Numerical Enzymology

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1 Numerical Enzymology Generalized Treatment of inetics & Equilibria Petr uzmič, Ph.D. Bioin, Ltd. DYNAFIT OFTWARE PACAGE 1. Overview of recent applications 2. elected examples - ATPase cycle of Hsp90 Analog Trap1 (Lesovar et al., 2008) - Nucleotide binding to ClpB (Werbec et al., 2009) - Clathrin uncoating (Rothnie et al., 2011) 3. Recent enhancements - Optimal Experimental Design DYNAFIT software NUMERICAL ENZYME INETIC AND LIGAND BINDING uzmic (1996) Anal. Biochem. 237, Numerical Enzyme inetics 2 1

2 DynaFit: Citation analysis JULY 2011: 683 BIBLIOGRAPHIC REFERENCE ( WEB OF CIENCE ) DYNAFIT paper - cumulative citations Biochemistry (UA) ~65% J. Biol. Chem. ~20% Numerical Enzyme inetics 3 A "inetic Compiler" HOW DYNAFIT PROCEE YOUR BIOCHEMICAL EQUATION 1 E + E. E + P 2 3 Input (plain text file): Rate terms: Rate equations: E + ---> E : 1 1 E de / dt - 1 E + 2 E + 3 E E ---> E + E ---> E + P : 2 : 3 2 E 3 E de / dt + 1 E - 2 E - 3 E imilarly for other species... Numerical Enzyme inetics 4 2

3 ystem of imple, imultaneous Equations HOW DYNAFIT PROCEE YOUR BIOCHEMICAL EQUATION 1 E + E. E + P 2 3 "The LEGO method" of deriving rate equations Input (plain text file): Rate terms: Rate equations: E + ---> E : 1 1 E E ---> E + : 2 2 E E ---> E + P : 3 3 E Numerical Enzyme inetics 5 DynaFit can analyze many types of experiments MA ACTION LAW AND MA CONERVATION LAW I APPLIED TO DERIVE DIFFERENT MODEL EXPERIMENT DYNAFIT DERIVE A YTEM OF... Reaction progress Initial rates Equilibrium binding First-order ordinary differential equations Nonlinear algebraic equations Nonlinear algebraic equations Numerical Enzyme inetics 6 3

4 DynaFit: Recent enhancements REVIEW (2009) uzmic, P. (2009) Meth. Enzymol. 467, Numerical Enzyme inetics 7 DynaFit Example 1: Trap1 ATPase cycle EXCELLENT EXAMPLE OF COMBINING TRADITIONAL (ALGEBRAIC) AND NUMERICAL (DYNAFIT) MODEL E o E c open conformation closed conformation Lesovar et al. (2008) "The ATPase cycle of the mitochondrial Hsp90 analog Trap1" J. Biol. Chem. 283, Numerical Enzyme inetics 8 4

5 DynaFit Example 1: Trap1 ATPase cycle - experiments THREE DIFFERENT TYPE OF EXPERIMENT COMBINED varied ATP analog stopped flow fluorescence varied ADP analog stopped flow fluorescence single-turnover ATPase assay Lesovar et al. (2008) J. Biol. Chem. 283, Numerical Enzyme inetics 9 DynaFit Example 1: Trap1 ATPase cycle - script MECHANIM INCLUDE PHOTO-BLEACHING (ARTIFACT) tas data progress tas fit mechanism Eo + ATP <> Eo.ATP : at dt Eo.ATP <> Ec.ATP : oc co Ec.ATP ----> Ec.ADP : hy Ec.ADP <> Eo + ADP : dd ad PbT ----> PbT* : bt PbD ----> PbD* : bd photo-bleaching is a first-order process data directory extension plot monitor./users/edu/de/mpimf/lesovar_a/... txt logarithmic Eo, Eo.ATP, Ec.ATP, Ec.ADP, PbT*, PbD* show concentrations of these species over time Lesovar et al. (2008) J. Biol. Chem. 283, Numerical Enzyme inetics 10 5

