Humboldt State University ® Department of Chemistry

Richard A. Paselk

Chem 431	Biochemistry	Fall 2007
Lecture Notes: 1 October	© R. Paselk 2007
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Enzyme Kinetics, cont.

simple, one-substrate enzymes, cont.:

Last time finished with the steady state (steady state assumption: d[ES]/dt= 0):

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] = [E_t] - [ES]; then

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

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 k₃[E_t] = V_max and therefore 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: (text Figure 6-12)

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. (text Table 6-08) 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].

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

. (text Box 6-01 figure 6-1)

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). We will see others in the laboratory discussion.

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.

Plots: (text Box 6-2 figure 1)

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: (text Figure 6-15a)

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! (text Figure 6-15c)

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):

Mixed: Like non-competitive above,

Model: ; and

but with different values of K_i. This is in fact the more common situation, with a L-B plot with an intersection on the plot above the 1/v_o axis. (text Box 6-2 figure 3)

Uncompetitive: In uncompetitive inhibition the inhibitor binds ONLY to the ES complex (text Figure 6-15b).

Model: ; and .

For double reciprocal plots get parallel lines! (text Box 6-2 figure 2) This is not generally found for single substrate enzymes, but is found in multi-substrate systems.

Pathway Diagrams

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Last modified 1 October 2007