CZ3253: Computer Aided Drug design Drug Design Methods I: QSAR Prof. Chen Yu Zong Tel: 6874-6877 Email: [email protected] Room

CZ3253: Computer Aided Drug designCZ3253: Computer Aided Drug design

Drug Design Methods I: QSARDrug Design Methods I: QSAR

Prof. Chen Yu ZongProf. Chen Yu Zong

Tel: 6874-6877Tel: 6874-6877Email: Email: [email protected]@nus.edu.sghttp://xin.cz3.nus.edu.sghttp://xin.cz3.nus.edu.sg

Room 07-24, level 7, SOC1, Room 07-24, level 7, SOC1, National University of SingaporeNational University of Singapore

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TerminologyTerminology• SAR (Structure-Activity Relationships)

– Circa 19th century?

• QSAR (Quantitative Structure Activity Relationships)– Specific to some biological/pharmaceutical function of

molecule (Absorption, Distribution/Digestion, Metabolism, Excretion)

– Brown and Frazer (1868-9)• ‘constitution’ related to biological response

– LogP

• QSPR (Quantitative Structure Property Relationships)– Relate structure to any physical-chemical property of

molecule

33

Statistical ModelsStatistical Models

• Simple– Mean, median and variation– Regression

• Advanced– Validation methods– Principal components, co-variance– Multiple Regression

QSAR,QSPR

44

Modern QSARModern QSAR

– Hansch et. Al. (1963)• Activity ‘travel through body’ partitioning

between varied solvents

– C (minimum dosage required)– (hydrophobicity)– (electronic)– Es (steric)

1/C a b 2 c dE s const.

55

Choosing DescriptorsChoosing Descriptors

• Buffon’s Problem

– Needle Length?– Needle Color?– Needle Composition?– Needle Sheen?– Needle Orientation?

66

Choosing DescriptorsChoosing Descriptors• Constitutional

– MW, Natoms of element

• Topological– Connectivity,Weiner index (sums of bond distances)– 2D Fingerprints (bit-strings)– 3D topographical indices, pharmacophore keys

• Electrostatic – Polarity, polarizability, partial charges

• Geometrical Descriptors– Length, width, Molecular volume

77

Choosing DescriptorsChoosing Descriptors• Chemical

– Hydrophobicity (LogP)– HOMO and LUMO energies– Vibrational frequencies– Bond orders– Energy total– GSH

88

Statistical MethodsStatistical Methods

• 1-D analysis• Large dimension sets require decomposition

techniques– Multiple Regression– PCA– PLS

• Connecting a descriptor with a structural element so as to interpolate and extrapolate data

99

Simple Error Analysis(1-D)Simple Error Analysis(1-D)

• Given N data points

– Mean

– Variance

– Regression

ycalc

yobs

xcalc

xobs

)()(

),(

YStdXStd

YXCovR

1010


• Given N data points– Regression

residualy

yyy obsi

calci

obscalc

obscalc

xx

yy

1111


• Given N data points– (Poor 0<R2<1(Good)

2

)()(

),(

)(

N

icalc yySSR

YStdXStd

YXCov

YStd

SSRR

nsfluctuatiobetween n Correlatio

1),(

1

YYXXN

YXCov i

N

ii

2

1

1)(

N

ii YY

NYStd

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Correlation vs. Dependence?Correlation vs. Dependence?

• Correlation– Two or more variables/descriptors may correlate to

the same property of a system

• Dependence– When the correlation can be shown to be due to one

changing caused by the change of the other

• Example: Elephants head and legs– Correlation exists between size of head and legs– The size of one does not depend on the size of the other

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Quantitative Structure Quantitative Structure Activity/Property Relationships Activity/Property Relationships

(QSAR,QSPR)(QSAR,QSPR)

• Discern relationships between multiple variables (descriptors)

• Identify connections between structural traits (type of subunits, bond angles local components) and descriptor values (e.g. activity, LogP, % denatured)

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Pre-QualificationsPre-Qualifications

• Size– Minimum of FIVE samples per descriptor

• Verification– Variance– Scaling– Correlations

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QSAR/QSPRQSAR/QSPRPre-QualificationsPre-Qualifications

• Variance– Coefficient of Variation

Standard Deviation

Mean

x

x

"Spread"

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• Scaling – Standardizing or normalizing descriptors to

ensure they have equal weight (in terms of magnitude) in subsequent analysis

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• Scaling – Unit Variance (Auto Scaling)– Ensures equal statistical weights (initially)

– Mean Centering

x i' x i

' 1

x i' x i x

x ' 0

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• Correlations

– Remove correlated descriptors

– Keep correlated descriptors so as to reduce data set size

– Apply math operation to remove correlation (PCR)

n)correlatio positive (100% 1

n)correlatio negative (100% 1

:

11

ij

ij

r

ENTATIONOVERREPRES

r

2

,

2

,

,,

,

thth,

descriptor j and ibetween n Correlatio)()(

),(

M

kjkj

M

kiki

jkj

M

kiki

ji

ji

jiji

XXXX

XXXXR

YStdXStd

XXCovR

1919


• Correlations

2020

QSAR/QSPR SchemeQSAR/QSPR Scheme

• Goal– Predict what happens next (extrapolate)!– Predict what happens between data points

(interpolate)!

