steady-state derivation of michaelis-menton equation (briggs & haldane)

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teady-State Derivation f Michaelis-Menton Equation Briggs & Haldane) time S ES E + P k 1 k -1 k 2 Upon mixing of enzyme and substrate [ES] rises rapidly and reaches a steady state, where the rate of formation and breakdown of [ES] are equal, i.e. v1= k1 [E][S] and v2= k-1 [ES] + k2[ES] so that at steady state v1 = v2 Free enzyme conc. [E]= [Et] -[ES]; note that [E] cannot be measured, but [Et] is known, as is [S] since initial velocities, and [P] can be measured. Now solve for the unknown [ES]. v1= k1 ([Et] -[ ES])[S] = v2 = k- 1[ES] + k2[ES] rearrange ([Et] -[ES])[S]/[ES] = (k-1+ k2) /k1 = Km Solving for [ES] gives [ES] = [Et][S]/ (Km + [S]) the velocity (v) of the reaction will be proportional to the formation of [ES], so v= k2 [ES] (substitute in the value for ES in red above) to get: v=k2 [Et][S]/ (Km + [S]) Note importance of initial velocities (v o )

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k 1. k 2. E + S ES E + P. k -1. Steady-State Derivation of Michaelis-Menton Equation (Briggs & Haldane). Upon mixing of enzyme and substrate [ES] rises rapidly and reaches a steady state, where the rate of formation and breakdown of [ES] are equal, i.e.   - PowerPoint PPT Presentation

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Page 1: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Steady-State Derivation of Michaelis-Menton Equation (Briggs & Haldane)

time

E + S ES E + Pk1

k-1

k2

Upon mixing of enzyme and substrate [ES] rises rapidly and reaches a steady state, where the rate of formation and breakdown of [ES] are equal, i.e.  

v1= k1 [E][S] and v2= k-1 [ES] + k2[ES] so that at steady state v1 = v2

Free enzyme conc. [E]= [Et] -[ES]; note that [E] cannot be measured, but [Et] is known, as is [S] since initial velocities, and [P] can be measured. Now solve for the unknown [ES].

v1= k1 ([Et] -[ ES])[S] = v2 = k-1[ES] + k2[ES]

rearrange ([Et] -[ES])[S]/[ES] = (k-1+ k2) /k1 = Km

Solving for [ES] gives [ES] = [Et][S]/ (Km + [S])

the velocity (v) of the reaction will be proportional to the formation of [ES], so v= k2 [ES] (substitute in the value for ES in red above) to get:

v=k2 [Et][S]/ (Km + [S])

and at saturating [S], Vmax =  k2[Et] (substitute Vmax for the k2 [Et] in the above equation), then   v= Vmax [S]/ (Km + [S]) which is the same as the Michaelis-Menton equation

Note importance ofinitial velocities (vo)

Page 2: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

The Meaning of Km & V

Recall from the Briggs & Haldane derivation (Lecture 1) that for the 2-step reaction

the velocity at steady-state is given by the equation

v = k2[Et][S] / ((k-1 + k2)/k1 + [S]) = V[S] / (Km + [S])

where V = k2[Et] and is thus a function of total enzyme concentration,

and Km = (k-1 + k2)/k1 is derived from multiple rate constants. The complexity of this term increases for more complicated kinetic mechanisms.

The value of Km is taken as an indicator of an enzyme’s affinity for substrate, but it is not a true dissociation constant.

For step one of the two-step mechanism above, Kd = k-1/k1 and Km only closelyapproximates this value when k-1 >> k2, i.e. when substrate binding is in rapidequilibrium w.r.t. the slow step of catalysis and product release.THIS IS NOT ALWAYS THE CASE

E + S ES E + Pk1

k-1

k2

Page 3: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Enzyme Inhibitors in Steady-State Kinetics

-- Drugs, poisons, mechanistic probes-- Reversible, irreversible, suicide-- Competitive, noncompetitive, mixed, uncompetitive-- Product inhibition-- Substrate inhibition-- Transition state analogs

Page 4: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

On-Line References for Steady-State Enzyme Kinetics

Dr. Peter Birch, University of Paisleyhttp://www-biol.paisley.ac.uk/kinetics/contents.html

University of Texashttp://www.cm.utexas.edu/academic/courses/Fall2001/CH369/LEC05/Lec5.htm

Terre Haute Medical Collegehttp://web.indstate.edu:80/thcme/mwking/enzyme-kinetics.html

Page 5: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Reversible Inhibitors:

E + I EI

Fast acting.Generally non-covalent EI complex.Removal restores enzyme activity.

