Leiden University. The university to discover. Desirability Indexes for Soft Constraint Modeling in Drug Design Johannes Kruisselbrink E-mail: [email protected]

Leiden University. The university to discover.

Desirability Indexes for Soft Constraint Modeling in Drug Design

Johannes KruisselbrinkE-mail: [email protected]


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


Uncertainty and noise can arise in various parts of the optimization model:

Uncertainty and noise in optimization problems

Input X

External (uncontrollable) parameters A

Output YSystem(Model)

GOALS

f1max / min

f2max / min

|fmmax / min

g1 ≤ 0

g2 ≤ 0

|gn ≤ 0

Objectives

Constraints

A) Uncertainty and noise in the design variables

B) Uncertainty and noise environmental parameters

C) Uncertain and/or noisy system output

D) Vagueness / fuzziness in the constraints


Our setup for Automated Molecule Evolution


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, drug-

likeness, etc.- Aim: provide a set of molecular structures that

can be promising candidates for further research


Molecule Evolution

Fragments extracted fromFrom Drug Databases

While not terminate do

Generate offspring O from PPt+1= select from (P U O)

Evaluate O

Initialize population P0

- ‘Normal’ evolution cycle- Graph based mutation and

recombination operators- Deterministic elitist (μ+λ) parent

selection (NSGA-II with Niching)

“The molecule evoluator. An interactive evolutionaryalgorithm 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.


Objectives and constraintsObjectives- Activity predictors based on support vector machines:

- f1: activity predictor based on ECFP6 fingerprints

- f2: activity predictor based on AlogP2 Estate Counts

- f3: 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

- AlogP value

- Minimized energy


Soft constraints in drug design


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


Molecules failing Lipinski

Atorvastatin

Liothyronine

Ethopropazine

Olmesartan

Doxycycline

Bexarotene

Acarbose

MW

MW

MW / HA

log Plog P

(5.088)

HA / HDMW / HA


Modeling constraints using desirability functions


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


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

0

violatedviolated satisfiedgrey area grey area

1

0

violated satisfiedgrey area

One-sided Two-sided

There are multiple ways to create such mappings!

Cutoff bound

Constraint bound


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

ALogP 0 1 4 5

Minimized energy NA NA 80 150

* Bounds settings were determined based on chemical intuition


Harrington Desirability Functions

One-sided: Two-sided:

))'exp(exp()'( YYd

YbbdY 10)lnln('

)'exp()'(n

YYd

LU

LUYY

)(2

'


Example one-sided Harrington DF

Molecular solubility:- Soft constraint: Y > -4- Absolute cutoff: Y < -6

))6(exp(exp(01.0

))4(exp(exp(99.0

10

10

bb

bb

6))01.0ln(ln(

4))99.0ln(ln(

10

10

bb

bb

10

10

61.5272-

44.6001

bb

bb

0637.3

8548.16

1

0

b

b

violated grey area satisfied

)))0637.38548.16exp(exp()( YYd ))'exp(exp()'( YYd

YbbdY 10)lnln('


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

)'exp()'(n

YYd

LU

LUYY

)(2

'


violated grey area

satisfied violatedgrey area

8273.7

150600

)150600(2exp)(

YYd

)'exp()'(n

YYd

LU

LUYY

)(2

'


One possibility:- Make symmetric- Base d(Y) on cutoff bounds- Tune n using a constraint bound

7.827399.0lnlog

5556.099.0ln

5556.0exp99.0

150600

)150600(2502exp)250(

5556.0

n

d

n

n

n


)'exp()'(n

YYd

LU

LUYY

)(2

'


Or:- Make symmetric- Base d(Y) on constraint bounds- Tune n using a cutoff bound

2.203301.0lnlog

201.0ln

2exp01.0

250450

)250450(1502exp)150(

2

n

d

n

n

n

violated grey area


2033.2

250450

)250450(2exp)(

YYd


violated grey area


5.6927

200525

)200525(2exp)(

YYd

)'exp()'(n

YYd

LU

LUYY

)(2

'


Or:- Make symmetric- Base d(Y) on average between

constraint bounds and cutoff bounds- Tune n using a cutoff bound

5.6927

01.0lnlog

3077.101.0ln

3077.1exp01.0

200525

)200525(1502exp)150(

3077.1n

d

n

n

n


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


Derringer Desirability Functions

One-sided: Two-sided:

