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Keywords:

  • enhanced replacement method;
  • multiple linear regression;
  • p53–MDM2 interaction inhibitors;
  • QSAR;
  • support vector machine regression

Abstract

  1. Top of page
  2. Abstract
  3. Materials and Methods
  4. Results and Discussion
  5. Conclusions
  6. Acknowledgments
  7. References

The design and optimization of p53–MDM2 interaction inhibitors has attracted a great deal of interest in the development of new anticancer agents. Systematical 2D-QSAR studies on 98 isoindolinone-based p53–MDM2 interaction inhibitors were carried out using linear and the non-linear mathematical methods. At first, a forward stepwise-multiple linear regression model (FS-MLR) was proposed with reasonable statistical parameters (Rtrain2 = 0.881, Qloo2 = 0.847, Rtest2 = 0.854). Then, enhanced replacement method–multiple linear regression (ERM-MLR) and support vector machine regression (SVMR) were applied to set up more accurate models (ERM-MLR: Rtrain2 = 0.914, Qloo2 = 0.894 and Rtest2 = 0.903; SVMR: Rtrain2 = 0.924, Qloo2 = 0.920 and Rtest2 of 0.874). Furthermore, the reliability and application value of the ERM and SVMR model was also validated in virtual screening through receiver operating characteristic studies.

p53 plays a crucial role in regulating cell cycle, DNA repair proteins, and apoptosis in multi-cellular organisms and thus functions as a tumor suppressor (1,2). Under normal conditions, p53 is maintained at a low level regulated by MDM2 (mouse double minute-2) protein that acts as an E3 ubiquitin ligase and thus marks p53 for degradation by the proteasome (3). The interaction between MDM2 and p53 has been widely studied and validated as a key target for the development of new cancer therapeutic strategies (4–7). During the past few years, a number of compounds have been identified as p53–MDM2 interaction inhibitors, such as 4-phenyl-piperazines (8,9), norcamphanes (10,11), imidazolines (12,13), 1,4-benzodiazepine-2,5-diones (14). However, the mechanism(s) for their actions and structure–activities relationship (SAR) have not been clearly understood yet.

Quantitative structure–activities relationship (QSAR), an effective tool in computational chemistry, is extensively used in rational drug design (15,16). Several models have been established to analyze the QSAR for p53–MDM2 interaction inhibitors, leading to identification of several common features that are very useful in designing novel inhibitors. Galatin et al. (17) reported the generation of non-covalent interaction measurements between peptide inhibitors and MDM2 using hydropathic interactions (HINT) program. Hu and colleagues (18) also reported the identification of a pharmacophore model, including one ring aromatic, three hydrophobic group, and one H-bond acceptor, and demonstrated that these topological features are essential for the interaction between MDM2 and its inhibitors. Recently, molecular descriptors, which are used to describe the structure and physico-chemical properties of molecules, are applied increasingly in helping predict the biological activity of molecules. It should be noted that the selection of appropriate molecular descriptors and computational techniques are essential factors to the successful construction of QSAR models (19,20). Several methods such as forward stepwise regression (FSR) (21), genetic algorithms (GA) (22), and replacement method (RM) (23) are available to search an optimal set of descriptors. Besides, a certain number of mathematical algorithms including linear and non-linear methods are useful for probing the relationships between descriptors and activities (24). For these methods, the meaning of the equations obtained by linear model is clear and easy to be handled, while the flexibility of non-linear models enable them to discover more complex relationships within experimental data. Thus, there is a growing interest in applying both the linear and the non-linear mathematical methods to systemically study the quantitative relationship between structure and activities of the molecules in a chemical series.

Herein, forward stepwise multiple linear regression (FS-MLR), enhanced replacement method multiple linear regression (ERM-MLR), and support vector machine regression (SVMR) were used to establish both linear and non-linear QSAR models of 98 isoindolinone-based p53–MDM2 interaction inhibitors (25–27), the SAR studies of which had been most extensively studied in all identified p53–MDM2 interaction inhibitors. Intercorrelative analysis, forward stepwise regression (FSR), enhanced replacement method (ERM), and support vector machine (SVM) feature selection tool were used to confirm the optimal descriptors. Then, receiver operating characteristic (ROC) studies were performed to validate the reliability and application value of the established models.

