ENZYME KINETICSENZYME KINETICS

By:By:

Engr. Vera Marie L. LanariaEngr. Vera Marie L. LanariaChE DepartmentChE Department

CIT UniversityCIT University

Kinetics of Enzyme Kinetics of Enzyme ReactionsReactions deals with the rate of enzyme reaction and deals with the rate of enzyme reaction and

how it is affected by various chemical and how it is affected by various chemical and physical conditionsphysical conditions

it provides information about the basic it provides information about the basic mechanism of the enzyme reaction and mechanism of the enzyme reaction and other parameters that characterize the other parameters that characterize the properties of the enzymeproperties of the enzyme

rate equations can be applied in calculating rate equations can be applied in calculating reaction time, yields, & optimum economic reaction time, yields, & optimum economic conditions needed in designing bioreactorsconditions needed in designing bioreactors

Let S – be the substrate (reactant)Let S – be the substrate (reactant)

E – be the enzymeE – be the enzyme

P – be the productP – be the product

A simple reaction would be:A simple reaction would be:

S + E S + E → P→ P

Rate of reaction can be expressed in terms Rate of reaction can be expressed in terms of: r = vof: r = vss = - dS/dt = - dS/dt

or: vor: vpp = dP/dt = dP/dt

Victor Henri (1902, a French physical Victor Henri (1902, a French physical chemist) proposed a quantitative theory of chemist) proposed a quantitative theory of enzyme kinetics and formulated the rate enzyme kinetics and formulated the rate equation:equation:

v = v = vvmaxmax S S

KKMM + S + S

In 1913, Leonor Michaelis (German bio-In 1913, Leonor Michaelis (German bio-chemist) and Maud Menten (Canadian chemist) and Maud Menten (Canadian physician) continued the work of Henri in physician) continued the work of Henri in which later on it becomes the which later on it becomes the Michaelis-Michaelis-Menten modelMenten model

Michaelis-Menten ModelMichaelis-Menten Model

Lock-and-Key ModelLock-and-Key Model(Emil Fischer – 1894)(Emil Fischer – 1894)

Induced-fit ModelInduced-fit Model(Daniel Koshland – 1958)(Daniel Koshland – 1958)

Derivation of Reaction Derivation of Reaction Rate EquationRate Equation

Assumptions:Assumptions:The total enzyme concentration stays The total enzyme concentration stays

constant during reaction, that is, constant during reaction, that is, CCEoEo = C = CESES + C + CEE

The amount of an enzyme is very small The amount of an enzyme is very small compared to the amount of substrate; so compared to the amount of substrate; so the formation of enzyme-substrate complex the formation of enzyme-substrate complex does not significantly deplete the substrate.does not significantly deplete the substrate.

The product concentration is so low that The product concentration is so low that product inhibition may be considered product inhibition may be considered negligible.negligible.

Linear Forms of Linear Forms of Michaelis-Menten Michaelis-Menten

Equation Equation Langmuir plot (or Hanes Woolf plot)Langmuir plot (or Hanes Woolf plot) Lineweaver-Burk plotLineweaver-Burk plot Eadie-Hofstee plotEadie-Hofstee plot

Langmuir PlotLangmuir Plot

Lineweaver-Burk PlotLineweaver-Burk Plot

Eadie-Hofstee PlotEadie-Hofstee Plot

Sample Problem:Sample Problem:From a series of batch runs with a constant From a series of batch runs with a constant

enzyme concentrations, the following initial enzyme concentrations, the following initial rate data were obtained as a function of rate data were obtained as a function of initial substrate concentration. (Refer to the initial substrate concentration. (Refer to the next slide for the data.) Evaluate the next slide for the data.) Evaluate the Michaelis-Menten kinetic parameters by Michaelis-Menten kinetic parameters by employing the 3 linear forms or plots. In employing the 3 linear forms or plots. In evaluating the parameters do not include evaluating the parameters do not include data points which deviate systematically data points which deviate systematically from the Michaelis-Menten model. from the Michaelis-Menten model.

