1 Non-competitive inhibition - A possible mechanism Non-competitive - - PDF document

1

Bio/ Chem ical Kinetics Made Easy

A Numerical Approach

Petr Kuzmič, Ph.D.

BioKin, Ltd.

• 1. Case study: Inhibition of LF protease from B. anthracis
• 2. Method: Num erical Enzym e Kinetics

ier

Bio/Chemical Kinetics Made Easy 2

Anthrax bacillus

CUTANEOUS AND INHALATION ANTHRAX DISEASE Bio/Chemical Kinetics Made Easy 3

Lethal Factor (LF) protease from B. anthracis

CLEAVES MITOGEN ACTIVATED PROTEIN KINASE KINASE (MAPKK)

I nhibitor?

Bio/Chemical Kinetics Made Easy 4

Neomycin B: an aminoglycoside inhibitor

PRESUMABLY A "COMPETITIVE" INHIBITOR OF LF PROTEASE

Fridman et al. (2004) Angew. Chem. Int. Ed. Eng. 4 4 , 447-452

• Bio/Chemical Kinetics Made Easy

5

Competitive inhibition - Possible mechanisms

Segel, I. (1975) Enzyme Kinetics, John Wiley, New York, p. 102

MUTUALLY EXCLUSIVE BINDING TO ENZYME Bio/Chemical Kinetics Made Easy 6

Competitive inhibition - Kinetics

AT VERY HIGH [SUBSTRATE], ANZYME ACTIVITY IS COMPLETELY RESTORED

log [S]

• 3
• 2
• 1

1 2 3

enzyme activity

0.0 0.2 0.4 0.6 0.8 1.0 [I] = 0 [I] = 1 [I] = 2 [I] = 4 [I] = 8 [I] = 16

same V∞ ! increase [I]

2

Bio/Chemical Kinetics Made Easy 7

Non-competitive inhibition - A possible mechanism

Segel, I. (1975) Enzyme Kinetics, John Wiley, New York, p. 126

NON-EXCLUSIVE BINDING, BUT TERNARY COMPLEX HAS NO CATALYTIC ACTIVITY Bio/Chemical Kinetics Made Easy 8

Non-competitive inhibition - Kinetics

EVEN AT VERY HIGH [SUBSTRATE], ANZYME ACTIVITY IS NEVER FULLY RESTORED

log [S]

• 3
• 2
• 1

1 2 3

enzyme activity

0.0 0.2 0.4 0.6 0.8 1.0

increase [I]

Bio/Chemical Kinetics Made Easy 9

Compare saturation curves

DIAGNOSIS OF MECHANISMS: SAME OR DIFFERENT RATE AT VERY LARGE [S]?

[S]

2 4 6 8 10

activity

0.0 0.2 0.4 0.6 0.8 1.0

[S]

2 4 6 8 10

activity

0.0 0.2 0.4 0.6 0.8 1.0

?

COMPETI TI VE NON-COMPETI TI VE

Bio/Chemical Kinetics Made Easy 10

Compare "double-reciprocal" plots

DIAGNOSIS OF MECHANISMS: STRAIGHT LINES INTERCEPT ON VERTICAL AXIS?

1 / [S]

0.0 0.5 1.0 1.5 2.0

1 / activity

5 10 15 20 25 30 [I] = 0 [I] = 1 [I] = 2 [I] = 4 [I] = 8

1 / [S]

0.0 0.5 1.0 1.5 2.0

1 / activity

5 10 15 20 [I] = 0 [I] = 1 [I] = 2 [I] = 4 [I] = 8

COMPETI TI VE NON-COMPETI TI VE

Bio/Chemical Kinetics Made Easy 11

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 com petitive

Fridman et al. (2004) Angew. Chem. Int. Ed. Eng. 4 4 , 447-452

Bio/Chemical Kinetics Made Easy 12

Collect experimental data at varied [S] and [I]

