Steady-State Enzyme Kinetics
Scott Morrical
smorrica@uvm.edu
B407 Given Bldg.
General References:
Textbook: Frey & Hegeman, Chapter 2, pp. 69-128
Reserve:
Cornish-Bowden (1979) Enzyme Kinetics, Butterworth & Co., London
Websites:
Dr. Peter Birch, University of Paisley
http://www-biol.paisley.ac.uk/kinetics/contents.html
Terre Haute Medical College
http://web.indstate.edu:80/thcme/mwking/enzyme-kinetics.html
Steady-State Enzyme Kinetics
Lecture 1:
Single-Substrate
Michaelis-Menton Kinetics
Enzymes-- Biological Catalysts
Catalyst- a substance that increases the rate of a
reaction without itself being changed or
consumed in the overall process
Turnover- the catalyst may be reused in
subsequent reactions
X* transition state
Free energy
of activation
The position of equilibrium is determined by the
free energy change, DGo
DG*
Free Energy
Chemical Kinetics & Equilibria:
DGo = -RTlnKo
reactants
The rate of a reaction depends on the free energy
of activation, DG*
DGo
products
Free energy
change
Free Energy
Reaction Progress
uncatalyzed
Catalysts:
catalyzed
Speed up reactions by lowering the free energy
of activation, DG*
Do not affect the position of equilibrium
(DGo unchanged)
reactants
products
Reaction Progress
Why do we need enzymes?
* A chemical reaction occurs only if the molecules possess a minimum
amount of energy---Activation Energy
* Chemical reactions require initial input of energy--usually in the form of
increased heat
* Raising the temperature increases the rate of (vibrational, translational)
movement of the molecules and the chance of collision
* An increase in the concentration of reactants can also increase the chances
of a chemical reaction occurring
* HEAT and MORE REACTANTS can increase chance of chemical reaction
occurring
* Biological systems cannot raise heat or concentrations at will
How do enzymes do that?
* Provide alternate pathway by lowering energy of
activation, stabilization of transition state--same as adding
heat
* Lowers activation energy, but does not change free energy
required for the reaction to occur (alters the rate, but not
thermodynamics)
* Provide a surface for the reaction to occur, bringing
reactants into close proximity to each other--functional
equivalent of increasing concentration
Enzyme catalysts contain unique active sites---where the
substrates bind and the reaction takes place
* Lock and key model--substrate fits exactly into active site
* Induced fit model--substrate causes change in enzyme's
active site shape to make substrate fit
Once bound, the substrate reaches the transition state and
bonds are rearranged. The enzyme active site:
* Places atoms in close proximity to each other
* Orients substrate correctly
* These two effects facilitate the breaking and reforming of
bonds
Enzyme-Substrate Binding Specificity
Lock & Key
Induced Fit
Classes of enzymes
* Oxidoreductases--oxidation/reduction; requires a co-factor such as NAD or FAD
A: + B => A + B:
* Transferases--transfer of a functional group
A-B + C => A + B-C
* Hydrolases--hydrolysis of functional group by water
A-B + H20 => A-H + B-OH
* Lyases--elimination to form double bond; or addition to a double bond
X-A-B-Y => A=B + X-Y
* Isomerases--isometric interconversions
X-A-B-Y => Y-A-B-X
* Ligases--ATP dependent joining of two molecules
A + B +ATP => A-B + ADP + Pi
Enzymes:
Compilations of Databases & Online Resources
http://restools.sdsc.edu/biotools/biotools12.html
FRONTIERS IN BIOSCIENCES
http://www.bioscience.org/urllists/protdb.htm
ExPASy Molecular Biology Server
http://us.expasy.org/
Enzyme Kinetics:
Studies of the rate or velocity of enzyme-catalyzed
reactions, and factors influencing these rates.
