Algebraic Model by 33GRVMI

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									                              ier
Bio/Chemical Kinetics Made Easy
           A Numerical Approach
                   Petr Kuzmič, Ph.D.
                       BioKin, Ltd.




   1. Case study: Inhibition LF protease from B. anthracis

   2. Method: Numerical Enzyme Kinetics
Anthrax bacillus
CUTANEOUS AND INHALATION ANTHRAX DISEASE




                           Bio/Chemical Kinetics Made Easy   2
Lethal Factor (LF) protease from B. anthracis
CLEAVES MITOGEN ACTIVATED PROTEIN KINASE KINASE (MAPKK)




                                                             Inhibitor?




                           Bio/Chemical Kinetics Made Easy                3
Neomycin B: an aminoglycoside inhibitor
PRESUMABLY A "COMPETITIVE" INHIBITOR OF LF PROTEASE




                     
                              

                    Fridman et al. (2004) Angew. Chem. Int. Ed. Eng. 44, 447-452
                            Bio/Chemical Kinetics Made Easy                   4
Competitive inhibition - Possible mechanisms
MUTUALLY EXCLUSIVE BINDING TO ENZYME




                   Segel, I. (1975) Enzyme Kinetics, John Wiley, New York, p. 102
                           Bio/Chemical Kinetics Made Easy                      5
Competitive inhibition - Kinetics
AT VERY HIGH [SUBSTRATE], ANZYME ACTIVITY IS COMPLETELY RESTORED




                                                                                           same V !
                    1.0




                    0.8
  enzyme activity




                    0.6

                                    increase [I]                            [I] = 0
                                                                            [I] = 1
                    0.4
                                                                            [I] = 2
                                                                            [I] = 4
                                                                            [I] = 8
                                                                            [I] = 16
                    0.2




                    0.0
                          -3   -2         -1       0          1         2              3

                                                log [S]


                                               Bio/Chemical Kinetics Made Easy                         6
Non-competitive inhibition - A possible mechanism
NON-EXCLUSIVE BINDING, BUT TERNARY COMPLEX HAS NO CATALYTIC ACTIVITY




                    Segel, I. (1975) Enzyme Kinetics, John Wiley, New York, p. 126
                            Bio/Chemical Kinetics Made Easy                      7
Non-competitive inhibition - Kinetics
EVEN AT VERY HIGH [SUBSTRATE], ANZYME ACTIVITY IS NEVER FULLY RESTORED




                    1.0




                    0.8
  enzyme activity




                    0.6             increase [I]

                    0.4




                    0.2




                    0.0
                          -3   -2       -1         0        1         2        3

                                              log [S]


                                             Bio/Chemical Kinetics Made Easy       8
Compare saturation curves
DIAGNOSIS OF MECHANISMS: SAME OR DIFFERENT RATE AT VERY LARGE [S]?
                     COMPETITIVE                                                   NON-COMPETITIVE




           1.0                                                           1.0




           0.8                                                           0.8


                                                     ?
           0.6                                                           0.6
activity




                                                              activity
           0.4                                                           0.4




           0.2                                                           0.2




           0.0                                                           0.0
                 0   2   4         6   8      10                               0    2   4         6   8   10

                             [S]                                                            [S]




                                           Bio/Chemical Kinetics Made Easy                                     9
     Compare "double-reciprocal" plots
          DIAGNOSIS OF MECHANISMS: STRAIGHT LINES INTERCEPT ON VERTICAL AXIS?
                          COMPETITIVE                                                                NON-COMPETITIVE




               20                                                                         30



                           [I] = 0                                                                       [I] = 0
                           [I] = 1                                                        25             [I] = 1
                           [I] = 2                                                                       [I] = 2
               15
                           [I] = 4                                                                       [I] = 4
                           [I] = 8                                                                       [I] = 8
                                                                                          20
1 / activity




                                                                           1 / activity
               10                                                                         15




                                                                                          10

               5

                                                                                          5




               0                                                                          0
                    0.0   0.5          1.0     1.5       2.0                                   0.0      0.5          1.0     1.5   2.0

                                     1 / [S]                                                                       1 / [S]




                                                     Bio/Chemical Kinetics Made Easy                                                10
Traditional plan to determine inhibition mechanism
THE TRADITIONAL APPROACH