THE RAW DATA

[S] (μM)

20 40 60 80

V (a.u./sec)

0.0 0.2 0.4 0.6 0.8

[I] = 0 [I] = 0.5 μM [I] = 1.0 μM [I] = 2.0 μM

3

Bio/Chemical Kinetics Made Easy 13

Check for intersection of double-reciprocal plots

[I] = 0 [I] = 0.5 μM [I] = 1.0 μM [I] = 2.0 μM

1 / [S]

0.00 0.02 0.04 0.06 0.08 0.10 0.12

1 / V

2 4 6 8 10 12

DO LINEWEAVER-BURK PLOTS INTERSECT?

• COMPETI TI VE

Bio/Chemical Kinetics Made Easy 14

Doubts begin to appear...

[I] = 0

I S THI S A STRAI GHT LI NE?

1 / [S]

0.00 0.02 0.04 0.06 0.08 0.10 0.12

1 / V

1.0 1.2 1.4 1.6 1.8 2.0 2.2 Bio/Chemical Kinetics Made Easy 15

Mysterious substrate saturation data

[I] = 0

MI CHAELI S-MENTEN KI NETI CS I S NOT SUPPOSED TO SHOW A MAXI MUM !

[S] (μM)

20 40 60 80

V (a.u./sec)

0.4 0.5 0.6 0.7 0.8

Throw these out? Bio/Chemical Kinetics Made Easy 16

Repeat substrate experiment at higher [S]

[I] = 0

SEE IF MAXIMUM HOLDS UP AT HIGHER [S] [S] (μM)

20 40 60 80 100 120

V (a.u./sec)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

1 / [S]

0.0 0.1 0.2 0.3 0.4

1 / V

1 2

Bio/Chemical Kinetics Made Easy 17

Substrate inhibition in LF protease is real

HAS ANYONE ELSE SEEN IT?

Tonello et al. (2003) J. Biol. Chem. 2 7 8 , 40075-78.

Bio/Chemical Kinetics Made Easy 18

Rate equation for inhibition by substrate

WHAT DOES THE "BIG BLUE BOOK" SAY?

Segel, I. (1975) Enzyme Kinetics, John Wiley, New York, p. 126

4

Bio/Chemical Kinetics Made Easy 19

Rate equation for inhibition by substrate + inhibitor

WHAT DOES THE "BIG BLUE BOOK" SAY?

?

Bio/ Chem ical Kinetics Made Easy

A Numerical Approach

Petr Kuzmič, Ph.D.

BioKin, Ltd.

ier

• 1. Case study: Inhibition LF protease from B. anthracis
• 2. Method: Num erical Enzym e Kinetics

Bio/Chemical Kinetics Made Easy 21

The task of mechanistic enzyme kinetics

SELECT AMONG MULTIPLE CANDIDATE MECHANISMS

concentration initial rate

DATA

computer Select most plausible model

MECHANISMS

competitive ?

E + S E.S E + P E + I E.I

uncompetitive ? mixed type ? competitive ?

Bio/Chemical Kinetics Made Easy 22

From mechanistic to mathematical models

DERIVE A MATHEMATICAL MODEL FROM BIOCHEMICAL IDEAS

concentration initial rate

DATA

computer

MATHEMATICAL MODEL E + S E.S E + P E + I E.I k +1 k -1 k +2 k +3 k -3

] )[ ( ] [ ) ( ] [ ] [

2 1 3 1 3 2 1 3 3 1 2

I k k k S k k k k k S k k E k v

+ − + + − + − − − + +

+ + + + =

MECHANISM Bio/Chemical Kinetics Made Easy 23

Problem: Simple mechanisms ...

MERELY FIVE REACTIONS ...

• 2

reactants (A, B)

• 1

product (P)

• 5

reversible reactions

• 10

rate constant

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

"RANDOM BI -UNI " MECHANISM Bio/Chemical Kinetics Made Easy 24

... lead to complex algebraic models

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

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

"RANDOM BI - UNI " MECHANISM

MERELY FIVE REACTIONS ...