Irreversible Inhibitors:

E + I EI

Often slow, time-dependent inactivation.Often covalent EI complex.Enzyme permanently disabled.

Suicide Inhibitors:

E + I* EI* EI

Enzyme converts precursor into irreversible inhibitor.

Page 6: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Competitive Inhibitors

1. Competitive Inhibition by Active Site Binding

-- reversible-- inhibitor (usually) structurally similar to substrate-- inhibitor competes directly for substrate binding to active site

(mutually exclusive binding)-- effects can be overcome by increasing substrate concentration

QuickTime™ and aGIF decompressorare needed to see this picture.

Page 7: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Competitive Inhibitors

2. Competitive Inhibition by Conformational Change

-- reversible-- substrate and inhibitor may be dissimilar -- inhibitor binds to remote site on enzyme, but causes a conformational

change that precludes substrate binding to the active site-- likewise, substrate binding to active site causes a conformational change

that precludes inhibitor binding to its site-- binding is still mutually exclusive-- effects of inhibitor can still be overcome by increasing substrate concentration

Page 8: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Kinetics of Competitive InhibitorsRemember: Inhibitor & substrate binding are mutually exclusive, also rapid & reversible.

High [I] competes out substrate, so enzyme isalmost completely inhibited.

High [S] competes out inhibitor, so enzyme is almost fully active.

Effect on Km. -- Km is an indicator of enzyme-substrate affinity (like a dissociation constant).-- With inhibitor present, both free enzyme (E) and EI complex exist.-- E has normal affinity for S; EI has no affinity for S.-- Solution average affinity decreases, therefore Km increases.

Effect on V.-- V is the velocity at very high [S]; i.e., conditions that compete out inhibitor.-- Thus V is unchanged.

E + S ES

E + I EI

EI + S EIS

ES + I EIS

Page 9: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Kinetics of Competitive InhibitorsEffect on V/Km. --V/Km is the rate constant at low [S]. Why?

At [S] << Km, the Michaelis-Menten equationsimplifies from v = V[S] / (Km + [S]) to:

v = (V/Km)[S] = k[S]

E + S ES

E + I EI

EI + S EIS

ES + I EIS

v

[S]

Slope = V/Km

Anything that affectsV or Km affects V/Km.

-- Km increases, V unchanged.-- Therefore V/Km decreases.

+ inhibitor

Page 10: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Effects of Competitive Inhibitor on Lineweaver-Burk Plot

= Km/V, i.e. reciprocal of rate constant V/Km

1/v = (Km/V)(1/[S]) + 1/V

Page 11: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Non-Competitive / Mixed Inhibitors

-- Binds to site on enzyme remote from active site.-- Causes conformational change in enzyme that prevents conversion of substrate to product, but does not prevent substrate binding to enzyme.-- I, S binding is not mutually exclusive.-- Comes in 2 varieties: Classic & Mixed

1. Classic Non-Competitive Inhibitors (Rare).-- do not alter affinity of substrate binding.

2. Mixed Inhibitors (Common).

-- typically lower the affinity of substrate binding.

Page 12: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Kinetics of Non-Competitive / Mixed Inhibitors

Remember: Inhibitor & substrate binding are NOT mutually exclusive.

EIS complex forms by either of two routes, but cannot convert substrate to product.

Substrate cannot compete out the inhibitor, so inhibitor works well at low and high [S].

Effect on Km. -- CLASSIC Non-Competitive Inhibitor: no effect on substrate affinity; Km unchanged.-- MIXED Inhibitor: allows substrate binding but lowers affinity; Km increases.

Effect on V.-- Both CLASSIC & MIXED inhibitors work at high [S], so V decreases.

Effect on V/Km.-- Both CLASSIC & MIXED inhibitors also work at low [S], so V/Km decreases.