6 DynaFit Example 1: Trap1 species concentrations UEFUL WAY TO GAIN INIGHT INTO THE MECHANIM E o E o.atp E c.atp E c.adp photo-bleaching Lesovar et al. (2008) J. Biol. Chem. 283, Numerical Enzyme inetics 11 DynaFit Example 2: Nucleotide binding to ClpB FROM THE AME LAB (MAX-PLANC INTITUTE FOR MEDICAL REEARCH, HEIDELBERG) NBD2-C protein MANT-dNt labeled nucleotide Nt unlabeled nucleotide determine on and off rate constants for unlabeled nucleotides from competition with labeled analogs Werbec et al. (2009) Nucleotide binding and allosteric modulation of the second AAA+ domain of ClpB probed by transient inetic studies Biochemistry 48, Numerical Enzyme inetics 12 6

7 DynaFit Example 2: Nucleotide binding to ClpB - data AGAIN COMBINE TWO DIFFERENT EXPERIMENT (ONLY LABELED NUCLEOTIDE HERE) variable constant ADP* ClpB constant variable ADP* ClpB Werbec et al. (2009) Biochemistry 48, Numerical Enzyme inetics 13 DynaFit Example 2: Nucleotide binding to ClpB script THE DEVIL I ALWAY IN THE DETAIL Residuals tas data progress tas fit model simplest mechanism P + madp <> P.mADP : > drift : v variable ADP* constant ClpB constants 1 5? ? v 0.1? drift in the machine constant ADP* variable ClpB Werbec et al. (2009) Biochemistry 48, Numerical Enzyme inetics 14 7

8 DynaFit Example 3: Clathrin uncoating inetics IN COLLABORATION WITH GU CAMERON (BRITOL) Rothnie et al. (2011) A sequential mechanism for clathrin cage disassembly by 70-Da heat-shoc cognate protein (Hsc70) and auxilin Proc. Natl. Acad. ci UA 108, Numerical Enzyme inetics 15 DynaFit Example 3: Clathrin uncoating - script MODEL DICRIMINATION ANALYI tas tas fit data progress model AAAH? mechanism CA + T ---> CAT : a CAT + T ---> CATT : a CATT + T ---> CATTT : a CATTT ---> CADDD : r CADDD ---> Prods : d... tas tas fit data progress model AHAHAH? mechanism CA + T ---> CAT : a CAT ---> CAD + Pi : r CAD + T ---> CADT : a CADT ---> CADD + Pi : r CADD + T ---> CADDT : a CADDT ---> CADDD + Pi : r CADDD ---> Prods : d... Arbitrary number of models to compare Model selection based on two criteria: - Aaie Information Criterion (AIC) - F-test for nested models Extreme caution is required for interpretation - Both AIC and F-test are far from perfect - Both are based on many assumptions - One must use common sense - Loo at the results only for guidance Rothnie et al. (2011) Proc. Natl. Acad. ci UA 108, Numerical Enzyme inetics 16 8

9 Optimal Experimental Design: Boos DOZEN OF BOO Fedorov, V.V. (1972) Theory of Optimal Experiments Fedorov, V.V. & Hacl, P. (1997) Model-Oriented Design of Experiments Atinson, A.C & Donev, A.N. (1992) Optimum Experimental Designs Endrenyi, L., Ed. (1981) Design and Analysis of Enzyme and Pharmacoinetics Experiments Numerical Enzyme inetics 17 Optimal Experimental Design: Articles HUNDRED OF ARTICLE, INCLUDING IN ENZYMOLOGY Numerical Enzyme inetics 18 9