5

Bio/Chemical Kinetics Made Easy 25

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 26

Theoretical foundations: Mass Action Law

RATE IS PROPORTIONAL TO CONCENTRATION(S)

A products MONOMOLECULAR REACTIONS rate is proportional to [A] A + B products BIMOLECULAR REACTIONS rate is proportional to [A] × [B]

• d [A] / d t = k [A]

monomolecular rate constant 1 / time

• d [A] / d t = - d [B] / d t = k [A] × [B]

bimolecular rate constant 1 / (concentration × time)

“rate” … “derivative”

Bio/Chemical Kinetics Made Easy 27

Theoretical foundations: Mass Conservation Law

PRODUCTS ARE FORMED WITH THE SAME RATE AS REACTANTS DISAPPEAR

• d [A] / d t =

A P + Q EXAMPLE COMPOSITION RULE ADDITIVITY OF TERMS FROM SEPARATE REACTIONS mechanism: d [B] / d t = A B B C k1 k2 + d [P] / d t = + d [Q] / d t

• k2 [B]

+ k1 [A]

Bio/Chemical Kinetics Made Easy 28

Composition Rule: Example

E + A k+1 k-1 EA EA + B k+2 k-2 EAB E + B k+3 k-3 EB EB + A k+4 k-4 EAB EAB k+5 E + P + Q EXAMPLE MECHANISM RATE EQUATIONS d[P] / d t = d[EAB] / d t = Similarly for other species... + k+5 [EAB]

• k+5 [EAB]

+ k+2 [EA]×[B]

• k-2 [EAB]

+ k+4 [EB]×[A]

• k-4 [EAB]

Bio/Chemical Kinetics Made Easy 29

Program DYNAFIT (1996)

http://www.biokin.com/dynafit

Kuzmic P. (1996) Anal. Biochem. 2 3 7 , 260-273.

50 100 150 200 250 300 350 400 1997 1999 2001 2003 2005 DYNAFIT paper - cumulative citations

375

Bio/Chemical Kinetics Made Easy 30

A "Kinetic Compiler"

E + S ---> ES : k1 ES ---> E + S : k2 ES ---> E + P : k3 Input (plain text file): d[E ] / dt = - k1 × [E] × [S]

HOW DYNAFIT PROCESSES YOUR BIOCHEMICAL EQUATIONS

E + S E.S E + P k1 k2 k3 k1 × [E] × [S] k2 × [ES] k3 × [ES] Rate terms: Rate equations: + k2 × [ES] + k3 × [ES] d[ES ] / dt = + k1 × [E] × [S]

• k2 × [ES]
• k3 × [ES]

Similarly for other species...

6

Bio/Chemical Kinetics Made Easy 31

System of Simple, Simultaneous Equations

E + S ---> ES : k1 ES ---> E + S : k2 ES ---> E + P : k3 Input (plain text file):

HOW DYNAFIT PROCESSES YOUR BIOCHEMICAL EQUATIONS

E + S E.S E + P k1 k2 k3 k1 × [E] × [S] k2 × [ES] k3 × [ES] Rate terms: Rate equations: "The LEGO method"

• f deriving rate equations

Bio/Chemical Kinetics Made Easy 32

Initial rate kinetics

TWO BASIC APPROXIMATIONS

• 1. Rapid-Equilibrium Approximation
• 2. Steady-State Approximation

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

assumed very much slow er than k1, k2

• no assumptions made about relative magnitude of k1, k2, k3
• concentrations of enzyme forms are unchanging

New in DynaFit Bio/Chemical Kinetics Made Easy 33

Initial rate kinetics - Traditional approach

DERIVE A MATHEMATICAL MODEL FROM BIOCHEMICAL IDEAS

concentration initial rate

DATA

computer

MATHEMATI CAL MODEL E + S E.S E + P E + I E.I k +1 k -1 k +2 k +3 k -3

] )[ ( ] [ ) ( ] [ ] [

2 1 3 1 3 2 1 3 3 1 2

I k k k S k k k k k S k k E k v

+ − + + − + − − − + +

+ + + + =

MECHANI SM

Think!

