Capillary electrophoresis (CE) has become an important separation technique and alternate to other analytical methods like high performance liquid chromatography (HPLC) due to its simple preprocessing,high separation efficiency,low operating costs and solvent consumption. It has been applied widely in the analysis of both small and large molecules,such as inorganic ions, organic acids,carbohydrates,pharmaceuticals,and even living cells ^[1]. Generally,the migration behavior in CE is denoted by electrophoretic mobility (μ),which is dependent on both the molecular structure and separation conditions. Prediction of the mobility of solutes by theoretical methods will relieve analysts of a large number of costly and time-consuming experiments in order to develop a faster optimization process in CE. Many investigators have paid attention to this problem,some of whom have contributed to the study of the quantitative relationship between molecular structures and electrophoretic mobilities. Based upon the published reports,two principal methods can be summarized: the mechanistic and the statistical methods. The mechanistic models are closely related to the mechanisms of electrophoretic separation. The basic expression for such method is Max Born’s model ^[2]:

Electrophoretic mobility is one of the most important parameters governing the separation of solutes in CE. It is possible to have a sufficient understanding of the separation mechanism and provide a useful tool for a faster method optimization process in CE if a model correlating the migration behavior of an analyte under certain practical conditions to its structural parameters can be developed. During methods development in high-performance capillary electrophoresis (HPCE),the analysts generally have to examine a large number of experiments,which are often costly and time-consuming. Alternatively,quantitative structure-migration relationship (QSMR) provides a promising method for the estimation of analyte mobilities based upon the descriptors derived solely from the molecular structures fit to the experimental data. The advantage of this approach over other methods lies in the fact that it requires only the knowledge of chemical structures, and is not dependent on any experimental properties. The main steps involved in QSMR include data collection,molecular geometry optimization,molecular descriptor generation,descriptor selection,model development,and finally,model performance evaluation ^[3]. This study can be used to develop a method for prediction of the properties of the new compounds that have not yet been synthesized or found. It can also identify and describe important structural features of the molecules that are relevant to variations in molecular properties,and thus gain some insight into the structural factors affecting solute migration.

Although methods based on the quantitative structure- property relationship have been successfully used to predict many physico-chemical properties,only a few research groups have investigated the quantitative correlation between the analytical parameters and the responses obtained in HPCE [4- 7]. Goudarzi et al.have used least square-support vector machine (LS-SVM) to predict the logarithmic n-octanol/water partition coefficient (log P) of some phenolic derivatives ^[8]. They have also studied the support vector machine (SVM),principal component regression (PCR),partial least squares (PLS),and multiple linear regression (MLR) to predict pK_a of some organic compounds ^[9]. Fatemi and Goudarzi have utilized MLR and ANN to predict the electrophoretic mobilities of some benzoic acid derivatives in different solvents ^[10]. There are also several other reports based on quantitative structure-property/activity relationship studies ^{[11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]}

Recently,quantitative structure-property relationship has been involved in modeling to investigate mobility as a property of compounds and the prediction of mobility of new compounds, which is valuable due to low operating costs and the savings of time. In an obvious manner,researchers have paid attention to these studies,especially for drug design. However,many of these studies have been aimed at predicting the ability of the model and not for its validation. Altogether,two parameters are very important for the prediction ability of QSPR studies,one is the descriptors which carry enough information about structures,and the other is the modeling method employed. In this investigation, we introduced SPA (successive projections algorithm) ^[27] as a feature selection,due to its ability in solving the descriptor selection problems in the QSPR model development.

SPA is a technique specifically designed to select subsets of variables with small collinearity to improve the conditioning of the MLR models. This algorithm was originally proposed for wavelength selection in spectroscopic data sets,especially under conditions of strong spectral overlapping ^[27]. The MLR models generated using SPA have been shown to be superior,in terms of prediction ability,to full-spectrum PLS (partial least squares) models in a variety of applications including UV-vis [27- 30],ICP-OES ^[31],FT-IR ^[32],and NIR spectrometry ^{[33, 34]}. SPA has also been successfully employed in various classification studies ^{[35, 36]}.