Mathematical analyses of the relationships between
substrate (or inhibitor, activator) concentrations and
reaction rates yield:
-- characteristic properties of enzymes
or classes of enzymes
-- insights into enzyme mechanisms and
physiology
The Effects of Substrate Concentration on Reaction Rate
S
E
P
S = substrate
P = product
E = enzyme
Uncatalyzed
Reaction:
At a fixed concentration of
enzyme, the velocity
reachs a maximum, which
fits the equation of a
rectangular hyperbolic
curve:
v
y=(ax)/(b + x)
[S]
Velocity v = d[P]/dt = -d[S]/dt
Michaelis-Menton Equation
For enzyme-catalyzed reaction
E+S
ES
E+P
v = V[S] / (Km + [S]) (rectangular hyperbola)
They made the following assumptions:
* [E] is always much less than [S]
* formation of [ES] is required to obtain [P]
* therefore, at high [S], all the [E] will be
saturated and the reaction cannot proceed any
faster by adding more [S], this velocity is V or
Vmax (“a” in general equation for hyperbolic
curve)
* velocity of reaction as a function of [S] is
only dependent on dissociation of [ES] into [E]
and [S]. The equilibrium constant, Km, is "b"
in the equation for a hyperbolic curve
When v= Vmax/2 (the velocity at 1/2 maximal
velocity), then [S] = Km
Km is called the Michaelis constant
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.
v1= k1 [E][S] and v2= k-1 [ES] + k2[ES] so that at
steady state v1 = v2
E+S
k1
k-1
ES
k2
E+P
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])
time
Note importance of
initial velocities (vo)
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
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
The Effects of Enzyme Concentration
Vmax is directly proportional
to enzyme concentration
At saturating substrate
([S] >> Km):
Km is independent of
enzyme concentration
Measuring Kinetic Parameters:
Graphical & Computational Methods
*
*
*
*
*
*
*
Lineweaver-Burk Plot
Eadie-Hofstee Plot
Hanes Plot
Direct Linear Plot
Direct Fitting of v vs. [S] Curve
Single Progress Curve
Statistics
Lineweaver-Burk Plot:
Rearrangement of Michaelis-Menten equation to linear form
1/v = (Km/V)(1/[S]) + 1/V
Plot for
hypothetical
enzyme with:
V = 10
Km = 4
Disadvantages of Lineweaver-Burk Plot
Still working with hypothetical enzyme with
V = 10, Km = 4…
1/v = (Km/V)(1/[S]) + 1/V
…only here random error has
been introduced into multiple
data sets, and results plotted to
illustrate how greatly estimates
of Km and V can vary from
plot to plot depending on data
quality.
How to improve:
--averaging
--weighting
--choose another method
Eadie-Hofstee Plot:
Rearrangement of Michaelis-Menten equation to another linear form
v = -Km(v/[S]) + V
Same
hypothetical
enzyme with:
V = 10
Km = 4
Disadvantages of Eadie-Hofstee Plot
Still working with hypothetical enzyme with
V = 10, Km = 4
v = -Km(v/[S]) + V
Once again random error has
been introduced to demonstrate
the scatter which can skew
estimates of Km and V; almost
as bad as Lineweaver-Burk
plot.
Another problem:
Both axes are functions of the
dependent variable (v)
Hanes Plot:
Rearrangement of Michaelis-Menten equation to still another linear form
[S]/v = (1/V)[S] + Km/V
Same
hypothetical
enzyme with:
V = 10
Km = 4
Error Issues in the Hanes Plot
Still working with hypothetical enzyme with
V = 10, Km = 4
[S]/v = (1/V)[S] + Km/V
Once again random error has
been introduced… generally
scatter is improved relative to
LH and EH plots, which can
improve the accuracy of
Km and V estimates.
Avoids the other problem of
EH plots in that dependent
variable (v) does not influence
the independent (horizontal)
axis.
Direct Linear Plot
Here, using two data points (w/o error)
from same hypothetical enzyme with
V = 10 and Km = 4
For both ([S], v) data points, plot
-[S] on horizontal axis and v on
vertical axis, then draw a line
connecting the two values. The
lines intersect at coordinates
(Km, V) allowing direct read-out
of these parameters.
