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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)

Enzyme catalysis: basics

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

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

Michaelis-Menten Mechanism:

dynamic simulation

for the validity of the qssa:

e0<

e0<

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<

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

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

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.

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