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:

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:

gathering constants: ,

Now define

Then , where K_M 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 v_i = k₃[ES], and dividing both sides by k₃, v_i/k₃ = [ES]

Substituting: and ,

But maximum possible velocity must = k₃[E_t] = V_max

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] >> K_M; then v_i = V_max and get Zero order (r = k)
• [S] << K_M; then , and get First order (r = k [S])
• [S] = K_M; then v_i = V_max/2 This is definition of K_M, 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, k_cat (k₃), is also known as the turnover number, that is the maximum number of substrate molecules processed/active site (moles substrate/mole active site): k_cat=V_max / [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:

v_o = (k_cat / K_M)([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 K_M intercept ­ this is not as much of a concern now with computer statistical packages, but you still have to understand the statistics).

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. 

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© R. A. Paselk 2010;

Last modified 22 February 2013