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


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 



active membrane area m2


Vm, Vm E V1/(K h ·q) (dimensionless)


Vm E V1/(Ks·q) (dimensionless)


refer to feed conditions


hydrogen ions


dimensionless concentration of hydrogen ions


constants characteristic the enzyme


equilibrium constant of water (Kmol2/m6)


hydroxyl ions


volumetric flow rate (m3/s)


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


acetate (CH3COO-)


rate of reaction Kmol/m3 s


rate of water formation Kmol/m3 s


dimensionless rate of reaction




dimensionless substrate concentration


t.q/V1 dimensionless time


maximum reaction rate (Kmol/Kmol·s)

V1 V2

volume of compartments (1) and (2)


membrane permeability for hydrogen ions (m/s)


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


membrane permeability for substrate (m/s)


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

α ’oh

membrane permeability for hydroxyl ions (m/s)


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


dimensionless substrate inhibition constant [Ks/Ki]


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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