10 10 Numerical Enzyme inetics 19 ome theory: Fisher information matrix D-OPTIMAL DEIGN: MAXIMIZE DETERMINANT OF THE FIHER INFORMATION MATRIX Fisher information matrix: EXAMPLE: Michaelis-Menten inetics V v + Model: two parameters (M2) Design: four concentrations (N4) 1, 2, 3, 4 Derivatives: ( sensitivities ) ( ) 2 V v s + V v s V + Numerical Enzyme inetics 20 ome theory: Fisher information matrix (contd.) D-OPTIMAL DEIGN: MAXIMIZE DETERMINANT OF THE FIHER INFORMATION MATRIX Approximate Fisher information matrix (M M): ) ( ) ( 1, N j i j i s F s EXAMPLE: Michaelis-Menten inetics D-Optimal Design: Maximize determinant of F over design points 1, det F F F F F determinant N N N N V V V ) ( ) ( ) ( F

11 Optimal Design for Michaelis-Menten inetics DUGGLEBY, R. (1979) J. THEOR. BIOL. 81, max Model: v V + V 1 1 opt max + 2 max opt is assumed to be nown! Numerical Enzyme inetics 21 Optimal Design: Basic assumptions OPTIMAL DEIGN FOR ETIMATING PARAMETER IN THE GIVEN MODEL TWO FAIRLY TRONG AUMPTION: 1. Assumed mathematical model is correct for the experiment 2. A fairly good estimate already exists for the model parameters Designed experiments are most suitable for follow-up (verification) experiments. Numerical Enzyme inetics 22 11

12 Optimal Experimental Design: Initial conditions IN MANY INETIC EXPERIMENT THE OBERVATION TIME CANNOT BE CHOEN CONVENTIONAL EXPERIMENTAL DEIGN: Mae an optimal choice of the independent variable: - Equilibrium experiments: concentrations of varied species - inetic experiments: observation time DYNAFIT MODIFICATION: Mae an optimal choice of the initial conditions: - inetic experiments: initial concentrations of reactants Assume that the time points are given by instrument setup. Numerical Enzyme inetics 23 Optimal Experimental Design: DynaFit input file EXAMPLE: CLATHRIN UNCOATING INETIC tas tas design data progress mechanism CA + T > CAT CAT > CAD + Pi CAD + T > CADT CADT > CADD + Pi CADD + T > CADDT CADDT > CADDD + Pi CADDD > Prods : a : r : a : r : a : r : d constants a 0.69? r 6.51? d 0.38? Choose eight initial concentration of T such that the rate constants a, r, d are determined most precisely. data file run01 concentration CA 0.1, T 1?? ( ) file run02 concentration CA 0.1, T 1?? ( ) file run03 concentration CA 0.1, T 1?? ( ) file run04 concentration CA 0.1, T 1?? ( ) file run05 concentration CA 0.1, T 1?? ( ) file run06 concentration CA 0.1, T 1?? ( ) file run07 concentration CA 0.1, T 1?? ( ) file run08 concentration CA 0.1, T 1?? ( ) Numerical Enzyme inetics 24 12

13 Optimal Experimental Design: Preliminary experiment EXAMPLE: CLATHRIN UNCOATING INETIC ACTUAL DATA Rothnie et al. (2011) Proc. Natl. Acad. ci UA 108, Actual concentrations of T (µm) ix different experiments 4 Numerical Enzyme inetics 25 Optimal Experimental Design: DynaFit results EXAMPLE: CLATHRIN UNCOATING INETIC D-Optimal initial concentrations: T 0.70 µm, 0.73 µm T 2.4 µm, 2.5 µm, 2.5 µm T 76 µm, 81 µm, 90 µm maximum feasible concentration upswing phase no longer seen Just three experiments would be sufficient for follow-up Numerical Enzyme inetics 26 13

14 Optimal Experimental Design in DynaFit: ummary NOT A ILVER BULLET! Useful for follow-up (verification) experiments only - Mechanistic model must be nown already - Parameter estimates must also be nown Taes a very long time to compute - Constrained global optimization: Differential Evolution algorithm - Clathrin design too minutes - Many design problems tae multiple hours of computation Critically depends on assumptions about variance - Usually we assume constant variance ( noise ) of the signal - Must verify this by plotting residuals against signal (not the usual way) Numerical Enzyme inetics 27 14