SPA comprises three phases ^[37]. Initially,the algorithm builds candidate subsets of variables on the basis of a co-linearity minimization criterion. Such subsets are constructed according to a sequence of vector projection operations applied to the columns of the matrix of available predictor data. In the second phase,the best candidate subset is chosen according to a criterion that evaluates the prediction ability of the resulting MLR model,such as the root mean square error,defined in a validation set ^[38]. In the third phase,the selected subset is subjected to an elimination procedure to determine whether any variable can be removed without a significant loss of the prediction ability. Each of these phases has been explained in detail elsewhere ^[39].

Although SPA was initially designed for use with MLR models,it may be worth investigating whether it could be employed with different modeling techniques. In the present work,the variables selected by SPA will be used to build the MLR and SVM models.

The experimental electrophoretic mobility (μ) data for some organic and inorganic compounds was used in this work ^[40]. QSPR model for the estimation of the electrophoretic mobility of these compounds is established in the following six steps: the molecular structure input and generation of the files containing the chemical structures is stored in a computer-readable format; quantum mechanics geometry is optimized with a semi-empirical (AM1) method; structural descriptors are computed; structural descriptors are selected; and the structure-electrophoretic mobility model is generated by multiple linear regression (MLR),support vector machine (SVM) and statistical analysis. The names of the compounds and their experimental and calculated electrophoretic mobility values are shown in Table 1. As it can be seen in Table 1, this set contains a total of 40 sets of electrophoretic mobility data. The electrophoretic mobility values for these compounds were obtained under the same instrumental conditions. The data set was randomly split into a training set and a test set consisting of 30 and 10 members,respectively. The training set was used to adjust the parameters of the model,and the test set of 10 compounds was used to evaluate its predictive ability.

Table 1

Table 1 Data set with experimental and calculated electrophoretic mobility.

Table 1 Data set with experimental and calculated electrophoretic mobility.

The electrophoretic mobilities of solutes are related to some of their structural,electronic,and geometric properties. The values for these properties can be encoded quantitatively by numerical values,named molecular descriptors. These molecular parameters are to be used to search for the best QSPR model of the electrophoretic mobilities. The 2D structures of the molecules were drawn using the HyperChem 7 software ^[41]. The final geometries were obtained using the semi-empirical AM1 method in the HyperChem program. The molecular structures were optimized using the Polak-Ribiere algorithm until the root mean square gradient was 0.001 Kcal/mol. The resulting geometries were transferred into the Dragon program package,developed by Milano chemometrics and QSPR group ^[42],to calculate 1481 descriptors in the constitutional,topological,geometrical,charge, GETAWAY (geometry,topology and atoms-weighted assembly), WHIM (weighted holistic invariant molecular descriptors),3DMoRSE (3D-molecular representation of structure based on electron diffraction),molecular walk count,BCUT,2D-autocorrelation,aromaticity index,randic molecular profile,radial distribution function,functional group,and atom-centered fragment classes.

It is worth mentioning that in the first pre-selected analyses, we removed 647 descriptors because many of them included zero or other constant/near constant values and did not include enough information about structure. On the other hand,to decrease the redundancy existing in the descriptor data matrix, the correlation coefficient,r,of the descriptors with each other was examined,and the co-linear descriptors (with r>0.9) were removed.

Upon application of SPA,eight descriptors were selected for model building. They were mean information index on atomic composition (AAC),3D-MoRSE-signal 18/unweighted (Mor18u),3D-MoRSE-signal 07/weighted by atomic Sanderson electronegativities (Mor07e),Moran autocorrelation-lag1/ weighted by atomic polarizabilities (MATS1p),3D-MoRSEsignal 26/weighted by atomic masses (Mor26m),number of chlorine atoms (nCL),leverage-weighted autocorrelation of lag 4/weighted by atomic masses (HATS4m),and the 2nd component symmetry directional WHIM index/weighted by atomic polarizabilities (G2p).

Prediction ability of the QSAR/QSPR models is affected by two factors. One is the descriptors,which should carry enough information about the molecular structures for interpretation of the activity/ property. The other is the modeling method employed ^[17]. The number of descriptors available for QSAR/QSPR studies is often so large that it is difficult to obtain a model including all of them. Therefore,identifying important descriptors certainly plays an important role in QSAR/QSPR. Descriptors should represent the maximum information in activity variations,and co-linearity among them must be kept to a minimum.

