Algebraic solution for time course of enzyme assays. Enzyme kinetics in the real world. Progress curvature at low initial [substrate]

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1 1 Algebraic solution for course of enzyme assays DynaFit in the Analysis of Enzyme Progress Curves Irreversible enzyme inhibition ONLY THE SIMPLEST REACTION MECHANISMS CAN BE TREATED IN THIS WAY EXAMPLE: slow binding inhibition BioKin, Ltd. WATERTOWN, MASSACHUSETTS, U.S.A. TOPICS TASK: compute [P] over 1. Numerical integration vs. algebraic models: DYNAFIT 2. Case study: Caliper assay of irreversible inhibitors 3. Devil is in the details: Problems to be aware of SIMPLIFYING ASSUMPTIONS: 1. No substrate depletion 2. No tight binding Kuzmic (2008) Anal. Biochem. 380, 5-12 DynaFit: Enzyme Progress Curves 2 Enzyme kinetics in the real world Progress curvature at low initial [substrate] SUBSTRATE DEPLETION USUALLY CANNOT BE NEGLECTED SUBSTRATE DEPLETION IS MOST IMPORTANT AT [S] 0 << K M [S] final 0 = 10 µm, K M = 90 µm slope 1.34 ~40% decrease in rate 8% systematic error initial slope 0.82 residuals not random linear fit: 10% systematic error Sexton, Kuzmic, et al. (2009) Biochem. J. 422, DynaFit: Enzyme Progress Curves 3 Kuzmic, Sexton, Martik (2010) Anal. Biochem., submitted DynaFit: Enzyme Progress Curves 4 Problems with algebraic models in enzyme kinetics THERE ARE MANY SERIOUS PROBLEMS AND LIMITATIONS Can be derived for only a limited number of simplest mechanisms Based on many restrictive assumptions: - no substrate depletion - weak inhibition only (no tight binding ) Quite complicated when they do exist Numerical solution of ODE systems: Euler method COMPLETE REACTION PROGRESS IS COMPUTED IN TINY LINEAR INCREMENTS k mechanism: A B d [A] / d t = - k [A] Δ [A] / Δ t = - k [A] differential rate equation [A] [A] 0 practically useful methods are much more complex! The solution: numerical models Δ [A] = - k [A] Δ t [A] t straight line segment Can be derived for an arbitrary mechanism No restrictions on the experiment (e.g., no excess of inhibitor over enzyme) [A] t + Δ t -[A] t = - k [A] t Δ t [A] t+δt No restrictions on the system itself ( tight binding, slow binding, etc.) [A] t + Δ t = [A] t - k [A] t Δ t Δ t Very simple to derive DynaFit: Enzyme Progress Curves 5 DynaFit: Enzyme Progress Curves 6

2 2 Automatic derivation of differential equations IT IS SO SIMPLE THAT EVEN A DUMB MACHINE (THE COMPUTER) CAN DO IT Software DYNAFIT ( ) PRACTICAL IMPLEMENTATION OF NUMERICAL ENZYME KINETICS Example input (plain text file): k1 E + S ---> ES Rate terms: k 1 [E] [S] Rate equations: E disappears d[e ]/dt = - k 1 [E] [S] multiply [E] [S] k2 ES ---> E + S k3 ES ---> E + P k 2 [ES] k 3 [ES] E is formed + k 2 [ES] + k 3 [ES] similarly for other species (S, ES, and P) 2009 DOWNLOAD Kuzmic (2009) Meth. Enzymol., 467, DynaFit: Enzyme Progress Curves 7 DynaFit: Enzyme Progress Curves 8 DYNAFIT: What can you do with it? ANALYZE/SIMULATE MANY TYPES OF EXPERIMENTAL DATA ARISING IN BIOCHEMICAL LABORATORIES DYNAFIT applications: mostly biochemical kinetics BUT NOT NECESSARILY: ANY SYSTEM THAT CAN BE DESCRIBED BY A FIRST-ORDER ODEs Basic tasks: - simulate artificial data (assay design and optimization) - fit experimental data (determine inhibition constants) - design optimal experiments (in preparation) Experiment types: - course of enzyme assays - initial rates in enzyme kinetics - equilibrium binding assays (pharmacology) Advanced features: - confidence intervals for kinetic constants * Monte-Carlo intervals * profile-t method (Bates & Watts) - goodness of fit - residual analysis (Runs-of-Signs Test) - model discrimination analysis (Akaike Information Criterion) - robust initial estimates (Differential Evolution) - robust regression estimates (Huber s Mini-Max) ~ 650 journal articles total DynaFit: Enzyme Progress Curves 9 DynaFit: Enzyme Progress Curves 10 Traditional analysis of irreversible inhibition DynaFit in the Analysis of Enzyme Progress Curves Irreversible enzyme inhibition BEFORE 1981 (IBM-PC) ALL LABORATORY DATA MUST BE CONVERTED TO STRAIGHT LINES 1962 Kitz-Wilson plot BioKin, Ltd. WATERTOWN, MASSACHUSETTS, U.S.A. log (enzyme activity/control) 1 / slope of straight line (k obs ) TOPICS increasing [inhibitor] 1. Numerical integration vs. algebraic models: DYNAFIT 2. Case study: Caliper assay of irreversible inhibitors 3. Devil is in the details: Problems to be aware of 1/ 1/ 1 / [inhibitor] Kitz & Wilson (1962) J. Biol. Chem. 237, DynaFit: Enzyme Progress Curves 12