      1. Measure enzyme activity at increasing [S]

         Collect multiple substrate-saturation curves at varied [I]

      2. Convert [S] vs. activity data to double-reciprocal coordinates

      3. Perform a linear fit of transformed (double-reciprocal) data

      4. Check if resulting straight lines intersect on the vertical axis

         If yes, declare the inhibition mechanism competitive




                    Fridman et al. (2004) Angew. Chem. Int. Ed. Eng. 44, 447-452
                           Bio/Chemical Kinetics Made Easy                   11
Collect experimental data at varied [S] and [I]
THE RAW DATA




                     0.8


                                                                      [I] = 0

                     0.6                                              [I] = 0.5 M
      V (a.u./sec)




                                                                      [I] = 1.0 M

                                                                      [I] = 2.0 M
                     0.4




                     0.2




                     0.0
                           0   20   40         60         80

                                    [S] (M)


                                    Bio/Chemical Kinetics Made Easy                  12
 Check for intersection of double-reciprocal plots
 DO LINEWEAVER-BURK PLOTS INTERSECT?



              12



              10
                                                                      [I] = 0
              8
                                                                      [I] = 0.5 M

                                                                      [I] = 1.0 M
        1/V




              6

                                                                      [I] = 2.0 M
              4




    
              2



COMPETITIVE
              0
               0.00   0.02   0.04     0.06    0.08    0.10    0.12

                                    1 / [S]


                                    Bio/Chemical Kinetics Made Easy                  13
Doubts begin to appear...
IS THIS A STRAIGHT LINE?



           2.2



           2.0
                                                                     [I] = 0
           1.8
     1/V




           1.6



           1.4



           1.2



           1.0
              0.00   0.02   0.04     0.06    0.08    0.10    0.12

                                   1 / [S]


                                   Bio/Chemical Kinetics Made Easy             14
Mysterious substrate saturation data
MICHAELIS-MENTEN KINETICS IS NOT SUPPOSED TO SHOW A MAXIMUM !



                    0.8




                    0.7                                              [I] = 0

                                              Throw these out?
     V (a.u./sec)




                    0.6




                    0.5




                    0.4
                          0   20   40           60       80

                                   [S] (M)


                                   Bio/Chemical Kinetics Made Easy             15
Repeat substrate experiment at higher [S]
SEE IF MAXIMUM HOLDS UP AT HIGHER [S]



                  1.4




                  1.2

                                                                                        [I] = 0
                  1.0
   V (a.u./sec)




                  0.8



                                  2
                  0.6
                            1/V




                                  1
                  0.4




                  0.2             0
                                      0.0   0.1     0.2       0.3   0.4
                                                  1 / [S]

                  0.0
                        0         20        40      60        80    100     120

                                                   [S] (M)


                                                      Bio/Chemical Kinetics Made Easy             16
Substrate inhibition in LF protease is real
HAS ANYONE ELSE SEEN IT?




                               Tonello et al. (2003) J. Biol. Chem. 278, 40075-78.
                           Bio/Chemical Kinetics Made Easy                      17
Rate equation for inhibition by substrate
WHAT DOES THE "BIG BLUE BOOK" SAY?




                    Segel, I. (1975) Enzyme Kinetics, John Wiley, New York, p. 126
                            Bio/Chemical Kinetics Made Easy                     18
Rate equation for inhibition by substrate + inhibitor
WHAT DOES THE "BIG BLUE BOOK" SAY?




                            Bio/Chemical Kinetics Made Easy   19
                              ier
Bio/Chemical Kinetics Made Easy
           A Numerical Approach
                   Petr Kuzmič, Ph.D.
                       BioKin, Ltd.




   1. Case study: Inhibition LF protease from B. anthracis

   2. Method: Numerical Enzyme Kinetics
The task of mechanistic enzyme kinetics
SELECT AMONG MULTIPLE CANDIDATE MECHANISMS




 E+S      E.S     E+P                                                initial rate
  E+I     E.I



         competitive ?

        uncompetitive ?                                                     concentration
             mixed type ?
                                 computer                     DATA
MECHANISMS




                         Select most plausible model




                            Bio/Chemical Kinetics Made Easy                                 21
   From mechanistic to mathematical models
   DERIVE A MATHEMATICAL MODEL FROM BIOCHEMICAL IDEAS


         k +1          k +2                                                                 initial rate
  E+S            E.S          E+P
         k -1

         k +3
 E+I             E.I
         k -3

MECHANISM
                                                                                                   concentration

                                                                                     DATA
                                         k 1k 3 [ S ]
v  k2 [ E ]
                k 3 (k 1  k  2 )  k 3 k 1[ S ]  k 3 (k 1  k  2 )[ I ]

MATHEMATICAL MODEL




                                                                        computer


                                                   Bio/Chemical Kinetics Made Easy                            22
Problem: Simple mechanisms ...
MERELY FIVE REACTIONS ...




