OlaposiIdowu Omotuyi^{*} and Hiroshi Ueda  
Department of Molecular Pharmacology and Neuroscience, Nagasaki University Graduate School of Biomedical Sciences, 8528521, Nagasaki, Japan  
Corresponding Author :  Omotuyi IO Department of Molecular Pharmacology and Neuroscience Nagasaki University Graduate School of Biomedical Sciences 8528521, Nagasaki, Japan Email: [email protected] 
Received May 31, 2013; Accepted July 26, 2013; Published July 29, 2013  
Citation: Omotuyi O, Ueda H (2013) Descriptorbased Fitting of Structurally Diverse LPA1 Inhibitors into a Single predictive Mathematical Model. J Phys Chem Biophys 3:121. doi: 10.4172/21610398.1000121  
Copyright: © 2013 Omotuyi O, et al. This is an openaccess article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. 
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120 structurally diverse compounds previously reported as LPA1 inhibitors have been used to derive a mathematical model based on their descriptors. The pre and postcrossvalidated correlation coefficient (R2) is 0.79168 (RMSE=0.61459) and 0.70939 (RMSE=0.72938) respectively. Principal component analysis (PCA) was also used to reduce the dimension and linearly transform the raw data. PCA results showed that nine (9) principal components sufficiently accounts for more than 98% of the variance of the dataset with a fitting mathematical equation. Our model accurately predicted ~86% of the compounds tested regardless of their structural diversities.
Keywords 
LPA1; Antagonists; Mathematical model; PCA; Descriptors 
Introduction 
In situations where protein Xray structure or related structures for templatebased homology modeling are unavailable for structurebased virtual screening, computational methods for drug design rely principally on ligandbased approaches. Ligandbased approach depends on at least one known active compound; which serves as the query for searching library of compounds using predefined molecular descriptor parameters [1,2]. Three categories of chemical descriptors have been characterized till date; physical properties descriptors (1Ddescriptor), molecular topology and pharmacophore descriptors (2Ddescriptors) and geometrical descriptors (3Ddescriptors, often requires prior knowledge of target protein bindingpocket) [35]. When there are multiple bioactive compounds for a given target, quantitative structure activity relationships (QSARs) method is more beneficial. QSAR method provides predictive mathematical model for biological activities using statistical clustering of multiple descriptors variables [6,7]. We sought to derive a mathematical equation from minimal set of ligand descriptors for set of Lysophosphatidic acid receptor (LPA1) inhibitors. With this equation, we hope to accurately predict the activity of a test set and hopefully used in ligandbased virtual screening for new highaffinity LPA1 antagonists. 
Materials and Methods 
Here, using Molecular Operating Environment (MOE) [8], multiple descriptors (SlogP (SlogP_VSA06), SMR (SMR_VSA04), a_acc, ASA, E_stb, a_hyd, and Kier (Kier12, KierA12)) [8] have been generated for training set of compounds (CHEMBL3819) in order to establish a mathematical equation to model LPA1 inhibition (antagonism). PCA analysis was also conducted to determine the principle components of the equation using scientific vector language (SVL) programming built into the MOE. 
Results and Discussion 
First, The IC50 values of 134 unique entries (LPA1 inhibitors) from ChemBL database (CHEMBL3819) were converted to Gibb’s free energy of binding using ChengPrusoff equation [9] {Equation1} at S<<<Km {Equation 2} approximation at 298K. 
(1) 
(2) 
The library was randomly and unbiasedly grouped OCHEM server [10] into the training (120 compounds, Supplementary Figure 1) and test (14 compounds) sets. The training set was initially fitted using partial least square (PLS) method into all Chemical descriptors implemented in MOE [8]. The descriptors were pruned in order of their relative importance until a mathematical model (Equation 3) was obtained. 
dGExpt = 3.0345 0.39537 x a_acc 0.02183 x ASA 0.36027 x a_hyd 0.01028 x E_stb +0.64979 x Kier1 +0.21026 x Kier2 +0.08358 x KierA1 0.47849 x KierA2 +0.03617 x SlogP_VSA0 +0.01945 x SlogP_ VSA1 +0.00494 x SlogP_VSA2 0.00339 x SlogP_ VSA3 +0.01846 x SlogP_VSA4 +0.05076 x SlogP_VSA6 0.06603 x SMR_VSA0 0.05469 x SMR_VSA1 0.03451 x SMR_VSA2 +0.00294 x SMR_VSA3 0.01021 x SMR_VSA4 (Equation 3) 
This model gives a high probabilistic (r^{2}=0.79168 with RMSE of 0.61459 dGexpt) Gibb’s free energy prediction using minimal set of descriptors (Figure 1). A crossvalidated correlation coefficient value of 0.70939 (RMSE = 0.72938) was also obtained for the model. 
