Enzymatic Reactions 2 Many ways of illustrating the steps - - PDF document

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CEE 670

TRANSPORT PROCESSES IN ENVIRONMENTAL AND WATER RESOURCES ENGINEERING

Enzyme Kinetics

David A. Reckhow

CEE 670 Kinetics Lecture #9 1

Updated: 8 December 2011

Print version

Kinetics Lecture #9

Enzyme Kinetics: Basic Models Kinetic Theory: Encounter & Transition Model & IS Effects Clark, 9.6 Brezonik, pp. 130-158

Enzymatic Reactions

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 Many ways of illustrating the steps  Substrate(s) bond to active site  Product(s) form via transition state  Product(s) are released

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Basic Enzyme Kinetics

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 Irreversible  Single intermediate  The overall rate is determined by the RLS, k2  But we don’t know [ES], so we can get it by the SS mass

balance

 Again, we only know [Eo] or [Etot], not free [E], so:

] [ ] [ ] ][ [ ] [

2 1 1

ES k ES k S E k dt ES d    



E + S ES  E + P → ←

k1 k-1 k2

] [ ] [ ] [

2 ES

k dt P d dt S d r    

 

] [ ] [ ] [ ] [ ] [

2 1 1

ES k ES k S ES E k

• 

  

 Note that some references use k2 for k-1, and k3 for k2

Reactants, products and Intermediates

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 Simple Progression of

components for simple single intermediate enzyme reaction

 Shaded block shows steady

state intermediates

 Assumes [S]>>[E]t  From Segel, 1975; Enzyme

Kinetics

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Basic Enzyme Kinetics II

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 And solving for [ES],

] ][ [ ] [ ] [ ] ][ [

1 2 1 1

S E k ES k ES k S ES k

• 

 

 2 1 1 1

] [ ] ][ [ ] [ k k S k S E k ES

• 

 



1 2 1

] [ ] ][ [ ] [

k k k

• S

S E ES





 

Michaelis-Menten

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 Irreversible  Single intermediate

] [ ] [ ] [ ] ][ [ ] [

max 2

1 2 1

S K S r S S E k dt P d r

s k k k

• 

   





E + S ES  E + P → ←

k1 k-1 k2

] [ ] [

2 ES

k dt P d r  

1 2 1

] [ ] ][ [ ] [

k k k

• S

S E ES





 

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Michaelis Menten Kinetics

David A. Reckhow

CEE 670 Kinetics Lecture #9

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Substrate Concentration

20 40 60 80 100 120

Reaction Rate

20 40 60 80 100

rmax

0.5rmax

Ks

 Classical substrate plot

] [ ] [ ] [

max

S K S r dt P d r

s 

 

Substrate and growth

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 If we consider Y  We can define a microorganism-specific substrate

utilization rate, U

 And the maximum rates are then

] [ ] [ ] [ 1

max

S K S dt X d X

s 

    dt dX Y dt S d dt P d r 1 ] [ ] [    

Y YX dt dX X r U     Y k U

max max

  

] [ ] [ ] [ 1 S K S k dt S d X U

s 

 

and

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Linearizations

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 Lineweaver-Burke  Double reciprocal plot Wikipedia version Voet & Voet version

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 das

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David A. Reckhow

CEE 670 Kinetics Lecture #9

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 Lineweaver Burk  Hanes  Eadie-Hofstee

3 types Compare predictions

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 ad

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Multi-step

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 Double intermediate  Also gives:  But now:  Note what happens when: k3 >> k2

] [ ] [ ] [

max

S K S r dt P d r

s 

 

E + S ES  EP2  2E + P2 → ←

k1 k-1 k2

3 2 3 2 max

] [ k k E k k r

• 



P1

k3

    1

3 2 2 1 3

k k k k k k Ks   



Activation Energy

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 Activation Energy must always be positive  Unlike ∆H, which may be positive or negative  Differing reaction rates Energy Reaction Coordinate

reactants products Activated Complex

Ea Energy Reaction Coordinate

reactants products Activated Complex

Ea

f

H E

   f

H E

  

