Steady-State Enzyme Kinetics

Scott Morrical smorrica@uvm.eduB407 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 Paisleyhttp://www-biol.paisley.ac.uk/kinetics/contents.html

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

Steady-State Enzyme Kinetics

Lecture 1:

Single-SubstrateMichaelis-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

reactants

products

Fre

e E

nerg

y

Reaction Progress

G*

Go

Free energyof activation

Free energychange

uncatalyzed

reactants

products

Fre

e E

nerg

y

Reaction Progress

catalyzed

Chemical Kinetics & Equilibria:

The position of equilibrium is determined by the free energy change, Go

Go = -RTlnKo

The rate of a reaction depends on the free energyof activation, G*

Catalysts:

Speed up reactions by lowering the free energyof activation, G*

Do not affect the position of equilibrium(Go unchanged)

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 FADA: + B => A + B:

* Transferases--transfer of a functional groupA-B + C => A + B-C

* Hydrolases--hydrolysis of functional group by waterA-B + H20 => A-H + B-OH

* Lyases--elimination to form double bond; or addition to a double bondX-A-B-Y => A=B + X-Y

* Isomerases--isometric interconversionsX-A-B-Y => Y-A-B-X

* Ligases--ATP dependent joining of two moleculesA + B +ATP => A-B + ADP + Pi

Enzymes:Compilations of Databases & Online Resources

http://restools.sdsc.edu/biotools/biotools12.html

FRONTIERS IN BIOSCIENCEShttp://www.bioscience.org/urllists/protdb.htm

ExPASy Molecular Biology Serverhttp://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 PE S = substrate

P = productE = enzyme

UncatalyzedReaction:

v

[S]

At a fixed concentration of enzyme, the velocity reachs a maximum, which fits the equation of a rectangular hyperbolic curve:

y=(ax)/(b + x)

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)

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)

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

The Effects of Enzyme Concentration

Vmax is directly proportionalto enzyme concentration

Km is independent ofenzyme concentration

At saturating substrate([S] >> Km):

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 hypotheticalenzyme with:

V = 10 Km = 4

Disadvantages of Lineweaver-Burk Plot

Still working with hypothetical enzyme withV = 10, Km = 4…

…only here random error has been introduced into multiple data sets, and results plotted toillustrate how greatly estimatesof Km and V can vary from plot to plot depending on dataquality.

How to improve:--averaging--weighting--choose another method

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

Eadie-Hofstee Plot:

Rearrangement of Michaelis-Menten equation to another linear formv = -Km(v/[S]) + V

Same hypotheticalenzyme with:

V = 10 Km = 4

Disadvantages of Eadie-Hofstee Plot

Still working with hypothetical enzyme withV = 10, Km = 4

Once again random error has been introduced to demonstratethe 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)

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

Hanes Plot:

Rearrangement of Michaelis-Menten equation to still another linear form[S]/v = (1/V)[S] + Km/V

Same hypotheticalenzyme with:

V = 10 Km = 4

Error Issues in the Hanes Plot

Still working with hypothetical enzyme withV = 10, Km = 4

Once again random error has been introduced… generallyscatter is improved relative toLH and EH plots, which can improve the accuracy of Km and V estimates.

Avoids the other problem of EH plots in that dependentvariable (v) does not influencethe independent (horizontal)axis.

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

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

Since there is no error, all of the lines intersect at a common point withcoordinates (Km, V).

V

Km

Effects of Error on Direct Linear PlotNow looking at 5 ([S], v) points from adata set including random error, againfrom our hypothetical enzyme withV = 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 totake the median (not the mean) values of Kmand V, which minimizes the contributions ofspurious outliers (such outliers could badlyskew the mean values). See Birch’s websitefor 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 Vwhich 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 thetrue ones (subject to experimental error of course).

Direct Fitting of v vs. [S] Curves

A number of commercially available non-linear regression packages exist whichwill 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 statisticalarguments such as variance and other indicators of goodness-of-fit.-- CAUTION: Always compare fitting results with standardgraphical methods of analysis. Be aware that different statistical packages can skew Km and V values depending on how data isweighted and how outliers are treated.

U. Paisley Website Includes InstructionsFor Setting Up a Global Fitting Routine forMichaelis-Menten Enzyme Kinetics Data inMicrosoft Excelhttp://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 improvedthrough the use of statistical weighting methods.

-- Error is usually greatest in reaction velocities measured at lowsubstrate 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 Plotby linear regression.-- The correct weighting factor is v4, which is the reciprocal of the variance.

References:

Cleland (1967) Adv. Enzymol. 29, 1-32.

Wilkinson (1961) Biochem. J. 80, 324-332.

Magar (1972) Data Analysis in BiochemistryAnd Biophysics, Academic Press, New York, pp. 429-432.

Atkinson et al. (1961) Biochem. J. 80, 318-323.

Barnes & Waring (1980) Pocket ProgrammableCalculators in Biochemistry, John Wiley & Sons, New York, pp. 204-235.

1/v

1/[S]

weighted

unweighted

Km and V From a Single Progress Curve

Substrate concentration is monitoredthroughout a reaction timecourse;

At any given time (substrate conc.),the instantaneous velocity is determinedfrom the slope of the curve’s tangent line.

([S], v) data pairs so generated are analyzedvia 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.

SiestaTime!