    E +A        E.A
                      +B                                  •2         reactants (A, B)
                                                          •1         product (P)
                           E.A.B          E +P
                 +A
                                                          •5         reversible reactions
    E +B        E.B
                                                          • 10       rate constant
 "RANDOM BI-UNI" MECHANISM




                                   Bio/Chemical Kinetics Made Easy                          23
   ... lead to complex algebraic models
    MERELY FIVE REACTIONS ...




         Segel, I. (1975) Enzyme Kinetics.
         John Wiley, New York, p. 646.




  E +A      E.A
                  +B
                       E.A.B   E +P
             +A
  E +B      E.B


"RANDOM BI-UNI" MECHANISM




                                      Bio/Chemical Kinetics Made Easy   24
A solution: Forget about algebra
POSSIBLE STRATEGY FOR MECHANISTIC MODEL BUILDING




       • Do not even try to derive complex algebraic equations

       • Instead, derive systems of simple, simultaneous equations

       • Solve these systems using numerical methods




                           Bio/Chemical Kinetics Made Easy           25
Theoretical foundations: Mass Action Law
RATE IS PROPORTIONAL TO CONCENTRATION(S)
                                                                   “rate” … “derivative”

MONOMOLECULAR REACTIONS

A           products                 rate is proportional to [A]
                                     - d [A] / d t = k [A]


                                                          monomolecular rate constant
                                                          1 / time

BIMOLECULAR REACTIONS

A+B            products              rate is proportional to [A]  [B]
                                     - d [A] / d t = - d [B] / d t = k [A]  [B]


                                                                        bimolecular rate constant
                                                                        1 / (concentration  time)



                            Bio/Chemical Kinetics Made Easy                              26
Theoretical foundations: Mass Conservation Law

PRODUCTS ARE FORMED WITH THE SAME RATE AS REACTANTS DISAPPEAR

EXAMPLE

 A           P+Q        - d [A] / d t = + d [P] / d t = + d [Q] / d t




COMPOSITION RULE       ADDITIVITY OF TERMS FROM SEPARATE REACTIONS

mechanism:

             k1
       A           B                 d [B] / d t = + k1 [A] - k2 [B]

             k2
       B           C

                         Bio/Chemical Kinetics Made Easy                27
Composition Rule: Example

 EXAMPLE MECHANISM                 RATE EQUATIONS


         k+1                        d[P] / d t = + k+5 [EAB]
 E+A           EA
         k-1
         k+2
EA + B         EAB                   d[EAB] / d t = + k+2 [EA][B]
         k-2                                             - k-2 [EAB]
         k+3                                             + k+4 [EB][A]
 E+B           EB
                                                         - k-4 [EAB]
         k-3
         k+4                                             - k+5 [EAB]
EB + A         EAB
         k-4

         k+5
                                    Similarly for other species...
 EAB           E+P+Q

                       Bio/Chemical Kinetics Made Easy                    28
Program DYNAFIT (1996)




                   http://www.biokin.com/dynafit




                        Kuzmic P. (1996) Anal. Biochem. 237, 260-273.
                Bio/Chemical Kinetics Made Easy                    29
A "Kinetic Compiler"
HOW DYNAFIT PROCESSES YOUR BIOCHEMICAL EQUATIONS


         k1           k3
E +S          E.S              E +P
         k2


   Input (plain text file):                   Rate terms:             Rate equations:


                                                                    d[E ] / dt = - k1  [E]  [S]
 E + S ---> ES             :   k1             k1  [E]  [S]                     + k2  [ES]
                                                                                 + k3  [ES]

 ES ---> E + S             :   k2             k2  [ES]            d[ES ] / dt = + k1  [E]  [S]
                                                                                 - k2  [ES]
                                                                                 - k3  [ES]
 ES ---> E + P             :   k3             k3  [ES]


                                                                   Similarly for other species...