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QSAR/QSPR SchemeQSAR/QSPR Scheme

• Types of Variable– Continuous

• Concentration, occupied volume, partition coefficient, hydrophobicity

– Discrete• Structural (1: Methyl group substituted, 0: no

methyl group substitution)

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QSAR/QSPRQSAR/QSPRPrincipal Components AnalysisPrincipal Components Analysis

• Reduces dimensionality of descriptors

• Principle components are a set of vectors representing the variance in the original data

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Principal components – Principal components – reducing the dimensionality of a datasetreducing the dimensionality of a dataset

x

y

Clearly there is a relationship between x and y- a high correlation.We can define a new variable z = x+y suchthat we can express most of the variation inthe data as the new variable z.This new variable is a principal component.

v

j

jjii xcp1

,pi is the ith principalcomponent and ci,j is the coefficient of the variable xj.There are v such variables.

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QSAR/QSPR-Principal QSAR/QSPR-Principal Components AnalysisComponents Analysis

• Geometric Analogy (3-D to 2-D PCA)

y

z

x

x1 x2 ....xNy1 y2 ....yNz1 z2 ....zN

O

~

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PCA is the transformation of a set of correlated variablesto a set of orthogonal uncorrelated variables called principalcomponents. These new variables are a linear combination of theoriginal variables in decreasing order of importance.

ikpkipiik tbYYr

p

.1

data matrix loadings (measure of the variation betweenvariables)

scores (measure of the variation between samples)

eigenvalue

Principal componentsPrincipal components

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QSAR/QSPRQSAR/QSPRPrincipal Components AnalysisPrincipal Components Analysis

• Formulate matrix

• Diagonalize matrix

• Eigenvectors are the principal components – These principal components (new descriptors) are a linear

combination of the original descriptors

• Eigenvalues represent variance– Largest accounts for greatest % of data variance– Next corresponds to second greatest and so on

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• Formulate matrix (Several types)

– Correlation or covariance (N x P)• N is number of molecules• P is number of descriptors

– Variance-Covariance matrix (N x N)

• Diagonalize (Rotate) matrix

r11 r12 ....r1pr21 r22 ....r2p rn1 rn2 ....rnp

A~

AA

T Avc

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• Eigenvectors (Loadings) – Represents contribution from each original descriptor

to PC (new descriptor)• # columns = # of descriptors• # rows = # of descriptors OR # of molecules

• Eigenvalues– Indicate which PC most important (representative of

original descriptors)• Benzene has 2 non-zero and 1 zero eigenvalue (planar)

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• Scores

– Graphing each object/molecule in space of 2 or more PCs

• # rows = # of objects/molecules• # columns = # of descriptors OR # of molecules

For benzene corresponds to graph in PC1 (x’) and PC2 (y’) system

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PC1

PC2

x

y

The PC’s each maximise the variancein the data in orthogonal directions andare ordered by size.

Usually only a few components are neededto explain (>90%) of the variance in thedata – or the properties are not relevant

The first step is to calculate the varience-covarience matrix from the data


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PC1

PC2

x

y

If there are s observations each of which contains v values, the data can be represented by a matrix D with v rows and s columns.

The varience-covariance matrix is Z = DTD.

The eigenvectors of Z are the principal components. Z is a square symmetric matrix so the eigenvectors are orthogonal. Usually the matrix is diagonalised to obtain the eigenvectors (the weightings for the properties) and eigenvalues (the explained variance).


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80 10 5 3 2

p1 .2 .3 .4 .1 .1 p2 .01 .02 .3 .4 .5p3 .02 .03 .1 .2 .4p5 .03 .4 .4 .04 .3p5 .3 .5 .5 .05 .3

eigenvalues – explain % variance

Properties

Multiply the property valuefor molecule by this for eacheigenvalue

Can do regression on the PC’s, egV = 0.3PC1(0.1) + 0.2PC2(0.1) + 0.4(0.2)

so, we’ve reduced a 5 property problem to a two property problem

The output looks like this :


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QSAR on SYBYL (Tripos Inc.)QSAR on SYBYL (Tripos Inc.)

