Enzyme Kinetics IINov. 11, 2008

Robert Nakamoto2-0279380 Snyderrkn3c@virginia.edu

Studying the photograph of a racehorse cannot tell you how fast it can run.

Jeremy Knowles

Eadweard Muybridge, 1878

Why bother with kinetics?

The rates at which a reaction occurs, compared to other reactions in a pathway, will determine the rate limiting and controlling reaction

A → B → C → D → E if the reaction C→D is the slowest then

regulating the enzyme carrying out this reaction will control the amount of E made

[C] will accumulate

A → B → C → D → E

If only the production of E is followed then one cannot tell which enzyme is controlling the overall rate

Or if only the disappearance of A were followed, then one cannot tell how fast E is made

Lots of information in a reaction time course

If only one time point is taken, then many important aspects may be missed. A non-linear rate A lag before the steady state Running out of substrate A competing activity or build up of product inhibition

time

[product]

Enzyme velocity at steady state- Michaelis-Menton considerations

[E]o [S] kcat

Ks + [S]V =

Where kcat[E]o = Vmax, then

The [S] at which v=1/2 Vmax is the Km

[S] Vmax

Km + [S]V =

E + S ES E + PkcatKs

Valid when binding is much faster than kcat

[S]

Vmax/2

Km

v = Vmax = kcat [E]o

Linear vs Log activity plots

log [S]

Km at ½ saturatingKm at ½ saturating

[S]

v/V

max

0.5

1.0

Linear plot is hyperbolic. In log plot, it takes two orders of magnitude in [S] to go from 10-90% saturation.

0-1 +1

Vmax is estimated from asymptote of maximal measured binding

Note: you use the same mathematical considerations for ligand binding to a receptor

What does the Km mean?

E + S ES E + PkcatKs

Km =k+2 + k-1

k+1

Ks approximates Km if k+2 << k-1

Valid when if k+2 << k-1

E + S ES E + Pk+2k+1

k-1

More general form

Elementary rate constants depend on the energy and entropy of activation

Transition state theory: temperature and the activation energy

enthalpyactivatedcomplex transition state

E’A reverse reaction

products

reactantsEA forward reaction

H of reaction

Reaction coordinate

EA is the activation energy for the forward reaction.E’A is the activation energy for the reverse reaction.EA- E’A = H, enthalpy change for the reaction.

Temperature and activation energy: the Arrhenius relationship

d lnKdT

P=

H°

RT2

Van’t Hoff equation shows the changewith temperature of an equilibrium constant.

A similar relationship holds for a reaction rate constant.

d lnkdT

P=

EA

RT2

This equation is rearranged to give: d lnk = EA dT

R T2

And integrated to give: lnk = lnA - and finally k = A eEA

RT

-EA

RT

A = integration factor

What does the Arrhenius eq. mean?

k = A eA is the frequency of collisions with the proper orientation to produce a chemical reaction. Can be as fast as 1013sec-1, which is about the frequency of collision in liquids.

Thus, Arrhenius theory says that the rate constant is determined by i) the ratio of EA to T and ii) by the frequency of collisions

-EA

RT

The Arrhenius plot

log v

1/T

slope = -EA

R

Note that a lower slope means a lower activation energy EA and that the reaction goes faster.

The “better enzyme” will reduce EA to a greater extent.

H‡ = EA - RT

S‡=Rln(ANh/RT)-R

G‡ = H‡ + T(S‡)

Example: amino acid substitutions can affect

EA, better or worse catalyst.

The Assay

If you want to understand the kinetics of a reaction,

like the binding of a ligand to a receptor, or

an enzymatic reaction, like a phosphorylation or

dephosphorylation of a signaling protein, or

transport of an ion across a membrane, or

transcriptional activation of a gene,

You need an assay with the proper “time constant”

Time domains of various techniques

10210-10 10010-5

seconds

Spectroscopic methods Hand mixingFlash and T jump

EPR and NMR

Pressure jump

Dielectric relaxation and electric dichroism

Laser scatter

Fl polarization

Ultrasound absorption and electric field jump Stopped flow and continuous flow

Specificity of the reaction

Is the reaction you are measuring carried out by only one enzyme?

