# Enzyme Kinetics, cont.

Recall basic reaction model from last time:

d[ES]/dt= 0; Thus: 0 = d[ES]/dt= k1[E][S] - k2[ES] - k3[ES].

Continuing we can now substitute for E (free enzyme), because hard to find, and gather constants:

[E] = [Et] - [ES]; then

d[ES]/dt= k1([Et] [S] - [ES][S]) - k2[ES] - k3[ES],

gathering constants: ,

Now define

Then , 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 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.

### 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: [Figure 5.4b]

### 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].

[Table 5.1]

## 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.

## Multi-substrate Enzymes

Look at three common and easily understood types. We will use Cleland Nomenclature. [Figure 5.7a, b]

• Ordered Sequential Bi Bi mechanism (two on; two off); Note: A must bind first, Q is released last.
• Random Sequential Bi Bi (two on; two off); Note: A or B may bind first, P or Q may be released last.
• Ping Pong Bi Bi or Double-Displacement for the simplest case (one on, one off; one on, one off); Note: have some sort of modified enzyme intermediate (often covalent intermediate)

### Kinetic Mechanism

Ordered Sequential Bi Bi

Random Sequential Bi Bi

Ping pong Bi Bi

Note that in the Ping Pong and Ordered sequential mechanisms the enzyme forms before & after are the same and thus Q is a competitive inhibitor of A (that is A can displace Q) where as in the random sequential P & Q will be noncompetitive since neither can be disiplaced by substrate because of the different forms allowed by the random binding. Such inhibition differences can be used to distinguish between the different mechanisms.

# 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. [Figure 5.8a, b]

#### Plots: [Figure 5.9]

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.

### NEXT

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