S (mmol/L)S (mmol/L) -- v (mmo/L-min)v (mmo/L-min)

11 -- 0.200.20

22 -- 0.220.22

33 -- 0.300.30

55 -- 0.450.45

77 -- 0.410.41

1010 -- 0.500.50

1515 -- 0.400.40

2020 -- 0.330.33

Solution:Solution:

Examination of the data reveals that as the Examination of the data reveals that as the substrate concentration (S) increased up to substrate concentration (S) increased up to 10 mmo/L, the rate increased. However, the 10 mmo/L, the rate increased. However, the further increases in the S to 15 mmol/L, the further increases in the S to 15 mmol/L, the initial reaction rate decreased. This behavior initial reaction rate decreased. This behavior may be due to substrate or product inhibition. may be due to substrate or product inhibition. Since the Michaelis-Menten equation does Since the Michaelis-Menten equation does not incorporate the inhibition effects, thus not incorporate the inhibition effects, thus these two data points will be included.these two data points will be included.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0 5 10 15 20 25

S (mmo/L)

v (m

mol/

Lmin)

Langmuir Plot y = 1.5866x + 4.6417

R2 = 0.9497

0

5

10

15

20

25

0 2 4 6 8 10 12

S (mmol/L

S/v

(min

)

From the line equation:From the line equation:

y = 1.5866x + 4.6417y = 1.5866x + 4.6417

slope = 1/vslope = 1/vmaxmax = 1.5866 = 1.5866

vvmaxmax = 1/1.5866 = 1/1.5866

vvmaxmax = 0.63 min = 0.63 min-1-1

y-intercept = Ky-intercept = KMM/v/vmaxmax = 4.6417 = 4.6417

KKMM = (4.6417)(0.63) = (4.6417)(0.63)

KKMM = 2.92 mmol/Lmin = 2.92 mmol/Lmin22

Lineweaver-Burk Plot y = 3.4575x + 1.945

R2 = 0.8463

0

2

4

6

0 0.2 0.4 0.6 0.8 1 1.2

1/S

1/v

From the line equation:From the line equation: y = 3.4575x + 1.945y = 3.4575x + 1.945

y-intercept = 1/vy-intercept = 1/vmaxmax = 1.945 = 1.945

vvmaxmax = 1/1.945 = 1/1.945

vvmaxmax = 0.514 min = 0.514 min-1-1

slope = Kslope = KMM/v/vmaxmax = 3.4575 = 3.4575

KKMM = 3.4575(0.514) = 3.4575(0.514)

KKMM = 1.78 = 1.78 mmol/Lminmmol/Lmin22

Eadie-Hofstee Plot y = -1.8923x + 0.5386

R2 = 0.6618

0.000.100.200.300.400.500.60

0 0.05 0.1 0.15 0.2 0.25

v/S

V

From the line equation:From the line equation:

y = -1.8923x + 0.5386y = -1.8923x + 0.5386

y-intercept = vy-intercept = vmaxmax = 0.5386 = 0.5386

vvmax max ≈ 0.54 min≈ 0.54 min-1-1

slope = -Kslope = -KMM = -1.8923 = -1.8923

KKMM = 1.8923 = 1.8923

KKMM ≈ 1.89 ≈ 1.89 mmol/Lminmmol/Lmin22

Enzyme Reactor with Enzyme Reactor with Simple KineticSimple Kinetic

BioreactorBioreactor – is a device/equipment within – is a device/equipment within which biochemical transformation are which biochemical transformation are caused by the action of enzyme or living caused by the action of enzyme or living cellscells

Classifications of bioreactor:Classifications of bioreactor:1)1) Batch Batch 2)2) Steady-State Plug-Flow Reactor (PFR)Steady-State Plug-Flow Reactor (PFR)3)3) Continuous Stirred-Tank Reactor (CSTR)Continuous Stirred-Tank Reactor (CSTR)

Batch ReactorBatch Reactor

is normally equipped with agitatoris normally equipped with agitator pH is maintained by using either a buffer pH is maintained by using either a buffer

solution or a pH controllersolution or a pH controller an ideal batch reactor is assumed to be well an ideal batch reactor is assumed to be well

mixed so that the contents are uniform in mixed so that the contents are uniform in composition at all timescomposition at all times