The correlation matrix for the eight descriptors selected using SPA is shown in Table 2. As it can be seen in this table, there is no significant correlation between the selected descriptors. These descriptors were used for both the linear and non-linear models.

Table 2

Table 2 Variations of RMSE with different values of C,γ,and ε.

Table 2 Variations of RMSE with different values of C,γ,and ε.

In order to build and test the models,the data set comprising 40 compounds was separated into a training set of 30 compounds and a test set of 10 compounds. Using the training set with the eight selected descriptors,the following linear model was obtained:

It is clear from the molecular descriptors selected in the model that the three parameters included in the descriptors contain polarizability,electronegativity,and atomic masses that affect the mobility of a species present in the descriptors. Also presence of some atoms,such as Cl,can affect the mobility of compounds. The mobilities of compounds in the data set calculated by this equation are shown in Table 1.

The next step was to use SVM as a non-linear modeling tool. The prediction power of SVM is affected by three parameters:γ, C,and ε. These parameters were optimized using the 5-fold cross-validation procedure and the optimal values for these parameters were C=32,γ=0.50,and ε= 0.06. Actually the optimal value for ε is depended on the type of noise in the data, which is usually unknown but there will be some workable careful attention of the number of support vector if enough knowledge of the noise is attainable to select an optimal value for ε. As a matter of fact,choosing the ε value is a critical step because it prevents the entire training set from meeting the boundary conditions,and allows the possibility of sparsity in the dual formulations solution.

To find an optimal value for ε,RMSE for the SVR models with different ε values were calculated,and its optimum value was selected based on minimizing RMSE on the 5-fold cross-validation procedure. The other parameter is the regularized constant,C,that controls the trade-off between maximizing the margin and minimizing the training error. As we have mentioned in our previous research work ^[8],if C is too large,the SVR model will overfit the training set; likewise,if C is too small,deficient stress will be placed on fitting the training set and it will cause an underfit state on the training set. In order to find an optimal value for C,RMSE for the SVR models with different C values were calculated and the best value for C was selected based on minimizing RMSE on the 5-fold cross-validation procedure. Finally, the γ value was optimized based on high accuracy of 5-fold crossvalidation and minimizing the RMSE. The results obtained indicate that the optimized epsilon value does not differ at this stage, meaning that C and ε are independent from each other. Optimization of the SVM parameters was performed by systemically changing their values in the training step and calculating RMSE of the model. The variations of RMSE with different values of these parameters are shown in Table 3.

Table 3

Table 3 Correlation matrix for the eight selected descriptors.

Table 3 Correlation matrix for the eight selected descriptors.

The optimized SVM method was then used to calculate the mobility of the data set compounds,and the results were shown in Table 1. We tested the SPA-MLR and SPA-SVM models by using statistical parameters,such as determination coefficients (R²),root mean square error of prediction (RMSEP),relative standard error of prediction (RSEP),and mean absolute error (MAE) values. The statistical results described in Table 4 show that the SPA-SVM model is more reliable than the SPA-MLR one for predicting the mobilities of these class compounds. Fig. 1 shows the plot of experimental values versus predicted values by SVM.

Table 4

Table 4 Statistical parameters for MLR and SVM models.

Table 4 Statistical parameters for MLR and SVM models.

Fig. 1

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Fig. 1. Plot of the SVM predicting the electrophoretic mobility of training and test sets against the experimental values.

In this work,a set of simple and reasonable models was established to estimate the electrophoretic mobilities of some organic and inorganic anions. Also,the successive projection algorithm feature selection (SPA) strategy was used as the descriptor selection. The different structural descriptors employed are very easy to calculate,and the models are physicochemically interpretable. MLR and SVM were used as feature mapping techniques for the prediction of the electrophoretic mobility of these compounds. The results demonstrate the superiority of the SVM model over the MLR one. This is due to the ability of SVM, unlike regression analysis,to allow for flexible mapping of the selected features by implicitly manipulating their functional dependence. Also,the SVM method handles both the linear and non-linear relationships without adding complexity to the model. The descriptors appearing in these QSPR models provide information related to different molecular properties,which can participate in the physicochemical process that affects the electrophoretic mobility of the solute in CE.