Introduction to Enzyme Kinetics

• Enzyme kinetics is the study of the chemical reactions that are catalysed by enzymes, with a focus on their reaction rates • The rate at which an enzyme works is influenced by several factors – the

concentration of substrate

molecules, [S] – the

temperature

. – the presence of

inhibitors

–

pH

• The study of an enzyme's kinetics reveals: – the catalytic mechanism of this enzyme – its role in metabolism – how its activity is controlled – how a drug or a poison might inhibit the enzyme

Tools of enzymology-1

Spectroscopic techniques (structure and reactivity in solution) • Optical (Circular dichroism, UV-visible, fluorescence) • Vibrational (Infrared, Raman) Electrochemical methods • Conductometry (kinetic analysis) • Potentiometric techniques Enthalpimetry (microcalorimetry) • Very sensitive and free of interference Radiochemical methods • Far more sensitive than photometric ones but...

Tools of enzymology-2

X-ray crystallography • First crystallized enzyme, urease (J. Sumner, in 1926)  crystals are proteins and their dissolution led to enzymatic activity • Within 20 years: >130 enzymes crystals documented • 3-D structure of a protein, myoglobin, was deduced (Kendrew, 1957) Multidimensional nuclear magnetic resonance (NMR) and X-ray crystallography are now commonly used: – to explain the mechanistic details of enzyme catalysis – to design new ligands Molecular Biology • Clone and express enzymes in foreign hosts ( minute quantity) • Manipulate the a.acid sequence ( overexpression site-directed mutagenesis  purification and characterization of enzymes occuring naturally in and deletional mutagenesis  chemical groups in ligand binding)

Introduction to Enzyme Kinetics

Kinetic data

• [S], pH, T or [modifying ligands] could be changed to see their effect in product formation rate – Discontinuous assay (stop & sample) – Continuous assay (usually by

coupled assay procedure

) Glucose + ATP Glucose-6-P + ADP

NADP + & G6P dehydrogenase NADPH (abs: 340 nm)

Glucano-lactone-6-P • Purity of all reagents used and stabilization of medium conditions are important • Initial rate of the reaction should be measured to avoid product inhibition, occurence of reverse reaction and depletion of substrate

Effect of substrate concentration on velocity

• Substrate depletion is a linear function up to the time when ca. 10 % of the [S O ] is converted to product • For this period, the initial velocity (

v O

) can be approximated from the slope of graph: v O = Δ[S]/Δt = Δ[P]/Δt • From now on, when reaction velocity is mentioned it means v O • When v vs. [S] curve is plotted, it can be seen that velocity is saturable at high [S]. (NO linear relationship as expected from first order kinetics)

Single-substrate enzyme reactions

• Hydrolases (H 2 O in large excess...), isomerases, most lyases • Mix enzyme with its substrate in solution, measure [S] or [P] over time 

progress curves

Three distinct regions:

• Low [S]  first order kinetics • Intermediate region • High [S]  zero-order

Single-substrate enzyme reactions

Brown (1902) suggested the below reaction scheme:

(the enzyme and substrate react rapidly to form an ES complex) E + S k 1 k -1 ES k 2 E + P • Low [S]  [ES]  [S] • High [S]  all E will be in the form of ES complex, increasing [S] will not affect a change in velocity • The equation to describe the kinetics of these scheme can be derived by two different types of assumption: 1. The rapid equilibrium 2. The steady-state

Single-substrate enzyme reactions

The rapid equilibrium assumption

• In 1913 L. Michaelis & M. Menten proposed the mathematical modeling of enzyme reactions (based on early work of Henri (1903)) • M&M assumptions 1) rapid equilibrium is established btw the reactants and the ES complex 2) rate LIMITING step is disassociation of ES to E + P  k 2 << k -1 3) [S] >> [E]  [S] o = [S] 4) Three different states of the ENZYME are present bound enzyme total enzyme free enzyme = = = [ES] [E] t [E] t – [ES]

Single-substrate enzyme reactions

The rapid equilibrium assumption

E + S

K s

ES

k cat

E + P • A 1st order reaction: the rate limiting equation thus becomes v = (dP/dt) = k cat [ES] BUT we can not measure [ES]: The real purpose of the derivation of M&M kinetics is to be able to express [ES] in terms of E & S alone

Single-substrate enzyme reactions

Derivation of MM equation: 1.

