Michaelis-Menten Enzyme Kinetics: Understanding the Rate of Biochemical Reactions, Study notes of Physics

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College of the Redwoods

Math 55, Differential Equations

Michaelis-Menten Enzyme Kinetics

The Jigman and The SauceMan

e-mail: [email protected] [email protected]

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Introduction

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

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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 an Enzyme

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Typical Enzymatic Reactions

E 0 + S

k 1 −↽−−⇀− k− 1

E 1

E 1 −^ k→^2 E 0 + P

S the concentration of the substrate (the unreacted molecules) P the concentration of product (the reacted molecules) E 0 the concentration of the unoccupied enzymes E 1 the concentration of occupied enzymes. k 1 , k− 1 , k 2 the rate constants

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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

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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 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 one other 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 a time - 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

E 0 + S

−−k⇀^1 ↽−− k− 1 E^1 (1) E 1 k 2 −→ E 0 + P (2)

• The rate at which reaction (1) occurs is derived as follows:
  • The number of possible contacts between S and E 0 is directly proportional to SE 0.
  • 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 SE 0 : Rate 1 = k 1 SE 0. where k 1 is the rate constant.

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E 0 + S

−−k⇀^1 ↽−− k− 1 E^1 (1)

E 1

k 2 −→ E 0 + P (2)

• The rate at which the reverse of reaction (1) occurs is derived as follows: - A certain proportion of E 1 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 E 1 :

Rate− 1 = k− 1 E 1

where k− 1 is the rate constant.

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E 0 + S

−−k⇀^1 ↽−− k− 1 E^1 (1)

E 1

k 2 −→ E 0 + P (2)

• The rate at which reaction (2) occurs is derived as follows:
  • A certain proportion of E 1 will produce P over a certain amount of time.
  • The rate of production of P is directly proportional to E 1 :

Rate 2 = k 2 E 1

where k 2 is the rate constant.

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Specific Rates of Reactions for each Compound

E 0 + S

−−k⇀^1 ↽−− k− 1 E^1 (1)

E 1 −^ k→^2 E 0 + P (2)

Rate 1 = k 1 SE 0 , Rate− 1 = k− 1 E 1 , and Rate 2 = k 2 E 1 The rate equations associated with each reaction determines the rate of change of S, E 0 , E 1 , and P. Realizing this, we can write the follow- ing:

dS dt = −Rate 1 + Rate− 1 = −k 1 SE 0 + k− 1 E 1

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Specific Rates of Reactions for each Compound

E 0 + S

−−k⇀^1 ↽−− k− 1 E^1 (1)

E 1 −^ k→^2 E 0 + P (2)

Rate 1 = k 1 SE 0 , Rate− 1 = k− 1 E 1 , and Rate 2 = k 2 E 1 The rate equations associated with each reaction determines the rate of change of S, E 0 , E 1 , and P. Realizing this, we can write the follow- ing:

dE 0 dt = −Rate 1 + Rate− 1 + Rate 2 = −k 1 SE 0 + k− 1 E 1 + k 2 E 1

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Specific Rates of Reactions for each Compound

E 0 + S

−−k⇀^1 ↽−− k− 1 E^1 (1)

E 1 −^ k→^2 E 0 + P (2)

Rate 1 = k 1 SE 0 , Rate− 1 = k− 1 E 1 , and Rate 2 = k 2 E 1 The rate equations associated with each reaction determines the rate of change of S, E 0 , E 1 , and P. Realizing this, we can write the follow- ing:

dE 1 dt = Rate 1 − Rate− 1 − Rate 2 = k 1 SE 0 − k− 1 E 1 − k 2 E 1

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Specific Rates of Reactions for each Compound

E 0 + S

−−k⇀^1 ↽−− k− 1 E^1 (1)

E 1 −^ k→^2 E 0 + P (2)

