enzyme kinetics

Enzyme Kinetics

Enzyme Kinetics

• Enzyme Kinetics – Quantitative measurement of the rates of enzyme catalyzed reactions

&• The systematic study of factors that affect these rates

• Enzyme kinetics began in 1902 when Adrina Brown reported an investigation of the rate of hydrolysis of sucrose as catalyzed by the yeast enzyme inveratase.

• Brown demonstrated – when sucrose concentration is much higher than that of the enzyme, reaction rate becomes independent of sucrose concentration

Enzyme Kinetics

• Brown proposal – overall reaction is composed of two elementary reactions in which the substrate forms a complex with the enzyme that subsequently decomposes to products and enzymes.

• Here E, S, ES and P symbolize the enzyme, substrate, enzyme-substrate complex and products

k1 k2

E + S ES P + E k-1

Enzyme Kinetics

• According to this model• When the substrate concentration becomes high enough

to entirely convert the enzyme to the ES form, the second step of the reaction becomes rate limiting step.

• The overall reaction rate becomes insensitive to further increase in substrate concentration.

• The general expression of the velocity (rate) of this reaction is

][][

2ESk

dtPd

v

Enzyme Kinetics

• The overall rate of production of [ES] – Difference between the rates of elementary reactions leading to its appearance and those resulting in its disappearance.

• At this point, an assumption is required to achieve an analytical solution.• The rapid equilibrium assumption

• Michaelis - Menten Approach.• The steady-state assumption.

• Briggs and Haldane Approach.

][2][1]][[1][

ESkESkSEkdtESd

EPkES 2E+SK-1

K1

Michaelis - Menten Approach

The rapid equilibrium assumption:

• Assumes a rapid equilibrium between the enzyme and substrate to form an [ES] complex.

• The equilibrium constant Km can be expressed by the following equation in a dilute system.

EPkES 2E+SK-1

K1

][1]][[1 ESkSEk

][]][[

1

1

ESSE

kkKm


• Since the enzyme is not consumed, the conservation equation on the enzyme yields

• Then rearrange the equilibrium constant equation

• Substituting [E] in the above equation with enzyme mass conservation equation

][]0[][ ESEE

][]][[

1

1

ESSE

kkKm

mKSEES ]][[][

mKSESEES ]])[[]([][ 0


mKSESEES ]])[[]([][ 0

]][[]][[][ 0 SESSEKES m

]][[]][[][ 0 SESESKES m

]][[])[]([ 0 SESKES m

][]][[][ 0

SKSEES

m


• Then the rate of production formation v can be expressed in terms of [S]

• Where

][][

][]][[

][][ 02

2 SKSV

SKSEk

ESkdtPdv

mm

max

][ 02EkV max

Steady State Assumption (SSA)• Progress curve for the

components of a simple michaelis-Menten reaction

• Except the transition phase of the reaction (before shaded block) [ES] remains constant until the substrate is nearly exhausted.

• Hence synthesis of ES must equals to its consumption over the course of reaction i.e. ES maintain steady state

•Now: Base on steady state assumption, d[ES]/dt = 0

•d[ES]/dt = k1[E][S] –k-1[ES] – k2[ES] = 0(steady state assumption)

•solve for [ES] (do some algebra)

•[ES] = [E][S] k1/(k-1 + k2)

•Define KM (Michealis Constant)•KM = (k-1 + k2)/k1 => [ES] = [E][S]/KM

•rearrange to give KM = [E][S]/[ES]

SSA and Rate Equation

• Substitute in KM = [E][S]/[ES]

][]0[][ ESEE

][]])[[]([ 0

ESSESE

Km

];])[[]([][ 0 SESEESKm ]][[]][[][ 0 SESSEKES m

]][[]][[][ 0 SESESKES m

]][[])[]([ 0 SESKES m

][]][[][ 0

SKSEES

m

SSA and Rate Equation

SSA lead to Michaelis - Menten

• Then the rate of production formation v can be expressed in terms of [S]

• Where

• Michaelis Menten Equation

][][

][]][[

][][ 02

2 SKSV

SKSEk

ESkdtPdv

mm

max

][ 02EkV max

][][SKSVv

m max

Michaelis Menten Equation• Michaelis-Menten equation, the rate equation for

a one-substrate enzyme-catalyzed reaction.

• It is a statement of the quantitative relationship between the initial velocity V0, the maximum velocity Vmax, and the initial substrate concentration [S], all related through the Michaelis constant Km.

Michaelis Menten Equation• Numerical relationship emerges from the Michaelis-

Menten equation in the special case when V0 is exactly one-half of Vmax

• On dividing by Vmax we obtained

• Solving for Km, we get Km + [S] = 2[S]

Km = [S] whenmaxVv

21

0

Km

• KM is the substrate concentration required to reach half-maximal velocity (vmax/2).

• KM is a measure of a substrate’s affinity for the enzyme.

• A small KM means the substrate binds tightly to the enzyme and saturates the enzyme

Vmax

• Considering the total enzyme concentration the maximal rate, that the enzyme can attain is Vmax,.

• Vmax is equal to the product of the catalytic rate constant (kcat) and the concentration of the enzyme.

• The Michaelis-Menten equation can then be rewritten as V= Kcat [Enzyme] [S] / (Km + [S]).

• Kcat is equal to K2, and it measures the number of substrate molecules "turned over" by enzyme per second.

• The higher the Kcat is, the more substrates get turned over in one second.

