Biochemical Kinetics Made Easier Petr Kuzmič Ph D

Biochemical Kinetics Made Easier Petr Kuzmič, Ph. D. Bio. Kin, Ltd. 1. Theory: differential equations - DYNAFIT software 2. Example I: Initial rate experiment - p 56 lck kinase / “ATP analog” inhibitor 3. Example II: Time course experiment - p 38 a kinase / desatinib / competitive ligand displacement Bio/Chemical Kinetics Made Easy

The task of mechanistic enzyme kinetics SELECT AMONG MULTIPLE CANDIDATE MECHANISMS initial rate competitive ? uncompetitive ? mixed type ? concentration computer DATA MECHANISMS Select most plausible model Bio/Chemical Kinetics Made Easy 2

From mechanistic to mathematical models DERIVE A MATHEMATICAL MODEL FROM BIOCHEMICAL IDEAS initial rate MECHANISM concentration DATA MATHEMATICAL MODEL computer Bio/Chemical Kinetics Made Easy 3

Problem: Simple mechanisms. . . MERELY FIVE REACTIONS. . . • 2 • 1 reactants (A, B) product (P) • 5 • 10 reversible reactions rate constant "RANDOM BI-UNI" MECHANISM Bio/Chemical Kinetics Made Easy 4

. . . lead to complex algebraic models MERELY FIVE REACTIONS. . . Segel, I. (1975) Enzyme Kinetics. John Wiley, New York, p. 646. "RANDOM BI-UNI" MECHANISM Bio/Chemical Kinetics Made Easy 5

New approach: Numerical Enzyme Kinetics NO MORE ALGEBRA: LET THE COMPUTER DEAL WITH IT ! Bio/Chemical Kinetics Made Easy 6

Theoretical foundations: Mass Action Law RATE IS PROPORTIONAL TO CONCENTRATION(S) MONOMOLECULAR REACTIONS rate is proportional to [A] - d [A] / d t = k [A] BIMOLECULAR REACTIONS rate is proportional to [A] [B] - d [A] / d t = - d [B] / d t = k [A] [B] Bio/Chemical Kinetics Made Easy 7

Theoretical foundations: Mass Conservation Law PRODUCTS ARE FORMED WITH THE SAME RATE AS REACTANTS DISAPPEAR EXAMPLE - d[ A]/dt = + d[ P]/dt = + d[ Q]/dt COMPOSITION RULE ADDITIVITY OF TERMS FROM SEPARATE REACTIONS mechanism: d [B] / d t = + k 1 [A] - k 2 [B] Bio/Chemical Kinetics Made Easy 8

Program DYNAFIT REFERENCES 1. Kuzmic P. (1996) Anal. Biochem. 237, 260 -273. “Program DYNAFIT for the analysis of enzyme kinetic data” 2. Kuzmic P. (2009) Methods in Enzymology, in press “DYNAFIT – A software package for enzymology” Ref. [1] – total citations: FREE TO ACADEMIC USERS www. biokin. com Bio/Chemical Kinetics Made Easy 9

Initial rate kinetics TWO BASIC APPROXIMATIONS 1. Rapid-Equilibrium Approximation assumed very much slower than k 1, k 2 2. Steady-State Approximation New in Dyna. Fit Mathematical details in BBA – Proteins & Proteomics, submitted • no assumptions made about relative magnitude of k 1, k 2, k 3 Bio/Chemical Kinetics Made Easy 10

Initial rate kinetics - Traditional approach DERIVE A MATHEMATICAL MODEL FROM BIOCHEMICAL IDEAS initial rate MECHANISM derive equations concentration DATA MATHEMATICAL MODEL computer Bio/Chemical Kinetics Made Easy 11

Initial rate kinetics in Dyna. Fit GOOD NEWS: MODEL DERIVATION CAN BE FULLY AUTOMATED! Dyna. Fit 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. B] – [A]tot 0 = [B] + [E. A. B] – [B]tot 0 = + k 1[E][A] – k 2[E. A] – k 3 [E. A][B] + k 4 [E. A. B] 0 = + k 5[E][B] – k 6[E. B] – k 7 [E. B][A] + k 8 [E. A. B] [mechanism] E + A <==> E. A + B <==> E. A. B E + B <==> E. B + A <==> E. A. B <==> E + P : : : k 1 k 3 k 5 k 7 k 9 k 2 k 4 k 6 k 8 k 10 push button 0 = + k 3 [E. A][B] + k 7 [E. B][A] + k 10 [E][P] – (k 4+k 8+k 9)[E. A. B] initial rate [constants]. . . concentration DATA MECHANISM computer Bio/Chemical Kinetics Made Easy 12

