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)

ENZYME CATALYSIS Some basic information

Enzyme catalysis: basics

Enzyme catalysis: basics

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

Details for specific cases are available

DERIVING RATE LAWS Mathematical description of catalytic activity

Deriving Enzymatic Rate Laws from Postulated Reaction Mechanisms 1.Formulate mass balances on elementary reactions 2.Identify mass balances/time invariants 3.Reduce to the dynamically independent variables 4.Apply simplifying assumptions: The QSSA or the QEA 5.Use numerical integration to determine when the assumptions apply 6.Scale equations and form dimensionless numbers (optional; advanced analysis)

MICHAELIS-MENTEN KINETICS

Michaelis-Menten Reaction Mechanism substrate free enzyme intermediate complex product fastslow (dynamic degree of freedom) const 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 =v m KmKm choose independent variables Applying the QSSA --, ODEsAEs

The Michaelis-Menten Rate Law vmvm vmvm 2 K m =s s (0 th order) (1 st order)

Finding Sensitivity (Advanced) chain rule

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

full and qss-solution are indistinguishable for the validity of the qssa: e 0 <<s 0 literature e 0 <<K m accurate Michaelis-Menten Mechanism: dynamic simulation

Applicability of the QEA, QSSA When k 2 << k -1 then the QEA works When e t << K m then the QSSA works When K m << s t then the QSSA works S+E ES P+E k -1 k2k2 fast slow k 2 <<k -1 ( see Chem. Eng. Sci., 42, )

Key Dimensionless Groups (Advanced) a = k 2 /k -1 <<1QEA b = e t /K m <<1QSSA c = s t /K m >>1QSSA stickiness number

Numerical simulation vs. Pooling for understanding

Regulatory Enzymes

HILL KINETICS Originally used to describe oxygen binding to hemoglobin

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

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

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

Sensitivity (advanced) The Hill rate law has a sigmoidal shape with sensitivity of the reaction rate to the end product concentration as: which has a maximum at the inflection point The values at the inflection point

Dynamic Simulation of Hill Kinetics Phase portraits Dynamic responses fastslow distribution of enzyme states catalysis

Key Dimensionless Groups (Advanced)

THE SYMMETRY MODEL And now, chemically realistic mechanisms

The Symmetry Model (R form) (T form)

Deriving the Rate Law Mass balance Combine QEA

Deriving the Rate Law (Cont) Similar equation for activators and substrates 4 4

Dynamic Response of the Symmetry Model Phase planes Dynamic responses fastslow distribution of enzyme states catalysis

Key Dimensionless Groups (Advanced) L:The allosteric constant t i =K i /v m time constant of inhibition b=e t /K i relative times of catalysis to inhibition

Scaling Equations (Advanced)

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.