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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
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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 k2 [ 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 k2 [ 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
