Enzyme Kinetics

 

ENZYME KINETICS By: Engr. Vera Marie L. Lanaria ChE Department CIT Universit University y

 

Kinetics of Enzyme Reactions deals with the rate of enzyme reaction and how it is affected by various chemical and physical conditions  it provides information about the basic mechanism of the enzyme reaction and other parameters that characterize the properties of the enzyme  rate equations can be applied in calculating reaction time, yields, & optimum economic 

conditions needed in designing bioreactors

 

S – be the substrate (reactant) Let S – E – be the enzyme P – be the product  A simple reaction reaction would would be: S + E → P  P 

Rate can be expressed in terms of: ofr reaction = vs = - dS/dt or:

vp = dP/dt

 

Victor Henri (1902, a French physical chemist) proposed a quantitative theory of  enzyme kinetics and formulated the rate equation: v = vmax S KM + S In 1913, Leonor Michaelis (German biochemist) and Maud Menten (Canadian physician) continued continued the work of Henri in which later on it becomes the MichaelisMenten model

 

 

 

Michaelis-Menten Model

 

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

 

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

 

Derivation of Reaction Rate Equation

 Assumptions:  Assumptio ns: 



The total during enzyme concentration constant reaction, that is, stays CEo = CES + CE  The amount anamount enzyme very small compared to of the ofissubstrate; so the formation of enzyme-substrate complex does not significantly deplete the substrate.

 

 The

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

 

Linear Forms of Michaelis  

Menten Equation

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

 

Langmuir Plot

 

Lineweaver-Burk Plot

 

Eadie-Hofstee Plot

 

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

 

  S (mmol/L) 1

-

v (mmo/L-min) 0.20

2

-

0.22

3 5

-

0.30 0.45

7

-

0.41

10

-

0.50

15

-

0.40

20

-

0.33

 

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

 

0.60

   ) 0.50   n   i   m0.40   L    /   l   o0.30   m0.20   m    (   v0.10 0.00 0

5

10

15

S (mmo/L (mm o/L))

20

25

 

y = 1.5866x + 4.6417

Langmuir Plot

R2 = 0.9497 25    ) 20

15    i   n   m    (   v 10    /    S 5 0 0

2

4

6

S (mmol/L

8

10

12

 

From the line equation: y = 1.5866x + 4.6417 slope = 1/vmax = 1.5866 vmax = 1/1.5866 vmax = 0.63 min-1  y-intercept = KM/vmax = 4.6417 KM = (4.6417)(0.63) KM = 2.92 mmol/Lmin2 

 

y = 3.4575x + 1.945

Lineweaver-Burk LineweaverBurk Plot

R2 = 0.8463 6   v 4    /    1

2 0 0

0.2

0.4

0.6

1/S

0.8

1

1.2

 

From the line equation: y = 3.4575x + 1.945 y-intercept = 1/vmax = 1.945 vmax = 1/1.945 vmax = 0.514 min-1  slope = KM/vmax = 3.4575 KM = 3.4575(0.514) KM = 1.78  1.78 mmol/Lmin2

 

Eadie-Hofstee Eadie-Hofst ee Plot

y = -1.8923x + 0.5386 2

R = 0.6618

0.60 0.50 0.40    V0.30 0.20 0.10 0.00 0

0.05

0.1

0.15

v/S

0.2

0.25

 

From the line equation: y = -1.8923x + 0.5386 y-intercept = vmax = 0.5386 vmax ≈ 0.54 min-1 

slope = -KM = -1.8923 KM = 1.8923 KM  ≈ 1.89  1.89 mmol/Lmin2

 

Enzyme Reactor with Simple Kinetic

Bioreactor  –  – is a device/equipment within which biochemical transformation are caused by the action of enzyme or living cells Classifications Classificatio ns of bioreactor: 1) Batch 2) Steady-State Plug-Flow Reactor (PFR) 3) Continuous Stirred-Tank Reactor (CSTR)

 

Batch Reactor 

is normally equipped with agitator 



pH is maintained by using either a buffer  solution or a pH controller 



an ideal batch reactor is assumed to be well mixed so thatatthe are uniform in composition all contents times

 

Reaction Mechanism: - dS = vmax S dt KM + S rearranging & integrating:

 -(KM+S).dS/S =  vmax.dt passing the limits: limits: at t=0 ; S = So  at t=t ; S = S - KM ln(S/So) – (S (S –  – So) = vmaxt KM ln(So/S) + (So –  – S) = vmaxt

 

PFR Reactor (or Tubularflow Enzyme Reactor) 

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



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

 







since the variation in the radial direction is small compared to that in the longitudinal direction, it’s called plug-flow reactor   if PFR is operated at steady-state, the properties will be constant with respect to time equation in batch reactor can be applied to an ideal steady-state PFR, however, the time, t, should residence time,be   replaced with the So –  – S = -KM + vmax  . ln(So/S)

ln(So/S)

 

CSTR 

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



continuous operation can increase the productivity significantly by eliminating the downtime



easy to automate

 



substrate balance can be set up as follows: Input - Output + Generation = Acc. F(So) - F(S) + r sV = V(dS/dt) where: F = flow rate V = volume of the reactor  r s = rate of substrate consumption

but steady-state the concentration offor substrate shouldCSTR, be constant, thus dS/dt = 0

 

and if Michaelis-Menten equation can be used for the rate of substrate consumption, then the equation can be arranged as: F = D = 1/ =

vmax S

.

V (So –  – S)(KM + S) where: D = is known as dilution rate (Note: It’s common in biochemic biochemical al reaction to use the term dilution rate, than the term residence time.) S = -K + (v M

S)(S  –  – S) max

o

 

Inhibition of Enzyme Reaction Inhibitor  –  – can decrease the rate of reaction either competitively, non-competitively, partially competitively, or mixed  mixed 

 

Other Factors that influences Enzyme Activity   temperature 

 pH  effect of shear