UY

UYBUB

UYBY

Ydl

,0

,

,1

)(

UY

UYTUT

UY

TYLLT

LYLY

Yd u

l

,0

,

,

,0

)(


violated grey area satisfied

Example one-sided Derringer DF

Molecular solubility:- Soft constraint: Y > -4- Absolute cutoff: Y < -6

6,0

64,64

64,1

)(

Y

YY

Y

Ydl

Note: l=1linear

UY

UYBUB

UYBY

Ydl

,0

,

,1

)(


Example two-sided Derringer DF

Molecular weight:- Absolute cutoff: Y < 150- Soft constraint: Y > 250- Soft constraint: Y < 450- Absolute cutoff: Y > 600

600,0

600450,600450

600450250,1

250150,150250

150150,0

)(

Y

YY

Y

YY

Y

Ydu

l

violated grey area


UY

UYTUT

UY

TYLLT

LYLY

Yd u

l

,0

,

,

,0

)(


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


Aggregating the Desirability Functions into score functions


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


Aggregation

- Desirability functions can be easily integrated into one single scoring function, e.g.:- Weighted sum- Min performance- Geometrical mean- Average

kk

iii xgDxF

1

1

xgDxF iiki ...1

min

k

iii xgD

kxF

1

1

k

iiii xgDaxF

1

The Desirability Index


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:

- Or…, use Harrington or Derringer DFs

maxexpˆ * xffdxf iiii

Original objective function minimization


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)


A case study


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

antagonistsEA settings:

- NSGA-II for the multi-objective test-cases- 80 parents / 120 offspring- 1000 generations- No niching


4D Pareto fronts

The Pareto fronts obtained using three different scoring methods

Optimization direction

Complete aggregation (1 objective) Only aggregate constraint scores (4 objectives)

Aggregate constraints and objectives separately (2 objectives)


Random subsets of the results


Separate constraints and objectives

Color: constraint scores(white = 0 black = 1)

f3: MDL max (=1)

f2: ECFP max (=1)

f1: AlogP2 EC max (=1)

Tamoxifen


Conclusions


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)

1

0

violated satisfiedgrey area

An ideal DF?

Never 0 (distinction on the degree of constraint)

When satisfied 1 (no distinction between satisfied regions)


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


Thank you!

Johannes KruisselbrinkNatural Computing GroupLIACS, Universiteit Leidene-mail: [email protected]://natcomp.liacs.nl


Matlab codes(no presentation stuff, just for creating the DF plots)


Harrington one-sided example

clfx = [0:.1:10];y = exp(-exp(-(-8 + 2 * x)));plot(x, y)ylim([-.1 1.1])xlabel('Y')ylabel('d(Y)')


Harrington two-sided example

clfx = [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)')


One-sided Harrington DF in MATLAB

clfx = [-8:.1:-2];y = exp(-exp(-(16.8548 + 3.0637 * x)));plot(x, y)hold onplot([-8 -6 -4 -2],[0 0 1 1], '-.r')ylim([-.1 1.1])xlabel('Y')ylabel('d(Y)')legend('Harrington DF', 'Linear DF', 'Location', 'NorthWest')


Two-sided Harrington DF 1 in MATLAB

clfx = [0:1:800];y = exp(-abs((2 * x - (600 + 150))/(600 - 150)).^(7.8273));plot(x, y)hold onplot([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', 'NorthEast')








One-sided Derringer DF in MATLAB

clfhold onx = [-8:.01:-2];y1 = (x >= -4) * 1 + (x < -4) .* (x >= -6) .* ((x + 6)/(-4 + 6)).^0.5;plot(x, y1, '-.b')y2 = (x >= -4) * 1 + (x < -4) .* (x >= -6) .* ((x + 6)/(-4 + 6)).^1;plot(x, y2, '--r')y3 = (x >= -4) * 1 + (x < -4) .* (x >= -6) .* ((x + 6)/(-4 + 6)).^2;plot(x, y3, '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', 'NorthWest')


Two-sided Derringer DF in MATLAB

clfhold onx = [0:.1:800];y1 = (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, y1, '-.b')y2 = (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, y2, '--r')y3 = (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, y3, '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', 'NorthEast')

Documents

Leiden University. The university to discover. Desirability Indexes for Soft Constraint Modeling in Drug Design Johannes Kruisselbrink E-mail: [email protected]