The Meaning of Km & V
Recall from the Briggs & Haldane derivation (Lecture 1) that for the 2-step reaction
E+S
k1
k-1
ES
k2
E+P
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 closely
approximates this value when k-1 >> k2, i.e. when substrate binding is in rapid
equilibrium w.r.t. the slow step of catalysis and product release.
THIS IS NOT ALWAYS THE CASE
Enzyme Inhibitors in Steady-State Kinetics
-- Drugs, poisons, mechanistic probes
-- Reversible, irreversible, suicide
-- Competitive, noncompetitive, mixed, uncompetitive
-- Product inhibition
-- Substrate inhibition
-- Transition state analogs
On-Line References for Steady-State Enzyme Kinetics
Dr. Peter Birch, University of Paisley
http://www-biol.paisley.ac.uk/kinetics/contents.html
University of Texas
http://www.cm.utexas.edu/academic/courses/Fall2001/CH369/LEC05/Lec5.htm
Terre Haute Medical College
http://web.indstate.edu:80/thcme/mwking/enzyme-kinetics.html
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.
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 a GIF decompressor are needed to see this picture.
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
Kinetics of Competitive Inhibitors
Remember: Inhibitor & substrate binding are
mutually exclusive, also rapid & reversible.
High [I] competes out substrate, so enzyme is
almost completely inhibited.
High [S] competes out inhibitor, so enzyme is
almost fully active.
E+S
ES
E+I
EI
EI + S
EIS
ES + I
EIS
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.
Kinetics of Competitive Inhibitors
Effect on V/Km.
--V/Km is the rate constant at low [S]. Why?
At [S] << Km, the Michaelis-Menten equation
simplifies 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
Slope = V/Km
v
Anything that affects
V or Km affects V/Km.
+ inhibitor
-- Km increases, V unchanged.
-- Therefore V/Km decreases.
[S]
Effects of Competitive Inhibitor on Lineweaver-Burk Plot
1/v = (Km/V)(1/[S]) + 1/V
= Km/V, i.e. reciprocal
of rate constant V/Km
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.
Kinetics of Non-Competitive / Mixed Inhibitors
Remember: Inhibitor & substrate binding are NOT
mutually exclusive.
E+S
ES
EIS complex forms by either of two routes, but cannot
convert substrate to product.
E+I
EI
Substrate cannot compete out the inhibitor, so inhibitor
works well at low and high [S].
ES + I
EIS
EI + S
EIS
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.
Effects of CLASSICAL Non-Competitive Inhibitor
on Lineweaver-Burk Plot
1/v = (Km/V)(1/[S]) + 1/V
= Km/V, i.e. reciprocal
of rate constant V/Km
Effects of MIXED Inhibitor on Lineweaver-Burk Plot
1/v = (Km/V)(1/[S]) + 1/V
= Km/V, i.e. reciprocal
of rate constant V/Km
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.
Kinetics of Uncompetitive Inhibitors
Remember: For uncompetitive inhibitor to work,
FIRST substrate must bind to enzyme
THEN 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.
E+S
ES
ES + I
EIS
E+I
EI
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)
Effects of Uncompetitive Inhibitor on Lineweaver-Burk Plot
1/v = (Km/V)(1/[S]) + 1/V
= Km/V, i.e. reciprocal
of rate constant V/Km
Analysis of Inhibition Constants
Consider the following schematic for enzyme binding to substrate and inhibitor:
E
KS
1
KI 2
EI
ES
P
3 KI’
4
KS’
EIS
P
1
2
3
4
Always possible
Not possible with uncompetitive inhibitors
Not possible with competitive inhibitors
Not possible with competitive or uncompetitive
inhibitors
A noncompetitive inhibitor is capable of all four reactions, but the classical noncompetitive
inhibitor, as opposed to a mixed one, is a special case. With these inhibitors Ks (of which
Km 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
Calculation.
Please note: equations on Paisley
website are incorrect!!
Interconversions between apparent Km and V values
(those observed in presence of inhibitor) and the true values involve multiplication or
division by the term (1 + i/Ki) or (1 + i/Ki’), where i = free inhibitor concentration.
* Substitute these terms into M-M equation to derive full velocity equations for each
inhibition model.
Cornish-Bowden (1979) Enzyme Kinetics, Butterworth & Co., London, p. 79
Primary Plot:
Lineweaver-Burk Plots of kinetics experiments performed at multiple, fixed
concentrations of a MIXED-type non-competitive inhibitor.
V/Km decreases
V decreases
Km increases
Secondary Plot #1: 1/Vapp vs. Inhibitor Concentration
MIXED non-competitive
Vapp = V / (1 + i/Ki’)
rearranges to:
1/ Vapp = (1/V Ki’)i + 1/V
-
Secondary Plot #2: 1/(V/Km)app vs. Inhibitor Concentration
MIXED non-competitive
= Kapp / Vapp
Vapp/Kapp = (V/K) / (1 + i/Ki)
rearranges to:
Kapp/Vapp = (K/VKi)i + K/V
-
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.
Use of Dixon Plot to Estimate Ki of Inhibitor
-- velocities measured at muliple 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 -Ki
* Uncompetitive: lines are parallel, cannot calculate Ki which is irrelevant anyway.
Restrictive: cannot
Calculate Km, V, or Ki’
Mathematical Basis of Dixon Plot for Mixed Inhibition
Cornish-Bowden (1979) Enzyme Kinetics,
Butterworth & Co., London, p. 81
Product Inhibition
EP
E+P
Products bind to the enzyme active site using the same bonds, or at least a subset
of 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 an
enzyme. 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.
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.
Hypothetical V
When you see something like this, first
check 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 likely
systematic errors and artifacts, then you
might need to consider substrate inhibition
in your kinetic model.
How Does True Substrate Inhibition Occur?
Example: Invertase, a.k.a. b-fructofuranosidase
Catalyzes: sucrose (disccharide) + 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 site
that leads to catalysis.
For inhibition to occur, the binding of
the second substrate molecule must
follow very rapidly upon binding of the
first, otherwise the first substrate would
be hydolyzed.
This is only likley to occur at very high
[substrate].
Can you think of other ways that
substrates could inhibit their enzymes?
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.
Rate Equations Considering Substrate Inhibition
Direct fitting of v vs. [S] curve
is potentially another way to
extract kinetic parameters from
substrate-inhibited enzymes.
Cornish-Bowden (1979) Enzyme Kinetics,
Butterworth & Co., London, pp. 93-94
Siesta
Time!