Society for Biomolecular Screening 10th Annual Conference, Orlando, FL, September 11-15, 2004 Advanced Methods in Dose-Response Screening of Enzyme Inhibitors Petr Kuzmi č , Ph.D. BioKin, Ltd. TOPICS: 1. Fitting model : Four-parameter logistic (IC 50 ) vs. Morrison equation (K i *) 2. Robust regression : Implementing outlier exclusion in practice 3. Confidence intervals : What should we store in activity databases? Acknowledgements : Craig Hill & Jim Janc Celera Genomics, Department of Enzymology and HTS Assumptions • We need a portable measure of inhibitory potency. • Failing portability, at least we need to rank compounds correctly. • For correct ranking, we need both precision and accuracy. • No measurement is perfectly accurate: confidence intervals . • Few experiments are designed ideally and executed flawlessly. Reminder: PRECISION ACCURACY PRECISION & ACCURACY Dose-response screening of enzyme 2 inhibitors 1
Measures of inhibitory potency INTRINSIC MEASURE OF POTENCY: Δ G = -RT log K i Depends on Example: DEPENDENCE ON EXPERIMENTAL CONDITIONS Competitive inhibitor [S] [E] 1. Inhibition constant NO NO K i * = K i (1 + [S]/K M ) K i 2. Apparent K i YES NO YES YES 3. IC 50 IC 50 = K i (1 + [S]/K M ) + [E]/2 [E] « K i : IC 50 ≈ K i * "CLASSICAL" INHIBITORS: [E] ≈ K i : IC 50 ≠ K i * "TIGHT BINDING" INHIBITORS: Dose-response screening of enzyme 3 inhibitors Tight binding inhibitors : [E] ≈ K i HOW PREVALENT IS "TIGHT BINDING"? A typical data set: ~ 10,000 compounds Completely inactive: ~ 1,100 ... NOT SHOWN Tight binding: ~ 400 2000 1500 N 1000 500 0 -12 -9 -6 -3 0 Data courtesy of log K i * Celera Genomics Dose-response screening of enzyme 4 inhibitors 2
Problem: Negative K i from IC 50 FIT TO FOUR-PARAMETER LOGISTIC: * = IC 50 - [E] / 2 K i 1.4 1.2 n Hill 1.4 1.0 IC 50 2.9 nM 0.8 rate 0.6 [E] = 7.0 nM 0.4 K i * = 2.9 - 7.0 / 2 = - 0.6 nM 0.2 0.0 -inf -11 -10 -9 -8 -7 -6 log [I] Data courtesy of Celera Genomics Dose-response screening of enzyme 5 inhibitors Solution: Do not use four-parameter logistic FIT TO MODIFIED MORRISON EQUATION: P. Kuzmic et al. (2000) Anal. Biochem. 281, 62-67. P. Kuzmic et al. (2000) Anal. Biochem. 286, 45-50. 1.4 1.2 1.0 0.8 [E] nominal = 7.0 nM rate 0.6 [E] fitted = 4.5 nM 0.4 K i * = 0.9 nM 0.2 0.0 -inf -11 -10 -9 -8 -7 -6 log [I] Data courtesy of Celera Genomics Dose-response screening of enzyme 6 inhibitors 3
Fitting model for enzyme inhibition: Summary MEASURE OF INHIBITORY POTENCY * is preferred over IC 50 • Apparent inhibition constant K i MATHEMATICAL MODEL • Modified Morrison equation is preferred over four-parameter logistic ( ) 2 − − + − − + [ E ] [ I ] K * [ E ] [ I ] K * 4 [ E ] K * = + i i i v V V b 0 2 [ E ] METHODOLOGY • Optionally, adjust the enzyme concentration in fitting K i * Dose-response screening of enzyme 7 inhibitors TOPICS: 1. Fitting model : Four-parameter logistic (IC 50 ) vs. Morrison equation (K i *) 2. Robust regression : Implementing outlier exclusion in practice 3. Confidence intervals : What should we store in activity databases? 4
Problem: Occasional "outlier" points LEAST-SQUARES FIT P. Kuzmic et al. (2004) Meth. Enzymol. 383, 66-81. 160 140 K i * = 43 μ M 120 100 80 rate 60 40 20 0 -inf -9 -8 -7 -6 -5 -4 log [I] Dose-response screening of enzyme 9 inhibitors Solution: Robust regression ("IRLS") HUBER'S "MINIMAX" METHOD P. Kuzmic et al. (2004) Meth. Enzymol. 383, 66-81. 160 140 K i * = 130 μ M 120 100 80 rate 60 40 20 0 -inf -9 -8 -7 -6 -5 -4 log [I] Dose-response screening of enzyme 10 inhibitors 5
