Enzyme Inhibitors Petr Kuzmi , Ph.D. BioKin, Ltd. TOPICS: 1. - - PDF document

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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 Screening 10th 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

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Dose-response screening of enzyme inhibitors 3

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 ON EXPERIMENTAL CONDITIONS

[E] « K i: IC50 ≈ K i

*

ΔG = -RT log K i [E] ≈ K i: IC50 ≠ K i

* "CLASSICAL" INHIBITORS: "TIGHT BINDING" INHIBITORS: Dose-response screening of enzyme inhibitors 4

Tight binding inhibitors : [E] ≈ K i

HOW PREVALENT IS "TIGHT BINDING"?

... NOT SHOWN

log K i *

• 12
• 9
• 6
• 3

N 500 1000 1500 2000

A typical data set: Completely inactive: Tight binding: ~ 10,000 compounds ~ 1,100 ~ 400

Data courtesy of Celera Genomics

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Dose-response screening of enzyme inhibitors 5

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 of Celera Genomics

Dose-response screening of enzyme inhibitors 6

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 of Celera Genomics

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Dose-response screening of enzyme inhibitors 7

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 * *

E K E K I E K I E V V v

i i i 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:

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Dose-response screening of enzyme inhibitors 9

Problem: Occasional "outlier" points

log [I]

• 9
• 8
• 7
• 6
• 5
• 4

rate 20 40 60 80 100 120 140 160

• inf

K i* = 43 μM

LEAST-SQUARES FIT

• P. Kuzmic et al. (2004) Meth. Enzymol. 383, 66-81.

Dose-response screening of enzyme inhibitors 10

Solution: Robust regression ("IRLS")

log [I]

• 9
• 8
• 7
• 6
• 5
• 4

rate 20 40 60 80 100 120 140 160

• inf

K i* = 130 μM

HUBER'S "MINIMAX" METHOD

• P. Kuzmic et al. (2004) Meth. Enzymol. 383, 66-81.

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Dose-response screening of enzyme inhibitors 11

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 12

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 of Celera Genomics

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Dose-response screening of enzyme inhibitors 13

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 2 100 88 58 50 91 79 100 IRLS weights

I.R.L.S.: AT MOST ONE HALF OF DATA POINTS WITH NON-UNIT WEIGHTS

Data courtesy of Celera Genomics

Dose-response screening of enzyme inhibitors 14

Robust fit: Productivity and objectivity gains

A CASE STUDY "BEFORE AND AFTER" IMPLEMENTING ROBUST REGRESSION

10 20 30 40 50 60 70 80 90

before after robust fit % repeat deletions

Data courtesy of Celera Genomics

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Dose-response screening of enzyme inhibitors 15

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:

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Dose-response screening of enzyme inhibitors 17

What is the "true" value of an inhibition constant?

experiment no.

50 60 70 80 90

K i* , μM

10 15 20 AVERAGE & STANDARD DEVIATION FROM 43 REPLICATES

Average:

• Std. Dev.: 0.9 μM

13.7 μM

#76 : Ki = 11.5 μM

Data courtesy of Celera Genomics

Dose-response screening of enzyme inhibitors 18

Formal standard errors are too narrow

EXPERIMENT #76

K i* = (11.5 ± 1.2) μM Formal standard error

INTERVAL DOES NOT INCLUDE "TRUE" VALUE 13.7 μM

Data courtesy of Celera Genomics

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Dose-response screening of enzyme inhibitors 19

Symmetrical confidence intervals are better

K i* = (8.6 ... 14.4) μM Symmetrical 95% confidence interval

INTERVAL DOES INCLUDE "TRUE" VALUE 13.7 μM

Data courtesy of Celera Genomics

EXPERIMENT #76 Dose-response screening of enzyme inhibitors 20

Nonsymmetrical confidence intervals are the best

experiment no.

50 60 70 80 90

K i* , μM

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 of Celera Genomics

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Dose-response screening of enzyme inhibitors 21

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:

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Dose-response screening of enzyme inhibitors 23

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