ISSN 0012�5008, Doklady Chemistry, 2009, Vol. 429, Part 1, pp. 273–276. © Pleiades Publishing, Ltd., 2009.Original Russian Text © N.I. Zhokhova, I.I. Baskin, D.K. Bakhronov, V.A. Palyulin, N.S. Zefirov, 2009, published in Doklady Akademii Nauk, 2009, Vol. 429, No. 2, pp. 201–205.

273

The rapid development of methods of design ofnew pharmaceuticals calls for new efficient computa�tional approaches that can reliably predict varioustypes of biological activity of organic compounds to besynthesized. This is due to the fact that the availablemethods widely used to search for quantitative struc�ture–activity relationships (QSARs) have significantdrawbacks. In particular, common methods of con�structing 3D QSARs, such as comparative molecularfield analysis (CoMFA) and comparative molecularsimilarity indices analysis (CoMSIA), underlying thestate�of�the�art approaches to the design of new phar�maceuticals are very sensitive to the dimensions, reso�lution, and spatial alignment of the hypothetical gridconstructed around a molecule and used for approxi�mating the electrostatic, steric, and hydrophobicmolecular fields, the potentials of the latter being cal�culated at the grid points as molecular structuredescriptors [1–3]. This leads to the ambiguity of theresulting 3D QSAR models and, hence, to the unreli�ability of the prediction based on these models.

In this work, we propose a new method for con�structing 3D QSAR models, namely, the method ofcontinuous molecular fields (MCMF). The basic ideaof this approach is in direct analysis of continuousmolecular fields rather than a discrete array of theirpotentials calculated at the points of the discretegrid of finite size (as in the standard CoMFA andCoMSIA methods). Such a description better cor�responds to the physical nature of molecular fields;therefore, we can expect better statistical character�istics of 3D QSAR models upon such a substitution.Until recently, it was impossible to use continuousmolecular fields in the framework of statistical analysissince common statistical procedures are intended tooperate only with finite and limited number of molec�ular descriptors. Therefore, we were interested to real�ize this idea on the basis of the latest statisticalapproaches, for example, the support vector machines

(SVM), which is free of this limitation and can operatewith an infinite number of variables [4]. This isachieved by using so�called kernels [5]. As is known,for any kernel, there must exist a linear vector space(referred to as the reproducing kernel Hilbert space) inwhich the former can be uniquely represented as thescalar product of the corresponding vectors. It is evi�dent that the scalar product of the molecular fieldpotential values at the grid points is also a kernel (bydefinition). Inasmuch as an increase in the griddimensions and a decrease in the grid cell size do notviolate this property, the integral of the product ofmolecular fields taken over the entire physical spacealso remains a kernel, which can be used in appropri�ate statistical methods, such as support vector regres�sion. Thus, this offers possibilities for constructing sta�tistical models based on the description of molecularobjects as continuous molecular fields.

The basic element of the MCMF proposed in thiswork is the procedure of calculation of kernels. Theuse of these kernels will be exemplified by constructingQSARs in the framework of the statistical method ofsupport vector regression [5].

In the proposed method, the kernel K(Mi, Mj),which describes the similarity between molecules Mi

and Mj, is calculated as a linear combination of ker�nels calculated for each of the Nf types of molecularfields:

(1)

where hk is the mixing coefficient of molecular fields,Kk(Mi, Mj) is the kernel describing the similaritybetween the molecular fields of the kth type of the ithand jth molecules. That K(Mi, Mj) is a correctly con�structed kernel follows from the fact that a linear com�bination of kernels is a kernel. To assign to the mixingparameters the meaning of the contribution made by a

K Mi Mj,( ) hkKk Mi Mj,( ),

k 1=

Nf

∑=

CHEMISTRY

Method of Continuous Molecular Fields in the Search for Quantitative Structure–Activity Relationships

