Enzyme Kinetics. Michaelis-Menten Theory Dehaloperoxidase: Multi-functional Enzyme. NC State University

Transcription

1 Enzyme Kinetics Michaelis-Menten Theory Dehaloperoxidase: Multi-functional Enzyme NC State University

2 Michaelis-Menton kinetics The rate of an enzyme catalyzed reaction in which substrate S is converted into products P depends on the concentration of the enzyme E even though the enzyme does not undergo any net change. k a on k b cat E + S ES P + E k a off

3 Michaelis-Menton rate equations k a k b E + S ES P + E k a

4 Steps in the Michaelis-Menton mechanism Step 1. Bimolecular formation of the enzyme E and and substrate S: E + S ES rate of formation of ES = k on [E][S] Step 2. Unimolecular decomposition of the complex: ES E + S rate of decomposition of ES = -k off [ES] Step 3. Formation of products and release from the enzyme: ES P + E rate of formation of P = k cat [ES] The rate law of interest is the formation of the product in terms of E and S.

5 The enzyme substrate complex can be eliminated The enzyme substrate complex is formed transiently and can be approximated using the steady state approximation. The result of this approximation is

6 Pseudo-first order Michaelis-Menton kinetics In an experiment we know the total enzyme concentration [E] 0 and not the unbound enzyme [E]. The total concentration of enzyme [E] 0 = [E] + [ES]. which rearranges to

7 Pseudo-first order Michaelis-Menton kinetics At this point it is convenient to define the Michaelis constant and to rearrange the equations as

8 Michaelis-Menton parameters The rate of formation of product can be written where K m is the Michaelis constant and k cat is the maximum turnover number. We often make the definitions which permit us to write the equation as

9 Limiting conditions of enzyme reactivity Maximal rate: If there is excess substrate present the rate is limited by the rate at which the ES complex falls apart. The rate of formation of products is a maximum and v max = k cat [E] 0 is called the maximum velocity. Second order regime: If [S] << K M then the rate of formation of products is d[p]/dt = k cat /K m [E] 0 [S]. The rate depends on [S] as well as [E] 0. A plot of 1/k yields k cat and K m but not the rate constants k on and k off. The latter rate constants can be obtained from stopped-flow experiments.

10 General expression for reaction velocity Based on the previous analysis the velocity at an arbitrary substrate concentration is:

11 Lineweaver-Burke Plots The Michaelis-Menton expression is non-linear. The Lineweaver-Burke plot is linearized plot of data. 1 v = K M + [S ] [S ]v max = 1 v max + K M v max 1 [S ] This expression has the form of an equation for a line: y = intercept + slope x Such plots are not necessary today with common non-linear fitting programs.

12 Transition State Stabilization The original idea of the enzyme having maximum complementarity to the TS was put forward by Linus Pauling in It wasn't until the early 70's that the idea was put on a more solid grounding. As put forward by Lienhard and Wolfenden the idea is as follows: K n k n E + S E + S E + P K s K c ES ES E + P K t k c

13 Transition State Stabilization Defining the equilibrium constants as association constants: K n = [S ]/[S], K t = [ES ]/[E][S ] from TS theory: G = -RT ln K and k obs = (k B T/h)e - G /RT Thus, k n = (k B T/h)K n and k c = (k B T/h)K c where c means catalyzed and n means uncatalyzed. From the scheme you can see that K s K c = K n K t hence K t /K s = K c /K n however, k c /k n = K c /K n Therefore the observed rate enhancement k c /k n = K t /K s >> 1 Therefore the transition state geometry S must bind more tightly than the substrate S in its equilibrium geometry!

14 Transition State Analogs The transition state stabilization hypothesis was tested by designing so-called transition state analogs, molecules which mimick the real TS as closely as possible. One of the first enzymes examined was proline racemase: N COO - H H COO - N N COO - H H The compound on the right is a planar TS state analog. This molecule was found to be a good inhibitor, with K i some two orders of magnitude smaller than K m. H

15 The Role of Entropy In a seminal paper Page and Jencks showed that the loss in entropy in going from a bimolecular to a unimolecular reaction, i.e. E + S <=> ES, could account for as much as 10 8 of the observed rate enhancement. In other words, this much free energy would come from the intrinsic binding energy. The entropy loss arises from the loss of translational and rotational degrees of freedom when the substrate is bound. The configurational entropy is: S = k B lnw where W is the number of degrees of freedom available to a molecule.

16 Inhibition An inhibitor is any compound that causes a decrease in the catalytic rate. We will consider non-covalent ligands that can bind to the enzyme. The general scheme is shown below: S k c I = inhibitor E ES E + P Inhibition occurs if k i [EIS] < k c [ES] I I S k i EI EIS EI + P

17 Competitive Inhibition Competitive inhibition results from the direct competition between the I and S for the substrate binding site. There is an additional equilibrium constant: EI E + I K I = [E][I ] [EI ] The velocity under these conditions turns out to be: v = [S ]v max αk M + [S ] α = 1 + [I ] K I

18 Uncompetitive Inhibition Uncompetitive inhibition arises when I can bind at site that is not the same as the substrate binding site. There is an additional equilibrium constant: EI E + I K I = [E][I ] [EI ] Here the complex IE indicates that the inhibitor does not bind in the same site as the substrate. The velocity under these conditions is: v = v max [S] K M + α[s] α = 1 + [I ] K I

19 DHP has a natural peroxidase function Engineered globin peroxidases Mauk group Watanabe group DHP O - O X X + H 2 O DHP X X 2 + H 2 O + X - Trihalogenated Phenol X (X = I, Br, Cl, F) Horseradish Peroxidase O Dihalogenated Quinone

20 Structural model of inhibitor and substrate binding based on X-ray, NMR and resonance Raman. Inhibitor Bound Resting State Substrate Bound H55 (open) 5cHS Heme a 4BP cHS ν 3 ν 3 ν cHS 4-XP (Int) H55 (equilibrium) 5cHS/6cHS b DHP cHS TXP H55 (closed) 6cHS Heme c TCP cHS Wavenumber (cm -1 ) Wavenumber (cm -1 ) Wavenumber (cm -1 )

21 Kinetic analysis showing competitive inhibition under native conditions Product/4-BP Absorbance Product 2,4,6-TBP 2,4,6-TBP Wavelength (nm) Wavelength (nm) Br OH Br + H 2 O 2 Br OH Br + OH + H 2 O 2 Br Br Br

22 a Absorbance TCP Kinetic analysis showing competitive inhibition mechanism DHP + Substrate 2,4,6-TCP b Absorbance DHP + Substrate + Inhibitor Product/4-BP 2,4,6-TCP c Absorbance ,6-DCQ Wavelength (nm) Wavelength (nm) Fe(IV)=O inhibited DHP DHP Cmp ES Cmp ES + 20µM 4BP Cmp ES + 40µM 4BP Cmp ES + 100µM 4BP Cmp ES + 200µM 4BP Cmp ES + 400µM 4BP Wavelength (nm) d V o V max 1/V o Substrate Substrate + Inhibitor 1/[S] Concentration TCP (mm)