Initial rate kinetics in Dyna. Fit 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 + B <==> E. B + A <==> E. A. B <==> E + P : : : k 1 k 3 k 5 k 7 k 9 k 2 k 4 k 6 k 8 k 10 [constants]. . . [concentrations]. . . Bio/Chemical Kinetics Made Easy 13

Biochemical Kinetics Made Easier Dyna. Fit applications to protein kinases Case study #1: INITIAL RATES OF ENZYME REACTIONS inhibition constants and kinetic mechanism Bio/Chemical Kinetics Made Easy

WIN-61651: Presumably an ATP analog? TRADITIONAL STUDY: KINASE INHIBITOR ‘WIN-61651’ IS COMPETITIVE WITH ATP Faltynek et al. (1995) J. Enz. Inhib. 9, 111 -122. Bio/Chemical Kinetics Made Easy 15

Lineweaver-Burk plots for WIN-61651 LINEWEAVER-BURK PLOTS AT VARIED [PEPTIDE] AND FIXED [ATP] ARE NONLINEAR Faltynek et al. (1995) J. Enz. Inhib. 9, 111 -122. [I] = 0 [I] = 80 m. M Bio/Chemical Kinetics Made Easy 16

Direct plot for WIN-61651: Initial rate vs. [peptide] MIXED-TYPE INHIBITION MECHANISM: WHICH IS SMALLER, Kis or Kii? Faltynek et al. (1995) J. Enz. Inhib. 9, 111 -122. – FIGURE 1 B [mechanism] E + S <===> ES ES ---> E + P E + I <===> EI ES + I <===> ESI Bio/Chemical Kinetics Made Easy 17

Adding a substrate inhibition term improves fit GLOBAL NUMERICAL FIT IS BOTH MORE PRECISE AND MORE ACCURATE [I] = 0 [mechanism] E + S <===> ES ES ---> E + P ES + S <===> ES 2 E + I <===> EI ES + I <===> ESI Bio/Chemical Kinetics Made Easy 18

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) Bio/Chemical Kinetics Made Easy 19

WIN-61651 summary: Comparison of methods WIN-61651 IS A MIXED-TYPE INHIBITOR, NOT COMPETITIVE WITH ATP parameter (m. M) competitive: uncompetitive: Dyna. Fit Faltynek et al. (1995) Ks Ks 2 Kis Kii 9100 3700 990 140 1100 450 — 28 2 18 4 14 5 67 18 residual squares 2. 1 Bio/Chemical Kinetics Made Easy 19. 5 20

Biochemical Kinetics Made Easier Dyna. Fit applications to protein kinases Case study #2: REACTION PROGRESS rate constants for kinase-inhibitor interactions competitive ligand displacement FRET assay Preliminary experimental data: Bryan Marks, Invitrogen (life Technologies) Bio/Chemical Kinetics Made Easy

Kinase – Antibody – Tracer – Inhibitor assay A FOUR-COMPONENT MIXTURE 1 2 3 4 Bio/Chemical Kinetics Made Easy 22

Kinase – Antibody – Tracer – Inhibitor: mechanism PURPOSE: OBTAIN RATE CONSTANTS FOR INHIBITOR ASSOCIATION & DISSOCIATION E A T I . . . enzyme antibody (FRET donor) tracer (FRET acceptor) inhibitor • four components • five complexes (3 binary, 2 ternary) • six unique rate constants Bio/Chemical Kinetics Made Easy 23

Rate constants and receptor-ligand residence time IS IT WORTH CHASING AFTER RATE CONSTANTS? Mbalaviele et al. (2009) J. Pharm. Exp. Ther. 329, 14 -25 “PHA-408 is an ATP competitive inhibitor, which binds IKK-2 tightly with a relatively slow off rate. ” Puttini et al. (2008) haematologica 93, 653 -61 “The present results suggest a slower off-rate (dissociation rate) of [a novel Abl kinase inhibitor] compared to imatinib as an explanation for the increased cellular activity of the former. ” Tummino & Copeland (2008) Biochemistry 47, 5481 -92 “. . . the extent and duration of responses to receptor-ligand interactions depend greatly on the time period over which the ligand is in residence on its receptor. ” Bio/Chemical Kinetics Made Easy 24

