Desirability Indexes for Soft Constraint Modeling in Drug Design Johannes Kruisselbrink E-mail: jkruisse@liacs. nl Leiden University. The university to discover.
Scope Context: - Quality measures for candidate molecular structures for automated optimization Contents: - Using the concept of Desirability for modeling soft or fuzzy constraints - The applicability in automated drug design and examples for integration within a scoring function Leiden University. The university to discover.
Uncertainty and noise in optimization problems Uncertainty and noise can arise in various parts of the optimization model: Input X System (Model) Output Y External (uncontrollable) parameters A G O A L S f 1 max / min f 2 max / min | fm max / min Objectives g 1 ≤ 0 g 2 ≤ 0 | gn ≤ 0 Constraints A) Uncertainty and noise in the design variables C) Uncertain and/or noisy system output B) Uncertainty and noise environmental parameters D) Vagueness / fuzziness in the constraints Leiden University. The university to discover.
Our setup for Automated Molecule Evolution Leiden University. The university to discover.
Automated molecule design - Search for molecular structures with specific pharmacological or biological activity - Objectives: Maximization of potency of drug (and minimization of side-effects) - Constraints: Stability, synthesizability, druglikeness, etc. - Aim: provide a set of molecular structures that can be promising candidates for further research Leiden University. The university to discover.
Molecule Evolution - ‘Normal’ evolution cycle - Graph based mutation and recombination operators - Deterministic elitist (μ+λ) parent selection (NSGA-II with Niching) Initialize population P 0 Fragments extracted from From Drug Databases While not terminate do Pt+1= select from (P U O) Evaluate O Generate offspring O from P “The molecule evoluator. An interactive evolutionary algorithm for the design of drug-like molecules. “, E. -W. Lameijer, J. N. Kok, T. Bäck, A. P. IJzerman, J. Chem. Inf. Model. , 2006, 46(2): 545 -552. Leiden University. The university to discover.
Objectives and constraints Objectives - Activity predictors based on support vector machines: - f 1: activity predictor based on ECFP 6 fingerprints - f 2: activity predictor based on Alog. P 2 Estate Counts - f 3: activity predictor based on MDL Constraints - Bounds based on Lipinski’s rule of five and the minimal energy confirmation: - Number of Hydrogen acceptors Number of Hydrogen donors Molecular solubility Molecular weight Alog. P value Minimized energy Leiden University. The university to discover.
Soft constraints in drug design Leiden University. The university to discover.
Soft constraints in Drug Design - Estimating the feasibility of candidate structures can be done using boundary values for certain molecule properties - Examples are Lipinski’s rule-of-five and estimations of the minimal energy conformations - But…, how strict are those rules? - Sometimes violations are easy to fix manually - Sometimes violations are not violations in practice Leiden University. The university to discover.
Molecules failing Lipinski MW Atorvastatin log P MW log P (5. 088) Ethopropazine Bexarotene Liothyronine HA / HD Doxycycline MW / HA Olmesartan MW / HA Acarbose Leiden University. The university to discover.
Modeling constraints using desirability functions Leiden University. The university to discover.
The real nature of the constraints The constraints are of the following forms: Where - x denotes a candidate structure g(x) denotes the property value of x Aj is the lower bound of the property filter Bj is the upper bound of the property filter reads: A is preferred to be smaller than B Leiden University. The university to discover.
Modeling constraints as objectives Constraints can be transformed into ‘objectives’ by mapping their values onto a function with the domain <0, 1> where: - Values close to 0 correspond to undesirable results - Values close to 1 correspond to desirable results - Values between 0 and 1 fall into the grey area 1 1 Cutoff bound Constraint bound 0 0 violated grey area One-sided satisfied violated grey area satisfied grey area violated Two-sided There are multiple ways to create such mappings! Leiden University. The university to discover.
Constraints in our studies Fuzzy constraint scores based on Lipinski’s rule of five and bounds on the minimal energy confirmation: Descriptor LB A B UB Num H-acceptors 0 1 6 10 Num H-donors 0 1 3 5 Molecular solubility -6 -4 NA NA Molecular weight 150 250 450 600 ALog. P 0 1 4 5 Minimized energy NA NA 80 150 * Bounds settings were determined based on chemical intuition Leiden University. The university to discover.
