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CHEM 2880 - Kinetics 118 Enzyme Kinetics Tinoco Chapter 8

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Page 1: CHEM 2880 - Kinetics Enzyme Kinetics Kinetics.pdf · to E interfering with the enzyme activity. These are inhibitors • there are two types of inhibition: reversible and irreversible

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Enzyme Kinetics

Tinoco Chapter 8

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Enzymes

• homogenous, biological catalysts that act by reducing

aE• contain an active site where substrates (reactants)

bind and are transformed to products• with the release of the product, the active site is

returned to its original state• usually consist of protein, some have organic or

inorganic co-factors• RNA can also act as a biological catalysts, these

enzymes are known as ribozymes

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2 2An Example - Decomposition of H O

2 2 2 2The decomposition of H O is typically 1 order in H Ost

2 2and a catalyst, though at higher [H O ], it is pseudo-1storder in the catalyst:

2 2v = k[H O ][catalyst]• there are a number of different species which can

catalyze this reaction

• the uncatalyzed reaction is probably “catalyzed” bydust particles which are difficult to completelyremove from a reaction system

• metals and halides increase the reaction rate by 4-5orders of magnitude

• the enzyme catalase causes an increase of 15 orders,corresponding to the decomposition of 10 millionmolecules per second

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Catalase is a hemoprotein with a ferriprotoporphyrin(hematin) prosthetic group.• Fe alone acts as a catalyst for the decomposition of

2 2H O as does hematin, but neither approach thecatalytic effect of the enzyme

Thermodynamically, the decomposition is highlyfavoured:

298)G° = -103.10 kJ mol-1

a highly exergonic reaction.

This is mostly an enthalpic effect with:

298 )H° = -94.64 kJ mol -1

• the reaction is highly favoured thermodynamicallyand should go to completion

• the activation energy must be high for the reactionrates to be so low

aE = 71 kJ mol-1

• assuming first order kinetics, calculations yield

A < 1 x 10 s5 -1

a• a large E and small A combine to yield a small rate

aconstant, but the effect is largely from the E

aNote: k = A exp(-E /RT), so for a fast reaction,

aneed large A and small E .

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For the enzyme catalyzed reaction:

aE = 8 kJ mol-1

A = 1.6 x 10 M s8 -1 -1

(A is now calculated for 2 order kinetics)nd

• the reaction rate increases 3 orders for the increase in

aA and 12 orders for the decrease in E

forward• because )G° is so largely negative, the reversereaction is very slow even with the enzyme present

reverse forward forward)G° = )G° - )G°‡

forward• the enzyme reduces )G° but does not effect‡

forward)G°

rxn• for most biochemical reactions, )G° � 0 and bothforward and reverse reactions are appreciablyeffected by an enzyme

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Typical Enzyme Behaviour

Most enzyme reactions with a single substrate follow thereaction scheme:

Enzyme + Substrate 6 Products

v = k[enzyme][substrate]

• these reactions show first order kinetics in substrateand enzyme at low [substrate]

• at high [substrate] for a single [enzyme] the reactionis pseudo-1st order in enzyme

v = k’[enzyme]

• v reaches a maximum with increasing [substrate]. This is the “catalytic constant” or “turnover number”

cat maxk = v / [enzyme active sites] (s )-1

• the [enzyme active sites] is used rather than[enzyme] for fair comparison to enzymes withmultiple catalytic sites (eg. catalase has fouractive sites)

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Comparing catalytic effect of different enzymes:

• catalase is not typical - faster by seven orders of

catmagnitude with k = 9 x 10 s6 -1

• more typical is 10 s ± 1 order of magnitude3 -1

cat• even with lower k values, enzymes are still veryeffective:

eg. The reaction: fumarate º L-malate is catalyzed byfumarase. The enzyme catalyzed reaction is 10 x faster9

compared to the same reaction catalyzed by acid or base.