Materials and Methods

  1. Top of page
  2. Abstract
  3. Materials and Methods
  4. Results and Discussion
  5. Conclusions
  6. Acknowledgments
  7. References

Data preparation

For the QSAR studies, a total of 98 isoindolinone-based p53–MDM2 interaction inhibitors with available IC50 were selected from literatures (25–27). The data set was randomly split into a training set of 75 compounds and a test set of 23 compounds. Also, a negative control set of 979 compounds used in ROC studies were randomly retrieved from the Available Chemical Directory (ACD) database (Symyx Technologies, Santa Clara, CA, USA) using the ‘Random Percent Filter protocol’ by Pipeline Pilot software (SciTegic, Inc., San Diego, CA, USA). All the compounds were minimized using CHARMm force field implemented in Discovery Studio 2.0 software (Accelrys, Inc. San Diego, CA, USA). The resulted geometries of molecules were input to Dragon software,a which can calculate constitutional descriptors, topological descriptors, walk and path counts, information indices, 2D autocorrelations, edge adjacency indices, Burden eigenvalue descriptors, etc. Then, a total of 1664 molecular descriptors of different types were calculated to describe each molecule. The descriptors stayed constant for all molecules were eliminated. The pairs of variables with a correlation coefficient greater than 0.75 were classified as intercorrelated, and one of each correlated pair was deleted. As a result, a pool of 486 descriptors were selected for further QSAR model development.

Mathematical algorithms

With the aim to establish both the linear and non-linear QSAR models, three different statistical methods were applied, including FS-MLR, ERM-MLR, and SVMR.

FS-MLR method

Multiple linear regression is a statistical tool that correlates independent variables against a dependent variable. The objective of MLR is to find a linear model of interest in terms of properties, which takes the form below:

y = a0 + a1x1 + a2x2… + anxn

where y is the dependent variable, ai (i: 1,…, n) represent the coefficients of those descriptors, xi (i: 1,…, n) represents molecular descriptors, and a0 is the intercept of the equation.

ERM-MLR method

Enhanced replacement method proposed by Mercader et al. (23) is a modified version of replacement method (28,29). The purpose of both algorithms is to choose an optimal subset of d (d < D) descriptors from the pool of D descriptors with minimum standard deviation SD:

  • image

where N is the number of molecules in the training set and resi the residual for molecule I (difference between the experimental and predicted property).

Considering that SD(dn) is a distribution on a dicrete space of D!/d!(D − d) disordered points dn, ERM produces linear models that are quite similar with the full search (FS), but with much less computational work. First, an initial set of descriptors dk was selected at random. And one of the descriptors, say Xki, with all the remaining descriptors (D − d) was replaced by other descriptor, one by one, and the set with the smallest value of SD was kept. Second, from the resulting set, the descriptor with the greatest SD in its coefficient is chosen and all the remaining D − d descriptors, one by one, until the set remains unmodified. More detailed information about these algorithms can be found in reference (23).

SVMR method

A detailed description of SVM theory can be found in several excellent books and tutorials (30,31). Here, only a brief description is given. The basic idea of SVM regression is to map the data x into a higher-dimensional feature space F via a nonlinear mapping φ and then to do linear regression in this space. Therefore, regression approximation addresses the problem of estimating a function based on a given data set inline image (xi is the input vector, di is the desired value, and n is the total number of data patterns). SVM regression approximates the function in the following form:

  • image(1)

where φ(x) is the high-dimensional feature space, which is non-linearly mapped from the input space x, and wi and b are coefficients. They are estimated by minimizing the regularized risk function R (C)

  • image(2)

where

  • image(3)

The first term inline image is the so-called empirical error (risk), which is measured by the ε-insensitive loss function (3). The second term inline image, on the other hand, is called the regularized term. ε is called the tube size of SVM, and C is the regularization constant determining the trade-off between the empirical error and the regularized term. Introduction of slack variables ‘ξ’ leads eqn 2 to the following constrained function:

  • image(4)

Thus, decision function of eqn 1 changes to the following form:

  • image(5)

where αi and inline image are the introduced Lagrange multipliers and K(x, xi) is the kernel function. The value is equal to the inner product of two vectors x and xi in the feature space Φ(x) and Φ(xi), that is, K(x, xi) = Φ(x)TΦ(xi). And the radial basis function (RBF) kernel inline image is commonly used.

