Transcript ier Bio/Chemical Kinetics Made Easy A Numerical Approach Petr Kuzmič, Ph.D. BioKin, Ltd. 1. Case study: Inhibition of LF protease from B.

ier
Bio/Chemical Kinetics Made Easy
A Numerical Approach
Petr Kuzmič, Ph.D.
BioKin, Ltd.
1. Case study: Inhibition of LF protease from B. anthracis
2. Method: Numerical Enzyme Kinetics
Anthrax bacillus
CUTANEOUS AND INHALATION ANTHRAX DISEASE
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Lethal Factor (LF) protease from B. anthracis
CLEAVES MITOGEN ACTIVATED PROTEIN KINASE KINASE (MAPKK)
Inhibitor?
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Neomycin B: an aminoglycoside inhibitor
PRESUMABLY A "COMPETITIVE" INHIBITOR OF LF PROTEASE


Fridman et al. (2004) Angew. Chem. Int. Ed. Eng. 44, 447-452
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Competitive inhibition - Possible mechanisms
MUTUALLY EXCLUSIVE BINDING TO ENZYME
Segel, I. (1975) Enzyme Kinetics, John Wiley, New York, p. 102
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Competitive inhibition - Kinetics
AT VERY HIGH [SUBSTRATE], ANZYME ACTIVITY IS COMPLETELY RESTORED
same V !
1.0
enzyme activity
0.8
0.6
increase [I]
[I] = 0
[I] = 1
[I] = 2
[I] = 4
[I] = 8
[I] = 16
0.4
0.2
0.0
-3
-2
-1
0
1
2
3
log [S]
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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
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Non-competitive inhibition - Kinetics
EVEN AT VERY HIGH [SUBSTRATE], ANZYME ACTIVITY IS NEVER FULLY RESTORED
1.0
enzyme activity
0.8
increase [I]
0.6
0.4
0.2
0.0
-3
-2
-1
0
1
2
3
log [S]
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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
?
activity
0.6
activity
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0
2
4
6
8
10
0
[S]
2
4
6
8
10
[S]
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Compare "double-reciprocal" plots
DIAGNOSIS OF MECHANISMS: STRAIGHT LINES INTERCEPT ON VERTICAL AXIS?
COMPETITIVE
NON-COMPETITIVE
20
30
[I] = 0
[I] = 1
[I] = 2
[I] = 4
[I] = 8
20
1 / activity
1 / activity
15
[I] = 0
[I] = 1
[I] = 2
[I] = 4
[I] = 8
25
10
15
10
5
5
0
0
0.0
0.5
1.0
1.5
2.0
0.0
1 / [S]
0.5
1.0
1.5
2.0
1 / [S]
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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
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Collect experimental data at varied [S] and [I]
THE RAW DATA
0.8
[I] = 0
[I] = 0.5 M
V (a.u./sec)
0.6
[I] = 1.0 M
[I] = 2.0 M
0.4
0.2
0.0
0
20
40
60
80
[S] (M)
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Check for intersection of double-reciprocal plots
DO LINEWEAVER-BURK PLOTS INTERSECT?
12
10
[I] = 0
[I] = 0.5 M
1/V
8
[I] = 1.0 M
6
[I] = 2.0 M
4

COMPETITIVE
2
0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
1 / [S]
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Doubts begin to appear...
IS THIS A STRAIGHT LINE?
2.2
2.0
[I] = 0
1/V
1.8
1.6
1.4
1.2
1.0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
1 / [S]
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Mysterious substrate saturation data
MICHAELIS-MENTEN KINETICS IS NOT SUPPOSED TO SHOW A MAXIMUM !
0.8
[I] = 0
0.7
V (a.u./sec)
Throw these out?
0.6
0.5
0.4
0
20
40
60
80
[S] (M)
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Repeat substrate experiment at higher [S]
SEE IF MAXIMUM HOLDS UP AT HIGHER [S]
1.4
1.2
[I] = 0
0.8
2
0.6
1/V
V (a.u./sec)
1.0
1
0.4
0.2
0
0.0
0.1
0.2
0.3
0.4
80
100
1 / [S]
0.0
0
20
40
60
120
[S] (M)
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Substrate inhibition in LF protease is real
HAS ANYONE ELSE SEEN IT?
