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
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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.