The Michaelis-Menton Model

• For non-allosteric enzymes, the most widely used kinetic model is based upon work done by Leonor Michaelis and Maud Menton

For a given enzyme-catalyzed reaction:

Substrate --> Product

E+S ES PE+k1

k-1

k2

Note: There is no appreciable back reaction of Product to substrate!

E +S ES PE +k1

k-1

k2

k1 = Rate constant for the Formation of the enzyme/substrate complex

k-1 = Rate constant for the Dissociation of the enzyme/substrate complex

k2 = Rate constant for the formation of product and its release

The enzyme is specifically (or explicitly) described in the mechanism

•This means it appears in the rate equation

•When we carry out an experiment to characterize the kinetics of an enzyme, we do so at several concentrations of substrate

•We want to mix the solution and then immediately measure the initial velocity (V0)

•Why?

•The [P] is low, so we do not have to be concerned with the back-conversion of product to substrate

We plot V0 versus the [S]

•We can see two distinct portions in the curve:

First Order Kinetics:

Increasing [S] yields an increased V0

Zero Order Kientics:

The enzyme molecules are saturated with substrate and increasing [S] does not yield an increased V0

Two useful terms can be extracted from this plot:

Vm or Vmax: The maximum velocity of the enzyme (where do we find it?)

Km: The concentration of substrate that gives (1/2)Vm

Deriving Km and the Steady State Assumption

E +S ES PE +

k1

k-1

k2

In the first step of the reaction, 2 competing processes are taking place.

Rate of Formation of ES:

Rate of Breakdown of ES:

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Δ[ES]

Δt= k1[E][S]

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−Δ[ES]

Δt= k -1[ES] + k2[ES]

Note: The conversion of ES --> E+P pulls ES out of the pool, so we have to include it in the breakdown rate!

The Steady State Assumption1. When we measure the velocity of a reaction, there is no product

present in the solution during the initial time periods

• This means that E+P --> ES does not occur

2. We also know that enzymes are very efficient, so during the initial time periods, the lifetime of the ES complex is VERY short

3. Based upon these two facts, we can assume that within milliseconds (or microseconds or less), the reaction will reach a Steady State in which the rate of ES formation equals the rate of ES breakdown.

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Δ[ES]

Δt=

−Δ[ES]

Δt

k1[E][S] = k -1[ES] + k2[ES]

This is the Steady State Assumption

Km and the Steady State Assumption

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Δ[ES]

Δt=

−Δ[ES]

Δt

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k1[E][S] = k -1[ES] + k2[ES]

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k1([E T ] - [ES])[S] = k -1[ES] + k2[ES]

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([ET ] - [ES])[S]

[ES] =

k -1 + k2

k1

= KM (The Michaelis Constant)

What is [E]? (We know [S] from the experiment?)

At any given moment in the reaction, the amount of free enzyme is:

[E] = [ET] - [ES]

Plug this into the equation:

From the steady state assumption

Km and the Steady State Assumption

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([ET ] - [ES])[S]

[ES] = KM

([ET ] - [ES])[S] = KM[ES]

[ET ][S] - [ES][S] = KM[ES]

[ET ][S] = KM[ES] + [ES][S]

[ET ][S] = [ES] (KM + [S])

[ES] = [ET ][S]

(KM + [S])

Multiply by [ES]

Multiply the left side through

Reorganize

Factor out [ES]

Isolate [ES]

Now we have a term to describe [ES], we can use it to describe V0 in terms of Km and Vm

V0, Km and Vm

1. Part of our steady state assumption is that the E+P --> ES back reaction doesn’t occur

2. We also know that the rate limiting step will be the conversion of ES to E+P going through the transition state– Does this make sense to

you?– If the rate limiting step

(energetically most expensive) step was the formation of the ES complex, would the reaction even occur?