These results suggest that the set of descriptors chosen can effectively cluster the minimal structural and molecular parameters required for the predicting relatively small differences in the ligand activity of structurally diverse compounds typifying the training set. 
Due to the relatively good mathematical correlation between the descriptors and the estimated free energy of ligand binding, we sought to further study the dataset descriptors long the principle components through the reduction of the dimensionality and linear transformation of the raw data (Principal component analysis (PCA)) [11]. Given the initial 120 compounds (represented as m) and for one of the compounds say ‘i’ its descriptors are represented by nvector of real numbers xi=(xi1,..,xin, where n=117). Assuming that each molecule ‘i’ has an associated importance weight ‘wi’, (nonnegative, real number) and that the weights is relative probability that the associated molecule ‘xi’ will be encountered (adding up to 1); If ‘W’ denotes the sum of all the weights then, the eigenvalues and eigenvectors for the final data are estimable from the raw data using equation (4) where S is a symmetric, semidefinite sample covariance matrix. S can be diagonalized such that S =Q^{T}DDQ (Q is orthogonal, D is diagonalsorted in descending order from top left to bottom right) [12]. 
(4) 
The effect of the each of the principal components (eigenvectors) on the condition and the variance (Supplementary Table 1) shows that nine (9) principal components sufficiently accounts for more than 98% of the variance in the dataset with a fitting mathematical equation (Equation 5). The 3Dscatter plot of the last three principal components (PCA7, PCA8 and PCA9) with respect to free energy is shown in Figure 2; each point in the plot corresponds to a molecule colored according to free energy values. 
PCA9 = 5.53218413e001 1.47174139e003 X ASA 5.28867555e 004 X E_stb  9.64502253e003 X Kier1 +2.92612997e002 X Kier2 9.05227786e004 X KierA1+2.57936088e002 X KierA2 +4.04361621e 002 X SMR_VSA02.37125484e002 X SMR_VSA1 +5.03998977e002 X SMR_VSA2 +8.13078695e003 X SlogP_VSA0  1.03630885e002 X SlogP_VSA1 5.72337043e002 X SlogP_VSA2 1.64177905e003 X SlogP_VSA3 7.55989243e002 X SlogP_VSA4 1.02026342e002 X SlogP_VSA6 +1.79553609e001 X a_acc 3.68295238e002 X a_hydPCA9 = 5.53218413e001 1.47174139e003 X ASA 5.28867555e 004 X E_stb  9.64502253e003 X Kier1 +2.92612997e002 X Kier2 9.05227786e004 X KierA1+2.57936088e002 X KierA2 +4.04361621e 002 X SMR_VSA02.37125484e002 X SMR_VSA1 +5.03998977e002 X SMR_VSA2 +8.13078695e003 X SlogP_VSA0  1.03630885e002 X SlogP_VSA1 5.72337043e002 X SlogP_VSA2 1.64177905e003 X SlogP_VSA3 7.55989243e002 X SlogP_VSA4 1.02026342e002 X SlogP_VSA6 +1.79553609e001 X a_acc 3.68295238e002 X a_hyd (Equation 5) 
When equation 3 was used to predict the Gibb’s free energy of the test set, it predicted accurately (residual free energy > +1.0) ~86% of the compounds regardless of their structural diversities (Figure 3). 
Conclusion 
Given the predictive finesse of this mathematical model, there is a question to be answered and two areas of potential applications to be exploited. Will this model sufficiently predict more chemically diverse compounds? If the yes, then we can predict a more robust interrelationship between statistics and ComputerAided Drug Discovery in the future. Also, descriptorbased mathematical model screening may be piped as confirmatory steps following structurebased screening for more successful hitcompound identification. 
Acknowledgements 
This work was supported by Platform for Drug Discovery, Informatics, and structural life Science from the ministry of Education, Culture, Sports, Science and Technology, Japan. 
References 

Figure 1  Figure 2  Figure 3 