Anthrax bacillus CUTANEOUS AND INHALATION ANTHRAX DISEASE ier Bio/ Chem ical Kinetics Made Easy A Numerical Approach Petr Kuzmi č , Ph.D. BioKin, Ltd. 1. Case study: Inhibition of LF protease from B. a nthracis 2. Method: Num erical Enzym e Kinetics Bio/Chemical Kinetics Made Easy 2 Lethal Factor (LF) protease from B. anthracis Neomycin B: an aminoglycoside inhibitor CLEAVES MITOGEN ACTIVATED PROTEIN KINASE KINASE ( MAPKK ) PRESUMABLY A "COMPETITIVE" INHIBITOR OF LF PROTEASE I nhibitor? � � Fridman et al. (2004) Angew. Chem. Int. Ed. Eng. 4 4 , 447-452 Bio/Chemical Kinetics Made Easy 3 Bio/Chemical Kinetics Made Easy 4 Competitive inhibition - Possible mechanisms Competitive inhibition - Kinetics MUTUALLY EXCLUSIVE BINDING TO ENZYME 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] Segel, I. (1975) Enzyme Kinetics , John Wiley, New York, p. 102 Bio/Chemical Kinetics Made Easy 5 Bio/Chemical Kinetics Made Easy 6 1
Non-competitive inhibition - A possible mechanism Non-competitive inhibition - Kinetics NON-EXCLUSIVE BINDING, BUT TERNARY COMPLEX HAS NO CATALYTIC ACTIVITY EVEN AT VERY HIGH [SUBSTRATE], ANZYME ACTIVITY IS NEVER FULLY RESTORED 1.0 0.8 enzyme activity increase [I] 0.6 0.4 0.2 0.0 -3 -2 -1 0 1 2 3 log [S] Segel, I. (1975) Enzyme Kinetics , John Wiley, New York, p. 126 Bio/Chemical Kinetics Made Easy 7 Bio/Chemical Kinetics Made Easy 8 Compare saturation curves Compare "double-reciprocal" plots DIAGNOSIS OF MECHANISMS: SAME OR DIFFERENT RATE AT VERY LARGE [S]? DIAGNOSIS OF MECHANISMS: STRAIGHT LINES INTERCEPT ON VERTICAL AXIS? COMPETI TI VE NON-COMPETI TI VE COMPETI TI VE NON-COMPETI TI VE 1.0 1.0 20 30 [I] = 0 [I] = 0 25 [I] = 1 [I] = 1 0.8 0.8 [I] = 2 [I] = 2 ? 15 [I] = 4 [I] = 4 [I] = 8 [I] = 8 20 0.6 0.6 1 / activity 1 / activity activity activity 10 15 0.4 0.4 10 5 0.2 0.2 5 0.0 0.0 0 0 0 2 4 6 8 10 0 2 4 6 8 10 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 [S] [S] 1 / [S] 1 / [S] Bio/Chemical Kinetics Made Easy 9 Bio/Chemical Kinetics Made Easy 10 Traditional plan to determine inhibition mechanism Collect experimental data at varied [S] and [I] THE TRADITIONAL APPROACH THE RAW DATA 1. Measure enzyme activity at increasing [S] Collect multiple substrate-saturation curves at varied [I] 0.8 2. Convert [S] vs. activity data to double-reciprocal coordinates [I] = 0 3. Perform a linear fit of transformed (double-reciprocal) data [I] = 0.5 μ M 0.6 V (a.u./sec) 4. Check if resulting straight lines intersect on the vertical axis [I] = 1.0 μ M 0.4 [I] = 2.0 μ M If yes, declare the inhibition mechanism com petitive 0.2 0.0 0 20 40 60 80 [S] ( μ M) Fridman et al. (2004) Angew. Chem. Int. Ed. Eng. 4 4 , 447-452 Bio/Chemical Kinetics Made Easy 11 Bio/Chemical Kinetics Made Easy 12 2
Check for intersection of double-reciprocal plots Doubts begin to appear... DO LINEWEAVER-BURK PLOTS INTERSECT? I S THI S A STRAI GHT LI NE? 12 2.2 10 2.0 [I] = 0 [I] = 0 [I] = 0.5 μ M 8 1.8 [I] = 1.0 μ M 1 / V 1 / V 6 1.6 [I] = 2.0 μ M 4 1.4 � 2 1.2 COMPETI TI VE 0 1.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.00 0.02 0.04 0.06 0.08 0.10 0.12 1 / [S] 1 / [S] Bio/Chemical Kinetics Made Easy 13 Bio/Chemical Kinetics Made Easy 14 Mysterious substrate saturation data Repeat substrate experiment at higher [S] SEE IF MAXIMUM HOLDS UP AT HIGHER [S] MI CHAELI S-MENTEN KI NETI CS I S NOT SUPPOSED TO SHOW A MAXI MUM ! 1.4 0.8 1.2 [I] = 0 [I] = 0 0.7 1.0 Throw these out? V (a.u./sec) V (a.u./sec) 0.8 0.6 2 0.6 1 / V 0.5 1 0.4 0.2 0 0.0 0.1 0.2 0.3 0.4 1 / [S] 0.4 0 20 40 60 80 0.0 0 20 40 60 80 100 120 [S] ( μ M) [S] ( μ M) Bio/Chemical Kinetics Made Easy 15 Bio/Chemical Kinetics Made Easy 16 Substrate inhibition in LF protease is real Rate equation for inhibition by substrate HAS ANYONE ELSE SEEN IT? WHAT DOES THE "BIG BLUE BOOK" SAY? Tonello et al. (2003) J. Biol. Chem. 2 7 8 , 40075-78. Segel, I. (1975) Enzyme Kinetics , John Wiley, New York, p. 126 Bio/Chemical Kinetics Made Easy 17 Bio/Chemical Kinetics Made Easy 18 3