N. I. Zhokhova, I. I. Baskin, D. K. Bakhronov, V. A. Palyulin, and Academician N. S. Zefirov

Received June 24, 2009

DOI: 10.1134/S0012500809110056

Moscow State University, Moscow, 119991 Russia

274

DOKLADY CHEMISTRY Vol. 429 Part 1 2009

ZHOKHOVA et al.

definite type of molecular field, we calculate the nor�malized version of the kernel:

(2)

In this case, the kernel value is represented by a lin�ear combination of the values of normalized kernelswith modified mixing coefficients:

(3)

For each kth type of molecular field, the kernel iscalculated by summation of the corresponding kernelsfor each pair of atoms of the ith and jth molecules:

(4)

where Kk( , ) is the kernel that shows the resem�blance between the molecular fields of the kth type ofthe lth atom in the ith molecule and the mth atom inthe jth molecule; Ni is the number of atoms in the ithmolecule; Nj is the number of atoms in the jth mole�cule. As in the previous case, that Kk(Mi, Mj) is a cor�rectly constructed kernel follows from the fact that alinear combination of kernels is a kernel. We suggest to

calculate the Kk( , ) kernel by integration of theproduct of the molecular fields of given atoms over theentire physical space �3:

(5)

where (x, y, z) is the potential of the molecular fieldof the kth type induced by the lth atom of the ith mol�ecule at the (x, y, z) point of the physical space, and

(x, y, z) is the same for the mth atom of the jth mol�ecule. To simplify integration, we approximate the

molecular field by the Gaussian function as is done inthe CoMSIA method:

(6)

where xil, yil, zil) are the Cartesian coordinates of the lthatom in the ith molecule, αk is the fitting parameter for

molecular fields of the kth type, and is the weight ofthe contribution of the lth atom of the ith molecule to the

molecular field of the kth type. The is taken to be thepartial charge on the lth atom of the ith molecule in thecase of an electrostatic field, the Lennard�Jones potentialparameters in the case of a steric field, and the contribu�tion of a given atom to the total hydrophobicity of a mol�ecule in the case of a lipophilic field. In this case, theabove integral is calculated analytically:

(7)

Thus, taking into account the above expressions,we can suggest the general formula for the calculationof the kernel for the pair of the Mi and Mj molecules,K(Mi, Mj):

(8)

In this case, the value of the predicted property yfor a new molecule M can be found by the formula

(9)

In addition to the set of parameters – and bdetermined by the method of support vector regres�sion, the MCMF method has a number of hyperpa�rameters. First of all, the method of support vectorregression ν�SVR itself contains two hyperparameters(ν and C), and their values should be optimized to

Kk' Mi Mj,( )Kk Mi Mj,( )

Kk Mi Mi,( )Kk Mj Mj,( )������������������������������������������������� .=

K Mi Mj,( ) hk' Kk' Mi Mj,( ).

k 1=

Nf

∑=

Kk Mi Mj,( ) Kk Ali

Amj,( ),

m 1=

Nj

∑l 1=

Ni

∑=

Ali

Amj

Ali

Amj

Kk Ali

Amj,( ) ρil

k x y z, ,( )ρjmk x y z, ,( ) xd yd z,d∫

�3

∫∫=

ρilk

ρjmk

ρilk x y z, ,( )

= wilk 1

2��αk x xil–( )2 y yil–( )2 z zil–( )2+ +( )–⎝ ⎠

⎛ ⎞ ,exp

wilk

wilk

Kk Ali

Amj,( ) ρil

k x y z, ,( )ρjmk x y z, ,( ) xd yd zd∫

�3

∫∫=

= wilkwjm

k 12��αk x xil–( )2 y yil–( )2 z zil–( )2+ +[ ]–

⎩ ⎭⎨ ⎬⎧ ⎫

exp∫�

3

∫∫

× 12��αk x xjm–( )2 y yjm–( )2 z zjm–( )2+ +[ ]–

⎩ ⎭⎨ ⎬⎧ ⎫

dxdydzexp

= wilk wjm

k π3

8αk3

�������

×αk

2���� xil xjm–( )2 yil yjm–( )2 zil zjm–( )2+ +[ ]–

⎩ ⎭⎨ ⎬⎧ ⎫

.exp

K Mi Mj,( ) hk'

wilk wjm

k αk

2���� xil xjm–( )2 yil yjm–( )2 zil zjm–( )2+ +[ ]–

⎩ ⎭⎨ ⎬⎧ ⎫

exp

m 1=

Nj

∑l 1=

Ni

∑

wilk( )