Leave-one-out cross-validation

The leave-one-out cross-validation (LOO-CV), a special case of the cross-validation technique (32), was employed to find the promising QSAR model. Given n samples available in a data set and m candidate models, each model is trained with n − 1 samples and then is tested on the sample that was left out. This process is repeated n times until every sample in the data set has been used once as a cross-validation instance. Finally, cross-validation correlation coefficient (inline image) of LOO-CV, a measure of the model generalization capability, for all candidate models can be obtained as below:

  • image

where yi is the desired output, inline image is the predicted value by model, and ym is the mean value of dependent variable.

Receiver operating characteristic studies

The generated models were further validated by ROC studies in screening an database that was retrieved randomly from ACD database (979 molecules as negative control) spiked with some known p53–MDM2 interaction inhibitors (15 molecules as positive control, Table 1). The ROC study (33) was performed to calculate true-positive rate (TPR) and false-positive rate (FPR) from a comparison between in vitro and in silico.

  • image
Table 1.   Structure, experimental, and estimated activities of isoindolinones-based p53-MDM2 interaction inhibitors Thumbnail image of

where TP is the number of true-positive compounds and FN is the number of false-negative compounds.

  • image

where TN is the number of true-negative compounds and FP is the number of false-positive compounds.

The ROC curve is a function of FPR versus the TPR, and the area under the ROC curve (AUC) is the important way of measuring the performance of the test.

  • image

where TPR(x) is the percent of the true positives versus the total positives at rank position x and FPR(x) is the percent of the false positives versus the total negatives at rank position x.

Results and Discussion

  1. Top of page
  2. Abstract
  3. Materials and Methods
  4. Results and Discussion
  5. Conclusions
  6. Acknowledgments
  7. References

FS-MLR model development

The relationship between p53–MDM2 inhibitory activities and 297 molecular descriptors of compounds 1–98 were initially analyzed by the FS-MLR. According to the number of variables selected should be kept lower than 20 percent of the number of molecules, the tolerance value in developing of FS-MLR model was adjusted to limit the number of descriptors. As a result, six preferable variables (Table 2) were selected to form the equation as given in eqn 6:

  • image(6)
Table 2.   The involved molecular descriptors and their corresponding definitions
ModelSymbolClassDefinition
  1. SVMR, support vector machine regression.

FS-MLRHOMTGeometrical descriptorsHOMA total
GATS1v2D autocorrelationsGeary autocorrelation – lag 1/weighted by atomic van der Waals volumes
CIC1Information indicesComplementary information content (neighborhood symmetry of 1-order)
Mor23v3D-MoRSE descriptors3D-MoRSE – signal 23/weighted by atomic van der Waals volumes
Mor23m3D-MoRSE descriptors3D-MoRSE – signal 23/unweighted by atomic masses
Mor25m3D-MoRSE descriptors3D-MoRSE – signal 25/unweighted by atomic masses
ERM-MLR (n = 6)H2uGETAWAY descriptorsH autocorrelation of lag 2/unweighted
HATS6uGETAWAY descriptorsLeverage-weighted autocorrelation of lag 6/unweighted
MATS7v2D autocorrelationsMoran autocorrelation – lag 7/weighted by atomic van der Waals volumes
RDF010pRDF descriptorsRadial Distribution Function – 1.0/weighted by atomic polarizabilities
RDF085mRDF descriptorsRadial Distribution Function – 8.5/weighted by atomic masses
Mor23e3D-MoRSE descriptors3D-MoRSE – signal 23/weighted by atomic Sanderson electronegativities
SVMRH2eGETAWAY descriptorsH autocorrelation of lag 2/weighted by atomic Sanderson electronegativities
HATS6uGETAWAY descriptorsLeverage-weighted autocorrelation of lag 6/unweighted
CIC3Information indicesComplementary information content (neighborhood symmetry of 3-order)
MATS7v2D autocorrelationsMoran autocorrelation – lag 7/weighted by atomic van der Waals volumes
RDF010pRDF descriptorsRadial Distribution Function – 1.0/weighted by atomic polarizabilities
RDF085mRDF descriptorsRadial Distribution Function – 8.5/weighted by atomic masses
Mor23e3D-MoRSE descriptors3D-MoRSE – signal 23/weighted by atomic Sanderson electronegativities