3. Based upon these observations, we can say that the observed velocity of the reaction is:

V0=k2[ES]

Deriving the Michaelis-Menton Equation

With the mathematical definition of [ES], V0, Vm and Km, we are ready to derive the Michaelis-Menton equation

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V0 = k2[ES]

but we know [ES] = [ET ][S]

KM + [S]

⎛

⎝ ⎜

⎞

⎠ ⎟, so :

V0 = k2

[ET ][S]

KM + [S]

⎛

⎝ ⎜

⎞

⎠ ⎟ =

k2[ET ][S]

KM + [S]

V0 = VM[S]

KM + [S] The Michaelis - Menton Equation

If we are in the zero order part of the V0 vs. [S] curve, then V0=VM=k2[ET]

Substituting Vm for k2[ET], we get:

The Michaelis-Menton Equation

• With the Michaelis-Menton equation, we can relate the observed enzymatic velocity at any substrate concentration to the value of Vm

• This allows us to analyze enzymes and:• Identify differences in enzymatic function between species• Compare the same enzyme from different tissues of the

same organism• Analyze the efficiency of newly discovered enzymes• Identify the effect of mutations on enzymatic function

Working with / Thinking about the Michaelis-Menton Equation

1. If [S] >>> Km, what does V0 equal?

2. If [S] = Km, what does V0 equal?

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V0 = VM[S]

KM + [S]

Working with Real Data to Calculate Km and Vm

• Using V0 vs. [S] plots to calculate Vm and Km can be difficult and inaccurate because of the asymptotic nature of the plot

• Two scientists, Hans Lineweaver and Dean Burk, realized that if you plotted the reciprocal of the Michaelis-Menton equation, you’d get a straight line:

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V0 = VM[S]

KM + [S] =

VM[S]

KM

+ VM[S]

[S] =

VM[S]

KM

+ VM

Take the reciprocal of everything :

1

V0

= KM

VM

⎛

⎝ ⎜

⎞

⎠ ⎟1

[S] +

1

VM

This has the form: y = mx + b

Lineweaver-Burk Plots

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1

V0

= KM

VM

⎛

⎝ ⎜

⎞

⎠ ⎟1

[S] +

1

VM

This has the form of a linear equation: y = mx + b

If we plot the data as 1/V0 vs 1/[S]:

The slope = ?

The y-intercept = ? (This occurs when 1/[S] = ?)

The x-intercept = ? (This occurs when 1/V0 = ?)

This is the common method to analyze non-allosteric Enzymological Data

Plot the data on p.156 of Campbell and Farrell on your own

Also: Check the Useful Links page on the course website!

What do Km and Vm Really mean?

• We know that V0=Vm/2 when [S] = Km

• We can use this to say that when [S] = Km, 50% of the active sites of the population of enzymes are occupied

• If an enzyme has a low Km, the enzyme achieves maximum efficiency at low [S]

• Km is unique for each and every enzyme-substrate pair

• Some enzymes catalyze reactions that utilize 2 substrates:

E + A EA + B EAB E + P + QThere is a unique Km value for A and one for B

Thinking about Km

• As long as k2 << k-1 , the Km can be viewed as a measure of the affinity of the enzyme for the substrate– If k-1 is high, what does that tell you?

• The Km value is also a function of pH and temperature.– Why?

Turnover number and kcat

• In order to better understand the catalytic efficiency of enzymes, we need to define another term: kcat

The catalytic efficiency of an enzyme will be defined as:€

k cat = VM

[ET ]

€

Catalytic Efficiency = k cat

KM

kcat and Catalytic Efficiency

• kcat/KM results in the rate constant that measures catalytic efficiency. – This measure of efficiency is helpful in determining whether the rate

is limited by the creation of product or the amount of substrate in the environment.

• If the rate of efficiency is:

• HIGH, kcat is much larger than KM, thus the enzyme complex turns more enzyme-substrate into product that the enzyme encounters substrate, which is defined as k1.

• LOW, this means that the rate of product turnover is much lower than the substrate concentration, thus the enzyme and substrate have high affinity for each other.

kcat and Catalytic Efficiency

• kcat/KM is only useful when the substrate concentration is much less than the KM. Looking at the enzyme/substrate catalytic reaction equation:

• And knowing that k2 could be considered to be equal to kcat, it is evident from

that even if kcat is much greater than k-1, the equation will still be limited by k1, which is the rate of ES formation.

• This tells us that kcat/KM has a limit placed on efficiency in that it cannot be faster than the diffusion controlled encounter of an enzyme and its substrate (k1).

E +S ES PE +k1

k-1

k2

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k cat

KM

= k cat

k -1 + k cat

⎛

⎝ ⎜

⎞

⎠ ⎟k1

kcat and Catalytic Efficiency

Therefore, enzymes that have high kcat/KM ratios have essentially attained kinetic perfection because they have come very close to reaching complete efficiency only being limited by the rate at which they encounter the substrate in solution.