E + S ES

E + I EI

ES + I EIS

EI + S EIS

Page 13: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Effects of CLASSICAL Non-Competitive Inhibitoron Lineweaver-Burk Plot

= Km/V, i.e. reciprocal of rate constant V/Km

1/v = (Km/V)(1/[S]) + 1/V

Page 14: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Effects of MIXED Inhibitor on Lineweaver-Burk Plot

= Km/V, i.e. reciprocal of rate constant V/Km

1/v = (Km/V)(1/[S]) + 1/V

Page 15: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Uncompetitive Inhibitors

-- Cannot bind to free enzyme.-- Binds only to enzyme-substrate complex (ES).

* substrate binds directly to inhibitor, or* substrate induces conformational change required for inhibitor binding.

-- S, I binding is not mutually exclusive, it is required.-- Once bound, inhibitor prevents enzyme from converting substrate to product.

Page 16: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Kinetics of Uncompetitive Inhibitors

Remember: For uncompetitive inhibitor to work,FIRST substrate must bind to enzymeTHEN inhibitor must bind to ES complex.Inhibitor binding to free enzyme is not allowed.

Uncompetitive inhibitors are not effective at low [S],because most of the enzyme exists as free enzyme.They are effective at high [S] because most of the enzyme exists as ES complex.

Effect on Km. -- Inhibitor binding to ES complex draws E + S <-> ES binding equilibrium to right via Law of Mass Action, thereby increasing the apparent affinity of enzyme for substrate, so Km decreases.

Effect on V.-- Inhibitor is most effective at high [S] where lots of ES complex forms, so V decreases.

Effect on V/Km.-- Inhibitor is least effective at low [S] where there is little ES complex, so V/Km is unchanged. (decrease in Km balances decrease in V, so ratio is insensitive to inhibitor)

E + S ES

ES + I EIS

E + I EI

Page 17: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Effects of Uncompetitive Inhibitor on Lineweaver-Burk Plot

= Km/V, i.e. reciprocal of rate constant V/Km

1/v = (Km/V)(1/[S]) + 1/V

Page 18: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

E ES P

EI EIS PKS’

KI’KI

KS

1

2 3

4

1 Always possible2 Not possible with uncompetitive inhibitors3 Not possible with competitive inhibitors4 Not possible with competitive or uncompetitive

inhibitors

A noncompetitive inhibitor is capable of all four reactions, but the classical noncompetitiveinhibitor, as opposed to a mixed one, is a special case. With these inhibitors Ks (of whichKm is usually a squishy approximation) and Ks' are equal to each other, as are Ki and Ki'.

Using MIXED INHIBITION as an example, we’ll consider 3 different ways to estimate Ki values:

-- calculation-- use of secondary plots-- Dixon plots

Analysis of Inhibition Constants

Consider the following schematic for enzyme binding to substrate and inhibitor:

Page 19: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Calculation. Interconversions between apparent Km and V values(those observed in presence of inhibitor) and the true values involve multiplication ordivision by the term (1 + i/Ki) or (1 + i/Ki’), where i = free inhibitor concentration.

Please note: equations on Paisleywebsite are incorrect!!

Cornish-Bowden (1979) Enzyme Kinetics, Butterworth & Co., London, p. 79

* Substitute these terms into M-M equation to derive full velocity equations for eachinhibition model.

Page 20: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Primary Plot:Lineweaver-Burk Plots of kinetics experiments performed at multiple, fixedconcentrations of a MIXED-type non-competitive inhibitor.

V decreases

Km increases

V/Km decreases

Page 21: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Secondary Plot #1: 1/Vapp vs. Inhibitor Concentration

MIXED non-competitive

Vapp = V / (1 + i/Ki’)

rearranges to:

1/ Vapp = (1/VKi’)i + 1/V

-

Page 22: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Secondary Plot #2: 1/(V/Km)app vs. Inhibitor Concentration

MIXED non-competitive

Vapp/Kapp = (V/K) / (1 + i/Ki)

rearranges to:

Kapp/Vapp = (K/VKi)i + K/V

= K

app /

Vap

p

-

Page 23: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Secondary plots with different inhibitor types The sample plots shown here were produced using a mixed inhibitor, as this is the kinetically most complex of the types that we've studied. For the other types matters are simplified as follows:

* Classical noncompetitive inhibitor

* The secondary plots are made as above but the two plots should give identical results as Ki and Ki' are equal for these inhibitors.

