Humboldt State University ® Department of Chemistry

Richard A. Paselk

Chem 438 - Introductory Biochemistry - Spring 2013

Lecture Notes:

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Enzyme Kinetics, cont.

Recall basic reaction model from last time:

chemical equation for  E +S in equilibrium with [ES] (showing rate constants k1 and k2) decaying to E + P  (with rate constants k3)

For steady state:

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: mathematical equation: (k2 + k3)/k1 = [S]([Et] - [ES])/[ES],

Now define mathematical equation: (k2 + k3)/k1 = Km

Then mathematical equation: Km = [S]([Et] - [ES])/[ES], 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]: mathematical equation: [ES] = [Et][S]/(Km + [S])

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

Substituting: mathematical equation: vi = k3[Et][S]/(Km + [S]) and mathematical equation: vi/k3 = [Et][S]/(Km + [S]),

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

So, M-M equation: vi = Vmax[S]/(Km + [S]) Which is known as the Michaelis-Menten Equation.

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

M-M equation: vi = Vmax[S]/(Km + [S])

Note predicted consequences of model:

M-M plot

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 right arrow 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

Lineweaver-Burke "double-reciprocal" equation. [Figure 5.6]

Lineweaver-Burke "double-reciprocal" plot

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.

Eadie-Hofstee (v vs. v/[S]) plot

Multi-substrate Enzymes

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

Kinetic Mechanism

ordered sequential Bi Bi Kinetic mechanism diagram

Ordered Sequential Bi Bi

Random Sequential Bi Bi Kinetic mechanism diagram

Random Sequential Bi Bi

Ping Pong Bi Bi Kinetic mechanism diagram

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?

Three major types of inhibition:

Competitive Inhibition

S & I are mutually exclusive, E can bind to one OR the other. [Figure 5.8a, b]

Plots: [Figure 5.9]

M-M Plot of  Competitive  InhibitionL-B Plot of  Competitive  Inhibition showing itersection of lines at 1/Vmax

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: Model of competitive inhibition with chemical equilibrium equations; and: M-M equation for competitive inhibition showing Km  multiplied by 1+ [I]/Ki; where equilibrium expression for Ki = [E][I]/[EI].

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. 

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Last modified 22 February 2013