Reaction MechanismReaction Mechanism::

- dS- dS = = vvmaxmax S S

dt Kdt KMM + S + S

rearranging & integrating: rearranging & integrating:

-(K-(KMM+S).dS/S = +S).dS/S = v vmaxmax.dt.dt

passing the limits: at t=0 ; S = Spassing the limits: at t=0 ; S = Soo

at t=t ; S = Sat t=t ; S = S

- K- KMM ln(S/S ln(S/Soo) – (S – S) – (S – Soo) = v) = vmaxmaxtt

KKMM ln(S ln(Soo/S) + (S/S) + (Soo – S) = v – S) = vmaxmaxtt

PFR Reactor (or Tubular-PFR Reactor (or Tubular-flow Enzyme Reactor)flow Enzyme Reactor)

the substrate enters one end of a cylindrical the substrate enters one end of a cylindrical tube which is packed with immobilized tube which is packed with immobilized enzyme and the product stream leaves at enzyme and the product stream leaves at the other end the other end

properties of flowing stream will vary in both properties of flowing stream will vary in both longitudinal and radial directions since there longitudinal and radial directions since there is no agitator usedis no agitator used

since the variation in the radial direction is since the variation in the radial direction is small compared to that in the longitudinal small compared to that in the longitudinal direction, it’s called direction, it’s called plug-flow reactorplug-flow reactor

if PFR is operated at steady-state, the if PFR is operated at steady-state, the properties will be constant with respect to properties will be constant with respect to timetime

equation in batch reactor can be applied to equation in batch reactor can be applied to an ideal steady-state PFR, however, the an ideal steady-state PFR, however, the time, t, should be replaced with the time, t, should be replaced with the residence time, residence time,

SSoo – S – S = -K = -KMM + + v vmaxmax . .

ln(Sln(Soo/S) ln(S/S) ln(Soo/S)/S)

CSTRCSTR

is an ideal reactor which is based on the is an ideal reactor which is based on the assumption that the reactor contents are assumption that the reactor contents are well mixedwell mixed

continuous operation can increase the continuous operation can increase the productivity significantly by eliminating the productivity significantly by eliminating the downtimedowntime

easy to automateeasy to automate

substrate balance can be set up as follows:substrate balance can be set up as follows:

Input - Output + Generation = Acc.Input - Output + Generation = Acc.

F(SF(Soo) - F(S) + r) - F(S) + rssV = V(dS/dt)V = V(dS/dt)

where: F = flow ratewhere: F = flow rate

V = volume of the reactorV = volume of the reactor

rrss = rate of substrate consumption = rate of substrate consumption

but for steady-state CSTR, the concentration but for steady-state CSTR, the concentration of substrate should be constant, thusof substrate should be constant, thus

dS/dt = 0dS/dt = 0

and if Michaelis-Menten equation can be and if Michaelis-Menten equation can be used for the rate of substrate consumption, used for the rate of substrate consumption, then the equation can be arranged as:then the equation can be arranged as:

FF = D = 1/ = D = 1/ = = v vmaxmax S . S .

V (SV (Soo – S)(K – S)(KMM + S) + S)

where: D = is known as dilution ratewhere: D = is known as dilution rate

(Note: It’s common in biochemical reaction to (Note: It’s common in biochemical reaction to use the term dilution rate, than the term use the term dilution rate, than the term residence time.)residence time.)

S = -KS = -KMM + (v + (vmaxmaxS)(SS)(Soo – S) – S)

Inhibition of Enzyme Inhibition of Enzyme ReactionReaction

Inhibitor Inhibitor – can decrease the rate of – can decrease the rate of reaction either reaction either competitivelycompetitively, , non-non-competitivelycompetitively, , partially competitivelypartially competitively, or , or mixedmixed

Other Factors that Other Factors that influences Enzyme Activityinfluences Enzyme Activity

temperaturetemperature pHpH effect of sheareffect of shear