Michaelis-Menten kinetics

• Michaelis-Menten equation

– Km = [S] at which reaction rate is ½ Vmax

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Michaelis curve

Michaelis-Menten Kinetics

• Subset of enzymes can be assumed to follow these kinetics– Fairly simplified way to model kinetics– Two-step reaction

• Assume steady-state– [ES] doesn’t change– k2 is rate limiting (ie. S→P (actual catalysis) not E + S →

ES (substrate binding to enzyme))

E + S ES E + Pk1

k-1

k2

Michaelis-Menten Equation

• Vo and [S] are experimental values (ie. measurable and/or defined by experimenter)

• Run experiments to determine Km and Vmax

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Km

• Conglomerate of rate constants

• If k2 << k-1 (ie. k2 is slow/rate-determining step), Km = k-1/k1 ( = Kd)

• How tightly does enzyme bind substrate?– Lower Km → stronger affinity

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kkkKm

E + S ES E + Pk1

k-1

k2

Km

• At low [S], Km doesn’t matter– Vo is well below Vmax

– Vo increases proportionally as [S] increases (pseudo-first order)

• At high [S], Km doesn’t matter– Vo ≈ Vmax

• Typically, Km will jive with [S]– In vivo and in experiments

Vmax

• How quickly does the enzyme ‘do the chemistry’?– ie. If the substrate is not limiting ([S]>>Km),

the reaction’s rate depends on k2 and [Et]

Vmax = k2[Et]

so Vmax gives a measure of k2 (rate of ES → E + P)

Vmax vs kcat

• kcat derived from Vmax, but more general usage

– Vmax ~ k2 for one-step reaction• One-step reactions: very simple case

– kcat ~ rate constant for rate-determining step in multi-step reaction

• Units: – Vmax mol/min (for example)

– kcat s-1 (so you also have to change min to s)

• kcat ~ rate constant for slowest step• kcat = turnover number

– How many substrate molecules are “turned over” per unit time by each enzyme molecule

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tcat

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Comparing kinetics

Which enzyme is more efficient?Which substrate is preferred?

• Km → Lower = “better”

• Vmax/kcat → Higher = “better”

• “Specificity constant”

– Units: M-1s-1

m

cat

Kk

How do you determine Km/Vmax?

• From Michaelis-Menten (saturation) plot][][max

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Estimate Km/Vmax

ORUse computer analysis

Vo and [S] are “raw” data

How do you determine Km/Vmax?

• ‘Linearize’ M-M curve– Lineweaver-Burk plot

• Reciprocal of both sides of MM equation

• y = mx + b• Y-intercept= 1/Vmax

• X-intercept= -1/Km

– Problem: Small errors at low [S] (ie. to the right) have major effects

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How do you determine Km/Vmax?

• ‘Linearize’ M-M curve (part 2)– Eadie-Hofstee plot

• Multiply both sides of MM equation by Km + [S] & simplify

• Plot Vo vs. Vo/[S]

• Slope = -Km

• y-int = Vmax • Equal weight to points from whole range of [S]

– Problem: both axes are dependent on Vo

• So any experimental error will be magnified

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Enzyme inhibition

• Drugs • Natural toxins • Natural regulators of enzyme function

• Use kinetic analyses to determine– Potency of inhibition– Mechanism of inhibition

Enzyme inhibition• Two simple/idealized ways an inhibitor can ‘slow down’

an enzyme

– Interfere with substrate binding• “Competitive” inhibitor• Increase the Km

– Apparent Km > Intrinsic Km

– Slow down catalysis• “Non-competitive” inhibitor• Decrease the Vmax

– “Real” reversible inhibitors often work by a combination of the two

• “Mixed” inhibitor