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Michaelis-Menten Kinetics

• the concentration of enzyme in a reaction is typicallylow, 10 - 10 M-10 -8

• in a complex mixture it is difficult to directlymeasure the [enzyme] so early studies of enzymekinetics focussed on the concentrations of thesubstrate and product which are typically 10 -10 M-6 -3

• these early studies produced the followingobservations;

• the rate increases linearly with [enzyme] - 1st

order in enzyme

• at fixed [enzyme], the rate is linear with[substrate] at low [substrate] = 1 order inst

substrate

• the rate approaches a maximum or saturation rateas [substrate] increases

0 maxv = v = k’[enzyme]

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• this behaviour is expected if the enzyme forms acomplex with the substrate

• at high [substrate] the enzyme is saturated withsubstrate, working at full capacity

cat max 0k = V / [E]

0[E] is the total concentration of enzyme sites

• at lower [susbtrate] the enzyme is not saturatedand turnover is limited by the availability of thesubstrate

Michaelis and Menten suggested the following kineticformulation:

• step two is also reversible, so this only applies in theearly stages of the reaction where [P] is very low

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Applying the steady-state approximation:

[E] and [S] are the concentrations of free enzyme andsubstrate respectively and are difficult to measure.

Using the mass balance equations:

0 0[E] = [E] + [ES] [E] = [E] - [ES]

0[S] = [S] + [ES] � [S] Assuming [S]>>[ES]

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The Michaelis-Menten equation

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The Michaelis-Menten Equation

maxV is the maximum rate of reaction

mK is the Michaelis constant

This equation matches well with the empirical data onenzyme catalyzed reactions:

• at low [S] the reaction is first order in substrate:

• at high [S] the rate approaches a maximum:

cat• the catalytic constant k is equal to the 1 -order ratest

constant for the rate-determining step

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m• when [S] = K :

0 maxv = ½V

m• K is equal to the substrate concentration thatproduces half the maximum rate for the enzyme

m• K reflects how tightly the enzyme binds thesubstrate

ES º E + S

and is an indication of which way the equilibriumbetween E and S lies

m• a small K indicates an enzyme that tightly bindsS and saturates at relatively low [S]

m• a large K indicates S is loosely bound to theenzyme and that a relatively high [S] is requiredto saturate the enzyme

m• K is equal to the dissociation constant for ES if

2 -1k <<k

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Enzyme reactions are generally reversible.

(Remember catalase was an exception)

• use the method of measuring initial rates, to avoidproblems with increasing [P]

• if the reverse reaction is not important (i.e. if anequilibrium heavily favours products), the Michaelis-Menten equation can be integrated

m max• there is no linear plot for [S] vs t, but K and Vcan be determined by computer

• few enzymes follow simple E+S mechanism, buttheir kinetic expressions often still fit the Michaelis-Menten equation

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Kinetic Data Analysis

• we always prefer a linear plot for analyzing data

Lineweaver-Burk

0• the reciprocal of the M-M equation, plot 1/v vs 1/[S]

m max• the slope is K /V and the x-int (by extrapolation)

mis -1/KFrom: Physical Chemistry for the Chemical and Biological Sciences, R. Chang, University Science Books, 2000, p. 519.

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Eadie-Hofstee

max 0 0 0• Lineweaver-Burk x V v , plot v vs v /[S]

m max• the slope is -K and the y-int is VFrom: Physical Chemistry for the Chemical and Biological Sciences, R. Chang, University Science Books, 2000, p. 519.

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mIdeally all plots should yield the same values for K and

maxV , but in practice this is not always true.

• often one plot may be preferable to another

• the E-H plot spreads the data at high [S]

• the L-B compresses data at high [S] and emphasizesdata at lower [S]

• the different plots may provide different values for

m maxK and V

Linear plots are usually analyzed using the method oflinear least squares to determine the equation of thestraight line.