Michaelis-Menten Kinetics

Features of Michaelis-Menten

• Assumes the formation of Enzyme substrate complex• Assumes that the ES complex is in rapid equilibrium

with free enzyme• Breakdown of ES to form products assumed to be

slower than 1. Formation of ES and 2. Breakdown of ES to reform E and S ][

][max0

SKSV

vm

Michaelis-Menten Kinetics

• KA is an equilibrium association constant (units: M-1)

• KD is an equilibrium dissociation constant (units: M)

• Tight binding implies a low dissociation constant and a high association constant

]][[][SE

ESKA

][]][[

ESSEKD

Transformations of the Michaelis-MentenEquation: The Double-Reciprocal Plot• The direct measurement of the numeric value of Vmax

and therefore the calculation of Km often requires impractically high concentrations of substrate to achieve saturating conditions

• The Michaelis-Menten equation can be algebraically transformed into equations that are more useful in plotting experimental data.

][][max

0SKSV

vm

Lineweaver-Burk Equation• Starting with the MM equation

• Reciprocal of MM equation

• Lineweaver-Burk Equation

• Equation is the equation for a straight line, y = ax + b, where y = 1/v0 and x = 1/[S].

][][max

0SKSV

vm

maxmax0

1][

1VSV

Kv

m

maxmax0

1][1)(1

VSVK

vm

Lineweaver-Burk Equation• A plot of 1/v0 as y as a function of 1/[S] as x therefore

gives a straight line whose y intercept is 1/Vmax and whose slope is Km/Vmax.

• Such a plot is called a double reciprocal orLineweaver-Burk plot

• Setting the y term of equationequal to zero and solving for x reveals that the x intercept is −1/Km

Lineweaver-Burk Equation• Lineweaver-Burk plot, has the great

advantage of allowing a more accurate determination of Vmax, which can only be approximated from a simple plot of V0 versus [S]

• The double-reciprocal plot of enzyme reaction rates is very useful in distinguishing between certain types of enzymatic reaction mechanisms.

Kinetics of Isosteric enzymes

• Isosteric enzymes (with only one enzyme conformation, 1), the efficiency of substrate binding (dashed curve) declines constantly with increasing [A], because the number of free binding sites is constantly decreasing.

Kinetics of allosteric enzymes• Allosteric enzymes, the

binding efficiency initially rises with increasing [A], because the free enzyme is present in a low-affinity conformation (square symbols), which is gradually converted into a higher-affinity form(round symbols) as a result of binding with A.

• It is only at high [A] values that a lack of free binding sites becomes noticeable and the binding strength decreases again.

Enzyme Kinetics - Factors• The catalytic properties of enzymes, and

consequently their activity, are influenced by numerous factors.

• These factors include • Physical quantities (temperature, pressure), • The chemical properties of the solution (pH value,

ionic strength), • The concentrations of the relevant substrates,

cofactors, and inhibitors.

pH Dependency of Enzyme Activity

• Effect of enzymes is strongly dependent on the pH

• Activity is plotted against pH, a bell-shaped curve is usually obtained

• Bell shape of the activity–pH profile results from the fact that amino acid residues with ionizable groups in the side chain are essential for catalysis.

pH Dependency of Enzyme Activity• a basic group B (pKa = 8),

which has to be protonated in order to become active.

• a second acidic amino acid AH (pKa = 6), which is only active in a dissociated state.

• At the optimum pH of 7, around 90% of both groups are present in the active form

• at higher and lower values, one or the other of the groups increasingly passes into the inactive state.

Temperature Dependency of Enzyme Activity• The temperature

dependency of enzymatic activity is usually asymmetric.

• With increasing temperature, the increased thermal movement of the molecules initially leads to a rate acceleration

• At a certain temperature, the enzyme then becomes unstable, and its activity is lost within a narrow temperature difference as a result of denaturation

Bisubstrate Kinetics• Most reactions in biological systems usually include two

substrates and two products A + B -> P + Q.

• In bisubstrate reactions transfer of a functional group, such as a phosphoryl or an ammonium group, from one substrate to the other

• In oxidation-reduction reactions, electrons are transferred between substrates

• Multiple substrate reactions can be divided into two classes: sequential displacement and double displacement.

Bisubstrate Kinetics Sequential Displacement

• In the sequential mechanism, all substrates must bind to the enzyme before any product is released.

• Sequential mechanisms are of two types: ordered, in which the substrates bind the enzyme in a defined sequence, and random.

• Many enzymes that have NAD+ or NADH as a substrate exhibit the sequential ordered mechanism

• Lactate dehydrogenase reduces pyruvate to lactate while oxidizing NADH to NAD+.


• In the ordered sequential mechanism, the coenzyme always binds first and the lactate is always released first.


• Random sequential mechanism, the order of addition of substrates and release of products is random.

• E.g. formation of phosphocreatine and ADP from ATP and creatine, a reaction catalyzed by creatine kinase

• Sequential random reactions can also be depicted in the notation.

Bisubstrate Kinetics – Ping-Pong• In double-displacement, or

Ping-Pong, reactions, one or more products are released before all substrates bind the enzyme.

• Mechanisms in which the first substrate A is bound and immediately cleaved.

• A part of this substrate remains bound to the enzyme, and is then transferred to the second substrate B after the first product C has been released. – Ping-Pong

Bisubstrate Kinetics – Ping-Pong• The enzyme aspartate aminotransferase catalyzes the transfer

of an amino group from aspartate to a-ketoglutarate.

• After aspartate binds to the enzyme, the enzyme removes aspartate's amino group to form the substituted enzyme intermediate.

• The first product, oxaloacetate, subsequently departs.

• The second substrate, a-ketoglutarate, binds to the enzyme, accepts the amino group from the modified enzyme, and is then released as the final product, glutamate.

enzyme kinetics

Food