Harrington Desirability Functions One-sided: Two-sided: Leiden University. The university to discover.
Example one-sided Harrington DF Molecular solubility: - Soft constraint: Y > -4 - Absolute cutoff: Y < -6 violated grey area satisfied Leiden University. The university to discover.
Example two-sided Harrington DF Molecular weight: - Absolute lower cutoff: Y < 150 Lower bound constraint: Y > 250 Upper bound constraint: Y < 450 Absolute upper cutoff: Y > 600 Problematic! - No support for non-symmetric boundaries - No explicit support for ‘completely satisfied’ intervals Leiden University. The university to discover.
Example two-sided Harrington DF One possibility: - Make symmetric - Base d(Y) on cutoff bounds - Tune n using a constraint bound violated grey area satisfied grey area violated Leiden University. The university to discover.
Example two-sided Harrington DF Or: - Make symmetric - Base d(Y) on constraint bounds - Tune n using a cutoff bound violated grey area satisfied grey area violated Leiden University. The university to discover.
Example two-sided Harrington DF Or: - Make symmetric Base d(Y) on average between constraint bounds and cutoff bounds Tune n using a cutoff bound violated grey area satisfied grey area violated Leiden University. The university to discover.
Harrington - Advantages: - Maps onto a continuous function - Strictly monotonous mapping - Distinction between completely violated points - Downsides: - Tuning the DF is somewhat arbitrary - Distinction between completely satisfied solutions - Not really suited for ‘completely satisfied intervals’ - Does not allow non-symmetric constraints Leiden University. The university to discover.
Derringer Desirability Functions One-sided: Two-sided: Leiden University. The university to discover.
Example one-sided Derringer DF Molecular solubility: - Soft constraint: Y > -4 Absolute cutoff: Y < -6 Note: l=1 linear violated grey area satisfied Leiden University. The university to discover.
Example two-sided Derringer DF Molecular weight: - Absolute cutoff: Y < 150 Soft constraint: Y > 250 Soft constraint: Y < 450 Absolute cutoff: Y > 600 violated grey area satisfied grey area violated Leiden University. The university to discover.
Derringer - Advantages: - Easy straightforward implementation - Control for modeling non-symmetric constraints - Easy integration for ‘completely satisfied’ intervals - No distinction between completely satisfied solutions - Downsides: - Maps onto a discontinuous function - Not strictly monotonous (just monotonous) - No distinction between solutions after lower cutoff Leiden University. The university to discover.
Aggregating the Desirability Functions into score functions Leiden University. The university to discover.
Many objective optimization - Modeling fuzzy constraints using DFs generates many additional objective functions - In our case: - 3 original objectives + 6 constraints 9 objectives - The possibilities: - Pareto optimization - Aggregation - A combination of the both Leiden University. The university to discover.
Aggregation - Desirability functions can be easily integrated into one single scoring function, e. g. : - Weighted sum Min performance Geometrical mean Average The Desirability Index Leiden University. The university to discover.
Remodeling the objectives - Desirability index aggregation of the objectives requires a normalization function that maps the objective function values to the interval [0, 1] - One possibility: Original objective function minimization - Or…, use Harrington or Derringer DFs Leiden University. The university to discover.
The aggregation possibilities - Full aggregation: - Aggregate the constraints and the objectives into one quality score (1 objective) - Partial aggregation: - Aggregate the constraints into one constraint score (1 extra objective 4 objectives) - Aggregate the constraints and the objectives into two separate scoring function (2 objectives) Leiden University. The university to discover.
A case study Leiden University. The university to discover.
Experiments Comparison of: - Complete aggregation (1 objective) - Separate aggregation of objectives and constraints (2 objectives) - Only aggregate constraint scores (4 objectives) Objectives: - three activity prediction models for estrogen receptor antagonists EA settings: - NSGA-II for the multi-objective test-cases 80 parents / 120 offspring 1000 generations No niching Leiden University. The university to discover.
4 D Pareto fronts Optimization direction Complete aggregation (1 objective) Only aggregate constraint scores (4 objectives) The Pareto fronts obtained using three different scoring methods Aggregate constraints and objectives separately (2 objectives) Leiden University. The university to discover.
