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ORDER OF REACTION AND MICHAELIS AND MENTEN EQUATION Binod Aryal M.Sc. Clinical Biochemistry First year

2.order of reaction & mm equ

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Page 1: 2.order of reaction & mm equ

ORDER OF REACTION AND MICHAELIS AND MENTEN EQUATION

Binod Aryal M.Sc. Clinical BiochemistryFirst year

Page 2: 2.order of reaction & mm equ

Reaction Rates and Reaction Mechanisms

Initial rate is found by determining the slope of a line tangent to the curve at time zero.

Initial rate is the rate of a chemical reaction at time zero.• products of the reaction are not present, so the reverse

reaction cannot occur• it is a more accurate method for studying the

relationship between concentration of reactant and reaction rate

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Graphing Reaction Rate in Terms of Concentration

To study the effects of concentration on reaction rate:• different starting concentrations of reactant are used• initial rates are calculated using the slopes of the tangent

lines from concentration vs time curves• initial rates are plotted against starting

concentration

Initial rates are determined (A) and these are plotted against concentration (B).

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Zero-Order Reactions

rate = -D[A]Dt rate = k [A]0 = k

[A] is the concentration of A at any time t[A]0 is the concentration of A at time t=0[A] - [A]0 = kt

rate is independent of reactant concentration

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First-order ReactionsThe initial rate vs starting concentration graph on the previous slide is a straight line.

• the equation of the line can be expressed as:

rate = k[A]• This represents a first-order reaction

For reactions with more than one reactant (e.g. A and B):• if experiments for each reactant produce straight lines,

the rate is “first order with respect to reactant A and first order with respect to reactant B.”

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Second-order Reactions

For chlorine dioxide in this reaction:• the initial rate vs concentration curve is parabolic• the reaction is proportional to the square of [ClO2]• it is a second order reaction

with respect to this reactant

rate = k[A]2

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The Rate LawThe rate law shows the relationship between reaction rates and concentration of reactants for the overall reaction.

rate = k[A]m[B]n

m: order of the reaction for reactant An: order of the reaction for reactant Bk: rate constantm + n: order of the overall reaction

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Reaction MechanismsA reaction mechanism is the series of elementary steps that occur as reactants are converted to products.

For example, oxygen and nitrogen are not formed directly from the decomposition of nitrogen dioxide:

It occurs in two elementary steps:

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Michaelis-Menten Kinetics

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Introduction to Enzyme Kinetics

E + S ESkfkr

E + Pkcat

Variables[E]: free enzyme molecules[S]: free substrate molecules[ES]: enzyme-substrate complexes[P]: free product molecules

Parameterskf , kr , kcat : reaction rates

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Michaelis Menten equation• ES complex is key to understand kinetic behavior• 1903- Victor Henri proposed – enzyme combines

with substrate mol to form ES complex- necessary step in enzyme catalysis

• This idea was expanded into general theory of enzyme action, esp by Leonor Michaelis and Maud Menton in 1913

• Their Postulation: reversible with its [S]

------breaks down to P

• Rxn 2 is slower and limits the rate of overall rxn• Therefore overall rate is proportional to ES• enzymes, at any point of rxn , exists in 2 forms: E

and ES• at low [S], predominant form is E(uncombined),

and rate proportional to [S]

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Michaelis Menten equation• Quantitative (algebraic) expression of relationship between substrate concn

and enzymatic reaction rate

• Imp terms: [S], Vo, Vmax, Km (Michaelis Menten constant)-all measured readily• Curve expressing this rxn is rectangular hyperbola (in case of most enzymes)• Based on hypothesis that rate limiting step in enzymatic reactions is break

down of ES complex and free enzymes

• DERIVATION OF MICHAELIS MENTEN EQUATION the derivation starts with 2 basic steps of formation and breakdown of ES

Assum: steady state[P]: negligibleP—S conversion is ignored

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MICHAELIS MENTEN EQUATION• Vo is determined by breakdown of ES to form Product which is determined by [ES]

• As ES is not easily measured, alternative expression is sought for• Et = total enzyme & free enzyme=Et –ES• With these conditions in mind, following steps are employed in expression of Vo in

terms of easily measured parametersStep 1

step 2. One important assumption is made : initial rate of reaction reflects a steady state in which ES is constant- rate of formation is equal to rate of break down (steady state assumtion)

step 3. This equation is now solved algebraically

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Km as substrate concn.• a very important derivation can be made from Michaelis Menten

equation when Vo is exactly half Vmax.Then,

• This derivation is very useful, practical definition of Km:Km is equivalent to substrate concn at which Vo is one half Vmax