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Encounter Theory I

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 Uncharged Solutes  Nature of diffusion in water  Encounter within a solvent cage  Random diffusion occurs through elementary jumps of

distance

 For a continuous medium:  For a semi-crystalline structure:

r 2  

Molecular diameter Molecular radius

  2

2

 D

• r

D 2

2

  

Average time between jumps

  6

2

 D

• r

D 6

2

   For water, D ~ 1x10-5 cm2s-1, and = 4x10-8 cm, so  ~ 2.5x10-11 s If time between vibrations is ~ 1.5x10-13 s, then the average water molecule vibrates 150 times (2.5x10-11/1.5x10-13) in its solvent cage before jumping to the next one.

Encounter Theory II

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 Probability of Encounter  If A and B are the same size as water  They will have 12 nearest neighbors  Probability that “A” will encounter “B” in a solvent cage

• f 12 neighbors is:

 Proportional to the mole fraction of “B”

B A B

X P 6 

With each new jump, “A’ has 6 new neighbors Where:

3

1         

B B

n X

# molecules of “B” per cm3 # molecules of solvent per cm3

Geometric packing factor Molecular volume (cm3)

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Encounter Theory III

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 And combining the rate of movement with the probability of

encountering “B”, we get an expression for the rate of encounter with “B”

 Then substituting in for the probability  For water, =0.74, and the effective diffusion coefficient, DAB = DA +

DB, and =rAB, the sum of the molecular radii

 Then we get:

 

B A AB

P D

2

6 1   

D n n D

B B AB

    36 ) 6 ( 6 1

2 3

 

B AB AB AB

n D r 25 1  

# of encounters/sec for each molecule of “A ”

Encounter Theory IV

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 Now the total # of encounters between “A” and “B” per cm3

per second is:

 In terms of moles of encounters (encounter frequency) this

becomes:

B A AB AB AB A

n n D r n 25  

B AB AB B A Mole molecules

• L

cm AB AB A Mole molecules

• L

cm AB e

n A D r n n N D r n N Z ] [ 25 1000 25 1000

3 3

,

                  ] ][ [ 10 5 . 2

2 ,

B A N D r x Z

AB AB AB e 



nB=[B]/N0/1000 #/Mole cm2s-1 cm L/cm3 M-1s-1

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Encounter Theory V

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 Frequency Factor  When Ea = 0, k=A

] ][ [ 10 5 . 2

2 ,

B A N D r x Z

AB AB AB e 



A Energy Reaction Coordinate

reactants products Activated Complex

Ea

RT Ea

Ae k

/ 



Transition State Theory I

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 Consider the simple bimolecular reaction  Even though it is an elementary reaction, we can break it down

into two steps

 Where the first “equilibrium” is:  So the forward rate is:

C B A

k

   C AB B A

k

   





] ][ [ ] [ B A AB K

  

] ][ [ ] [ B A K AB

  

] ][ [ ] [ ] [ B A K k AB k dt C d

   

  Energy Reaction Coordinate

reactants products Activated Complex

Ea

“Activated Complex”

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Transition State Theory II

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 Now the transition state is just one bond vibration away from

conversion to products

 Planks Law:  Bond energy must be in the thermal region:  So equating, we get:  And since conversion occurs on the next vibration:

 h E vib

vibrational energy Frequency of vibration (s-1) Planck’s constant (6.62 x 10-27 ergs·s)

kT Ebond

Bond energy Temperature (ºK) Boltzman constant (1.3807×10−16 ergs ºK-1)

kT h   h kT  

  

  K h kT K k k

Transition State Theory III

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 Now from basic thermodynamics:  And also  So:  And combining:  Recall:  And substituting back in:

K RT Go ln  



• r

RT Go

e K





 S T H Go

  

 