Equilibrium dissociation constant:

Ks

 [

E

][

S

] [

ES

] 2.

Mass balance equation expressing the distribution of the total enzyme [

E

0 ]  [

E

]  [

ES

] 3.

4.

Substitute 2 to 1

K S

 ([

E

0 ]  [

ES

])[

S

] [

ES

] Rearrange [

ES

]  [ [

S E

] 0 ][ 

S K

]

S

Single-substrate enzyme reactions

5.

Write the velocity-dependence equation: v is equal to [all product forming species], each multiplied by its catalytic rate constant

v



k cat

[

ES

] 6.

Substitute 4 into 5

v



k

[

cat S

[ ]

E

0  ][

K S S

]

v

 [

V

max

S

]  [

S K

]

S

Single-substrate enzyme reactions

Steady-state assumption

• Briggs and Haldane (1925) • [ES] is fairly constant for the majority of experimental measurements • This approach does not require k 2 << k -1 • Despite the differences with previous method, the final equation referred as HMM equation • Assumptions: – [E] t = [E] f + [ES] – [S] >> [E]  [S] 0 = [S] – After a pre-steady-state period  [ES] remain constant  d[ES]/dt = 0

Single-substrate enzyme reactions

The steady-state assumption

E + S k 1 k -1 ES k 2 E + P 1. Rate of formation of ES complex 2. Rate of destruction ES complex 3. Steady State Equilibrium

d[ES] -d[ES]

 

dt dt

=

k 1

[E] [S] =

(k -1

+

k 2 )

[ES]

k 1

([E]t- [ES]) [S] =

(k -1

+

k 2 )

[ES] 4. Abbreviate kinetic constants as 5. Solve for [ES ] 6. Substitute in above

Km k -1

+

k 2

= ([E]t- [ES]) [S]

k 1

[ES]

Km v = k 2 [ES]

[ES] = [E]t [S]

Km

+ [S]

v

=

k 2

[E]t [S]

Km

+ [S] 7. Substitute

Vmax

for

k 2 [E]t v

=

Vmax Km

[S] + [S]

Single-substrate enzyme reactions

Does MM equation agree with observed kinetics?

A. [S] << Km

v = Vmax [S] Km + [S]  v = Vmax [S] Km  v o depends on [S] first-order kinetics

B. [S] >> Km

v = Vmax [S] Km + [S]  v = Vmax  v indp from [S] zero-order kinetics

Single-substrate enzyme reactions

• For most of the reactions, k 2 << k -1 assumption is not valid but we still make extensive use of rapid equilibrium approach since – Simple and most direct way to derive equations in the absence of prior knowledge of the relative magnitudes of rate constants – For many situations, rapid equilibrium and steady state approaches yield the same final equation BUT definition of constants are different – St-st yields complex equations

Single-substrate enzyme reactions

Km - the MM Constant

• When

[S] = Km

v

=

Vmax [S] = Vmax [S] Km + [S] [S] + [S] 

v = Vmax /2

•

Definition:

Km is the substrate concentration that provides a reaction velocity that is half of the maximal velocity under saturating substrate conditions • Note that Km and Ks are different terms and NOT always the same...

Ks= k -1 / k when k -1 1

>>

k 2



Km = (k -1

+

k 2 ) / k 1 Ks = Km NOT

all enzymes are treatable by MM kinetics… most regulatory enzymes (multi-subunits) are not treatable by MM kinetics

Single-substrate enzyme reactions

Significance of Km

• a characteristic physical property for each and every different enzyme • it is independent of [E] • it measures "relative affinity" of an enzyme for its substrate  many enzymes have individual steps in a complex reaction sequences i.e., Km is a complex function of many individual rate constants  If there is more than 1 substrate, then each has its own Km !

