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•Michaelis – Menten equation

•Steady state vs pre-steady state

Today

Review derivation and interpretation

Graphical representation

Michaelis – Menten equations and parameters

[ES] = [ET][S]

Km + [S]

The Michaelis – Menten Equation

DerivationE + S ES E + P

k1 k2

k-1

d[P]dt

= k2[ET][S]Km + [S]

max

d[P]dt

=> k2[ET] = Vmax

d[P]dt

= v0 = Vmax [S]Km + [S]

v0 = d[P]/dt = k2 [ES]

The Michaelis – Menten Equation

The Michaelis – Menten Equation

v = Vmax [S]Km + [S]

Km = Michaelis constant: Concentration of Substrate needed to reach half maximum velocity – measure of substrate affinity

Vmax = maximum velocity – directly proportional to enzyme concentration

The Michaelis – Menten Equation

Km

E + S ES E + P k1 k2

k-1

[E][S][ES]

= k-1 + k2 = Km Eqn 1k1

Km represents the [S] when v = ½ Vmax (sub into eqn)

Therefore, a lower Km value indicates a higher affinity for the substrate

The Michaelis – Menten Equation

Interpretation

Obtain kinetic behaviour of an enzyme

E + S ES E + P k1 k2

k-1

k2 = kcat Catalytic constant of the reaction (first order)

kcat is also known as the turnover number of the enzyme –defining the maximum number of substrate molecules converted to product per unit of time

How is How is kcatkcat determined experimentally?determined experimentally?

The Michaelis – Menten Equation

kcat/Km

The constant part kcat/Km is referred to as the catalytic efficiency of the enzyme:

A direct measure of the efficiency of the enzyme in transforming Subst.

kcat/Km combines: the effectiveness of transformation of bound ES with the effectiveness of productive substrate binding

kcat/Km can be obtained directly from steady state expts

Unitsv0 = initial velocity of Product formation = moles Product formation (moles substrate loss)/litre x time

= mM s-1

Vmax = represents the maximum rate the enzyme reaction can achieve. Vmax occurs when all of the enzyme is in the ES complex.

Km = [S] at Vmax/2 = µM

kcat = first order rate constant = s-1

# of catalytic cycles active site undergoes per unit of time

kcat/Km = second order rate reaction of E and S = M-1 s-1

Plotting kinetic data

Michaelis-Menten kinetics (steady state kinetics)

Measure Initial rates (v0) at different substrate concentrations by detecting the absorbance difference over time and obtaining the slope of the line

Obtain various initial rates at different substrate concentrations and plot

Use Lineweaver-Burke plots to obtain kcat, Km, kcat/Km

Plotting kinetic data

Measuring initial rates:

•The initial velocity is the amount of product produced per minute

•Assay involves adding all of the components including substrate first

•Once enzyme is added the absorbance is continually monitored andrecorded with respect to time

•Abs vs time can be plotted

•Determine slope of tangentA

bsor

.

time

Enzyme added

Plotting kinetic data

Measuring initial rates:

•The slope of tangent = ∆Abs/unit of time (min)

•Recall A = εcl, therefore ∆A = ε∆cl (change in conc of Product)

•When ∆c occurs in a known time period then ∆c min-1 = v0

•v0 = ∆A min-1/ εl = ∆c min-1

•Has M/min after multiplying by vol. of enzyme assay solution theunits change to mol/min.

Plotting kinetic data

Determine v0 at different substrate concentrations and plot

The Michaelis – Menten Equation

Lineweaver-Burke transformation of the Michaelis-Menton equation.

•Velocity vs substrate plots are useful for visually estimating kinetic parameters

•Hyperbolic curves cannot properly determine the upper limit of the curve (Vmax)

•Transforming the data to a form that can be plotted as a line may improve accuracy.

Lineweaver-Burke transformation of the Michaelis-Menton equation.

v = Vmax [S]Km + [S]

Reciprocal of the equation is:

1

v Vmax [S]

Km + [S]=

Express reciprocal in the familiar form y = mx + b

1 V Vmax [S]

Km= + 1 Vmax

y = mx + b

Lineweaver-Burke transformation of the Michaelis-Menton equation.

Lineweaver-Burke equation represents a straight line with slope = Km/Vmax, y intercept = 1/Vmax, and x intercept = -1/Km

1 V Vmax [S]

Km= + 1 Vmax

1/[S] (mM)

1/v (mM/min)

0.3 0.25 0.20 0.15 0.10 0.1 0.15 0.2 0.25 0.30 0.35

0.6

0.5

0.4

0.3

0.2

0.1

A

B

Lineweaver-Burke transformation of the Michaelis-Menton equation.

•Most commonly used

•Magnitude of errors can become distorted

•Furthest point to the right (lowest [S]) influences where line is drawn

•e.g. small error in v = large error in 1/v

•Good for observing enzyme inhibition

Eadie-Hofstee Plot

•Multiply both sides of L.B. by Vmax

•Multiply both sides by v

•Rearrange for v

v0

[S]Km = -Vmax v0

Plot v0 vs v0[S]

Eadie-Hofstee Plot

v0 -Km = slope

Vmax

Vmax Km

v0/[S]

Eadie-Hofstee Plot

•Best for controlling slight deviations from linearity

•One disadvantage is that the least precise parameter (v0) is expressed in both sides of the equation and plot

Proximity Effects of Intramolecular CatalysisProximity Effect-is defined as a rate increase due to two reactants

being brought out of a dilute environment and placed closer together

Catalyst only: Large loss of entropy would be experienced by substrate

Enzymes (because they bind substrate; ES) provide a docking siteand a micro environment allowing proper substrate orientation for reaction (induced fit, strain, binding energy)

Increase in Effective Concentration of Catalytic Groups:

Increases chances of reaction --molecules are closer together within an ES complex (becomes an intramolecular process)

Proximity Effects of Intramolecular Catalysis

Proximity Effect accounts for only part of rate enhancement

Desolvating the substrate from water (or other solvent) eliminates energy barrier imposed by ordered solvent molecules (get some entropy back) – accelerating reaction

Remember: substitution of catalytic residues-not completely dead

As an ES complex, substrate does lose freedom of movement (loss of entropy) upon fixation into active site. However, binding energy gained by positioning substrate with respect to catalytic residues overcomes loss of entropy