3 3 Traditional analysis Take 2 : nonlinear AFTER 1981 STRAIGHT LINES ARE NO LONGER NECESSARY ( NONLINEAR REGRESSION ) 1981 IBM-PC (Intel 8086) Traditional analysis: Three assumptions (part 1) LINEAR OR NONLINEAR ANALYSIS THE SAME ASSUMPTIONS APPLY 1. Inhibitor binds only weakly to the enzyme enzyme activity/control k obs increasing [inhibitor] 0.5 no tight binding [inhibitor] [I], must not be comparable with [E] A = A 0 exp(-k obs t) k obs = / (1 + / [I]) DynaFit: Enzyme Progress Curves 13 DynaFit: Enzyme Progress Curves 14 Traditional analysis: Three assumptions (part 2) LINEAR OR NONLINEAR ANALYSIS THE SAME ASSUMPTIONS APPLY Traditional analysis: Three assumptions (part 3) LINEAR OR NONLINEAR ANALYSIS THE SAME ASSUMPTIONS APPLY 2. Enzyme activity over is measured directly In a substrate assay, plot of product [P] vs. must be a straight line at [I] = 0 3. Initial binding/dissociation is much faster than inactivation ( rapid equilibrium approximation ) ASSUMED MECHANISM: ACTUAL MECHANISM IN MANY CASES: K m k cat E + S E S E + P DynaFit: Enzyme Progress Curves 15 DynaFit: Enzyme Progress Curves 16 Simplifying assumptions: Requirements for data HOW MUST OUR DATA LOOK SO THAT WE CAN ANALYZE IT BY THE TRADITIONAL METHOD? Actual experimental data (COURTESY OF Art Wittwer, Pfizer) NEITHER OF THE TWO MAJOR SIMPLIFYING ASSUMPTION ARE SATISFIED! 1. control curve = straight line [I] = 0 SIMULATED: [E] = 1 nm 1. control curve [I] = 0 [I] = 100 nm [I] = 200 nm [S] = 10 μm K m = 1 μm [I] = 2.5 nm [E] = 0.3 nm [I] = 400 nm [I] = 800 nm 2. inhibitor concentrations must be much higher than enzyme concentration 2. inhibitor concentrations within the same order of magnitude DynaFit: Enzyme Progress Curves 17 DynaFit: Enzyme Progress Curves 18

4 4 Numerical model for Caliper assay data NO ASSUMPTIONS ARE MADE ABOUT EXPERIMENTAL CONDITIONS Caliper assay: Results of fit optimized [E] 0 THE ACTUAL ENZYME CONCENTRATION SEEMS HIGHER THAN THE NOMINAL VALUE DynaFit input: [mechanism] Automatically generated fitting model: units: μm, minutes Ki ~ [E] 0 tight binding ~ not rapid equilibrium = /k on = 1 nm k on E + S k cat /K m E + P k on M -1.sec sec sec -1 k cat /K m M-1.sec-1 [E] nm DynaFit: Enzyme Progress Curves 19 DynaFit: Enzyme Progress Curves 20 Caliper assay violates assumptions of classic analysis ALL THREE ASSUMPTIONS OF THE TRADITIONAL ANALYSIS WOULD BE VIOLATED Numerical model more informative than algebraic MORE INFORMATION EXTRACTED FROM THE SAME DATA 1. Enzyme concentration is not much lower than [I] 0 or Traditional model (Kitz & Wilson, 1962) General numerical model 2. The dissociation of the E I complex is not much faster than inactivation 3. The control progress curve ([I] = 0) is not a straight line (Substrate depletion is significant) k on very fast very slow no assumptions! If we used the traditional algebraic analysis, the results (, ) would likely be incorrect. Two model parameters Three model parameters Add another dimension (à la residence ) to the QSAR? DynaFit: Enzyme Progress Curves 21 DynaFit: Enzyme Progress Curves 22 Numerical modeling looks simple, but... DynaFit in the Analysis of Enzyme Progress Curves A RANDOM SELECTION OF A FEW TRAPS AND PITFALLS Irreversible enzyme inhibition BioKin, Ltd. WATERTOWN, MASSACHUSETTS, U.S.A. TOPICS 1. Numerical integration vs. algebraic models: DYNAFIT 2. Case study: Caliper assay of irreversible inhibitors Residual plots we must always look at them Adjustable concentrations we must always float some concentrations in a global fit Initial estimates: the false minimum problem nonlinear regression requires us to guess the solution beforehand Model discrimination: Use your judgment the theory of model discrimination is far from perfect 3. Devil is in the details: Problems to be aware of DynaFit: Enzyme Progress Curves 24