Rate equation for inhibition by substrate + inhibitor WHAT DOES THE "BIG BLUE BOOK" SAY? ier Bio/ Chem ical Kinetics Made Easy A Numerical Approach Petr Kuzmi č , Ph.D. BioKin, Ltd. ? 1. Case study: Inhibition LF protease from B. a nthracis 2. Method: Num erical Enzym e Kinetics Bio/Chemical Kinetics Made Easy 19 The task of mechanistic enzyme kinetics From mechanistic to mathematical models SELECT AMONG MULTIPLE CANDIDATE MECHANISMS DERIVE A MATHEMATICAL MODEL FROM BIOCHEMICAL IDEAS k +1 initial rate k +2 E + S E.S E + P initial rate k -1 E + S E.S E + P E + I E.I k +3 E + I E.I k -3 competitive ? competitive ? MECHANISM concentration uncompetitive ? concentration DATA mixed type ? k k [ S ] computer DATA = + − v k [ E ] 1 3 + 2 + + + + k ( k k ) k k [ S ] k ( k k )[ I ] MECHANISMS − − + − + + − + 3 1 2 3 1 3 1 2 MATHEMATICAL MODEL Select most plausible model computer Bio/Chemical Kinetics Made Easy 21 Bio/Chemical Kinetics Made Easy 22 Problem: Simple mechanisms ... ... lead to complex algebraic models MERELY FIVE REACTIONS ... MERELY FIVE REACTIONS ... Segel, I. (1975) Enzyme Kinetics . John Wiley, New York, p. 646. E + A E. A • 2 reactants (A, B) + B • 1 product (P) E. A . B E + P E. A E + A + A + B • 5 reversible reactions E. A . B E + P E. B E + B • 10 rate constant + A E + B E. B "RANDOM BI -UNI " MECHANISM "RANDOM BI - UNI " MECHANISM Bio/Chemical Kinetics Made Easy 23 Bio/Chemical Kinetics Made Easy 24 4
A solution: Forget about algebra Theoretical foundations: Mass Action Law POSSIBLE STRATEGY FOR MECHANISTIC MODEL BUILDING RATE IS PROPORTIONAL TO CONCENTRATION(S) “rate” … “derivative” MONOMOLECULAR REACTIONS A products rate is proportional to [A] - d [ A ] / d t = k [ A ] • Do not even try to derive complex algebraic equations • Instead, derive systems of simple, simultaneous equations monomolecular rate constant 1 / time • Solve these systems using numerical methods 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 25 Bio/Chemical Kinetics Made Easy 26 Theoretical foundations: Mass Conservation Law Composition Rule: Example EXAMPLE MECHANISM RATE EQUATIONS PRODUCTS ARE FORMED WITH THE SAME RATE AS REACTANTS DISAPPEAR EXAMPLE k + 1 d[ P ] / d t = + k +5 [ EAB ] E + A EA - d [ A ] / d t = + d [ P ] / d t = + d [ Q ] / d t k -1 A P + Q k + 2 + k +2 [ EA ] × [ B ] d[ EAB ] / d t = EAB EA + B k -2 - k -2 [ EAB ] + k +4 [ EB ] × [ A ] k + 3 E + B EB COMPOSITION RULE ADDITIVITY OF TERMS FROM SEPARATE REACTIONS - k -4 [ EAB ] k -3 - k +5 [ EAB ] k + 4 mechanism: EB + A EAB k 1 k -4 d [ B ] / d t = + k 1 [A] - k 2 [B] A B k + 5 k 2 Similarly for other species... EAB E + P + Q B C Bio/Chemical Kinetics Made Easy 27 Bio/Chemical Kinetics Made Easy 28 Program DYNAFIT (1996) DYNAFIT paper - cumulative citations A "Kinetic Compiler" 400 HOW DYNAFIT PROCESSES YOUR BIOCHEMICAL EQUATIONS 375 350 k 1 300 k 3 E. S E + S E + P 250 k 2 200 150 Rate terms: Rate equations: Input (plain text file): 100 50 d[ E ] / d t = - k 1 × [E] × [S] 0 + k 2 × [ES] 1997 1999 2001 2003 2005 k 1 × [E] × [S] E + S ---> ES : k1 + k 3 × [ES] http://www. biokin.com/dynafit k 2 × [ES] d[ ES ] / d t = + k 1 × [E] × [S] ES ---> E + S : k2 - k 2 × [ES] - k 3 × [ES] ES ---> E + P k 3 × [ES] : k3 Similarly for other species... Kuzmic P. (1996) Anal. Biochem. 2 3 7 , 260-273. Bio/Chemical Kinetics Made Easy 29 Bio/Chemical Kinetics Made Easy 30 5
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