Ciencies Bilogies

College of the Redwoods
Math 55, Differential Equations

Michaelis-Menten Enzyme Kinetics
The Jigman and The SauceMan
e-mail: thejigman@yahoo.com
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Introduction
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Figure 1: Triophosphate Enzyme
What is Enzyme Kinetics?
• Kinetics is the study of rates of chemical reactions
• Enzymes are little molecular machines that carry out
reactions in cells
•Enzyme kinetics is the study of rates of chemical reactions that
involve enzymes

Michaelis-Menten Equation
• The Michaelis-Menten Equation is a differential equation used to
model the rate at which enzymatic reactions occur
• This model allows scientist to predict how fast a reaction will take
place based on the concentrations of the chemicals being reacted.

Figure 2: A Model of anEnzyme

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Typical Enzymatic Reactions
k

1
−
−
E0 + S − − E1

k−1

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k

2
E1 − E0 + P
→

S
P
E0
E1
k1, k−1, k2

the concentration of the substrate
(the unreacted molecules)
the concentration of product
(the reacted molecules)
the concentration of the unoccupied enzymes
the concentration of occupied enzymes.
the rate constants

Conditions forMichaelis-Menten Modelling
• In order to model an enzymatic reaction, some conditions must be
maintained:
– Temperature, ionic strength, pH, and other physical
conditions that might affect the rate must remain constant

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Conditions for Michaelis-Menten Modelling
• In order to model an enzymatic reaction, some conditions must be
maintained:
– Temperature, ionic strength, pH, and other physicalconditions that might affect the rate must remain constant
– Each enzyme can act on only one other molecule at a time

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Conditions for Michaelis-Menten Modelling
• In order to model an enzymatic reaction, some conditions must be
maintained:
– Temperature, ionic strength, pH, and other physical
conditions that might affect the rate must remain constant
– Each enzyme can act on only oneother molecule at a time
– The enzyme must remain unchanged during the course of the
reaction.

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Conditions for Michaelis-Menten Modelling
• In order to model an enzymatic reaction, some conditions must be
maintained:
– Temperature, ionic strength, pH, and other physical
conditions that might affect the rate must remain constant
– Each enzyme can act on only one other molecule at atime
– The enzyme must remain unchanged during the course of the
reaction.
– The concentration of substrate must be much higher than the
concentration of enzyme

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Rate Equations
k1

−
−
E0 + S − − E1
k−1

k

2
E1 − E0 + P
→

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(1)
(2)

• The rate at which reaction (1) occurs is derived as follows:
– The number of possible contacts between S and E0 is directlyproportional to SE0.
– The number of successful contacts over a certain amount of time
is proportional to the number of possible contacts.
– Thus, the rate of reaction is directly proportional to SE0:
Rate1 = k1SE0.
where k1 is the rate constant.

k

1
−
−
E0 + S − − E1

k−1

(1)
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k

2
E1 − E0 + P
→

(2)

• The rate at which the reverse of reaction (1) occurs isderived as
follows:
– A certain proportion of E1 will release S over a certain amount
of time before the reaction is carried out.
– The rate of the reverse reaction is directly proportional to E1:
Rate−1 = k−1E1
where k−1 is the rate constant.

k

1
−
−
E0 + S − − E1

k−1

(1)
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k

2
E1 − E0 + P
→

(2)

• The rate at which reaction (2) occurs is derived asfollows:
– A certain proportion of E1 will produce P over a certain amount
of time.
– The rate of production of P is directly proportional to E1:
Rate2 = k2E1
where k2 is the rate constant.

Specific Rates of Reactions for each Compound
k

1
−
−
E0 + S − − E1

(1)

k−1

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k

2
→
E1 − E0 + P

Rate1 = k1SE0,

Rate−1 = k−1E1,

(2)

and Rate2 = k2E1

The rate...