Random subsets of the results Leiden University. The university to discover.
Separate constraints and objectives Tamoxifen Color: constraint scores (white = 0 black = 1) f 3: MDL max (=1) f 2: ECFP max (=1) f 1: Alog. P 2 EC max (=1) Leiden University. The university to discover.
Conclusions Leiden University. The university to discover.
Discussion - Ranking issues - DFs that can yield 0 values will generate 0 values for the performance when aggregating using the geometric mean - DFs that make distinctions between completely satisfied constraints might be involved in unnecessary further optimization (maximization while already satisfied) An ideal DF? Never 0 (distinction on the degree of constraint) 1 When satisfied 1 (no distinction between satisfied regions) 0 violated grey area satisfied Leiden University. The university to discover.
Conclusions - Desirability Functions and Desirability Indexes for modeling soft / fuzzy constraints: - Are intuitive and easy to incorporate - Allow for easy integration of additional constraints - Incorporate the concept of vagueness present in all rule-of-thumb measures - Prevent the optimization method from ruling out promising candate structures Leiden University. The university to discover.
Thank you! Johannes Kruisselbrink Natural Computing Group LIACS, Universiteit Leiden e-mail: jkruisse@liacs. nl http: //natcomp. liacs. nl Leiden University. The university to discover.
Matlab codes (no presentation stuff, just for creating the DF plots) Leiden University. The university to discover.
![Harrington one-sided example clf x = [0: . 1: 10]; y = exp(-(-8 +](https://present5.com/presentation/0399732a9cb3acf09ecb351f0249a7d7/image-41.jpg "Harrington one-sided example clf x = [0: . 1: 10]; y = exp(-(-8 +")
Harrington one-sided example clf x = [0: . 1: 10]; y = exp(-(-8 + 2 * x))); plot(x, y) ylim([-. 1 1. 1]) xlabel('Y') ylabel('d(Y)') Leiden University. The university to discover.
![Harrington two-sided example clf x = [0: . 01: 10]; y = exp(-abs((2 *](https://present5.com/presentation/0399732a9cb3acf09ecb351f0249a7d7/image-42.jpg "Harrington two-sided example clf x = [0: . 01: 10]; y = exp(-abs((2 *")
Harrington two-sided example clf x = [0: . 01: 10]; y = exp(-abs((2 * x - (6 + 4))/(6 - 4)). ^(3)); plot(x, y) ylim([-. 1 1. 1]) xlabel('Y') ylabel('d(Y)') Leiden University. The university to discover.
![One-sided Harrington DF in MATLAB clf x = [-8: . 1: -2]; y =](https://present5.com/presentation/0399732a9cb3acf09ecb351f0249a7d7/image-43.jpg "One-sided Harrington DF in MATLAB clf x = [-8: . 1: -2]; y =")
One-sided Harrington DF in MATLAB clf x = [-8: . 1: -2]; y = exp(-(16. 8548 + 3. 0637 * x))); plot(x, y) hold on plot([-8 -6 -4 -2], [0 0 1 1], '-. r') ylim([-. 1 1. 1]) xlabel('Y') ylabel('d(Y)') legend('Harrington DF', 'Linear DF', 'Location', 'North. West') Leiden University. The university to discover.
![Two-sided Harrington DF 1 in MATLAB clf x = [0: 1: 800]; y =](https://present5.com/presentation/0399732a9cb3acf09ecb351f0249a7d7/image-44.jpg "Two-sided Harrington DF 1 in MATLAB clf x = [0: 1: 800]; y =")
Two-sided Harrington DF 1 in MATLAB clf x = [0: 1: 800]; y = exp(-abs((2 * x - (600 + 150))/(600 - 150)). ^(7. 8273)); plot(x, y) hold on plot([0 150 250 450 600 850], [0 0 1 1 0 0], '-. r') ylim([-. 1 1. 1]) xlabel('Y') ylabel('d(Y)') legend('Harrington DF', 'Linear DF', 'Location', 'North. East') Leiden University. The university to discover.