The FS-MLR model shows the good correlation coefficient, inline image of 0.881 and inline image of 0.854, and cross-validation correlation coefficient inline image of 0.847. As shown in Table 1, most of the compounds were predicted correctly. Then, the predictions in function of the experimental coefficients are plotted in Figure 1B, suggesting that the 98 data point (overall data set) follow a straight line trend. However, it was found that the prediction residue of some compounds seems a little high, such as compounds 44, 57, 58, 74, and 98 (the deviation of FS-MLR for 44, 57, 58, 74, and 98 are 0.733, 0.689, −0.525, 0.996, and 0.841). It encourages us to develop more accurate QSAR model using ERM-MLR and SVMR.

image

Figure 1.  Predicted versus experimental activities (−LogIC50) of isoindolinone-based p53–MDM2 interaction inhibitors: (A) FS-MLR model, (B) EMR-MLR model, and (C) SVMR model.

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ERM-MLR model development

As shown in Table 3, seven developed ERM-MLR models, the descriptor number of which was between 4 and 10, were obtained. A set of six molecular descriptors in these models has an optimal influence on improving correlation, taking account of the good prediction for both training set and test set. The physical–chemical meanings of the molecular descriptors in the best model (n = 6) are listed in Table 2, forming the equation as given in eqn 7:

  • image(7)
Table 3.   The statistical parameters of obtained ERM-MLR models
No.Training setTest setinline imageFna
inline imageSDinline imageSD
  1. aThe number of molecular descriptors selected in ERM-MLR QSAR models.

10.8830.3320.8810.3630.863131.84
20.8970.3110.8910.3550.877120.45
30.9140.2850.9030.3410.894119.86
40.9250.2650.8630.4150.906118.57
50.9370.2430.8750.3970.916123.38
60.9440.2240.8860.4330.930128.39
70.9540.2050.8730.3700.941137.610

The quality of the fit for the resulting model (eqn 7) could be judged by the inline image of 0.913, the SD value of 0.285, and the inline image of 0.738 for training set. Its predictive ability is characterized by the inline image of 0.903 and the SD value of 0.341 for test set. The predictions by eqn 7 are shown in Table 1. Comparing the results of FS-MLR and ERM-MLR, models indicated that parameters of ERM-MLR are preferable than that of FS-MLR, may be because of the promising effect of enhanced replacement method in selecting the optimal set of molecular descriptors. However, the prediction of the compounds that are not estimated well by FS-MLR is not remarkably improved by ERM-MLR (the deviation of FS-MLR for 44, 57, 58, and 74 are 0.733, 0.689, −0.525, 0.996, and 0.841, respectively; the deviation of ERM-MLR for 44, 57, 58, and 74 are 0.355, 0.486, −0.385, 0.444, and 0.313, respectively). The reason for this problem may be that both of these models were developed using multiple linear regression method, resulting in the development of non-linear regression model to solve the problem.