2

l 1=

Ni

∑ wjmk( )

2

m 1=

Nj

∑

����������������������������������������������������������������������������������������������������������������������k 1=

Nf

∑= .

y λj– λj

+–( )K M Mj,( )j 1=

Nj

∑ b.+=

λj– λj

+

DOKLADY CHEMISTRY Vol. 429 Part 1 2009

METHOD OF CONTINUOUS MOLECULAR FIELDS IN THE SEARCH 275

improve the predictive power of the model [5]. Inaddition, for each molecular field of the kth type, twohyperparameters are introduced: αk (the fittingparameter that shows the molecular field “decay” rate,which is related to the width of the Gaussian func�tion), is the mixing coefficient that has the meaningof the relative contribution of a molecular field of agiven type. In this work, the optimal values of this setof hyperparameters were found by maximizing the q2

value by successive use of two nonlinear programmingmethods: the simulated annealing method [6] and theNelder–Mead simplex method [7]. The computa�tional procedure of the MCMF method was imple�mented by us as a program package in C and Pythonlanguages.

To check the possibilities of the proposed methodfor constructing 3D QSAR models, we used a databasecontaining information on 72 structures of 3�amidi�nophenylalanine derivatives, as well as the data ontheir inhibiting activity with respect to three serineprotease enzymes—trypsin, thrombin, and factor Xa[8]. The general structure of 3�amidinophenylalaninederivatives is

.

The proteins used as targets for modeling the inhib�iting activity of the set have similar structural features:in the active site, the hydroxyl group of the serineamino acid functions as a catalyst. These proteins areoften used as potential targets for design of physiolog�ically active compounds. The structures of organiccompounds were selected with taking into accounttheir ability to inhibit all three enzymes. In this work,we used the experimental inhibition constants deter�mined by the Dixon method [9] and expressed as the

logarithm of the dissociation constant рKi (– ),which characterizes the degree of binding of the inhib�itor with the enzyme (Ki, mol/L).The predictive powerof the models was estimated by the leave�one�outcross�validation procedure. The quality of the modelswas estimated on the basis of the statistical character�istics q2 and RMSE, which were calculated by Eqs. (10)and (11).

(10)

hk'

NHR1 R2

HN NH2

O

H

Kilog

q2 1 PRESSSS

���������������– 1

yi yipred–( )

2

i 1=

n

∑

yi y–( )2

i 1=

n

∑����������������������������,–= =

5.5

4.0 4.5

pKi (prediction)

pKi (experiment)

5.0

5.0 5.5

4.5

6.0 (c)

7

4 5

pKi (prediction)

pKi (experiment)

6

6 7

5

8

(b)

8

7

4 5

pKi (prediction)

pKi (experiment)

6

6 7

5

8

(a)

8 9

8

Correlations of the experimental and predicted logarithmsof inhibition constants (pKi) obtained for the independenttest set with respect to (a) thrombin, (b) trypsin, and(c) factor Xa with the use of the MCMF.

276

DOKLADY CHEMISTRY Vol. 429 Part 1 2009

ZHOKHOVA et al.

where PRESS is the predictive sum of squares of differ�ences between the experimental (yi) and predicted

( ) values of the ith biological activity over all ncompounds included in the cross�validation proce�dure; SS is the sum of squared deviations of the exper�imental values (yi) from their arithmetic mean ( ) foreach ith biological activity.

(11)

where yi is the experimental biological activity of the

ith compound, is the predicted biological activityof the ith compound, and n is the number of com�pounds in the set.