Direct Linear Plot
Here is the same type of plot only with
lines drawn for all 10 of the error-free
([S], v) data points from our hypothetical
enzyme with V = 10 and Km = 4
V
Since there is no error, all of the lines
intersect at a common point with
coordinates (Km, V).
Km
Effects of Error on
Direct Linear Plot
Now looking at 5 ([S], v) points from a
data set including random error, again
from our hypothetical enzyme with
V = 10 and Km = 4.
Lines no longer intersect at a common point,
thereby giving a large range of values for
Km and V from all of the different intersection
points. In fact, the number of intersections is
given by the simple equation: n(n-1)/2, where
n = the number of lines.
Statistically, the best way to deal with this is to
take the median (not the mean) values of Km
and V, which minimizes the contributions of
spurious outliers (such outliers could badly
skew the mean values). See Birch’s website
for a good description of this approach.
Theory Behind Direct Linear Plot
Yet another linear rearrangement of the Michaelis-Menten equation:
V = (v/[S])Km + v
…would give a straight line if “constants” V and Km plotted against each other;
slope and intercept would be “variables” v/[S] and v, respectively.
* In fact, lines in Direct Linear Plot represent infinite number of values of Km and V
which satisfy the Michaelis-Menten equation for a given ([S], v) data pair.
* But the intersection of two lines generated from two different ([S], v) data pairs
gives the unique values of Km and V that satisfy both conditions, so these must be the
true ones (subject to experimental error of course).
Direct Fitting of v vs. [S] Curves
A number of commercially available non-linear regression packages exist which
will perform global fits of the hyperbolic form of the Michaelis-Menten equation
to experimental data.
-- Enzfit, Kaleidograph, etc.
-- Outputs usually consist of Km and V parameters with statistical
arguments such as variance and other indicators of goodness-of-fit.
-- CAUTION: Always compare fitting results with standard
graphical methods of analysis. Be aware that different statistical
packages can skew Km and V values depending on how data is
weighted and how outliers are treated.
U. Paisley Website Includes Instructions
For Setting Up a Global Fitting Routine for
Michaelis-Menten Enzyme Kinetics Data in
Microsoft Excel
http://www-biol.paisley.ac.uk/kinetics/Chapter_2/contents-chap2.html
Statistical Weighting of Enzyme Kinetics Data
The utility of some methods, such as the Lineweaver-Burk Plot, can be improved
through the use of statistical weighting methods.
-- Error is usually greatest in reaction velocities measured at low
substrate concentrations (slow reactions)
-- Therefore you can give the higher-velocity data (containing less error)
more weight in, say, fitting a straight line to a Lineweaver-Burk Plot
by linear regression.
-- The correct weighting factor is v4, which is the reciprocal of the variance.
References:
unweighted
Cleland (1967) Adv. Enzymol. 29, 1-32.
Wilkinson (1961) Biochem. J. 80, 324-332.
Magar (1972) Data Analysis in Biochemistry
And Biophysics, Academic Press, New York,
pp. 429-432.
weighted
1/v
Atkinson et al. (1961) Biochem. J. 80, 318-323.
Barnes & Waring (1980) Pocket Programmable
Calculators in Biochemistry, John Wiley &
Sons, New York, pp. 204-235.
1/[S]
Km and V From a Single Progress Curve
Substrate concentration is monitored
throughout a reaction timecourse;
At any given time (substrate conc.),
the instantaneous velocity is determined
from the slope of the curve’s tangent line.
([S], v) data pairs so generated are analyzed
via any of the methods already discussed.
Pitfalls (there are many):
-- not based on initial velocities.
-- product inhibition.
-- noise affects slope.
-- enzyme and/or substrate lability.
-- more susceptible to changes in pH, ions, etc.
Siesta
Time!