# enzyme kinetics

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enzyme kinetics course notes

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• IBSB Industrial Biotechnology and Systems Biology Research Group

Marmara University, Department of Bioengineering, Istanbul, Turkey

Ebru Toksoy ner

• Reaction rate is the change in the concentration of a reactant or a product with time (M/s).

D[A, B, C, D] = change in concentration of A/B/C/D over time period Dt

a A + b B c C + d D

rate = -D[A]

Dt1

a= -

D[B]

Dt1

b=

D[C]

Dt1

c=

D[D]

Dt1

d

• The rate law expresses the relationship of the rate of a reaction to the rate constant and the concentrations of the reactants.

Rate = k [A]x [B] y

rxn is xth order in A

yth order in B

(x +y)th order overall

a A + b B c C + d D

• A product

rate = -

D[A]

Dt

rate = k [A]0

k = rate M/s= [A] is the concentration of A @ any t

D[A]

Dt= k-

• D[A]

Dt= k-

[A] is the concentration of A @ any t

[A]0 is the concentration of A @t=0

t = t when [A] = [A]0/2

t =[A]0

2k

The half-life, t, is the time required for the concentration of a reactant to

decrease to half of its initial

concentration.

[A] = [A]0 - kt

• A product

rate = -

D[A]

Dt

rate = k [A]

k = rate

[A]= 1/s or s-1

M/s

M= [A] is the concentration of A @ any t

D[A]

Dt= k [A]-

• D[A]

Dt= k [A]-

[A] is the concentration of A @ any t

[A]0 is the concentration of A @t=0

[A] = [A]0exp(-kt) ln[A] = ln[A]0 - kt

ln[A]0

[A]0/2

k=t

ln2

k=

0.693

k=

• A product

rate = -

D[A]

Dt

rate = k [A]2

k = rate

[A]2= 1/M s or M-1s-1

M/s

M2= [A] is the concentration of A @ any t

D[A]

Dt= k [A]2-

• D[A]

Dt= k [A]2-

[A] is the concentration of A @ any t

[A]0 is the concentration of A @t=0

1

[A]=

1

[A]0+ kt

t =1

k[A]0

• Order Rate Law

Concentration-Time

Equation Half-Life

0

1

2

rate = k

rate = k [A]

rate = k [A]2

ln[A] = ln[A]0 - kt

1

[A]=

1

[A]0+ kt

[A] = [A]0 - kt

tln2

k=

t =[A]02k

t =1

k[A]0

• In the situation where [S] >> [E] and at initial velocity rates, it is assumed that the

changes in the concentration of the intermediate ES complex are very small over time (vo).

This condition is termed a steady-state rate, and is referred to as steady-state kinetics.

Rate of ES formation will be equal to the rate ES breakdown.

• k1 : forward rate constant for substrate binding

k-1 : reverse rate constant for substrate binding

k2 : catalytic rate constant

The rate of the reaction is: v = d[P]/dt = k2[ES]

The change in [ES] as a function of time:

d[ES]/dt = k1[E][S] - k-1[ES] - k2[ES]

During the steady state: d[ES]/dt = 0

0 = k1[E][S] - k-1[ES] - k2[ES]

• 0 = k1[E][S] - k-1[ES] - k2[ES]

The goal is to relate this equation to readily measurable experimental parameters, such as:

The total amount of enzyme: ET = [E] + [ES] The concentration of substrate: [S] The measured steady state velocity (v = k2 [ES])

We do not have a suitable way to measure [E], so ET will be used in its place:

[E] = ET - [ES]

[ES](k-1 + k-2) = k1[S](ET -[ES])

[ES](k-1 + k2) = k1 ET[ES] - k1[ES][S]

[ES](k-1 + k2 + k[S]) = k1 ET[S]

[ES] = k1ET[S]/(k-1 + k2 + k1 [S])

• [ES] = k1ET[S]/(k-1 + k2 + k1 [S])

The rate (velocity) of the reaction:

v = k2[ES] = k1k2ET[S]/{k-1 + k2 + k1[S]}

= k2ET[S]/{(k-1 + k2)/ k1+ [S]}

Vmax = k2ET

highest reaction rate that can be attained because all (i.e. ET) of the enzyme is saturated with substrate.

