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Lecture #5

Lecture #5. Enzyme Kinetics. Outline. The principles of enzyme catalysis Deriving rate laws for enzymes Michaelis-Menten kinetics Hill kinetics The symmetry model Scaling equations (Advanced). Some basic information. ENZYME CATALYSIS. Enzyme catalysis: basics.

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Lecture #5

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  1. Lecture #5 Enzyme Kinetics

  2. Outline • The principles of enzyme catalysis • Deriving rate laws for enzymes • Michaelis-Menten kinetics • Hill kinetics • The symmetry model • Scaling equations (Advanced)

  3. Some basic information ENZYME CATALYSIS

  4. Enzyme catalysis: basics http://ebooklibrary.thieme.com/SID2502958536850/ebooklibrary/flexibook/pubid1619260736/index.html

  5. Enzyme catalysis: basics

  6. EC Classification of enzymes EC # = enzyme commission # EC x.x.x.x

  7. Details for specific casesare available

  8. Mathematical description of catalytic activity DERIVING RATE LAWS

  9. Deriving Enzymatic Rate Laws from Postulated Reaction Mechanisms • Formulate mass balances on elementary reactions • Identify mass balances/time invariants • Reduce to the dynamically independent variables • Apply simplifying assumptions: The QSSA or the QEA • Use numerical integration to determine when the assumptions apply • Scale equations and form dimensionless numbers (optional; advanced analysis)

  10. MICHAELIS-MENTEN KINETICS

  11. Michaelis-Menten Reaction Mechanism free enzyme intermediate complex product substrate fast slow const const (dynamic degree of freedom) the two time invariants

  12. Mass Action Kinetics: introduction of time-invariants to go from 4 variables to 2 dynamically independent variables

  13. The Quasi-steady State Assumption choose independent variables ODEs AEs Applying the QSSA =vm - - , Km

  14. The Michaelis-Menten Rate Law vm (0th order) vm 2 (1st order) Km=s s

  15. Michaelis-Menten Mechanism: dynamic simulation phase portrait fast response slow response error

  16. Michaelis-Menten Mechanism: dynamic simulation for the validity of the qssa: e0<<s0 literature e0<<Km accurate full and qss-solution are indistinguishable

  17. Applicability of the QEA, QSSA k2 S+E ES P+E • When k2 << k-1then the QEA works • When et << Km then the QSSA works • When Km << st then the QSSA works k-1 slow fast k2<<k-1 ( see Chem. Eng. Sci., 42, 447-458.)

  18. Regulatory Enzymes

  19. Originally used to describe oxygen binding to hemoglobin HILL KINETICS

  20. Hill Kinetics Inhibitor 1. Reaction mechanism catalytically inactive form of E 2. Mass balance “degree of cooperativity”, rarely an integer due to lumping effect of reaction (2) nHb~2.3-2.6, also called the Hill coefficient 3. QEA on reaction (2) conservation quantity “per site” binding constant 4. Reaction rate

  21. Applying Simplifying Assumptions mass balance: QEA Add e to the rate law: inhibition activation a: concentration of A

  22. Graphical Representation maximum sensitivity no sensitivity to effector molecule Activated form avm activation precursor example inflection point Normal form vm aa protein synth. inflection point inhibition i or a no sensitivity

  23. Dynamic Simulation of Hill Kinetics Phase portraits Dynamic responses fast slow distribution of enzyme states catalysis

  24. And now, chemically realistic mechanisms THE SYMMETRY MODEL

  25. The Symmetry Model (T form) (R form)

  26. Deriving the Rate Law Mass balance QEA Combine

  27. Deriving the Rate Law (Con’t) 4 4 Similar equation for activators and substrates

  28. Dynamic Response of the Symmetry Model Phase planes Dynamic responses fast slow distribution of enzyme states catalysis

  29. Summary • Enzymes are highly specialized catalysts that accelerate reaction rates • Reaction mechanisms are formulated for the chemical conversions carried out by enzymes in terms of elementary reactions. • Rate laws for enzyme reaction mechanisms are derived based on simplifying assumptions. • Two simplifying assumptions are commonly used: the quasi-steady state (QSSA) and the quasi-equilibrium assumptions (QEA). • The validity of the simplifying assumptions can be determined using scaling of the equations followed by mathematical and numerical analysis. • A number of rate laws have been developed for enzyme catalysis and for the regulation of enzymes. Only three reaction mechanisms were described in this chapter.

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