Robust fit: Practical considerations "The devil is in the details." • Treat negative controls in a special way (unit weight). • Allow only a certain maximum number of "outliers". Dose-response screening of enzyme 11 inhibitors Robust fit: Constant weighting of negative controls NEGATIVE CONTROL WELLS ([I] = 0) ARE EXCLUDED FROM ROBUST WEIGHTING SCHEME 1.4 1.2 Huber's method 1.0 Unit weight @ [I] = 0 0.8 rate 0.6 0.4 0.2 0.0 -inf -9 -8 -7 -6 -5 -4 log [I] Data courtesy of Celera Genomics Dose-response screening of enzyme 12 inhibitors 6
Robust fit: Limiting the number of "outliers" I.R.L.S. : AT MOST ONE HALF OF DATA POINTS WITH NON-UNIT WEIGHTS 2.5 100 Max 50% points with weight < 1.0 2.0 Huber's method 1.5 rate 1.0 IRLS weights 2 0.5 100 88 58 50 91 79 100 0.0 -inf -9 -8 -7 -6 -5 -4 log [I] Data courtesy of Celera Genomics Dose-response screening of enzyme 13 inhibitors Robust fit: Productivity and objectivity gains A CASE STUDY "BEFORE AND AFTER" IMPLEMENTING ROBUST REGRESSION 90 80 70 60 % 50 repeat deletions 40 30 20 10 0 before after robust fit Data courtesy of Celera Genomics Dose-response screening of enzyme 14 inhibitors 7
Robust fit: Summary • Tested on 10,000+ dose response curves • Huber's "Minimax method" proved most effective • Modifications for inhibitor screening: a. Handling of negative controls b. Prevent too many outliers • Increase in scientific objectivity & productivity Dose-response screening of enzyme 15 inhibitors TOPICS: 1. Fitting model : Four-parameter logistic (IC 50 ) vs. Morrison equation (K i *) 2. Robust regression : Implementing outlier exclusion in practice 3. Confidence intervals : What should we store in activity databases? 8
What is the "true" value of an inhibition constant? AVERAGE & STANDARD DEVIATION FROM 43 REPLICATES Average: 13.7 μ M 20 Std. Dev.: 0.9 μ M K i * , μ M 15 #76 : Ki = 11.5 μ M 10 Data courtesy of Celera Genomics 50 60 70 80 90 Dose-response screening of enzyme 17 experiment no. inhibitors Formal standard errors are too narrow EXPERIMENT #76 Formal standard error K i * = ( 11.5 ± 1.2 ) μ M INTERVAL DOES NOT INCLUDE "TRUE" VALUE 13.7 μ M Data courtesy of Celera Genomics Dose-response screening of enzyme 18 inhibitors 9
Symmetrical confidence intervals are better EXPERIMENT #76 Symmetrical 95% confidence interval K i * = ( 8.6 ... 14.4 ) μ M INTERVAL DOES INCLUDE "TRUE" VALUE 13.7 μ M Data courtesy of Celera Genomics Dose-response screening of enzyme 19 inhibitors Nonsymmetrical confidence intervals are the best NONSYMMETRICAL 99% C.I. Watts, D.G. (1994) Meth. Enzymol. 240, 23-36. Bates & Watts (1988) Nonlinear Regression , p. 207 20 K i * , μ M 15 10 Data courtesy of Celera Genomics 50 60 70 80 90 Dose-response screening of enzyme 20 experiment no. inhibitors 10
Confidence intervals (C.I.): Summary • Report two numbers for each compound: high and low end of the C.I. • If two C.I.'s overlap, the two inhibitory activities are indistinguishable . • Thus, many compounds can end up with identical rank ! Dose-response screening of enzyme 21 inhibitors TOPICS: 1. Fitting model : Four-parameter logistic (IC 50 ) vs. Morrison equation (K i *) 2. Robust regression : Implementing outlier exclusion in practice 3. Confidence intervals : What should we store in activity databases? Conclusions : Toward a "best-practice" standard in secondary screening 11
Toward "best-practice" in secondary screening DOSE-RESPONSE STUDIES OF ENZYME INHIBITORS • Measure K i * , not IC 50 (dependence on experimental conditions). • Use a mechanism-based model (Morrison equation), not the four-parameter logistic equation (no physical meaning). • Employ robust regression techniques, but very carefully. • Report a high/low range (confidence interval) for every K i * . Dose-response screening of enzyme 23 inhibitors 12
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