advanced methods in dose-response screening of enzyme inhibitors 1. fitting model: four-parameter...
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Advanced Methods in
Dose-Response Screening of
Enzyme Inhibitors
1. Fitting model: Four-parameter logistic (IC50) vs. Morrison equation (K i*)
2. Robust regression: Implementing outlier exclusion in practice
3. Confidence intervals: What should we store in activity databases?
TOPICS:
Acknowledgements:
Craig Hill & Jim Janc Celera Genomics, Department of Enzymology and HTS
Petr Kuzmič, Ph.D.BioKin, Ltd.
Society for Biomolecular Screening10th Annual Conference, Orlando, FL, September 11-15, 2004
Dose-response screening of enzyme inhibitors
2
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 inhibitors
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Measures of inhibitory potency
1. Inhibition constant
2. Apparent K i
3. IC50
Depends on
[S] [E]
NO
YES
YES
NO
NO
YES
K i
K i* = K i (1 + [S]/KM)
IC50 = K i (1 + [S]/KM) + [E]/2
Example:
Competitive inhibitor
INTRINSIC MEASURE OF POTENCY:
DEPENDENCE ONEXPERIMENTAL CONDITIONS
[E] « K i: IC50 K i*
G = -RT log K i
[E] K i: IC50 K i*
"CLASSICAL" INHIBITORS:
"TIGHT BINDING" INHIBITORS:
Dose-response screening of enzyme inhibitors
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Tight binding inhibitors : [E] K i
HOW PREVALENT IS "TIGHT BINDING"?
... NOT SHOWN
log K i *
-12 -9 -6 -3 0
N
0
500
1000
1500
2000
A typical data set: Completely inactive:
Tight binding:
~ 10,000 compounds
~ 1,100~ 400
Data courtesy ofCelera Genomics
Dose-response screening of enzyme inhibitors
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Problem: Negative Ki from IC50
log [I]
-11 -10 -9 -8 -7 -6
rate
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
-inf
nHill
IC50
1.4
2.9 nM
[E] = 7.0 nM
K i* = 2.9 - 7.0 / 2 = - 0.6 nM
FIT TO FOUR-PARAMETER LOGISTIC:K i
* = IC50 - [E] / 2
Data courtesy ofCelera Genomics
Dose-response screening of enzyme inhibitors
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Solution: Do not use four-parameter logistic
log [I]
-11 -10 -9 -8 -7 -6
rate
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
-inf
[E]nominal = 7.0 nM
[E]fitted = 4.5 nM
K i* = 0.9 nM
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.
Data courtesy ofCelera Genomics
Dose-response screening of enzyme inhibitors
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Fitting model for enzyme inhibition: Summary
• Apparent inhibition constant K i* is preferred over IC50
• Modified Morrison equation is preferred over four-parameter logistic
• Optionally, adjust the enzyme concentration in fitting K i*
MEASURE OF INHIBITORY POTENCY
MATHEMATICAL MODEL
METHODOLOGY
][2
][4][][][][ *2**
0 E
KEKIEKIEVVv iii
b
1. Fitting model: Four-parameter logistic (IC50) vs. Morrison equation (K i*)
2. Robust regression: Implementing outlier exclusion in practice
3. Confidence intervals: What should we store in activity databases?
TOPICS:
Dose-response screening of enzyme inhibitors
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Problem: Occasional "outlier" points
log [I]
-9 -8 -7 -6 -5 -4
rate
0
20
40
60
80
100
120
140
160
-inf
K i* = 43 M
LEAST-SQUARES FIT P. Kuzmic et al. (2004) Meth. Enzymol. 383, 66-81.
Dose-response screening of enzyme inhibitors
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Solution: Robust regression ("IRLS")
log [I]
-9 -8 -7 -6 -5 -4
rate
0
20
40
60
80
100
120
140
160
-inf
K i* = 130 M
HUBER'S "MINIMAX" METHOD P. Kuzmic et al. (2004) Meth. Enzymol. 383, 66-81.
Dose-response screening of enzyme inhibitors
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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 inhibitors
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Robust fit: Constant weighting of negative controls
log [I]
-9 -8 -7 -6 -5 -4
rate
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
-inf
Huber's method
Unit weight @ [I] = 0
NEGATIVE CONTROL WELLS ([I] = 0) ARE EXCLUDED FROM ROBUST WEIGHTING SCHEME
Data courtesy ofCelera Genomics
Dose-response screening of enzyme inhibitors
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Robust fit: Limiting the number of "outliers"
log [I]
-9 -8 -7 -6 -5 -4
rate
0.0
0.5
1.0
1.5
2.0
2.5
-inf
Max 50% points with weight < 1.0
Huber's method
100
2
10088 58 50 91 79 100
IRLS weights
I.R.L.S.: AT MOST ONE HALF OF DATA POINTS WITH NON-UNIT WEIGHTS
Data courtesy ofCelera Genomics
Dose-response screening of enzyme inhibitors
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Robust fit: Productivity and objectivity gains
A CASE STUDY "BEFORE AND AFTER" IMPLEMENTING ROBUST REGRESSION
0
10
20
30
40
50
60
70
80
90
before after
robust fit
%repeat
deletions
Data courtesy ofCelera Genomics
Dose-response screening of enzyme inhibitors
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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
1. Fitting model: Four-parameter logistic (IC50) vs. Morrison equation (K i*)
2. Robust regression: Implementing outlier exclusion in practice
3. Confidence intervals: What should we store in activity databases?
TOPICS:
Dose-response screening of enzyme inhibitors
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What is the "true" value of an inhibition constant?
experiment no.
50 60 70 80 90
K i*
, M
10
15
20
AVERAGE & STANDARD DEVIATION FROM 43 REPLICATES
Average:
Std. Dev.: 0.9 M
13.7 M
#76 : Ki = 11.5 M
Data courtesy ofCelera Genomics
Dose-response screening of enzyme inhibitors
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Formal standard errors are too narrow
EXPERIMENT #76
K i* = (11.5 ± 1.2) M
Formal standard error
INTERVAL DOES NOT INCLUDE "TRUE" VALUE 13.7 MData courtesy ofCelera Genomics
Dose-response screening of enzyme inhibitors
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Symmetrical confidence intervals are better
K i* = (8.6 ... 14.4) M
Symmetrical 95% confidence interval
INTERVAL DOES INCLUDE "TRUE" VALUE 13.7 MData courtesy ofCelera Genomics
EXPERIMENT #76
Dose-response screening of enzyme inhibitors
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Nonsymmetrical confidence intervals are the best
experiment no.
50 60 70 80 90
K i*
, M
10
15
20
NONSYMMETRICAL 99% C.I. Watts, D.G. (1994) Meth. Enzymol. 240, 23-36.Bates & Watts (1988) Nonlinear Regression, p. 207
Data courtesy ofCelera Genomics
Dose-response screening of enzyme inhibitors
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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!
1. Fitting model: Four-parameter logistic (IC50) 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
TOPICS:
Dose-response screening of enzyme inhibitors
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Toward "best-practice" in secondary screening
• Measure Ki*, not IC50 (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 Ki*.
DOSE-RESPONSE STUDIES OF ENZYME INHIBITORS