It should be noted that the kinetic constants are determined in highly purified solutions by in vitro laboratory experiments

– Isolated enzymes are not in subcellular organizations anymore...

– How the

in vitro

results can be used to understand enzyme action

in vivo??

Single-substrate enzyme reactions

in vitro vs in vivo

• Loss of organization and compartmentation • The dilution factor-dilution of cell constituents • Relative concentration of enzyme and substrate • Closed and open systems

Enzymes in membranes

• What membranes provide?

– Anchoring point  more economical use of NZs – Substrates and products with only limited solubility in water •e.g. NZs involved in lipid and glycolipid synthesis – Medium to organize multi enzyme complexes •e.g. acyl CoA desaturase: at least 3 protein which act together to desaturate fatty acids – Membranes may separate substrates, products and effectors and allow controlled transport •e.g. cAMP or Ca 2+

Single-substrate enzyme reactions

Significance of kcat

(=Vmax/[E]) • Sometimes referred as the turnover number (min -1 , s -1 ) (generally: molecules of P produced per unit time per molecules of E present) • In crude enzyme samples, it is impossible to know [E], ONLY  protein....

– Useful to compare different enzyme batches – BUT difficult to relate to kinetic constants like kcat • The rate of enzyme turnover is different

in vivo

([S] = 0.1-1.0Km) – BUT useful to compare rates of different enzymatic rxns • kcat relates to the chemical steps subsequent to formation of ES complex so kcat is effected by changes: – in the enzyme (different NZ or mutagenesis) – in solution conditions – in substrate identity • It reflects perturbations of chemical steps subsequent to initial substrate binding

Single-substrate enzyme reactions

kcat / Km

• The catalytic efficiency of an enzyme is best defined by this ratio • Useful to compare – The efficiencies of different enzymes – The utilization of different substrates – The efficiency with which an enzyme catalyses a particular rxn in the forward and reverse directions – Steady-state or equilibrium mechanism??

– Metabolic role of enzyme?? • E.g. different isozymes of glucokinases in liver and other tissues

Single-substrate enzyme reactions

Kinetic perfection

• The diffusion limit seems as the upper limit for kcat/Km that an NZ achieve... kcat/Km value at diffusion limit (10 8 -10 9 M -1 .s

-1 ) 

kinetic perfection

• Examples: – Acetylcholinesterase – Catalase – Fumarase acetylcholine H 2 O 2 fumarate malate 1.6 x 10 8 4.0 x 10 7 1.6 x 10 8 3.6 x 10 7 • Some NZs can overcome this limit....

Single-substrate enzyme reactions

Linear transformations of enzyme kinetic data

• Determination of kinetic constants from linearized data is more simple....

• These plots are extremely useful – in diagnosing the mechanistic details of multisubstrate NZs and – for determining the mode of interaction btw the NZ and its inhibitor • Some examples: – Lineweaver-Burk plot (1934) – Eadie-Hofstee and Hanes plots – Eisenthal-Cornish-Bowden direct plots

Single-substrate enzyme reactions

Lineweaver-Burk (double reciprocal)



1/v vs 1/[S]

• • Disadvantage-1 – Small errors in measured v are amplified by taking the reciprocal – The greatest percent of errors are likely to be associated with v at low [S]  Low [S]  high 1/[S], in linear regression, these data pts are weighted more heavily...

Disadvantage-2 – Deviations from linearity are less obvious than in some other plots  This is very important in investigation of reaction mechanism

V

max

v



K m

 [

S

]  [

S

]

K m

[

S

]  1 1

v



K m V

max 1 [

S

]  1

V

max

Single-substrate enzyme reactions

Lineweaver-Burk Substrate concentration range

• The concentrations of substrate chosen for reciprocal plot should be in the neighborhood of Km – If very high  curve will be essentially horizontal, slope ~ 0  difficult to determine Km – If very low  Intercept will be too close to the origin  difficult to determine both Km and Vmax • It is better to choose [S] that will give evenly spaced reciprocals – 1.0, 1.11, 1.25, .., 5, 10 in stead of 1.0, 2.0, 3.0, 4.0,..