Bio/Chemical Kinetics Made Easy 34

Initial rate kinetics in DynaFit

GOOD NEWS: MODEL DERIVATION CAN BE FULLY AUTOMATED!

[task] task = fit data = rates approximation = Steady-State [mechanism] E + A <==> E.A : k1 k2 E.A + B <==> E.A.B : k3 k4 E + B <==> E.B : k5 k6 E.B + A <==> E.A.B : k7 k8 E.A.B <==> E + P : k9 k10 [constants] ...

DynaFit input file

computer concentration initial rate

MATHEMATI CAL MODEL MECHANI SM DATA

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

CRANK!

Bio/Chemical Kinetics Made Easy 35

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

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

Bio/ Chem ical Kinetics Made Easy

A Numerical Approach

Petr Kuzmič, Ph.D.

BioKin, Ltd.

ier

• 1. Case study: Inhibition LF protease from B. anthracis
• 2. Method: Num erical Enzym e Kinetics

7

Bio/Chemical Kinetics Made Easy 37

DynaFit model for inhibition by substrate

ENZYME KINETICS MADE EASI ER

[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 38

DynaFit model for inhibition by substrate + inhibitor

ENZYME KINETICS MADE EASI ER

[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 ? Ki = 1 ?, Kis = 1 ? ... ...

initial estimate

• ptimization flag

Bio/Chemical Kinetics Made Easy 39

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 I nform ation Criterion

Review: Burnham & Anderson (2004)

Bio/Chemical Kinetics Made Easy 40

The best model: mixed-type noncompetitive

NEOMYCIN B IS NOT A COMPETITIVE INHBITOR OF LETHAL FACTOR PROTEASE

Kuzmic et al. (2006) FEBS J. 2 7 3 , 3054-3062.

Bio/Chemical Kinetics Made Easy 41

Direct plot: maximum on dose-response curves

Kuzmic et al. (2006) FEBS J. 2 7 3 , 3054-3062.

[S] (μM)

20 40 60 80 100

V (a.u./sec)

0.0 0.2 0.4 0.6 0.8

Bio/Chemical Kinetics Made Easy 42

Double-reciprocal plot is nonlinear

Kuzmic et al. (2006) FEBS J. 2 7 3 , 3054-3062.

1 / [S]

0.00 0.02 0.04 0.06 0.08 0.10

1 / V

2 4 6 8

8

Bio/Chemical Kinetics Made Easy 43

DR plot obscures deviations from the model

Kuzmic et al. (2006) FEBS J. 2 7 3 , 3054-3062.

1 / [S]

0.00 0.02 0.04 0.06 0.08 0.10

1 / V

2 4 6 8

Bio/Chemical Kinetics Made Easy 44

Direct plot makes model departures more visible

Kuzmic et al. (2006) FEBS J. 2 7 3 , 3054-3062.

[S] (μM)

20 40 60 80

V (a.u./sec)

0.0 0.2 0.4 0.6 0.8

Bio/Chemical Kinetics Made Easy 45

Summary: Enzyme kinetics made (almost) easy

HOW DO I BUI LD A MATHEMATI CAL MODEL FOR AN ENZYME MECHANI SM?

• Let the com puter 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:
• m ass action law
• m ass 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 46

Acknowledgements: Lethal Factor protease work

Haw aii Biotech

currently

Panthera BioPharma

Mark Goldman Sheri Millis Lynne Cregar

Aiea, Island of Oahu, Hawaii National Institutes of Health Grant No. R43 AI52587-02 U.S. Army Medical Research and Materials Command Contract No. V549P-6073