Advanced Enzyme Kinetics and

Metabolism

BOC 324 Part A

Dr. A. van Tonder (for 3rd quarter; Part B in 4th quarter with Dr. E. van Heerden)

BOC 324 Part A SOURCES

Textbook: “Biochemistry” Mathews et al : Ch 11 pp. 360-413

Internet Resources: http://www-biol.paisley.ac.uk/kinetics/contents.html

http://www.cf.ac.uk/biosi/staff/kille/dentals/dental5_99/

Articles: Articles 1-8 are in the study guide

Advanced Enzyme Kinetics

TOPICS: 1. Enzyme kinetics: Basics

2. Determination of kinetic constants

3. Kinetics of enzyme inhibitors

4. Kinetics of multisubstrate reactions

5. Kinetics of allosteric enzymes

MIND MAP

1. Enzyme Kinetics: Basics

Contents Revision : BOC226 work (Ch 11 in Mathews)

plus Internet sources plus Wikipedia (article #8)

Steady state models for 1S, 1P when [S]>>[E]

The effect of [S] on v

The effects of [E]

The meaning of kcat and kcat/Km

The significance of Km, kcat and kcat/Km

REVISION What is an enzyme? It is a BIOLOGICAL CATALYST!!!

The reaction catalysed by an enzyme uses

exactly the same reactants and produces exactly the same products as the uncatalysed reaction.

Like other catalysts, enzymes do not alter the position of equilibrium between substrates and products.

However, unlike uncatalysed chemical reactions, enzyme-catalysed reactions display saturation kinetics.

For a given enzyme concentration and for relatively low substrate concentrations, the reaction rate increases linearly with substrate concentration; the enzyme molecules are largely free to catalyze the reaction, and increasing substrate concentration means an increasing rate at which the enzyme and substrate molecules encounter one another:

What an enzyme does:

e.g.:

The reduction of activation energy (ΔG) increases the number of reactant molecules with enough energy to reach the activation energy and form the product.

By providing an alternative reaction route and by stabilizing intermediates the enzyme reduces the energy required to reach the highest energy transition state of the reaction.

Not so simple –

may look like this:

The favored model for the enzyme-substrate interaction is the induced fit model of Daniel Koshland (1958)....

This model proposes that the initial interaction between enzyme and substrate is relatively weak, but that these weak interactions rapidly induce conformational changes in the enzyme that strengthen binding.

Catalysis by induced fit - Stabilising effect of strong

enzyme binding.

- Two different mechanisms of substrate binding:

uniform binding: strong substrate binding, differential binding: strong transition state binding.

- The stabilizing effect of uniform binding increases both substrate and transition state binding affinity, while differential binding increases only transition state binding affinity.

These conformational changes also bring catalytic residues in the active site close to the chemical bonds in the substrate that will be altered in the reaction.

After binding takes place, one or more mechanisms of catalysis lowers the energy of the reaction's transition state, by providing an alternative chemical pathway for the reaction.

There are five possible mechanisms of "over the barrier" catalysis as well as a "through the barrier" mechanism (see Wikipedia article for detail):

- Catalysis by bond strain - Catalysis by proximity and orientation - Catalysis involving proton donors/acceptors (Acid/Base

Catalysis) - Electrostatic catalysis - Covalent catalysis - Quantum tunnelling

1. Catalysis by bond strain - The affinity of the enzyme to the transition state is

greater than to the substrate itself. - Induces structural rearrangements which strain substrate

bonds into a position closer to the conformation of the transition state, so lowering the energy difference between the substrate and transition state and helping catalyze the reaction.

2. Catalysis by proximity and orientation - Increases the rate of the reaction as enzyme-substrate

interactions align reactive chemical groups and hold them close together.

- This reduces the entropy of the reactants and thus makes reactions such as ligations or addition reactions more favourable

- There is a reduction in the overall loss of entropy when two reactants become a single product.

3. Catalysis involving proton donors/acceptors

(Acid / Base Catalysis) - Proton donors and acceptors, i.e. acids and bases, may

donate and accept protons in order to stabilize developing charges in the transition state.

- Typically has the effect of activating nucleophile and electrophile groups, or stabilizing leaving groups.

4. Electrostatic catalysis - Stabilization of charged transition states can also be by

residues in the active site forming ionic bonds (or partial ionic charge interactions) with the intermediate.

- These bonds can either come from acidic or basic side chains found on amino acids such as Lys, Arg, Asp or Glu or come from metal cofactors such as zinc.

5. Covalent catalysis - Involves the substrate forming a transient covalent bond

with residues in the active site. - Adds additional covalent intermediate to the reaction, and

helps to reduce the energy of later transition states of the reaction.

- Covalent bond must, at a later stage in the reaction, be broken to regenerate the enzyme.

- Found in enzymes such as proteases like chymotrypsin and trypsin, where an acyl-enzyme intermediate is formed.

6. Quantum tunnelling - Some enzymes operate with kinetics which are faster than

predicted. - In "through the barrier" models, a proton or an electron

can tunnel through activation barriers. - Quantum tunnelling for protons has been observed in

tryptamine oxidation by aromatic amine dehydrogenase. - Does not appear to provide a major catalytic advantage.

