and .

Experimental and theoretical dynamic phase diagrams. (a) . Experiment (symbols) in a CSTR. Black rhombs denote oscillations, “+” are high steady state, and “×” are low steady state. Concentrations: , , , , and . Curves 1 and 2 are result of linear stability analysis of Eqs. (32) and (33) with parameters , , , , , , , , , and . Theoretical oscillatory region is between curves 1 and 2; curve 2 is supercritical Hopf bifurcation and curve 1 is subcritical Hopf or saddle bifurcation. (b) . Symbols and parameters as in (a), except , , and . (c) . Symbols and parameters as in (a), except (experiment); 0.0025 and (curves 3 and 4) 0.006 (calculations); curves 2 and 4 show supercritical Hopf bifurcation; curves 1 and 3 show subcritical Hopf or saddle. Oscillatory region lies between curves 1 (3) and 2 (4). (d) Bistability region for experimental system GO-glucose-ferricyanide in a CSTR, Symbols: full circles, bistability between and ; +, low ; , high . and .

Linear stability analysis of Eqs. (32) and (33) at (curves 1 and 2 in and b) and eqs. (17) and (18) (, curves 3 and 4). Oscillatory regions are between curves 1 and 2 for model (32) and (33) and between curves 3 and 4 for for model (17) and (18). Curve 1 is subcritical Hopf line or saddle, curve 2 is supercritical Hopf bifurcation. Parameters for curves 1 and 2 in (a): , , , , , , , , and . Parameters for curves 3 and 4 in (a) , , and . (b) , , , , , , , , , and .

Linear stability analysis of Eqs. (32) and (33) at (curves 1 and 2 in and b) and eqs. (17) and (18) (, curves 3 and 4). Oscillatory regions are between curves 1 and 2 for model (32) and (33) and between curves 3 and 4 for for model (17) and (18). Curve 1 is subcritical Hopf line or saddle, curve 2 is supercritical Hopf bifurcation. Parameters for curves 1 and 2 in (a): , , , , , , , , and . Parameters for curves 3 and 4 in (a) , , and . (b) , , , , , , , , , and .

(a) Diagram for model (32) and (33). Oscillatory region is between curves 1 and 2. Curve 3 is frequency of oscillations for parametric points that are very close to curve 1. For curve 1, ; for curve 2, . (b) Typical dependence of the frequency (curve 3) and amplitude of pH oscillations for model (32) and (33) at constant . Curves 1 and 2 are the maximum and minimum of oscillations. Parameters: , , , , , , and ; (a) , , and ; (b) , , , and .

(a) Diagram for model (32) and (33). Oscillatory region is between curves 1 and 2. Curve 3 is frequency of oscillations for parametric points that are very close to curve 1. For curve 1, ; for curve 2, . (b) Typical dependence of the frequency (curve 3) and amplitude of pH oscillations for model (32) and (33) at constant . Curves 1 and 2 are the maximum and minimum of oscillations. Parameters: , , , , , , and ; (a) , , and ; (b) , , , and .

Examples of oscillations in model (32) and (33) at . Parameters: , , , , and ; (a) , , , , 4.1 and (curve 2) 4.05 and ; (b) , 0.0087 and (curve 2) 0.0084, , , , and .

Examples of oscillations in model (32) and (33) at . Parameters: , , , , and ; (a) , , , , 4.1 and (curve 2) 4.05 and ; (b) , 0.0087 and (curve 2) 0.0084, , , , and .

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