                                 Bio/Chemical Kinetics Made Easy                              30
System of Simple, Simultaneous Equations
HOW DYNAFIT PROCESSES YOUR BIOCHEMICAL EQUATIONS


         k1           k3
E +S          E.S              E +P                                "The LEGO method"
         k2
                                                                   of deriving rate equations

   Input (plain text file):                   Rate terms:            Rate equations:



E + S ---> ES              :   k1             k1  [E]  [S]


ES ---> E + S              :   k2             k2  [ES]


ES ---> E + P              :   k3             k3  [ES]




                                 Bio/Chemical Kinetics Made Easy                        31
Initial rate kinetics
 TWO BASIC APPROXIMATIONS



    1. Rapid-Equilibrium Approximation


                            k1           k3
                 E +S            E.S            E +P
                            k2
                                           assumed very much slower than k1, k2

    2. Steady-State Approximation
New in
DynaFit
          • no assumptions made about relative magnitude of k1, k2, k3

          • concentrations of enzyme forms are unchanging




                             Bio/Chemical Kinetics Made Easy               32
   Initial rate kinetics - Traditional approach
   DERIVE A MATHEMATICAL MODEL FROM BIOCHEMICAL IDEAS


         k +1          k +2                                                             initial rate
  E+S            E.S          E+P
         k -1

         k +3
 E+I             E.I
         k -3

MECHANISM                         Think!                                                       concentration

                                                                                     DATA
                                         k 1k 3 [ S ]
v  k2 [ E ]
                k 3 (k 1  k  2 )  k 3 k 1[ S ]  k 3 (k 1  k  2 )[ I ]

MATHEMATICAL MODEL




                                                                        computer


                                                   Bio/Chemical Kinetics Made Easy                        33
 Initial rate kinetics in DynaFit
 GOOD NEWS: MODEL DERIVATION CAN BE FULLY AUTOMATED!

 DynaFit input file                                                      MATHEMATICAL MODEL

[task]
   task = fit                                                  0 = [E] + [E.A] + [E.B] + [E.A.B] – [E]tot
   data = rates                                                0 = [A] + [E.A] + [E.A.B] – [A]tot
                                                               0 = [B] + [E.B] + [E.A.B] – [B]tot
   approximation = Steady-State
                                                               0 = + k1[E][A] – k2[E.A] – k3 [E.A][B] + k4 [E.A.B]
                                                               0 = + k5[E][B] – k6[E.B] – k7 [E.B][A] + k8 [E.A.B]

[mechanism]                                                    0 = + k3 [E.A][B] + k7 [E.B][A] + k10 [E][P] – (k4+k8+k9)[E.A.B]




   E + A   <==> E.A     :   k1    k2
   E.A + B <==> E.A.B   :   k3    k4             CRANK!
   E + B   <==> E.B     :   k5    k6                                                  initial rate

   E.B + A <==> E.A.B   :   k7    k8
   E.A.B   <==> E + P   :   k9    k10

[constants]                                                                                  concentration
   ...
                                                                                                    DATA
MECHANISM
                                                              computer




                            Bio/Chemical Kinetics Made Easy                                                  34
 Initial rate kinetics in DynaFit vs. traditional method
 WHICH DO YOU LIKE BETTER?
                                  [task]
                                     task = fit
                                     data = rates
                                     approximation = Steady-State

                                  [reaction]
                                     A + B --> P

                                  [mechanism]
E +A   E.A
             +B
                  E.A.B   E +P         E + A   <==> E.A            :   k1   k2
        +A                             E.A + B <==> E.A.B          :   k3   k4
E +B   E.B
                                       E + B   <==> E.B            :   k5   k6
                                       E.B + A <==> E.A.B          :   k7   k8
                                       E.A.B   <==> E + P          :   k9   k10

                                  [constants]
                                     ...

                                  [concentrations]
                                     ...
                                 Bio/Chemical Kinetics Made Easy                  35
                              ier
Bio/Chemical Kinetics Made Easy
           A Numerical Approach
                   Petr Kuzmič, Ph.D.
                       BioKin, Ltd.