The two most important kinetic properties of an enzyme are:

1. how quickly the enzyme becomes saturated with a particular substrate, and

2. the maximum rate it can achieve.

Knowing these properties suggests what an enzyme might do in the cell and can show how the enzyme will respond to changes in these conditions.

[E] + [S] [ES] [E] + [P] k1

k-1

kcat v0 = Vmax [S]

Km + [S]

Enzyme kinetics is the study of the chemical reactions that are catalysed by enzymes, with a focus on their reaction rates.

The study of an enzyme's kinetics reveals the catalytic mechanism of this enzyme, its role in metabolism, how its activity is controlled, and how a drug or a poison might inhibit the enzyme. Dihydrofolate reductase from E. coli with

its two substrates, dihydrofolate (right) and

NADPH (left), bound in the active site.

What is the most plentiful single enzyme on earth? Answer: ribulose bisphosphate carboxylase /

oxygenase (or RUBISCO)

- Catalyses the attachment of carbon dioxide to ribulose bisphosphate, a short sugar chain with five carbon atoms and then clips the lengthened chain into two identical phosphoglycerate pieces.

- Why so abundant? It fixes only about three carbon dioxide molecules per second so plants make more of it – half of the protein in chloroplasts is Rubisco. 2 x 8 protein chains

Steady State Models for 1S, 1P

Con

cent

ration

s

Time

Pre-steady

state

ES forming

Steady state

ES almost constant

[E]t

[E] [ES]

[P] [S]

[E] + [S] [ES] [E] + [P] k1

k-1

kcat

[S] (mM) >> [E]t (10-8 - 10-10M)

[S] changes, [E]t constant

Assumptions and Givens:

d[ES]/dt = O (Steady state)

[P] = 0 at t = 0

v = d[P]/dt = kcat [ES]

[E]t = [E] + [ES]

Vmax = kcat[E]t

Km = {k-1 + kcat}/k1

= [S]½ at V0 = ½Vmax

v0 = Vmax [S] = kcat[E]t[S]

Km + [S] Km + [S]

Cannot measure

Michaelis-Menten

0

0max0

][

][

SK

SVv

m

MICHEALIS-MENTEN KINETICS

Leonor Michaelis

(1875-1949)

Maud Menten (1879-1960)

1913

M-M: The Effect of [S] on v

Low [S] : [S] << Km V0 = {Vmax/Km}[S] V0 [S]

First order reaction

[S] = Km V0 = {Vmax[S]}/2[S] V0 = ½Vmax

[S] > Km : mixed order reaction

High [S] : [S] >> Km

V0 = {kcat[E]t[S] }/[S] = kcat[E]t = Vmax

Zero order reaction

E is saturated with S

v0 = Vmax [S]

Km + [S]

The Effects of [E]

High [S] : [S] >> Km

v0 = kcat[E]t = Vmax

Vmax [E]t

Km independent of [E]

Deviations from Michaelis-Menten kinetics

[S]o

vo

[S]o

vo

Substrate inhibition Positive co-operativity

Deviations from Michaelis-Menten kinetics

[S]o

vo

[S]o

vo

Negative co-operativity Alternative pathways

E EAB

EA

EB

Products

Deviations from Michaelis-Menten kinetics

[S]o

vo

Two or more molecules

of the same substrate

[S]o

vo

Failure to determine vo

Deviations from Michaelis-Menten kinetics

[S]o

vo

More than one enzyme catalysing

the same reaction

0

0max

0

0max

0][

][

][

][

SK

SV

SK

SVv

b

m

b

a

m

aVma

Kma

Vmb

Kmb

Vmtot

Deviations from Michaelis-Menten kinetics

[S]o

vo

Failure to subtract blank rate

Blank rate

Enzyme reaction plus blank rate

The meaning of kcat and kcat/Km Km is a useful kinetic constant

Indicates [S] at ½Vmax

Suggests putative [S] in vivo Not an independent constant

Independent constants obtained by extrapolating to low or high [S]

Vmax and kcat at very high [S]

Vmax/Km and kcat/Km at very low [S]

The catalytic constant kcat is the first order rate constant for the conversion of the ES complex to E + P.