Rate 1 = k 1 SE 0 , Rate− 1 = k− 1 E 1 , and Rate 2 = k 2 E 1 The rate equations associated with each reaction determines the rate of change of S, E 0 , E 1 , and P. Realizing this, we can write the follow- ing:

dP dt

= Rate 2 = k 2 E 1

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Thus the system of differential equations modelling the process is: dS dt = −k 1 SE 0 + k− 1 E 1 dE 0 dt

= −k 1 SE 0 + k− 1 E 1 + k 2 E 1 dE 1 dt

= k 1 SE 0 − k− 1 E 1 − k 2 E 1 dP dt

= k 2 E 1

The rate constants can be difficult or impossible to determine. For the purpose of seeing the behavior of the system, we give them the the values k 1 = 10, k− 1 = 1, and k 2 = 5, with initial conditions S = 1. 0 and an E 0 = 0. 08.

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0 2 4 6 8 0

1

Substrate/ Enzyme Reaction Model

Time

Concentration

Substrate, S Unoccupied Enzyme, E 0 Occupied Enzyme, E 1 Product, P

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Reducing the Four Equations to Two

By adding equations and doing some algebraic manipulation, we find that dS dt

= −k 1 SET + (k− 1 + k 1 S)E 1 dE 1 dt

= k 1 SET − (k 1 S + k− 1 + k 2 )E 1

Figure 3: A model of an enzyme

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The Quasi-Steady-State Assumption

As long as ET  S then we can assume that dE 1 /dt ≈ 0.

dS dt

= −k 1 SET + (k− 1 + k 1 S)E 1 0 ≈ k 1 SET − (k 1 S + k− 1 + k 2 )E 1

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The Quasi-Steady-State Assumption

As long as ET  S then we can assume that dE 1 /dt ≈ 0.

dS dt

= −k 1 SET + (k− 1 + k 1 S)E 1 0 ≈ k 1 SET − (k 1 S + k− 1 + k 2 )E 1

Solve dE 1 /dt for E 1

E 1 = k 1 SET (k− 1 + k 2 + k 1 S)

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The Quasi-Steady-State Assumption

As long as ET  S then we can assume that dE 1 /dt ≈ 0.

dS dt

= −k 1 SET + (k− 1 + k 1 S)E 1 0 ≈ k 1 SET − (k 1 S + k− 1 + k 2 )E 1

Solve dE 1 /dt for E 1

E 1 = k 1 SET (k− 1 + k 2 + k 1 S)

Plug this into dS/dt and evaluate various steps to obtain

dS dt

= −

VmaxS KM + S

where Vmax = k 2 ET and KM =

k− 1 + k 2 k 1

.

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dS dt

= −

VmaxS KM + S

where Vmax = k 2 ET and KM = k− 1 + k 2 k 1

.

This is the Michealis-Menten enzyme equation.

• This one equation replaces the system for the modelling the substrate rate equation.
• There are only two parameters, and they can both be determined experimentally.

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(^00 2 4 6 8)

0.2

0.4

0.6

0.8

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Comparing the Approximation

Time, t

Concentration of Substrate, S

Michaelis−Menten Approximation Original Rate Equation

Figure 4: The solutions for the Michaelis-Menten Equation and the Original dS/dt.

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Summary

• The overall process of converting a substrate to a product is given by the following two reactions:

E 0 + S −−k⇀^1 ↽−− k− 1 E^1 , and E 1 k 2 −→ E 0 + P.

• These reactions give rise to a system of differential equations: dS dt

= −k 1 SE 0 + k− 1 E 1 dE 0 dt = −k 1 SE 0 + k− 1 E 1 + k 2 E 1 dE 1 dt

= k 1 SE 0 − k− 1 E 1 − k 2 E 1 dP dt

= k 2 E 1.

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• By manipulating these equations we can derive the Michaelis-Menten equation for dS/dt. dS dt

= −

VmaxS KM + S

where Vmax = k 2 ET and KM = k− 1 + k 2 k 1

.

Figure 5: A model of an enzyme