Temperature? Co-factors? Competing activities?

Are there “non-enzymatic” pathways to the products?

Controls, controls, controls.

Example of kinetic analysis of a chemical reaction: ATP hydrolysis

Detection of ATP hydrolysis Pi production: How?

“Coupled” assays Colorimetric or Chromogenic assay Radioactivity

What are the variables? Sensitivity Time domain Background

Chromogenic reactions for Pi production

Acid Molybdate Taussky and Shorr (Fe2+ at acid pH) Fiske and SubbaRow (1-amino-2-naphthol-4-

sulfonic acid with sulfite buffer Lin and Morales (Vanadate at alkaline pH)

Malachite Green These assays stop the reaction, one time

point per sample.

Luciferase assay

ATP + luciferin ADP +

h

luciferase

Enzyme coupled assay

ATP ADP + Pi

ADP + Phosphoenol pyruvate Pyruvate + ATP

Pyruvate + NADH Lactate + NAD+

YFE

Pyruvate kinase

Lactate dehydrogenase

Follow absorbance change at 340 nm in the spectrophotometer.

Radioactivity detection of ATP hydrolysis

Labels: [-32P]ATP OR or labels or 14C (3H) labels on adenine Separation of labeled Pi from labeled ATP

Acid molybdate Organic extraction Selective precipitation

Extraction of ATP by charcoal Norit TLC to separate ATP, ADP and Pi

For fast reactions, rapid mixing and quench of the reaction is needed

Assay for production of radioactive Pi from [-32P]ATP

Reactions are prepared by having an enzyme solution and a substrate solution. [-32P]ATP is isotopically diluted with non-radioactive ATP.

Reactions are carried out. The reaction stopped with acid. One or more samples for each time point.

An acid molybdate solution is added to precipitate the Pi Samples are centrifuged to sediment precipitate,

supernatants are removed The pelleted precipitates are dissolved in alkali solution Radioactivity each sample is determined by scintillation

counting The amount (moles) of Pi is determined by comparison to

standards.

Fluorescence assay for Pi production

Phosphate binding protein modified with coumarin Fluorescence increase upon binding of Pi

Detects release of Pi from enzyme Fluorescence change can be followed

continuously in stopped flow

Mechanical mixing-spectroscopic observation

Usual deadtime ~ 1 ms; time resolution is less than 1 ms

Stopped-flow spectrometers

Pre-steady state measurements

ATP hydrolysis: production of 32Pi from

[-32P]ATP (rapid acid quench)

•Syringe A•1 M F1

•25 mM TES-KOH•0.244 mM MgCl2, •0.20 mM EDTA

Syringe B25 mM TES-KOH0.46 mM MgCl20.20 mM EDTA0.50 mM [32P]ATP

Final49 M Mg2+

free

107 M Mg·ATP0.5 M F1

pH 7.525 °C

Quench0.3 N PCA1 mM Pi

Pi release: fluorescence signal from MDCC-labeled PBP (stopped flow)

•Syringe A•1 M F1

•25 mM TES-KOH•0.244 mM MgCl2, •0.20 mM EDTA•10 M MDCC-PBP •“Pi mop”

Syringe B25 mM TES-KOH0.46 mM MgCl20.20 mM EDTA0.50 mM ATP10 M MDCC-PBP “Pi mop”

Final49 M Mg2+

free

107 M Mg·ATP0.5 M F1

pH 7.525 °C

hPMT

Pi mop: purine nucleoside phosphorylase (PNPase), phosphodeoxiribomutase (PDRM), 100 M 7-methylguanosine, 0.1 M -D-glucose 1,6-bis-phosphate

Pre-steady state: addition of 107 M ATP·Mg to F1 ATPase

E+ATP↔E·ATP↔E·ADP·Pi→E·ADP+Pi 1. ATP hydrolysis by production of 32Pi

from [-32P]ATP (rapid quench)2. Pi release by coumarin labeled Pi

binding protein (stopped flow)

Fit requires a slow step after hydrolysis and before Pi release

E+ATP↔E·ATP↔E·ADP·Pi ↓ krotation

E’·ADP·Pi→E”·ADP+Pi