Tonello et al. (2003) J. Biol. Chem. 278, 40075-78.
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Rate equation for inhibition by substrate
WHAT DOES THE "BIG BLUE BOOK" SAY?
Segel, I. (1975) Enzyme Kinetics, John Wiley, New York, p. 126
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Rate equation for inhibition by substrate + inhibitor
WHAT DOES THE "BIG BLUE BOOK" SAY?
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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+ I
E.I
initial rate
E+ P
competitive ?
uncompetitive ?
mixed type ?
concentration
computer
DATA
MECHANISMS
Select most plausible model
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From mechanistic to mathematical models
DERIVE A MATHEMATICAL MODEL FROM BIOCHEMICAL IDEAS
k +1
E.S
E+ S
initial rate
k +2
E+ P
k -1
k +3
E.I
E+ I
k -3
MECHANISM
concentration
DATA
v  k 2 [ E ]
k1k3[ S ]
k3 (k1  k 2 )  k3k1[ S ]  k3 (k1  k 2 )[I ]
MATHEMATICAL MODEL
computer
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Problem: Simple mechanisms ...
MERELY FIVE REACTIONS ...
E+A
E.A
+B
E.A.B
E+P
+A
E+B
E.B
•2
•1
reactants (A, B)
product (P)
•5
• 10
reversible reactions
rate constant
"RANDOM BI-UNI" MECHANISM
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... 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
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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
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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)
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Theoretical foundations: Mass Conservation Law
PRODUCTS ARE FORMED WITH THE SAME RATE AS REACTANTS DISAPPEAR
EXAMPLE
A
- d [A] / d t = + d [P] / d t = + d [Q] / d t
P+Q
COMPOSITION RULE
ADDITIVITY OF TERMS FROM SEPARATE REACTIONS
mechanism:
A
B
k1
k2
B
d [B] / d t = + k1 [A] - k2 [B]
C
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Composition Rule: Example
EXAMPLE MECHANISM
k+1
E+ A
EA
RATE EQUATIONS
d[P] / d t = + k+5 [EAB]
k-1
k+2
EAB
EA + B
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+5 [EAB]
k+4
EAB
EB + A
k-4
k+5
EAB
E+ P+ Q
Similarly for other species...
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Program DYNAFIT (1996)
DYNAFIT paper - cumulative citations
400
375
350
300
250
200
150
100
50
0
1997
1999
2001
2003
2005
http://www.biokin.com/dynafit
Kuzmic P. (1996) Anal. Biochem. 237, 260-273.
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A "Kinetic Compiler"
HOW DYNAFIT PROCESSES YOUR BIOCHEMICAL EQUATIONS
k1
E.S
E+S
k3
E+P
k2
Rate terms:
Input (plain text file):
E + S ---> ES
:
k1
k1  [E]  [S]
ES ---> E + S
:
k2
k2  [ES]
ES ---> E + P
:
k3
k3  [ES]
Rate equations:
d[E ] / dt = - k1  [E]  [S]
+ k2  [ES]
+ k3  [ES]
d[ES ] / dt = + k1  [E]  [S]
- k2  [ES]
- k3  [ES]
Similarly for other species...
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System of Simple, Simultaneous Equations
HOW DYNAFIT PROCESSES YOUR BIOCHEMICAL EQUATIONS
k1
E.S
E+S
k3
"The LEGO method"
E+P
k2
of deriving rate equations
Rate terms:
Input (plain text file):
E + S ---> ES
:
k1
k1  [E]  [S]
ES ---> E + S
:
k2
k2  [ES]
ES ---> E + P
:
k3
k3  [ES]
Bio/Chemical Kinetics Made Easy
Rate equations:
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Initial rate kinetics
TWO BASIC APPROXIMATIONS
1. Rapid-Equilibrium Approximation
k1
E.S
E+S
k3
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
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Initial rate kinetics - Traditional approach
DERIVE A MATHEMATICAL MODEL FROM BIOCHEMICAL IDEAS
k +1
E.S
E+ S
initial rate
k +2
E+ P
k -1
k +3
E.I
E+ I
k -3
MECHANISM
Think!
concentration
DATA
v  k 2 [ E ]
k1k3[ S ]
k3 (k1  k 2 )  k3k1[ S ]  k3 (k1  k 2 )[I ]
MATHEMATICAL MODEL
computer
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Initial rate kinetics in DynaFit
GOOD NEWS: MODEL DERIVATION CAN BE FULLY AUTOMATED!