Kinase - Antibody - Tracer - Inhibitor: data KINASE: p 38 a | ANTIBODY: anti-GST | TRACER: Invitrogen “Tracer-199” | INHIBITOR: desatinib Data: Bryan Marks, Invitrogen EXPERIMENT: 1. 0 incubate [E] = 4 n. M [Ab] = 40 n. M [In] = varied 56 167 500 n. M [In] 30 minutes 2. dilute 1: 20 with Tracer final concentrations [E] [Ab] [Tr] [In] = 0. 2 n. M = 100 n. M = varied Bio/Chemical Kinetics Made Easy 25

Kinase - Antibody - Tracer - Inhibitor: fitting model AUTOMATICALLY DERIVED BY DYNAFIT system of simultaneous ordinary differential equations [mechanism] Dyna. Fit Input d[E]/dt = - ka. I[E][In] + kd. I[E. In] - ka. T[E][Tr] + kd. T[E. Tr] - ka. A[E][Ab] + kd. A[E. Ab] d[In]/dt = - ka. I[E][In] + kd. I[E. In] - ka. I[E. Ab][In] + kd. I[E. In. Ab] E + In <===> E. In : ka. I kd. I d[E. In]/dt Tr ka. I[E][In] - kd. I[E. In] - ka. A[E. In][Ab] +: d. A[E. In. Ab] k E + = + <===> E. Tr ka. T kd. T d[Tr]/dt = - ka. T[E][Tr] + kd. T[E. Tr] - ka. T[E. Ab][Tr] + kd. T[E. Tr. Ab] E + Ab <===> E. Ab : ka. A kd. A d[E. Tr]/dt = + ka. T[E][Tr] - kd. T[E. Tr] - ka. A[E. Tr][Ab] + kd. A[E. Tr. Ab] d[Ab]/dt = - ka. A[E][Ab] + kd. A[E. Ab] - ka. A[E. In][Ab] + kd. A[E. In. Ab] - ka. A[E. Tr][Ab] + kd. A[E. Tr. Ab] E. In + Ab <===> E. In. Ab : ka. A kd. A d[E. Ab]/dt = + ka. A[E][Ab] - kd. A[E. Ab] - ka. I[E. Ab][In] + kd. I[E. In. Ab] - ka. T[E. Ab][Tr] + kd. T[E. Tr. Ab] E. Ab + + k <===> k [E. In. Ab] ka. I kd. I d[E. In. Ab]/dt = Ina. A[E. In][Ab] -E. In. Ab + ka. I: [E. Ab][In] - kd. I[E. In. Ab] d. A E. Tr + + k <===> - kd. A[E. Tr. Ab] : ka. A kd. A d[E. Tr. Ab]/dt = Aba. A[E. Tr][Ab] E. Tr. Ab + ka. T[E. Ab][Tr] - kd. T[E. Tr. Ab] E. Ab + Tr <===> E. Tr. Ab : ka. T Bio/Chemical Kinetics Made Easy kd. T 26

Kinase - Antibody - Tracer - Inhibitor: rate constants ASSUMPTION: INDEPDENT BINDING SITES – ONLY TWO ADDITIONAL RATE CONSTANTS DATA + MODEL LEAST-SQUARES FIT ka. I = 2. 1 109 kd. I = 19 M-1. s-1 PARAMETERS “RESIDENCE TIME” t = 0. 05 sec Bio/Chemical Kinetics Made Easy 27

Kinase - Antibody - Tracer - Inhibitor: state variables EVOLUTION OF SPECIES CONCENTRATIONS DURING THE KINETIC EXPERIMENT: 1. incubate [E] = 4 n. M [Ab] = 40 n. M [In] = 370 n. M 30 minutes 2. dilute 1: 20 with Tracer final concentrations [E] [Ab] [Tr] [In] = 0. 2 n. M = 100 n. M = 18. 5 n. M optimize design! Bio/Chemical Kinetics Made Easy 28

Acknowledgments ACADEMIC COLLABORATION: • Bryan Marks: all kinase experiments Invitrogen, a. k. a. Life Technologies, Madison, Wisconsin • Steve Riddle: project management Invitrogen, a. k. a. Life Technologies, Madison, Wisconsin INVITATION TO PRESENT: • IPK 2009 organizers, Jan Antosiewicz (IBB) Bio/Chemical Kinetics Made Easy 29

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