Enzyme Kinetics II Nov. 11, 2008 Robert Nakamoto Snyder

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] k cat K s + [S] V = Where k cat [E] o = Vmax, then The [S] at which v=1/2 Vmax is the Km [S] Vmax Km + [S] V = E + S ES E + P k cat KsKs Valid when binding is much faster than k cat

[S] Vmax/2 Km v = Vmax = k cat [E] o

Linear vs Log activity plots log [S] K m at ½ saturating [S] v/Vmax Linear plot is hyperbolic. In log plot, it takes two orders of magnitude in [S] to go from 10-90% saturation. 0+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 + P k cat KsKs Km = k +2 + k -1 k +1 K s approximates K m if k +2 << k -1 Valid when if k +2 << k -1 E + S ES E + P k +2 k +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 enthalpy activated complex transition state E’ A reverse reaction products reactants E A forward reaction  H of reaction Reaction coordinate E A is the activation energy for the forward reaction. E’ A is the activation energy for the reverse reaction. E A - E’ A =  H, enthalpy change for the reaction.

Temperature and activation energy: the Arrhenius relationship d lnK dTdT P =  H° RT 2 Van’t Hoff equation shows the change with temperature of an equilibrium constant. A similar relationship holds for a reaction rate constant. d lnk dTdT P = EAEA RT 2 This equation is rearranged to give: d lnk = E A dT R T 2 And integrated to give: lnk = lnA - and finally k = A e EAEA RT -EA-EA A = integration factor

What does the Arrhenius eq. mean? k = A e A is the frequency of collisions with the proper orientation to produce a chemical reaction. Can be as fast as sec -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 E A to T and ii) by the frequency of collisions -EA-EA RT

The Arrhenius plot log v 1/T slope = - EAEA R Note that a lower slope means a lower activation energy E A and that the reaction goes faster. The “better enzyme” will reduce E A to a greater extent.  H ‡ = E A - RT  S ‡ =Rln(ANh/RT)-R  G ‡ =  H ‡ + T(  S ‡ )

Example: amino acid substitutions can affect E A, 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 seconds Spectroscopic methodsHand mixingFlash and T jump EPR and NMR Pressure jump Dielectric relaxation and electric dichroism Laser scatter Fl polarization Ultrasound absorption and electric field jumpStopped 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 (Fe 2+ 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: [  - 32 P]ATP OR  or  labels or 14 C ( 3 H) 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 [  - 32 P]ATP Reactions are prepared by having an enzyme solution and a substrate solution. [  - 32 P]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 32 Pi from [  - 32 P]ATP (rapid acid quench) Syringe A 1  M F 1 25 mM TES-KOH mM MgCl 2, 0.20 mM EDTA Syringe B 25 mM TES-KOH 0.46 mM MgCl mM EDTA 0.50 mM [  32 P]ATP Final 49  M Mg 2+ free 107  M Mg·ATP 0.5  M F 1 pH °C Quench 0.3 N PCA 1 mM Pi

Pi release: fluorescence signal from MDCC-labeled PBP (stopped flow) Syringe A 1  M F 1 25 mM TES-KOH mM MgCl 2, 0.20 mM EDTA 10  M MDCC-PBP “Pi mop” Syringe B 25 mM TES-KOH 0.46 mM MgCl mM EDTA 0.50 mM ATP 10  M MDCC-PBP “Pi mop” Final 49  M Mg 2+ free 107  M Mg·ATP 0.5  M F 1 pH °C h PMT 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 F 1 ATPase E+ATP↔E·ATP↔E·ADP·Pi→E·ADP+Pi 1. ATP hydrolysis by production of 32 Pi from [  - 32 P]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 ↓ k rotation E’·ADP·Pi→E”·ADP+Pi