5 5 Residual plots RESIDUAL PLOTS SHOWS THE DIFFERENCE BETWEEN THE DATA AND THE BEST-FIT MODEL Direct plots of data: Example 1 THESE TWO PLOTS LOOK INDISTINGUISHABLE, DO THEY NOT? One-step inhibition Two-step inhibition signal data residual + model residual 0 - E + I X k on DynaFit: Enzyme Progress Curves 25 DynaFit: Enzyme Progress Curves 26 Residual plots: Example 1 THESE TWO PLOTS LOOK VERY DIFFERENT, DO THEY NOT? Residual plots: Runs-of-signs test WE DON T HAVE TO RELY ON VISUALS ( LOG VS. HORSESHOE ) One-step inhibition Two-step inhibition One-step inhibition Two-step inhibition +5 horseshoe +2.5 log -10 E + I X -5 k on probability 0% probability 31% passes p > 0.05 test DynaFit: Enzyme Progress Curves 27 DynaFit: Enzyme Progress Curves 28 Residual plots: Example 2 SOMETIMES IT S O.K. TO HAVE OUTLIERS USE YOUR JUDGMENT Relaxed inhibitor concentrations: Example 1 WE ALWAYS HAVE TITRATION ERROR! It s not always easy to judge just how good the residuals are: Residual plots: fixed [I] Residual plots: relaxed [I] something happened with the first three -points DynaFit: Enzyme Progress Curves 29 DynaFit: Enzyme Progress Curves 30

6 6 Relaxed inhibitor concentrations: Example 1 (detail) WE ALWAYS HAVE TITRATION ERROR! Relaxed inhibitor concentrations: not all of them ONE (USUALLY ANY ONE) OF THE INHIBITOR CONCENTRATIONS MUST BE KEPT FIXED Residual plots: fixed [I] Residual plots: relaxed [I] DynaFit script:... [data] directory./users/com/.../100514/c1/data extension txt file 0nM offset auto? file 2p5nM offset auto? conc I = ? file 5nM offset auto? conc I = ? file 10nM offset auto? conc I = file 20nM offset auto? conc I = ? file 40nM offset auto? conc I = ?... FIXED n D = 100, n P = 44 n R = 18 p < n D = 100, n P = 55 n R = 44 p = 0.08 DynaFit: Enzyme Progress Curves 31 DynaFit: Enzyme Progress Curves 32 Initial estimates: the false minimum problem NONLINEAR REGRESSION REQUIRES US TO GUESS THE SOLUTION BEFOREHAND Large effect of slight changes in initial estimates IN UNFAVORABLE CASES EVEN ONE ORDER OF MAGNITUDE DIFFERENCE IS IMPORTANT initial estimate iterative refimenement residuals best fit initial estimate iterative refimenement best fit residuals [constants] kdp = 10? kai = 10? kdi = 1? kx = 0.01? [concentrations] S = 0.85? kdp = 72 kai = 4.5 kdi = 1.8 kx = [S] = 0.22 DynaFit: Enzyme Progress Curves 33 [constants] kdp = 10? kai = 0.1? kdi = 0.01? kx = 0.1? [concentrations] S = 0.85? kdp = 96 kai = 0.36 kdi = kx = ~ 0 [S] = 0.27 DynaFit: Enzyme Progress Curves 34 Solution to initial estimate problem: systematic scan Model discrimination: Use your judgment DYNAFIT-4 ALLOWS HUNDREDS, OR EVEN THOUSANDS, OF DIFFERENT INITIAL ESTIMATES [mechanism] [constants] kdp = { 10, 1, 0.1, 0.01 }? kai = { 10, 1, 0.1, 0.01 }? kdi = { 10, 1, 0.1, 0.01 }? kx = { 10, 1, 0.1, 0.01 }? MEANS: Try all possible combinations of initial estimates. 4 rate constants 4 estimates for each rate constant DYNAFIT IMPLEMENTS TWO MODEL-DISCRIMINATION CRITERIA 1. Fischer s F-ratio for nested models relative sum of squares 2. Akaike Information Criterion for all models One-step model: Two-step model: = 4 4 = 256 initial estimates DynaFit: Enzyme Progress Curves 35 probability (0.. 1) DynaFit: Enzyme Progress Curves 36

7 7 Summary and Conclusions Questions? NUMERICAL MODELS ENABLE US TO DO MORE USEFUL EXPERIMENTS IN THE LABORATORY ADVANTAGES of Numerical Enzyme Kinetics (the new approach): MORE INFORMATION AND CONTACT: No constraints on experimental conditions EXAMPLE: large excess of [I] over [E] no longer required No constraints on the theoretical model EXAMPLE: dissociation rate can be comparable with deactivation rate Theoretical model is automatically derived by the computer No more algebraic rate equations Learn more from the same data EXAMPLE: Determine k ON and k OFF, not just equilibrium constant = k OFF/k ON DISADVANTAGES: Change in standard operating procedures Is it better stick with invalid but established methods? (Continuity problem) Training / Education required Where to find for continuing education? (Short-term vs. long-term view) BioKin Ltd. Software Development Consulting Employee Training Continuing Education since DynaFit: Enzyme Progress Curves 37 DynaFit: Enzyme Progress Curves 38