• the slope and intercept are chosen such that the sumof the squares of the differences between theexperimental and calculated y-values is minimized

• different x, y co-ordinates for the different plotsmeans different sums are minimized to getthe best fitequation

• each point is weighted equally for the linear fitthough the precision varies from point to point

• thus each plot has a different statistical bias

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Unbiased Method

An unbiased method was developed by Cornish-Bowdenand Wharton (1988)

0 m• each pair of v and [S] provides values for K and

maxV

• using a possible pairs of data points, a series of

m maxvalues for each of K and V are determined using:

m max• the median value for each of K and V are thendetermined

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Competition and Inhibition

• for a multi-step reaction:

• E binds S to form ES which then undergoes twotransformations before releasing P and free E

• the M-M equation still applies

cat m• k and K are functions of all the rate constants inthe mechanism

cat• if there is a rate-determining-step, k is the 1 -st

order k for that step

m• K can always be treated as an apparentdissociation constant for ES

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Specificity Constant

cat m• k /K = specificity constant

0• this relates v to [E] (free enzyme) not [E] (totalenzyme)

• for two competing substrates A and B

• the ratio of rates depends on the ratio ofconcentrations and the ratio of the specificityconstants

• in biological systems there are almost alwayscompeting substrates, so specificity is important

• enzymes are highly specific and selective catalysts,but there are often species other than S that can bindto E interfering with the enzyme activity. These areinhibitors

• there are two types of inhibition: reversible andirreversible

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Competitive Inhibition

• a reversible inhibition, the inhibitors resemble S instructure enough to bind to active sites on E but arenearly or completely unreactive

• they inhibit by preventing proper S from binding,they occupy active sites

• inhibition can be reversed by diluting I or addingexcess S

• EI is an inactive form of the enzyme

• the M-M equation is:

m m• K is larger for a system with I present (K ’) - need

maxmore S to reach ½V

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• the L-B equation is:

I• slope is increased by a factor of (1+ [I]/K ), the

maxintercept (and therefore V ) does not change

Example:

• succinate is oxidized to fumarate by succinicdehydrogenase

• malonate has a structure very similar to succinate(one carbon shorter) but cannot be oxidized - acts asa competitive inhibitor for the enzyme

I• K in yeast is 1 x 10 - [I] of 1 x 10 M decreases the-5 -5

affinity of E for S by 2

Products are often competitive inhibitors

• usually similar in structure to S

• a biochemical way of turning E off (or down)

• one of the most immediate methods for regulation ormetabolic control

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Noncompetitive Inhibition

• a reversible inhibition, the inhibitor binds the enzymeat a site distinct from that where the substrate binds

• cannot be overcome by increasing [S]

• the inhibitor can bind reversibly to E or ES, and doesnot effect the binding of the substrate

• neither EI nor ESI form products

• the M-M equation is

max I m• V is reduced by a factor of (1+[I]/K ), K isunchanged

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• the L-B equation is:

• both the slope and the intercept increase, though thex-intercept remains unchanged

• common in multi-substrate enzymes

Uncompetitive Inhibition

• a reversible inhibition, the inhibitor binds theenzyme-substrate complex at a site distinct from thatwhere the substrate binds

• cannot be reversed by increasing [S]

• the inhibitor can bind reversible only to ES

• ESI does not form products

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• M-M equation is:

m max• both K and V are reduced by a factor of

I(1+[I]/K )

• the L-B equation is:

• the slope remains the same (because the changes in

m maxK and V cancel out), the intercept increases by a

Ifactor of (1+[I]/K )

• uncommon in single substrate enzymes, often seen inmulti-substrate enzymes

The following two illustrations are from: Physical Chemistry for the Chemical and Biological Sciences, R. Chang, UniversityScience Books, 2000, pp. 528-529.

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Irreversible Inhibition

• the inhibitor covalently binds to the enzyme andcannot be removed

• M-M kinetics do not apply in this case

• the effectiveness of the inhibitor is dependant on therate at which I binds to E

• also called suicide inhibition