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Periodic and chaotic behavior of substrate-inhibited enzymatic reactions with hydrogen ions production

  • G. Ibrahim
  • F. A. Teymour
  • S. S. E. H. ElnashaieEmail author
Original Articles

Abstract

A two-compartment model of an enzyme system with substrate inhibition kinetics and hydrogen ion production is investigated. The model is used to study the bifurcation, instability, and chaotic behavior of the system. The investigation, although in a restricted region of the parameters’ space, has uncovered a good part of the rich dynamic characteristics of this system, including: period doubling sequences leading to chaos, banded chaos, fully developed chaos, interior crisis, tangent bifurcation leading to intermittency, periodic windows interrupting chaotic regions, and alternating periodic chaotic sequences. The results relate to the phenomena occurring in physiological experiments, such as the periodic stimulation of neural cells and the voltagegated ion channel dynamics.

Index Entries

Enzyme systems substrate-inhibition acetylcholinesterase bifurcation chaos 

Abbreviations

Am

active membrane area m2

Bh

Vm, Vm E V1/(K h ·q) (dimensionless)

Bs

Vm E V1/(Ks·q) (dimensionless)

f

refer to feed conditions

H

hydrogen ions

h

dimensionless concentration of hydrogen ions

Ks,Ki,Kh,K’h

constants characteristic the enzyme

Kw

equilibrium constant of water (Kmol2/m6)

OH

hydroxyl ions

q

volumetric flow rate (m3/s)

P1

choline [HO(CH2)N+(CH3)3]

P2-

acetate (CH3COO-)

R

rate of reaction Kmol/m3 s

Rw

rate of water formation Kmol/m3 s

r

dimensionless rate of reaction

S

substrate

s

dimensionless substrate concentration

T

t.q/V1 dimensionless time

Vm

maximum reaction rate (Kmol/Kmol·s)

V1 V2

volume of compartments (1) and (2)

α’h

membrane permeability for hydrogen ions (m/s)

αh

dimensionless membrane permeability for hydrogen ions [α’ohAm/q]

α’s

membrane permeability for substrate (m/s)

αs

dimensionless permeability for substrate(α’ s A m /q)

α ’oh

membrane permeability for hydroxyl ions (m/s)

αoh

dimensionless permeability for hydroxyl ions [α’ohA m /q]

αi

dimensionless substrate inhibition constant [Ks/Ki]

δ,Kh/K’h

γ, K w /K h2

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Copyright information

© Humana Press Inc 1995

Authors and Affiliations

  • G. Ibrahim
    • 1
    • 2
  • F. A. Teymour
    • 1
  • S. S. E. H. Elnashaie
    • 1
    • 3
    Email author
  1. 1.Non-Linear Dynamics Group (NLDG), Chemical Engineering DepartmentKing Saud UniversitySaudi Arabia
  2. 2.Faculty of EngineeringMenofia UniversityEgypt
  3. 3.Chemical Engineering DepartmentCairo UniversityEgypt

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