   1. Case study: Inhibition LF protease from B. anthracis

   2. Method: Numerical Enzyme Kinetics
DynaFit model for inhibition by substrate
ENZYME KINETICS MADE EASIER

 [reaction]     |   S ---> P
 [enzyme]       |   E
 [modifiers]    |   I

 [mechanism]

    E + S <===> E.S           :   Ks      dissociation
    E.S + S <===> E.S.S       :   Ks2     dissociation
    E.S ---> E + P            :   kcat
 ...




                              Bio/Chemical Kinetics Made Easy   37
DynaFit model for inhibition by substrate + inhibitor
ENZYME KINETICS MADE EASIER



             [reaction]        |    S ---> P
             [enzyme]          |    E
             [modifiers]       |    I

             [mechanism]

                E +   S <===> E.S             :   Ks       dissoc
                E.S   + S <===> E.S.S         :   Ks2      dissoc
                E.S   ---> E + P              :   kcat
                E +   I <===> E.I             :   Ki       dissoc
                E.S   + I <===> E.S.I         :   Kis      dissoc

             [constants]

                Ks = 1 ?, Ks2 = 1 ?, kcat = 1 ?                     optimization flag
                Ki = 1 ?, Kis = 1 ?

             ...
             ...


                                              initial estimate
                              Bio/Chemical Kinetics Made Easy                  38
How do we know which mechanism is "best"?
COMPARE ANY NUMBER OF MODELS IN A SINGLE RUN

 [task]
    task = fit | data = rates
    model = mixed-type ?

 [reaction]     |   S ---> P
 [enzyme]       |   E
 [modifiers]    |   I

 ...

 [task]

    task = fit | data = rates
    model = competitive ?

 ...

 [task]

    task = fit | data = rates
    model = uncompetitive ?
                                                 Akaike Information Criterion
 ...                                            Review: Burnham & Anderson (2004)
                           Bio/Chemical Kinetics Made Easy                    39
The best model: mixed-type noncompetitive
NEOMYCIN B IS NOT A COMPETITIVE INHBITOR OF LETHAL FACTOR PROTEASE




                                      Kuzmic et al. (1996) FEBS J. 273, 3054-3062.
                            Bio/Chemical Kinetics Made Easy                    40
Direct plot: maximum on dose-response curves



                        0.8




                        0.6
         V (a.u./sec)




                        0.4




                        0.2




                        0.0
                              0   20      40         60      80      100

                                                [S] (M)


                                               Kuzmic et al. (1996) FEBS J. 273, 3054-3062.
                                   Bio/Chemical Kinetics Made Easy                      41
Double-reciprocal plot is nonlinear



                 8




                 6
           1/V




                 4




                 2




                 0
                  0.00   0.02     0.04         0.06   0.08   0.10

                                           1 / [S]


                                         Kuzmic et al. (1996) FEBS J. 273, 3054-3062.
                           Bio/Chemical Kinetics Made Easy                        42
DR plot obscures deviations from the model



                8




                6
          1/V




                4




                2




                0
                 0.00   0.02     0.04         0.06   0.08   0.10

                                          1 / [S]


                                        Kuzmic et al. (1996) FEBS J. 273, 3054-3062.
                          Bio/Chemical Kinetics Made Easy                        43
Direct plot makes model departures more visible



                         0.8




                         0.6
          V (a.u./sec)




                         0.4




                         0.2




                         0.0
                               0     20         40          60       80

                                                [S] (M)


                                             Kuzmic et al. (1996) FEBS J. 273, 3054-3062.
                                   Bio/Chemical Kinetics Made Easy                    44
Summary: Enzyme kinetics made (almost) easy


           HOW DO I BUILD A MATHEMATICAL MODEL FOR AN ENZYME MECHANISM?




• Let the computer derive your model - don't bother with algebra.

• For many important mechanisms, algebraic models don't exist anyway.

• The theoretical foundation is simple and well understood:

  - mass action law
  - mass conservation law

• The same set of    -like rules apply to all types of kinetic models:

  - reaction progress curves
  - initial reaction rates


                         Bio/Chemical Kinetics Made Easy                 45
Acknowledgements: Lethal Factor protease work

                                          Hawaii Biotech Inc.
                                                                    currently

                                       PantheraPharma Inc.




                                                       Mark Goldman
                                                          Sheri Millis
                                                        Lynne Cregar



                                              Aiea, Island of Oahu, Hawaii




                 Bio/Chemical Kinetics Made Easy                                46

								
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