The KM or Michaelis constant: KM = (k-1 + k2)/ k1

Michaelis-Menten Equation

• Michaelis-Menten Equation

• Michaelis-Menten Equation

First order Zero order

• A. Low [S] B. 50% [S] or Km C. High, saturating [S]

• The significance of Km change based on the different rate

constants and which step is the slowest (also called the rate-

limiting step).

In the simplest assumption, the rate of ES breakdown to

product (k2) is the rate-determining step of the reaction

k -1 >> k2 and Km = k -1/k1.

This relation is also called a dissociation constant for the ES

complex and can be used as a relative measure of the affinity of

a substrate for an enzyme.

k2 >> k -1 or k2 and k-1 are similar, then Km remains more complex

and cannot be used as a measure of substrate affinity.

• Experimentally, Km is a useful parameter for

characterizing the number and/or types of substrates that a particular

enzyme will utilize

comparing similar enzymes from different tissues or different organisms

Km of the rate-limiting enzyme in many of the biochemical metabolic

pathways that determines the amount of product and overall regulation

of a given pathway.

Clinically, Km comparisons are useful for evaluating the effects mutations

have on protein function for some inherited genetic diseases.

• The catalytic constant of an enzyme is defined as: kcat = Vmax/[ET]

kcat = k2kcat = 1000 sec

-1 : the enzyme can convert 1000 molecules of substrate into

product each second at saturating [S].

kcat (units of sec-1), is also called the turnover number because under

saturating substrate conditions, it represents the number of substrate molecules

converted to product in a given unit of time on a single enzyme molecule.

In practice, kcat values (not Vmax) are most often used for comparing the catalytic

efficiencies of related enzyme classes or among different mutant forms of an

enzyme.

• Determination of KM and vmax by Lab experiments

Nonlinear Approach : optimization technique, where the constants are

adjusted so that the sum of square of the errors between the predicted

rate and the observed rate is minimum.

N is the number of data points collected during the experiments.

Nonparametric Approach : find rate at different substrate concentrations

and then solve the equations for the unknown parameters.

• Determination of KM and vmax by Lab experiments

Graphical Approach

• At equilibrium,

no net change of [S] & [P]

or of [ES] & [E]

At pre-steady-state,

[P] is low (close to zero

time), hence, V0 for

initial reaction velocity

At pre-steady state, we can ignore the back reactions

• Too much substrate

inhibitors bind to the enzyme substrate

complex but not the enzyme itself

inhibitor and substrate bind simultaneously

to enzyme, binding of one does not

influence the affinity of either species to

complex with the enzyme.

substrate and inhibitor

compete for the enzyme

• Increase [S] to overcome inhibition

Ki = dissociation constant

for inhibitor

Ki values are used to characterize and compare the

effectiveness of inhibitors.

This parameter is especially useful and important

in evaluating the potential therapeutic value of

inhibitors (drugs) of a given enzyme reaction.

For example, Ki values are used for

comparison of the different types of HIV

protease inhibitors.

In general, the lower the Ki value, the tighter

the binding, and hence the more effective

an inhibitor is.

• Increase [S] to overcome inhibition

Ki = dissociation constant

for inhibitor

Vmax unaltered, Km increased

• Km unaltered, Vmax decreased

• Km and Vmax both change

Uncompetitive Inhibition

Inhibitor binds ES complex.

Works best when S is high.

Rare.

• = noncovalent interaction away from the active site.

Protein-protein interactions.

Small molecules.

Common in multi-subunit protein complexes.

Feedback inhibition

• Feedback inhibition

• Feedback inhibition

• Allosteric enzymes often show sigmoid

kinetics (i.e., non-Michaelis-

Menten)

they are very sensitive to small

changes in substrate concentration

Sigmoid kinetics is a consequence of

interaction between sites due to the presence of sites that bind substrate other than the

active site.

• Thank you for your listening !

http://ibsb. marmara.edu.tr

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