Single-substrate enzyme reactions

Presentation of disadvantage of Lineweaver-Burk plot

Single-substrate enzyme reactions

Eadie-Hofstee (single reciprocal) and Hanes plots

v

 

K m v

[

S

] 

V

max [

S

]

v

 1

V

max [

S

] 

K m V

max

Single-substrate enzyme reactions

Eisenthal-Cornish-Bowden direct plots (1974)

v values  in y axis -[S] values  in x axis • •    Connect each pair and extrapolate these lines till their intersection point Coordinates of the intersection gives Km (x axis) and Vmax (y axis) It is considered to give best estimates among all linear transformation methods Not very suitable for multisubstrate reactions Not easy to detect departures from the basic equation

Single-substrate enzyme reactions

Some extensions to the simple model



Substrate inhibition

usually interpreted in terms of the existence of two types of substrate binding site in the enzyme e.g. Invertase by sucrose

Single-substrate enzyme reactions

Some extensions to the simple model



More than one intermediate -

E + S ES ES’ The King and Altman procedure E + P 

Multiple active sites

– Complex kinetics are observed for enzymes with multiple subunits and possessing more than one active site – Non-linearity in kinetic plots  positive or negative cooperativity 

Inhibition by products (

e.g. inhibition of lactase by galactose) 

Interference from reverse reactions

Enzyme Assays

Initial velocity

•

In vitro

assay conditions: the enzyme is present in limiting amounts • The initial velocity is directly proportional to [E] t • Since v varies with [S], the assay period must be short enough to ensure the usage of small fraction of S (ca 5% or less) • First thing to do is to establish the limits of linearity (max [P] that can accumulate before the [P] vs t and v vs [E] responses become nonlinear)

Enzyme Assays

Enzyme units and Specific activities

• In most preparations, the actual molar concentration of enzyme is unknown • Amount of enzyme expressed in terms of activity • Commission of Enzymes of the International Union of Biochemistry defined a standard unit: • 1 unit ACTIVITY= International unit (IU) amount enzyme which converts 1 μmole substrate per min at 25

o

C – e.g. IU= 10 μmole/min • 1 unit SPECIFIC ACTIVITY # IU of enzymatic activity per mg of total protein present – e.g. 10 μmole/min/mg protein or 10 IU/mg protein

Enzyme Assays

Turnover number

• the maximum number of moles of substrate that an enzyme can convert to product per catalytic site per unit time

Enzyme Assays

Determination of v

• A method to measure the +d[P]/dt or –d[S]/dt is required • In practice, the former is more precise  and a finite number the value is in between 0 • Assays –

Direct:

direct measurement of [S] or [P] as a function of time e.g. Cytochrome c oxidase  cytochrome c (550 nm) –

Indirect:

reaction P generation can be coupled with a non-enzymatic e.g. Redox active dyes –

Coupled:

A second enzymatic reaction is used; generally P of the 1st rxn is the S of the 2nd rxn e.g. Hexokinase activity (G-6-P dehydrogenase: NADP +  NADPH (340 nm))

Detection Methods

Spectrophotometry

• Cuvettes: – Disposable plastics  350-800 nm, no sample-to-sample cross contamination, convenient – Quartz  in UV range • In absorbance – Avoid concentrations in which there is deviation from Beer’s law; generally absorbance above 1.0 is not used – Give time to the lamps to warm up (15-30 min)

Detection Methods

Differential spectra

• If the substrate and product have overlapping absorption bands • The wavelenght at which the largest difference observed should be used

Detection Methods

Fluorescence measurements

• More sensitive than spectrophotometry (up to ca 100 x) BUT T control is important • Avoid inner filter effect • NADH and NADPH exhibit fluorescence (ex:340nm-em:460 nm) • Tyrosine and tryptophan fluoresce at 330-350 nm. They could give high background emission in UV region  It is advisable to detect substance which emit light in the visible region

Detection Methods

Electrochemical

• Ion-selective electrodes – pH meter: rarely used – pH-stat • Voltametry • Conductometry • Oxygen electrode