![Two-sided Harrington DF 2 in MATLAB clf x = [0: 1: 800]; y =](https://present5.com/presentation/0399732a9cb3acf09ecb351f0249a7d7/image-45.jpg "Two-sided Harrington DF 2 in MATLAB clf x = [0: 1: 800]; y =")
Two-sided Harrington DF 2 in MATLAB clf x = [0: 1: 800]; y = exp(-abs((2 * x - (450 + 250))/(450 - 250)). ^(2. 2033)); plot(x, y) hold on plot([0 150 250 450 600 850], [0 0 1 1 0 0], '-. r') ylim([-. 1 1. 1]) xlabel('Y') ylabel('d(Y)') legend('Harrington DF', 'Linear DF', 'Location', 'North. East') Leiden University. The university to discover.
![Two-sided Harrington DF 3 in MATLAB clf x = [0: 1: 800]; y =](https://present5.com/presentation/0399732a9cb3acf09ecb351f0249a7d7/image-46.jpg "Two-sided Harrington DF 3 in MATLAB clf x = [0: 1: 800]; y =")
Two-sided Harrington DF 3 in MATLAB clf x = [0: 1: 800]; y = exp(-abs((2 * x - (525 + 200))/(525 - 200)). ^(5. 6927)); plot(x, y) hold on plot([0 150 250 450 600 850], [0 0 1 1 0 0], '-. r') ylim([-. 1 1. 1]) xlabel('Y') ylabel('d(Y)') legend('Harrington DF', 'Linear DF', 'Location', 'North. East') Leiden University. The university to discover.
![One-sided Derringer DF in MATLAB clf hold on x = [-8: . 01: -2];](https://present5.com/presentation/0399732a9cb3acf09ecb351f0249a7d7/image-47.jpg "One-sided Derringer DF in MATLAB clf hold on x = [-8: . 01: -2];")
One-sided Derringer DF in MATLAB clf hold on x = [-8: . 01: -2]; y 1 = (x >= -4) * 1 + (x < -4). * (x >= -6). * ((x + 6)/(-4 + 6)). ^0. 5; plot(x, y 1, '-. b') y 2 = (x >= -4) * 1 + (x < -4). * (x >= -6). * ((x + 6)/(-4 + 6)). ^1; plot(x, y 2, '--r') y 3 = (x >= -4) * 1 + (x < -4). * (x >= -6). * ((x + 6)/(-4 + 6)). ^2; plot(x, y 3, 'g') ylim([-. 1 1. 1]) xlabel('Y') ylabel('d(Y)') legend('Derringer DF (l=0. 5)', 'Derringer DF (l=1)', 'Derringer DF (l=2)', 'Location', 'North. West') Leiden University. The university to discover.
![Two-sided Derringer DF in MATLAB clf hold on x = [0: . 1: 800];](https://present5.com/presentation/0399732a9cb3acf09ecb351f0249a7d7/image-48.jpg "Two-sided Derringer DF in MATLAB clf hold on x = [0: . 1: 800];")
Two-sided Derringer DF in MATLAB clf hold on x = [0: . 1: 800]; y 1 = (x >= 150). * (x < 250). * ((x - 150) / (250 - 150)). ^(0. 5) + (x >= 250). * (x <= 450). * 1 + (x > 450). * (x <= 600). * ((x - 600) / (450 - 600)). ^(0. 5); plot(x, y 1, '-. b') y 2 = (x >= 150). * (x < 250). * ((x - 150) / (250 - 150)). ^(1) + (x >= 250). * (x <= 450). * 1 + (x > 450). * (x <= 600). * ((x - 600) / (450 - 600)). ^(1); plot(x, y 2, '--r') y 3 = (x >= 150). * (x < 250). * ((x - 150) / (250 - 150)). ^(2) + (x >= 250). * (x <= 450). * 1 + (x > 450). * (x <= 600). * ((x - 600) / (450 - 600)). ^(2); plot(x, y 3, 'g') ylim([-. 1 1. 1]) xlabel('Y') ylabel('d(Y)') legend('Derringer DF (l=0. 5)', 'Derringer DF (l=1)', 'Derringer DF (l=2)', 'Location', 'North. East') Leiden University. The university to discover.
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