DynaFit input file
MATHEMATICAL MODEL
[task]
task = fit
data = rates
approximation = Steady-State
0 = [E] + [E.A] + [E.B] + [E.A.B] – [E]tot
0 = [A] + [E.A] + [E.A.B] – [A]tot
0 = [B] + [E.B] + [E.A.B] – [B]tot
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]
E + A
<==> E.A
E.A + B <==> E.A.B
E + B
<==> E.B
E.B + A <==> E.A.B
E.A.B
<==> E + P
0 = + k3 [E.A][B] + k7 [E.B][A] + k10 [E][P] – (k4+k8+k9)[E.A.B]
:
:
:
:
:
k1
k3
k5
k7
k9
k2
k4
k6
k8
k10
CRANK!
initial rate
[constants]
...
concentration
DATA
MECHANISM
computer
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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
+A
E+B
E.B
E+P
E + A
<==> E.A
E.A + B <==> E.A.B
E + B
<==> E.B
E.B + A <==> E.A.B
E.A.B
<==> E + P
:
:
:
:
:
k1
k3
k5
k7
k9
k2
k4
k6
k8
k10
[constants]
...
[concentrations]
...
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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
DynaFit model for inhibition by substrate
ENZYME KINETICS MADE EASIER
[reaction]
[enzyme]
[modifiers]
|
|
|
S ---> P
E
I
[mechanism]
E + S <===> E.S
E.S + S <===> E.S.S
E.S ---> E + P
:
:
:
Ks
Ks2
kcat
dissociation
dissociation
...
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DynaFit model for inhibition by substrate + inhibitor
ENZYME KINETICS MADE EASIER
[reaction]
[enzyme]
[modifiers]
|
|
|
S ---> P
E
I
[mechanism]
E +
E.S
E.S
E +
E.S
S <===> E.S
+ S <===> E.S.S
---> E + P
I <===> E.I
+ I <===> E.S.I
:
:
:
:
:
Ks
Ks2
kcat
Ki
Kis
dissoc
dissoc
dissoc
dissoc
[constants]
Ks = 1 ?, Ks2 = 1 ?, kcat = 1 ?
Ki = 1 ?, Kis = 1 ?
optimization flag
...
...
initial estimate
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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]
[enzyme]
[modifiers]
|
|
|
S ---> P
E
I
...
[task]
task = fit | data = rates
model = competitive ?
...
[task]
task = fit | data = rates
model = uncompetitive ?
...
Akaike Information Criterion
Review: Burnham & Anderson (2004)
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The best model: mixed-type noncompetitive
NEOMYCIN B IS NOT A COMPETITIVE INHBITOR OF LETHAL FACTOR PROTEASE
Kuzmic et al. (2006) FEBS J. 273, 3054-3062.
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Direct plot: maximum on dose-response curves
0.8
V (a.u./sec)
0.6
0.4
0.2
0.0
0
20
40
60
80
100
[S] (M)
Kuzmic et al. (2006) FEBS J. 273, 3054-3062.
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Double-reciprocal plot is nonlinear
8
1/V
6
4
2
0
0.00
0.02
0.04
0.06
0.08
0.10
1 / [S]
Kuzmic et al. (2006) FEBS J. 273, 3054-3062.
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DR plot obscures deviations from the model
8
1/V
6
4
2
0
0.00
0.02
0.04
0.06
0.08
0.10
1 / [S]
Kuzmic et al. (2006) FEBS J. 273, 3054-3062.
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Direct plot makes model departures more visible
0.8
V (a.u./sec)
0.6
0.4
0.2
0.0
0
20
40
60
80
[S] (M)
Kuzmic et al. (2006) FEBS J. 273, 3054-3062.
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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
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Acknowledgements: Lethal Factor protease work
Hawaii Biotech
currently
Panthera BioPharma
National Institutes of Health
Grant No. R43 AI52587-02
U.S. Army Medical Research and Materials Command
Contract No. V549P-6073
Mark Goldman
Sheri Millis
Lynne Cregar
Aiea, Island of Oahu, Hawaii
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