SVMR model development

At first, the molecular descriptors in SVMR model were selected by FSR and SVM selection tool, and their physical–chemical meanings are listed in Table 2. Then, the capacity parameter C, ε, and the corresponding parameters γ of RBF kernel were needed to be optimized. Initially, the optimal parameters were found by grid search (GS) method. A robust model was obtained by selecting those parameters that give the lowest error in a smooth area. To find the optimal parameters C, ε, and γ, a process of LOO-CV of training set was performed as shown in Figure 2. The best choices for C, ε, and γ are 100, 0.005, and 0.48, respectively, and the corresponding support vector number is 9. The predicted results of SVMR model are listed in Table 1, and the plots of predicted versus experimental values for training and test set are recorded in Figure 1C. The SVMR model shows the good correlation coefficient inline image of 0.924, the inline image of 0.920, and the inline image of 0.874, the result of which was also better than that of FS-MLR model. Interestingly, compounds 44, 57, 58, 74, and 98, which are not estimated well by FS-MLR and ERM-MLR models, could be predicted accurately by SVMR model (the deviation of FS-MLR for 44, 57, 58, 74, and 98 are 0.733, 0.689, −0.525, 0.996, and 0.841, respectively; the deviation of SVMR for 44, 57, 58, 74, and 98 are 0.100, 0.100, −0.172, 0.228, and 0.313, respectively). Therefore, combination utilization of linear and non-linear regression methods could further enhance the prediction ability.

image

Figure 2.  Selection of C, γ, and ε for the training set data in the support vector machine regression model development: (A) C versus inline image on LOO cross-validation. (B) γ versus inline image on LOO cross-validation. (C) ε versus inline image on LOO cross-validation.

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Description of the model descriptors in QSAR models

One of the purposes of QSAR analyses is to understand the forces governing the activity of a particular class of compounds and to assist drug design. Therefore, the evaluation of the descriptors relevance is proven quite interesting and useful to shed more light on the structure–activity relationship. In developed QSAR models, several class descriptors are found to be remarkably related to biological activities, such as 3D-MoRSE (relating to atomic polarizability of the molecules), 2D autocorrelations descriptors (describing the autocorrelation of topological structures), RDF descriptors (representing the molecular conformation in 3D), and GETAWAY descriptors (deduced by the centering all of the atomic coordinates). Among these descriptors, we also found that several ones are related to Van der Waals volumes (GATS1v, Mor23v and MATS7v) and polarizabilities (RDF010p), indicating that these two features play an important role in the p53–MDM2 interaction inhibitors. The result is consistent with the importance of hydrophobic interaction between MDM2 and its inhibitors (34). In addition, the 3D-MoRSE, 2D autocorrelations, and GETAWAY descriptors, which belong to the 2D or 3D topological features, are used to describe the conformational requirement of the hydrophobic moieties of the p53–MDM2 inhibitors.

Receiver operating characteristic studies

The ERM-MLR and SVMR models were further applied to evaluate its ability in picking up active molecules (p53–MDM2 interaction inhibitors) from an in-house database that included 979 inactive compounds (randomly from ACD database) and 15 known inhibitors (labeled in Table 1). The result of ROC study is shown in the Figure 3. The ERM-MLR and SVMR model picked up all 15 known inhibitors with high AUC value of 0.846 and 0.787, respectively, showing that the accurate rate of choosing true-positive compounds by ERM-MLR and SVMR model are much higher than that of choosing randomly.

image

Figure 3.  Receiver operating characteristic (ROC) studies of the ERM-MLR and support vector machine regression (SVMR) model in picking up positive compounds from in-house database: (A) ERM-MLR model with area under the ROC curve (AUC) value of 0.846; (B) SVMR model with AUC value of 0.787.

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Conclusions

  1. Top of page
  2. Abstract
  3. Materials and Methods
  4. Results and Discussion
  5. Conclusions
  6. Acknowledgments
  7. References

In this work, we applied both linear and non-linear methods to explore the relationships between molecular descriptors and biological activities of isoindolinone-based p53–MDM2 interaction inhibitors. Both of ERM-MLR and SVMR methods showed reliable and good prediction ability in establishing QSAR models. The preferable models were further validated as good tools in picking up active molecules from the in-house database using ROC methods. The prediction from the results showing good correlation could serve as a good guideline for structural modifications of isoindolinone-based compounds and should be of aid in designing p53–MDM2 interaction inhibitors with better activity.

