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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 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 http://ebooklibrary.thieme.com/SID2502958536850/ebooklibrary/flexibook/pubid1619260736/index.html
EC Classification of enzymes EC # = enzyme commission # EC x.x.x.x
Mathematical description of catalytic activity DERIVING RATE LAWS
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)
Michaelis-Menten Reaction Mechanism free enzyme intermediate complex product substrate fast slow const const (dynamic degree of freedom) the two time invariants
Mass Action Kinetics: introduction of time-invariants to go from 4 variables to 2 dynamically independent variables
The Quasi-steady State Assumption choose independent variables ODEs AEs Applying the QSSA =vm - - , Km
The Michaelis-Menten Rate Law vm (0th order) vm 2 (1st order) Km=s s
Michaelis-Menten Mechanism: dynamic simulation phase portrait fast response slow response error
Michaelis-Menten Mechanism: dynamic simulation for the validity of the qssa: e0<<s0 literature e0<<Km accurate full and qss-solution are indistinguishable
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.)
Originally used to describe oxygen binding to hemoglobin HILL KINETICS
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
Applying Simplifying Assumptions mass balance: QEA Add e to the rate law: inhibition activation a: concentration of A
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
Dynamic Simulation of Hill Kinetics Phase portraits Dynamic responses fast slow distribution of enzyme states catalysis
And now, chemically realistic mechanisms THE SYMMETRY MODEL
The Symmetry Model (T form) (R form)
Deriving the Rate Law Mass balance QEA Combine
Deriving the Rate Law (Con’t) 4 4 Similar equation for activators and substrates
Dynamic Response of the Symmetry Model Phase planes Dynamic responses fast slow distribution of enzyme states catalysis
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.