### Richard A. Paselk

Chem 438

Introductory Biochemistry

Spring 2010

Lecture Notes: 26 February

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# Enzyme Kinetics

Last time left off with: , where KM is the Michaelis-Menten constant.

{Note that if k2 >> k3 (that is the equil. of E+S with ES is rapid compared to breakdown of ES to P), then M-M const = 1/(affinity)= the dissociation constant, but only in these special conditions.}

Now a couple of tricks: Solve for [ES]:

and recall that k3[Et] = Vmax and therefore vi = k3[ES], and dividing both sides by k3, vi/k3 = [ES]

Substituting: and ,

But maximum possible velocity must = k3[Et] = Vmax

So, Which is known as the Michaelis-Menten Equation.

### For simple, one-substrate enzymes then, have Michaelis-Menten Equation as a model for enzyme activity.

Note predicted consequences of model:

• [S] >> KM; then vi = Vmax and get Zero order (r = k)
• [S] << KM; then , and get First order (r = k [S])
• [S] = KM; then vi = Vmax/2 This is definition of KM, the substrate concentration at half-saturation.
• Note consequences for a plot: start off with approximately linear slope with y = kx. Then at the limit of high concentrations have a horizontal line. This is exactly what we expect if we look at the general form of the equation: , the formula for a rectangular hyperbola:

Turnover Number. The rate constant (First order) for the breakdown of the [ES] complex, kcat (k3), is also known as the turnover number, that is the maximum number of substrate molecules processed/active site (moles substrate/mole active site): kcat=Vmax / [E]total. Note that this is best determined under saturating conditions. At very low concentrations of [S] can find the second-order rate constant for the conversion of E + S E + P: vo = (kcat / KM)([E][S].

## Linear plots for enzyme kinetic studies

Double Reciprocal or Lineweaver-Burke Plot: Need in form: y = ax + b , so take reciprocals of both sides and have

. (Figure 5.6)

Other linear plots are also available, and are better in terms of statistics (L-B one of worst, best quality points [high concentration] have least influence on slope, while low precision points [low concentration] are more spread out, and have a large moment, with a strong influence on the slope and the KM intercept ­ this is not as much of a concern now with computer statistical packages, but you still have to understand the statistics).

### FYI - The Eadie-Hofstee Plot

One common plot is shown below. Note that the data points are distributed much more evenly over the plot giving better statistics for the slope. In addition the value of KM is obtained from the slope, giving better precision.

# ENZYME KINETICS AND INHIBITION

### What's exciting about enzyme inhibition?

• Potential to tell us about enzyme.
• Potential uses as drugs and toxins.
• Understanding drugs and toxins to counter etc.

Three major types of inhibition:

• Competitive inhibition
• Noncompetitive inhibition
• Uncompetitive inhibition

Competitive Inhibition: S & I are mutually exclusive, E can bind to one OR the other. (overhead Horton: T-33, Fig 5.8)

Plots:

We can model this inhibition with chemical equations, keeping in mind that S & I are mutually exclusive, E can bind to one OR the other:

Model: ; and: ; where .

Classically assume binding to same site, but other possibilities also.

1. steric hindrance between S & I in different sites.
2. overlapping sites for S & I.
3. Partial sharing of sites.
4. Conformational change of enzyme with binding of either such that other can not bind.

Noncompetitive: the inhibitor can bind to either E or ES. S & I do not bind to the same sites!

Model: ; and .

Note that will have two inhibitor binding constants, they may be the same, as in the equation above, or could be different, leading to more complex behavior.

Plots for classic, simple situation (overhead MvH 11.5):

### FYI - Uncompetitive Inhibition

In uncompetitive inhibition the inhibitor binds ONLY to the ES complex (overhead P 6.10).

Model: ; and .

For double reciprocal plots get parallel lines! This is not generally found for single substrate enzymes, but is found in multi-substrate systems.

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