Footnotes

Acknowledgments

  1. Top of page
  2. Abstract
  3. Materials and Methods
  4. Results and Discussion
  5. Conclusions
  6. Acknowledgments
  7. References

The authors are grateful to the support provided by the National Natural Science Foundation of China (NO.30873163), the China Postdoctoral Science Foundation (NO. 20090460103, 201003734), and the Educational Commission of Zhejiang Province, China (No. Y201018915).

References

  1. Top of page
  2. Abstract
  3. Materials and Methods
  4. Results and Discussion
  5. Conclusions
  6. Acknowledgments
  7. References
  • 1
    Klein C., Vassilev L. (2004) Targeting the p53-MDM2 interaction to treat cancer. British J Cancer;91:14151419.
  • 2
    Vousden K.H., Lane D.P. (2007) p53 in health and disease. Nat Rev Mol Cell Biol;8:275283.
  • 3
    Harris S.L., Levine A.J. (2005) The p53 pathway: positive and negative feedback loops. Oncogene;24:28992908.
  • 4
    Toledo F., Wahl G.M., (2006) Regulating the p53 pathway : in vitro hypotheses, in vivo veritas. Nat Rev Cance;6:909923.
  • 5
    Chene P. (2003) Inhibiting the p53-MDM2 interaction: an important target for cancer therapy. Nat Rev Cancer;3:102109.
  • 6
    Hollstein M., Sidransky D., Vogelstein B., Harris C.C. (1991) p53 mutations in human cancers. Science;253:4953.
  • 7
    Zheleva D.I., Lane D.P., Fischer P.M. (2003) The p53-Mdm2 pathway: targets for the development of new anticancer therapeutics. Mini-Rev Med Chem;3:257270.
  • 8
    Luke R.W.A., Hudson K., Hayward D.F., Fielding C., Cotton R., Best R., Giles M.B., Veldman M.H., Griffiths L.A., Jewsbury P.J., Breeze A.L., Embrey K.J. (1999) Design and synthesis of small molecule inhibitors of the MDM2-p53 interaction as potential anti-tumor agents. Proc Am Assoc Cancer Res;40:abs.: 4099.
  • 9
    Luke R.W.A., Jewsbury P.J, Cotton R. (2000) Preparation of amino acid and peptidyl piperazine-4-phenyl derivatives as inhibitors of the interaction between MDM2 and p53. PCT Int. Pat. Appl. Publ. WO 2000015657. Zeneca Ltd., UK.
  • 10
    Zhao J.H., Liu Z.H., Yin D.L., Chen J., Luo A.P., Wu M. (2001) Initial Evaluation of mdm2 Inhibitors Based on p53- mdm2 Complex Structure. Chin J Cancer;20:354357.
  • 11
    Zhao J., Wang M., Chen J., Wang A.L.X., Wu M., Yin D., Liu Z. (2002) The initial evaluation of non-peptidic small-molecule HDM2 inhibitors based on p53-HDM2 complex structure. Cancer Lett;183:6977.
  • 12
    Tovar C., Rosinski J., Filipovic Z., Higgins B., Kolinsky K., Hilton H., Zhao X., Vu B.T., Qing W., Packman K., Myklebost O., Heimbrook D.C., Vassilev L.T. (2006) Small-molecule MDM2 antagonists reveal aberrant p53 signaling in cancer: implications for therapy. Proc Natl Acad Sci U S A;103:18881893.
  • 13
    LaRusch G.A., Jackson M.W., Dunbar J.D., Warren R.S., Donner D.B., Mayo L.D. (2007) Nutlin3 blocks vascular endothelial growth factor induction by preventing the interaction between hypoxia inducible factor 1alpha and Hdm2. Cancer Res;67:450454.
  • 14
    Pantoliano M.W., Petrella E.C., Kwasnoski J.D., Lobanov V.S., Myslik J., Graf E., Carver T., Asel E., Springer B.A., Lane P., Salemme F.R. (2001) High-density miniaturized thermal shift assays as a general strategy for drug discovery. J Biomol Screen;6:429440.