• Km is often used as denotation of enzyme affinity for a particular substrate

• Km : estimate of equilibrium constant of S binding to E

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Dependence of initial velocity on substrate concentrationshows kinetic parameters that define the limits of the

curve at high and low [S]

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Transformations of Michaelis Menten equation ( the double reciprocal plot)

• Algebraic transformation of Michaelis Menten equation into equations that are more useful in plotting experimental data

• One of them is simply taking reciprocal of both sides of M-M equation

• Separating the components on R.H.S,

• Simplification of this gives:

• This form of M-M equation is known as Lineweaver burk equation

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Double reciprocal plot contd..• A plot of 1/Vo vs. 1/[S] or double reciprocal plot yields

a straight line.• This line has:

slope: Km/ Vmax

intercept on 1/Vo axis : 1/Vmax

intercept on 1/[S] axis: -1/Km

• Advantages:- accurate determination of Vmax

- distinguish between certain types of enzymatic reaction mechanisms

-analysis of enzyme inhibitions

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Kinetic parameters used to compare enzyme activities

• Km:-vary greatly from enzyme to enzyme and even for different substrate of same enzyme-term sometimes used as indicator of affinity (inappropriate) of enzyme to substrate, ie, higher the Km lower is affinity and vice versa-for two step reaction -if k2<<k1, ie when k2 is rate limiting, Km=k-1/k1 which is dissociation constant, Kd of ES complex. In such case only, Km can be considered as measure of affinity

-sometimes, k2>>k1 or k2 andk-1 are comparable.

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Effect of substrate concentration on reaction velocity for 2 different enzymes, with different Km

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Kinetic parameters….• Kcat:

- k2 is used as rate constant only in 2 step reaction (M-M reaction) Vmax= K2 [Et]- if reaction is 3 step, k3 commonly is rate constantEg. In this case, Vmax= K3[Et]

So, a more general rate constant is used, kcat, that describes limiting rate of any enzyme catalysed reaction- turnover number

-direct measure of catalytic production of product under saturating substrate conditions

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Kinetic parameters….• Kcat/ Km:

-best way to compare catalytic efficiency of different enzymes or turnover of different substrate by same enzyme- also known as specificity constant-2nd order rate constant. Unit of m-1 s-1

- upper limit depends on rate at which E and S can diffuse together in aq. soln, 108 to 109 m-1 s-1

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Enzymes catalyzing reactions involving two or more products• Many enzymatic reactions involve two or more substrates binding to

enzyme

eg. ATP+ glucose ADP +glucose 6 phosphate

Rates of such bisubstrate reaction also can be analysed by M-M approach.

• Bisubstrate reactions usually involve transfer of an atom of a functional group from one substrate to other.

• It can follow any of the following common mechanisms.1.Enzyme rxn involving ternary complex (ordered or random)

2.Enzyme reaction in which no ternary complex is formed eg. Pingpong or double displacement mechanism

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Ordered sequentialmechanism

Random sequential mechanism

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Double displacement or Pingpong mechanism

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Limitations of Michaelis Menten model• Certain enzymes do not exhibit classic M-M saturation

kinetics.• When [S] is plotted vs. Vo, saturation curve is sigmoid- this

indicates cooperative binding of substrate to multiple sites (binding at 1side affects binding at another side)

• Double reciprocal plot for sigmoid- saturation kinetics (relation betn [S] & ½ Vmax) invalid, as straight lines are not produced. Instead, graphic representation ofHill’s equation is used

Here, k’ is a complex constantThis equation states that when [S] is low compared to k’ , the reaction velocity increases as the nth power of [S].

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Limitation…..

Representation of sigmoidSubstrate saturation kinetics

Graphic evaluation of hill equation to determine the substrate concentration that produces half maximal velocity when substrate saturation kinetics are sigmoid A Hill plot of kinetic data for an enzyme with

cooperative binding kineticn= Hill’s coefficient (depends upon the no. of binding sites +no &type of interactions)S50=substrate concentration resulting half maximal velocity

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The Hills equation• To evaluate sigmoid saturation kinetics• An eqn originally derived to describe the cooperative binding of O2 to Hb

• In this eqn when • n=1 ; the binding site act independently of one another and simple

saturation i.e. M-M kinetic behavior• n > 1 the sites are cooperative binding, > n value –stronger the

cooperative binding ( more sigmoid) • n<1 the sites are said to be exhibit negative cooperation

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References• Principle of biochemistry- Lehninger 5th

ed• Harper’s biochemistry- 26th ed.• Lippincot’s biochemistry-4th ed.• Stryer’s biochemistry• Color atlas of biochemistry 2005