It is measured when the enzyme is saturated with substrate (region A)

The ratio kcat/Km is the second-order rate constant for the conversion of E + S to E + P at very low [S] (region B)

Region A

V0 = kcat[E]1[S]0

Region B

V0 = (kcat/Km)[E]1[S]1

ES E + P kcat

kcat

Km

E + S E + P

A

B

The significance of Km, kcat and kcat/Km

Km (M) = (k-1 + kcat)/k1

If k-1 >> kcat, then Km k-1/k1 = ([E][S]) / [ES] = Kd

Km is an inverse measure of binding strength

Large Km can also be due to large kcat

Interpretation of Km as Kd for [ES] must be used with caution

Km is not an independent kinetic constant

kcat (s-1) -Turnover number

Measures the number of S molecules converted to P per E

molecule per second - the rate of the catalytic process

Compare catalytic productivity of different enzymes

1/ kcat (s) = time required for 1 E molecule to convert 1 S to P

kcat = Vmax/[E]t

kcat/Km (M-1.s-1)

If [S] << Km, then [E]t = [E] (most E is free)

V0 = (kcat/Km ) [E][S]

kcat/Km is a second order rate constant that is a direct measure of E efficiency

Compare enzyme specificity for different substrates

VA/VB = {kcat/Km }A/ {kcat/Km }B

Reaction rate cannot exceed rate of diffusion, = 108-1010

Enzymes such as carbonic anhydrase, acetylcholine esterase, fumarase with kcat/Km 108 have reached the highest level of

catalytic evolution

Enzyme assays follow changes in the concentration of either substrates or products to measure the rate of reaction.

There are many methods of measurement: - Spectrophotometric assays observe change in the

absorbance of light between products and reactants (most convenient since they allow the rate of the reaction to be measured continuously);

- Radiometric assays involve the incorporation or release of radioactivity to measure the amount of product made over time (i.e., they are discontinuous assays).

The most sensitive enzyme assays use lasers focused through a microscope to observe changes in single enzyme molecules as they catalyse their reactions.

How do we obtain V0 at different [S]? •Continuous assays (spectrophotometric)

•Fixed time assays (spectrophotometric or radiometric)

2. Determination of Kinetic Constants

Contents Graphical Methods

Lineweaver-Burk

Eadie Hofstee

Hanes/Woolf

Direct Linear i.e. linear transformations

Best fit curve to Michaelis equation

v0 = Vmax [S]

Km + [S]

y = mx + c

Graphical Methods:

1/v0 = {(Km/Vmax) (1/[S])} + 1/Vmax

1/v vs 1/S

ADVANTAGES:

- Straight line (y = mx +c) is easy to draw - Vmax does not have to be determined directly - Can read Km and Vmax easily from the graph

Lineweaver-Burk

Graph

Hans Lineweaver and Dean Burk in 1934

Lineweaver-Burk Graph: Disadvantages

Not acceptable: - This plot conceals a poor fit between the data and a straight line - Large error at low [S] where measurements are less accurate

Eadie-Hofstee Graph

v0 = -Km (v0 / [S] ) + Vmax

V = - Km + Vmax V

[S]

Advantages: - Error is not so severe as with the Lineweaver-Burk plot - Generally regarded as being a better technique than LB.

Eadie-Hofstee Graph: Disadvantages

•Scatter in the data resulting in values for Km and Vmax which are skewed away from the true values. •The dependent variable v0 occurs in both the x- and y-axis •Large error at low [S]

Slope = -Km

Vmax

Hanes/Woolf Graph

+[S]

V Vmax 1 [S]=

Km

Vmax

Advantages: - Direct readout of Km

- Calculate Vmax from Km/Vmax intercept - Safer to use- distortion of error bars is minimal - Less scatter than Lineweaver-Burke or Eadie-Hofstee - Velocity (dependent) data does not influence the data on the x-axis

[S]/v0 = (1/Vmax)[S] + Km/Vmax

Hanes/Woolf Graph: Disadvantages

Both axes represent an independent variable: [substrate] Still get errors at low [S]

Slope = 1/Vmax

-Km

Direct Linear Graph

Vmax = (v0/[S])Km + v0

Join –S and v data points on x and y axis and then extrapolate into positive quadrant. Intersect used to determine Vmax and Km

Advantages: - Reliable. - No calculations required - Kinetic constants read directly off plot - Recommended equally with least square fit to hyperbola.

Eisenthal & Cornish-Bowden, 1974

Direct Linear Graph: How it copes with errors

Km and Vmax are calculated from the median values

Disadvantage: no provision for replicate values of v

Best Curve Fit: non-linear regression (BEST METHOD OF ALL)

If the mechanism is known and complex then the data must be reconciled to the appropriate model (hypothesis) - usually by use of a computer-aided analysis involving a weighted least-squares fit.

Many such computer programs are currently available;

If the mechanism is not known, initial attempts are usually made to fit the data to the Michaelis-Menten kinetic model:

d[P] =

dt (eqn 1)

v0 = Vmax [S]

Km + [S]

Use of equation 1 involves the determination of the initial rate of reaction over a wide range of substrate concentrations.

Equation 1 can be utilised directly using a computer program,

involving a weighted least-squares fit, where the parameters for determining the hyperbolic relationship between the initial rate of reaction and initial substrate concentration (i.e.. Km and Vmax) are chosen in order to minimise the errors between the data and the model, and the assumption is made that the errors inherent in the practically determined data are normally distributed about their mean (error-free) value.

Example of such a program is GraphPad Prism (Article #7):

For programs such as Prism that easily do nonlinear regression, the best way to determine Km and Vmax is to fit a hyperbola directly to the substrate-velocity data:

This topic of non-linear regression will be expanded upon during the

practical sessions…

See also Article #7 for more detailed information