* Competitive inhibitor

* The first secondary plot cannot be made as there is no change in maximal velocity. This plot is not required though as it gives Ki' which is irrelevant for a competitive inhibitor.

* Uncompetitive inhibitor

* This is really the opposite of the competitive inhibitor. The second secondary plot can't be made as there is no change in slope. Again this is not required as the Ki is irrelevant to an uncompetitive inhibitor.

Page 24: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Use of Dixon Plot to Estimate Ki of Inhibitor

-- velocities measured at multiple fixed substrate concentrations, inhibitor concentration is varied.-- graph of 1/v vs. i gives intersecting lines; Ki is derived from the point of intersection.

* Classical NC: all lines intersect on horiz. axis --> read -Ki directly* Mixed or Competitive: lines intersect above horiz, axis; drop a line to -K i

* Uncompetitive: lines are parallel, cannot calculate Ki which is irrelevant anyway.

Restrictive: cannotCalculate Km, V, or Ki’

Page 25: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Cornish-Bowden (1979) Enzyme Kinetics, Butterworth & Co., London, p. 81

Mathematical Basis of Dixon Plot for Mixed Inhibition

Page 26: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Product Inhibition

Products bind to the enzyme active site using the same bonds, or at least a subsetof the bonds, used to bind substrate.

Frequently products are capable of binding to free enzyme, and do so rapidly and reversibly. -- for a single-substrate enzyme, this can lead to competitive inhibition since substrate and product binding are mutually exclusive. -- product inhibition is more complicated in multi-substrate enzymes.

Use of initial velocities avoids the effects of product inhibition on kinetics.(but see Single Progress Curve method)

Occasionally product release is slow and can limit the catalytic turnover of anenzyme. In this case there is usually some kind of exchange factor requirement.

Product inhibition can be a useful tool for understanding enzyme kinetics, especially of multi-substrate systems.

EP E + P

Page 27: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Hypothetical V

Substrate Inhibition (a.k.a. Excess Substrate Inhibition)

An odd kind of kinetic behavior in which velocities actually decrease, rather than continue to approach V asymptotically, at high substrate concentrations.This phenomenon can seriously complicate the analysis of kinetics data.

Seems to be most common in enzymes with large and complex substrates,such as nucleic acids, polysaccharides, etc., but is probably over-reported.

When you see something like this, firstcheck for problems/artifacts with your assay.

-- substrate or enzyme precipitation st high [S].-- metal ion chelation.-- pH or ionic strength changes, etc.

If you can eliminate all of the likelysystematic errors and artifacts, then you might need to consider substrate inhibitionin your kinetic model.

Page 28: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

How Does True Substrate Inhibition Occur?

Example: Invertase, a.k.a. -fructofuranosidaseCatalyzes: sucrose (disaccharide) + H2O --> glucose + fructose (monosaccharides)

Substrate inhibition of invertase probably occurs when 2 molecules of substrate (sucrose)bind to the active site simultaneously, in an improper end-on fashion.

Each sucrose molecule blocks the other from assuming the correct position in the active sitethat leads to catalysis.

For inhibition to occur, the binding of the second substrate molecule mustfollow very rapidly upon binding of the first, otherwise the first substrate wouldbe hydrolyzed.

This is only likely to occur at very high[substrate].

Can you think of other ways that substrates could inhibit their enzymes?

Page 29: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Determining Kinetic Parameters When Substrate Inhibition Occurs

-- Substrate inhibition introduces curvature at the lower end of a Lineweaver-Burk Plot.You wouldn’t want to use v4 weighted linear regression here!

-- You can still extrapolate to slope and intercept using low [S] data, but recall that this data contains the most error.

Page 30: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

Cornish-Bowden (1979) Enzyme Kinetics, Butterworth & Co., London, pp. 93-94

Rate Equations Considering Substrate Inhibition

Direct fitting of v vs. [S] curve is potentially another way toextract kinetic parameters fromsubstrate-inhibited enzymes.

Page 31: Steady-State Derivation  of Michaelis-Menton Equation  (Briggs & Haldane)

SiestaTime!