  • 15
    Yap C.W., Li H., Ji Z.L., Chen Y.Z. (2007) ???????. Mini-Rev Med Chem;7:10971107.
  • 16
    Estrada E. (2008) How the parts organize in the whole? A top-down view of molecular descriptors and properties for QSAR and drug design. Mini-Rev Med Chem;8:213221.
  • 17
    Galatin P.S., Abraham D.J. (2001) QSAR: hydropathic analysis of inhibitors of the p53-mdm2 interaction. Proteins;45:169175.
  • 18
    Rong S., Hu C., Huang W., Hu Y. (2007) Pharmacophore Model Construction of p53-MDM2 Binding Inhibitors. Acta Phy Chim Sin;23:18151820.
  • 19
    Hemmateenejad B., Sanchooli M. (2007) Substituent electronic descriptors for fast QSAR/QSPR. J Chemometrics;21:96107.
  • 20
    Wan J., Zhang L., Yang G., Zhan C.G. (2004) Quantitative structure-activity relationship for cyclic imide derivatives of protoporphyrinogen oxidase inhibitors: a study of quantum chemical descriptors from density functional theory. J Chem Inf Comput Sci;44:20992105.
  • 21
    Draper N.R., Smith H. (1981) Applied Regression Analysis. New York: John Wiley&Sons.
  • 22
    Kubinyi H. (1996) Evolutionary Variable Selection in Regression and PLS Analyses. J Chemometrics;10:119133.
  • 23
    Mercader A.G., Duchowicz P.R., Fernández F.M., Castro E.A. (2008) Modified and enhanced replacement method for the selection of molecular descriptors in QSAR and QSPR theories. Chemom Intell Lab Syst;92:138144.
  • 24
    Liu P., Long W. (2009) Current mathematical methods used in QSAR/QSPR studies. Int J Mol Sci;10:19781998.
  • 25
    Hardcastle I.R., Ahmed S.U., Atkins H., Calvert A.H., Curtin N.J., Farnie G., Golding B.T. et al. (2005) Isoindolinone-based inhibitors of the MDM2-p53 protein-protein interaction. Bioorg Med Chem Lett;15:15151520.
  • 26
    Hardcastle I.R., Ahmed S.U., Atkins H., Farnie G., Golding B.T., Griffin R.J., Guyenne S. et al. (2006) Small-molecule inhibitors of the MDM2-p53 protein-protein interaction based on an isoindolinone scaffold. J Med Chem;49:62096221.
  • 27
    Hardcastle I.R., Liu J., Valeur E., Watson A., Ahmed S.U., Blackburn T.J., Bennaceur K. et al. (2011) Isoindolinone inhibitors of the murine double minute 2 (MDM2)-p53 protein-protein interaction: structure-activity studies leading to improved potency. J Med Chem;54:12331243.
  • 28
    Duchowicz P.R., Castro E.A., Fernández F.M. (2006) Alternative algorithm for the search of an optimal set of descriptors in QSAR-QSPR studies. MATCH Commun Math Comput Chem;55:179192.
  • 29
    Duchowicz P.R., Fernández M., Caballero J., Castro E.A., Fernández F.M. (2006) QSAR for non-nucleoside inhibitors of HIV-1 reverse transcriptase. Bioorg Med Chem;14:58765889.
  • 30
    Vapnik V.N. (1995) The Nature of Statistical Learning Theory. Berlin: Springer.
  • 31
    Burges C.J.C. (1998) A Tutorial on Support Vector Machines for Pattern Recognition. Data Min Know Discovery;2:121167.
  • 32
    Gong G. (1986) Cross-validation, the Jackknife, and the bootstrap: Excess error estimation in forward logistic regression. J Am Stat Assoc;81:108113.
  • 33
    Fawcett T. (2006) An introduction to ROC analysis. Pattern Recogn Lett;27:861874.
  • 34
    Clackson T., Wells J. (1995) A hot spot of binding energy in a hormone-receptor interface. Science;267:3386.