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10D3D

3535


• Eigenvalues Explanation of variance in data

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• Each point corresponds to column (# points = # descriptors) in original data

Proximity correlation

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QSAR on SYBYL (Tripos Inc.)QSAR on SYBYL (Tripos Inc.)• Each point corresponds to row of original data

(i.e. #points = #molecules) or graph of molecules in PC space

HeNapthalene

H2O

Molecular Size

Small acting Big

Proximitysimilarity

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Outlier

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QSAR/QSPR-Regression TypesQSAR/QSPR-Regression Types

• Principal Component Analysis

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• Principal Component Analysis

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Non-Linear MappingsNon-Linear Mappings

• Calculate “distance” between points in N-dimensional descriptor/parameter space– Euclidean– City-block distances

• Randomly assign compounds in set to points on a 2-D or 3-D space

• Minimize Difference (Optimal N-d 2D plot)

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Non-Linear MappingsNon-Linear Mappings

• Advantages– Non-linear– No assumptions!– Chance groupings unlikely (2D group likely an

N-D group)

• Disadvantages– Dependence on initial guess (Use PCA scores

to improve)

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• Multiple Regression (MR)• PCR• PLS

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• Linear Regression– Minimize difference between calculated and

observed values (residuals)

Multiple Regression

y mx b

mx i x y i y

i1

N

x i x 2

i1

N

b y m x

y mi * x ii1

N

B

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• Principal Component Regression

– Regression but with Principal Components substituted for original descriptors/variables

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• Partial Least Squares

– Cross-validation determines number of descriptors/components to use

– Derive equation – Use bootstrapping and t-test to test

coefficients in QSAR regression

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• Partial Least Squares (a.k.a. Projection to Latent Structures)– Regression of a Regression

• Provides insight into variation in x’s(bi,j’s as in PCA) AND y’s (ai’s)

– The ti’s are orthogonal – M= (# of variables/descriptors OR

#observations/molecules whichever smaller)

y ai * tii

N

ti bij * x jj

M

4949


• PLS is NOT MR or PCR in practice

– PLS is MR w/cross-validation– PLS Faster

• couples the target representation (QSAR generation) and component generation while PCA and PCR are separate

• PLS well applied to multi-variants problems

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QSAR/QSPRQSAR/QSPRPost-QualificationsPost-Qualifications

• Confidence in Regression– TSS-Total Sum of Squares– ESS-Explained Sum of Squares– RSS-Residual Sum of Squares

TSSESS RSS

R2 ESS

TSS

1 (100% explaination of data)

0 (no explaination of data)

y i y 2

i

N

TSS

ycalc,i y 2ESS

i

N

y i ycalc,i 2

i

N

RSS

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QSAR/QSPRQSAR/QSPRPost-QualificationsPost-Qualifications

• Confidence in Prediction (Predictive Error Sum of Squares)

Q2 1PRESS

y i y 2

i1

N

, PRESS y i ycalc,i 2

i1

N

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QSAR/QSPRQSAR/QSPRPost-QualificationPost-Qualification

• Bias?– Bootstrapping

• Choosing best model?– Cross Validation

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• Bootstrapping

– ASSUME calculated data is experimental/observed data

– Randomly choose N data (allowing for a multiple picks of same data)

– Re-generate parameters/regression – Repeat M times– Average over M bootstraps– Compare (calculate residual)

• If close to zero then no bias• If large then bias exists

M is typically 50-100

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• Cross-Validation (used in PLS)– Remove one or more pieces of input data– Re-derive QSAR equation– Calculate omitted data– Compute root-mean-square error to evaluate efficacy of model

• Typically 20% of data is removed for each iteration• The model with the lowest RMS error has the optimal number of

components/descriptors

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QSPR ExampleQSPR Example

• Relation between musk odorant properties and benzenoid structure– Training set of 148 compounds (81 non-musk and 67 musk)– 47 chemical descriptors initially– Pre-qualifications

• Correlations (47-12=35)

– Post-qualifications• Bootstrapping • Test-set

– 6/6 musks, 8/9 non-musks

Narvaez, J. N., Lavine, B. K. and Jurs, P. C. Chemical Senses, 11, 145-156 (1986)

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Practical IssuesPractical Issues

• 10 times as many compounds as parameters fit

• 3-5 compounds per descriptor

• Traditional QSAR – Good for activity prediction– Not good for whether activity is due to binding

or transport

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Advanced MethodsAdvanced Methods

• Neural Networks• Support Vector Machines• Genetic/Evolutionary Algorithms• Monte Carlo• Alternate descriptors

– Reduced graphs– Molecular connectivity indices– Indicator variables (0 or 1)

• Combinatorics (e.g. multiple substituent sites)

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Tools AvailableTools Available

• Sybyl (Tripos Inc.)

• Insight II (Accelrys Inc.)

• Pole Bio-Informatique Lyonnais – http://pbil.univ-lyon1.fr/

• Molecular Biology– http://www.infobiogen.fr/services/deambulum/

english/logiciels.html

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SummarySummary

• QSAR/QSPR– Statistics connect structure/behavior w/ observables– Interpolate/Extrapolate

• Multi-Variate Analysis– Pre-Qualification– Regression

• PCA• PLS• MLS

– Post-Qualification

Documents

CZ3253: Computer Aided Drug design Drug Design Methods I: QSAR Prof. Chen Yu Zong Tel: 6874-6877 Email: [email protected] Room