Biol/Chem 5310

Lecture: 17

October 24, 2002

Summary of Michaelis-Menten Kinetics

If a plot of v vs. [S] is hyperbolic, revealing saturation ( or more easily recognized, if 1/v vs. 1/[S] is a straight line) the enzyme is said to obey Michaelis-Menten kinetics with respect to that substrate.

For a hyperbolic curve, mathematically:

i.e. the rate is equal to something "a" times [S] divided by something "b" plus [S]

- a is equal to V
_{max} - b is equal to K
_{m}, the concentration of substrate when v = 1/2 V_{max} - at High [S], where [S]>>K
_{m}, then v = V_{max}, a constant with no dependence on [S] - at low [S], where [S]<<Km, then v =(V
_{max}/K_{m})[S]- in this case, the rate is proportional to [S] with apparent first order rate constant of V
_{max}/K_{m}

- in this case, the rate is proportional to [S] with apparent first order rate constant of V

Remember, K_{m} is the substrate concentration at which the rate is half-maximal. In general it is a function of several individual rate constants. In the simple scheme we have considered, shown below,

In the event that k_{-1} >> k_{2} , equilibrium in binding will occur, and K_{m} will be equal to K_{s}, where

K_{m} is not in general equal to the dissociation constant of the substrate, although it correlates with it:

High K_{m} --> loose binding

Low K_{m} --> tight binding

The catalytic constant *k _{cat }* is defined as

V_{max} divided by the total enzyme concentration ([E]_{T}). It has units of (sec)^{-1}; , so it is a first order rate constant.
It is sometimes called the turnover number, because it reflects
how many times the enzyme can turnover per second. In the simple
scheme above, *k* _{cat} is equal to *k* _{2 , }because V_{max} = *k*_{ 2 }[E]_{T}

In general, k_{cat} will be a function of the rate
constants in the reaction scheme.

A special case exists when [S] << K_{m}

In this case [E] is approximately equal to [E]_{T}

because [ES] << 0

so,

so *k* _{cat} / K_{m} is an apparent second order rate constant for E and S

This is sometimes called the affinity constant, its value reflects the enzyme's catalytic efficiency.

High efficiency will be associated with k_{2} >>
k_{-1}

i.e. product formation over release of substrate (back reaction)

The limit of efficiency will be limited by the rate of enzyme
encountering the substrate. This is the "diffusion-controlled
limit" which is ~10^{8} - 10^{9} M^{-1}
s^{-1}

Enzymes with *k* _{cat} / K_{m }> 10^{8 }M^{-1}
s^{-1}, such as acetylcholinesterase, are sometimes called "perfect enzymes"

Reversible Inhibitors: 3 general categories

1) Competitive: binds at the active site, competes with substrate. Only one can occupy the binding site at a time. Competitive inhibitors must look like the substrate, but are unable to be converted to products. They are usually the same size or smaller than the substrate. Example, malonate is a competitive inhibitor of succinate dehydrogenase, which converts succinate to fumarate.

{see structures}

Here [E]_{T}
= [E] + [ES] + [EI]

where

EI cannot bind substrate.

Define a :

a is equal to or greater than 1

It can be seen that the effect of a competitive inhibitor is to raise the apparent K_{m}, but does not affect the V_{max}.

V_{max} is unaffected by a competitive inhibitor.

The apparent K_{m} is increased by a competitive inhibitor.

See the Animation of Fig.12-7

2) Mixed Inhibition (Noncompetitive is a special case)

A mixed inhibitor binds to both E and ES, not at the substrate binding site:

Note that there are 2 K_{i}'s. If they have the same value, i.e. the inhibitor binds equaly well to E and ES, then a mixed inhibitor is usually called a noncompetitive inhibitor.

and

In the presence of a mixed inhibitor, the following Lineweaver-Burk plot is obtained:

Both the apparent K_{m} and the apparent V_{max} are changed by the inhibitor:

or:

If , then the inhibitor is called
*noncompetitive*, and a different looking Lineweaver-Burk
plot is obtained:

In this case, the presence of the inhibitor does not affect the x-intercept.

But the V_{max} is reduced by the inhibitor:

See the Animation of Fig. 12-9

3) Uncompetitive inhibition occurs when the inhibitor binds only to the ES complex:

Both V_{max} and K_{m} are reduced by the same
factor, a'

In a Lineweaver-Burk plot, this leads to parallel lines:

See the Animation of Fig. 12-8

In general, K_{i}'s can be determined, if a values can be calculated from the plots, and if the
concentration of Inhibitor is known.

For example, if a parallel line is obtained, as shown above, and K_{m}=6mM, and K_{m(app)}=4mM:

then, a'=1.5

If [I]=10 mM, then solve 1.5 = (1+10/x)

so x=K_{i}' = 20 mM (Notice that the units of Ki's come out as the units of I that are put in.)

Regulation of Enzyme Activity

1) Control of enzyme levels

- gene expression, on or off
- protein degradation

2) Control of enzyme activity: fine tuning, up and down, example aspartate transcarbamoylase

carbamoyl phosphate + aspartate -------> N-carbamoyl aspartate + phosphate

N-carbamoyl aspartate is committed to go (in 6 steps) on to cytidine triphosphate (CTP), a DNA precursor.

A --> B --> C --> D --> E --> F --> G

- Each arrow represents an enzyme-catalyzed reaction
- The final product shown, G, inhibits the first reaction A --> B
- This is called Feedback Inhibition
- In a committed pathway it is more efficient to regulate at the first step.
- Under other conditions, it may be useful to activate the first enzyme,rather than inhibit it.

How does this occur? *Allostery*

Consider Aspartate transcarbamoylase, ATCase:

The crystal structure was first solved by William Lipscomb

ATCase has 12 subunits (see Chime link):

6 C-catalytic in 2 trimers 6 R-regulatory in 3 dimers

- ATCase shows positive cooperativity with respect to the substrate aspartate

- CTP and ATP are allosteric effectors
- CTP is an inhibitor
- ATP is an activator

- Because of base pairing in DNA, the levels of purines (A and G) and pyrimidines (C and T) must be equal.
- In the absence of regulatory subunits, the catalytic subunits are non-cooperative with respect to aspartate, and not affected by ATP and CTP

- ATP stabilizes the R-state form
- CTP stabilizes the T-state form

- In the R-state, 2 domains of the catalytic subunits are bought closer together, to facilitate catalysis
- See Chime link

Try the Chapter 12 quiz.

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Comments/questions: svik@mail.smu.edu

Copyright 2002, Steven B. Vik, Southern Methodist University