RT H R S e

e K

 





RT H R S

e e h kT k

   





H V P H E

   

  

RT E R S

a

e e h kT k



      

 

A

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Activation Energy

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 Activation Energy must always be positive  Unlike ∆H, which may be positive or negative  Differing reaction rates Energy Reaction Coordinate

reactants products Activated Complex

Ea Energy Reaction Coordinate

reactants products Activated Complex

Ea

f

H E

   f

H E

  

Temperature Effects

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 Arrhenius Equation Log k 1/T

Log A

Ea/2.3R

2

ln RT E dT k d

a



RT Ea

Ae k

/ 



 

2 1 1 2 1 2

ln T RT E T T k k

a

 

 

2 1 1 2 1 2

ln T RT H T T K K



 

Analogous to Van’t Hoff Equation for Equilibria

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Ionic Strength Effects

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 Ion-ion Reactions  Based on activated complex theory  So let’s look at the equilibrium constant  Which means:

C AB B A

k

   





] ][ [ ] [ ] [ B A K k AB k dt C d

   

 

B A AB

B A AB B A AB K    ] [ ] [ ] [ } }{ { } {



  

 

        



  AB B A

B A K AB    ] ][ [ ] [

• r

        



 AB B A

B A K h kT dt C d    ] ][ [ ] [

K2

• (for I=0)

Reactions with charged ions

David A. Reckhow

CEE 670 Kinetics Lecture #9

26

 Using the Debye-Huckel Equation

 I<0.005

 Using the Guntelberg Approximation

 I<0.01

5 . 2

55 . log I zi

i 

 

 

 

5 . 2 5 . 2 2 2 2 2

02 . 1 log 51 . 51 . 51 . log log I z z k I z z z z k k

B A

• B

A B Z

• 

      

2 2

2

B B A A

z z z z  

) 1 ( 55 . log

5 . 5 . 2

I I zi

i

   

) 1 ( 02 . 1 log log

5 . 5 . 2 2

I I z z k k

B A

• 

 

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I corrections (cont.)

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 Neutral species  Some case studies:

I bi

i 

  log

 I

b b b k k

AB B A

• 

   

2 2

log log

Case Study: TCP

David A. Reckhow

CEE 670 Kinetics Lecture #9

28 Reckhow & Singer, 1985

“Mechanisms of Organic Halide Formation During Fulvic Acid Chlorination and Implications with Respect to Preozonation”, In Jolley et al., Water Chlorination; Chemistry, Environmental Impact and Health Effect, Volume 5, Lewis.

 Observed loss of 1,1,1-

trichloropropanone in distribution systems

 Lab studies show that

chloroform is the product

 Logically presumed to be

a simple hydrolysis

Note: both TCP and TCAC refer to the 1,1,1-trichloropropanone

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TCP (cont.)

David A. Reckhow

CEE 670 Kinetics Lecture #9

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I kH 4 . 1 81 . 4 ln     

T T

HOCl k 32 024 .  

I kH 6 . 08 . 2 log   

 Ionic strength effects  Rate with chlorine

 Increases greatly  High intercept

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 Gurol & Suffet showed 10x

higher rate constants

 Phosphate?

Disagreement with prior study

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David A. Reckhow

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Putting it together Catalysis

David A. Reckhow

CEE 670 Kinetics Lecture #9

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 Homogeneous Catalysis  Definition

 Liquid-phase substances which react with the main reactants or

intermediates thereby providing an alternative pathway to products with a lower activation energy or a higher frequency factor. Catalysts are often regenerated over the course of the reaction.

3 2

2 2

   

   B A B A termolecular reaction? – be skeptical

3 3 3 2 2 2 2            

         B C B C C A C A C A C A

3 2

2 2

   

   B A B A What really happens:

“C” serves as a sort of charge- transfer facilitator, since “B” does not exist in a divalent state

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David A. Reckhow

CEE 670 Kinetics Lecture #9

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 To next lecture