The table presents the statistical characteristics ofthe models obtained for the inhibiting activity of thecompounds of the training set with respect to throm�bin, trypsin, and factor Xa by traditional 3D QSARmethods and the MCMF. For the CoMFA and CoM�SIA methods, the parameters of the models con�structed at the optimal grid cell size of 1.5, 1 and 1 Åfor thrombin, trypsin, and factor Xa, respectively, arepresented.

As follows from the table, all models obtained bythe MCMF have a better predictive power than themodels obtained by the common CoMFA and CoM�SIA methods. The largest contribution to the models ismade by steric molecular fields for thrombin, electro�static and hydrophobic molecular fields for trypsin,and hydrophobic molecular fields for factor Xa. Forfactor Xa, the range of the experimental pKi values israther small, which leads to significantly lower q2 val�ues for this enzyme as compared with analogous mod�els for thrombin and trypsin.

To independently estimate the predictive power ofthe models obtained by the proposed MCMF method,we used the test set comprising 16 3�amidinophenyl�alanine derivatives. These compounds were notincluded into the training set and were not used for

constructing 3D QSAR models. The scatter plots forthe experimental and predicted values of the inhibitingactivity for the compounds of the independent test setwith respect to thrombin, trypsin, and factor Xa areshown in the figure.

Thus, the models constructed with the use of theMCMF, especially when the thrombin� and trypsin�inhibiting activity is predicted, have a rather high pre�dictive power and allow one to reliably differentiatehigh�active compounds from low�active compounds.This is evidence that the models can be used for effi�cient design of novel biologically active structures. Theproposed method is free from some limitations inher�ent in the common 3D QSAR methods; i.e., the qual�ity of the models is independent of the alignmentinside a 3D grid and of the grid dimensions and gridcell size (the discreteness of estimated fields). Furtherdevelopment of the method will be focused on solvingthe problem of structure alignment and on choosingthe biologically active conformation of compounds.

REFERENCES

1. 3D QSAR in Drug Design. Theory, Methods and Applica�tions, Kubinyi, H., Ed, New York: Kluwer, 1997.

2. 3D QSAR in Drug Design, vol. 2: Ligand–Protein Inter�actions and Molecular Similarity, Kubinyi, H., Fol�kers, G., and Martin, Y.C., Eds, New York: Kluwer,1998.

3. 3D QSAR in Drug Design, vol. 3: Recent Advances, Kubi�nyi,H., Folkers, G., and Martin, Y.C., Eds, New York:Kluwer, 1998.

4. Ivanciuc, O., in Reviews in Computational Chemistry,Lipkowitz, K.B. and Cundari, T.R., Eds., Weinheim:Wiley–VCH, 2007, vol. 23, pp. 291–400.

5. Vapnik, V., Statistical Learning Theory, New York:Wiley�Interscience, 1998.

6. Kirkpatrick, S., Gelatt, C.D., and Vecchi, M.P., Sci�ence, New Ser., 1983, vol. 220, no. 4598, pp. 671–680.

7. Nelder, A. and Mead, R., Comput. J., 1965, vol. 7,pp. 308–313.

8. Böhm, M., Stürzebecher, J., and Klebe, G., J. Med.Chem., 1999, vol. 42, pp. 458–477.

9. Dixon, M., Biochem. J., 1953, vol. 55, pp. 170–171.

yipred

y

RMSE

yi yipred–( )

2

i 1=

n

∑n

����������������������������,=

yipred

Statistical characteristics of the models obtained by the CoMFA, CoMSIA, and MCMF methods for the compounds of thetraining set

ParameterThrombin Trypsin Factor Xa

CoMFA CoMSIA MCMF CoMFA CoMSIA MCMF CoMFA CoMSIA MCMF

q2 0.697 0.757 0.805 0.635 0.754 0.794 0.429 0.590 0.600

RMSE 0.378 0.244 0.210 0.204 0.159 0.137 0.294 0.194 0.189