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 CATALYSISSome basic information

Enzyme catalysis: basics

http://ebooklibrary.thieme.com/SID2502958536850/ebooklibrary/flexibook/pubid1619260736/index.html

Enzyme catalysis:

basics

EC Classification of enzymes

EC # = enzyme commission #

EC x.x.x.x

Details for

specific casesare

available

DERIVING RATE LAWSMathematical description of catalytic activity

Deriving Enzymatic Rate Laws from Postulated Reaction Mechanisms

1. Formulate mass balances on elementary reactions

2. Identify mass balances/time invariants3. Reduce to the dynamically independent variables4. Apply simplifying assumptions: The QSSA or the

QEA5. Use numerical integration to determine when

the assumptions apply6. Scale equations and form dimensionless

numbers (optional; advanced analysis)

MICHAELIS-MENTEN KINETICS

Michaelis-Menten Reaction Mechanism

subs

trate free

enzy

me

inter

med

iate

com

plex

prod

uct

fast slow

(dynamic degree of freedom)

constconst

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

=vm

Km

choose independentvariables

Applying the QSSA

- -,

ODEs AEs

The Michaelis-Menten Rate Law

vm

vm

2

Km=s s

(0th order)

(1st order)

phase portrait

fastresponse

slowresponse

error

Michaelis-Menten Mechanism:dynamic simulation

full and qss-solutionare indistinguishable

for the validity of the qssa:e0<<s0 literaturee0<<Km accurate

Michaelis-Menten Mechanism:dynamic simulation

Applicability of the QEA, QSSA

• When k2 << k-1 then the QEA works

• When et << Km then the QSSA works

• When Km << st then the QSSA works

S+E ES P+Ek-1

k2

fast

slow

k2<<k-1

( see Chem. Eng. Sci., 42, 447-458.)

Regulatory Enzymes

HILL KINETICSOriginally 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~2.3-2.6,

also called the Hill coefficient

“per site” binding constant

2. Mass balance

4. Reaction rate

1. Reaction mechanism

conservationquantity

Inhibitor

catalyticallyinactive form of E

Applying Simplifying Assumptions

mass balance: QEA

activation

a: concentration of A

Graphical Representation

vm

vm

no sensitivity

maximumsensitivity

no sensitivityto effector molecule

i or a

inflection point

activation

inflection pointinhibition

precursor

aa

protein synth.

example

Activated form

Normalform

Dynamic Simulation of Hill Kinetics

Pha

se p

ortr

aits

Dyn

amic

resp

onse

s

fast slowdistribution of enzyme states catalysis

THE SYMMETRY MODELAnd now, chemically realistic mechanisms

The Symmetry Model

(R form) (T form)

Deriving the Rate LawMass balance

Combine

QEA

Deriving the Rate Law (Con’t)

Similar equation for activators and substrates

4

4

Dynamic Response of the Symmetry Model

Pha

se p

lane

sD

ynam

icre

spon

ses

fast slowdistribution 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.