Radiochemical methods

• Radioactively labelled substrates: H-3, C-14, P-32, S-35, I-131 • Very sensitive but hazardous and no continuous monitoring

Calorimetric Methods

Bianconi,

Biophysical chemistry

, 2006 • Since most substrates and/or the products do not possess suitable properties, modified substrates or a coupled enzyme assay should be used • These strategies can introduce a number of experimental errors in the determination of

K

m and

k

cat • Virtually all chemical reactions, and therefore enzyme catalyzed reactions, occur with some heat effect  calorimetric techniques  Isothermal titration calorimetry (ITC)  Pioneer: Sturtevart (1937)

Calorimetric Methods

Bianconi,

Biophysical chemistry

, 2006 Rate vs [S] data can be fit to MM equation using non-linear least squares regression analysis and kinetic constants are determined

Q



n

.



H app

 [

P

]

T V



H app dQ



dt d

[

dt P

]

V



H app rate



d

[

P

]

dt

 1

V



H app dQ dt

Calorimetric Methods

Bianconi,

Biophysical chemistry

, 2006

Advantages

• Possibility of doing a direct assay without the requirement of modified substrates or coupled reactions • No need to have a clear sample since calorimetric measurements do not involve absorption or emission of light • Crowding effects can be studied

Experimental measurement of enzyme activity

Continuous vs discontinuous assays

• Continuous: signal monitored at discrete intervals over entire linear time period • Discontinuous: (e.g. detection with HPLC, electrophoresis) one reading was taken and rate is determined from the difference in signal btw t and t 0 can be misleading 

Experimental measurement of enzyme activity

A typical assay

• All but one component are added to the vessel  well mixed and equilibrated • At t=0, add the missing component – A small volume of concentrated solution – Unless all conditions are well-matched, it should not exceed 5-10 % of rxn mixture – Efficient BUT NOT vigorous shaking – For spectroscopic assays, rxn can be initiated in the cuvette • Two control measurements – All except the enzyme – All except the substrate • For non-spectroscopic methods requiring long times, the rxn should be stopped at a specific time – By rapid freezing – By denaturing the enzyme

Experimental measurement of enzyme activity

Running controls and error types

• We run controls to correct experimental data for – Any time dependent change that might happen – For any static signal due to other components • Random vs systematic error – Random error affects precision (duyarlık), whereas systematic error affects accuracy (doğruluk) – A result can be highly precise, without being accurate – A result cannot be highly accurate, unless it is highly precise

Factors affecting the rate

• We can change some parameters in a controlled manner and get some information...

Enzyme concentration

Curve a: v should be linearly proportional to [E] Curve b: in case of substrate depletion Curve c: reversible inhibitor present in the sample

Factors affecting the rate

pH

• Denaturation at high and low pH • Protein conformation can be maintained within a 4-5 pH units but activity different  ionizable groups in amino acids are affected Dixon and Webb, 1979

Factors affecting the rate

Temperature

• Chemical reaction rate generally doubles with every 10 O C increase in T • In case of the enzymes, denaturation should be considered • General enzyme assays are done at 25 O C and 37 O C • Rate of reaction is related to activation energy by Arrhenius equation

k cat



A

exp

E a RT

log(

k cat

)  

E a

1 2 .

3

RT

 log

A

Computers in Enzyme Kinetics

An Example: ENZYME KINETICS MODULE - SIGMAPLOT MODULE From the advertisement: • Simplified data management to organize your data and results • Select from a wide range of built-in models • Easily determine the best-fit inhibition model for your data • See results clearly with interactive graphs – Curve fitting and graphing capabilities to analyze and present the enzyme kinetics data – Import or enter your data, select the type of study and the equation you would like to fit, as well as the interactive graphs to display your results.

– Using non-linear curve-fitter, the module fits the selected equations to your data, as well as provides interactive graphs you need to see to study the kinetics mechanism – A detailed report complete with all statistical parameters for each model you fit so you can easily compare the different models to identify the best one for your data