Amin Bahmani
Ceumar Pee
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In this paper, multiple linear regression (MLR) was used to build quantitative structure property relationship (QSPR) of n-octanol-water partition coefficient (logPo/w) of 195 substituted aromatic drugs. The molecular descriptors were calculated for each compound by the VLifeMDS. By applying genetic algorithm/multiple linear regressions (GA/MLR) the most relevant descriptors were selected to build a QSPR model. The robustness of the model was characterized by the statistical validation and applicability domain (AD). The prediction results from MLR are in good agreement with the experimental values. The R2 and Q2 LOO for MLR are 0.9433, 0.9341. The AD of the model was analyzed based on the Williams plot. The effects of different selected descriptors are described.
Lipophilicity is the tendency of a compound to partition into a non-polar organic phase versus an aqueous phase. The typical quantitative descriptor of lipophilicity is the partition coefficient P of a given compound between two immiscible solvents1. Traditionally, n-octanol has been widely used as the non-polar phase and water as the polar phase. The partitioning value that is measured is termed logPo/w 2.
In this work we develop QSPR modeling of logPo/w of 195 substituted aromatic drugs. These drugs are very important in medicinal chemistry, such as: Alprazolam, that is mostly used to treat anxiety disorders, panic disorders, and nausea due to chemotherapy, Dapsone, that is commonly used in combination with Rifampicin and Clofazimine for the treatment of leprosy, Procaine, that is a local anesthetic drug of the amino ester group. It is used primarily to reduce the pain of intramuscular injection of penicillin, and it is also used in dentistry, Warfarin treatment can help prevent formation of future blood clots and help reduce the risk of embolism30. In this paper all of 195 drugs are homogeneous set of aromatic drugs.
Molecular descriptors are generated from molecular structures. Although different descriptors utilize different processing steps, still there are numerous steps common to these procedures. Molecular descriptors are powerful tools for the approximation of selected properties of chemical structures in an easy-to-handle form that allows efficient comparison and selection of compounds possessing required chemical, structural, pharmacological or biological features. In this study molecular descriptors were calculated for each compound by the VLifeMDS on the minimal energy conformations. VLifeMDS calculates about 500 different molecular descriptors from the categories: topological, electronic, electrostatic, E-state, information theory based, physicochemical and semi-empirical.
After descriptor generation a pool of the molecules with the corresponding descriptors become available for model calculation. But a limited number of modeling descriptors, related to the studied response, must be selected from the available pool. Descriptor selection is the process of selecting a subset of relevant variables for use in model construction. In QSARINS this is done using a GA/MLR procedure. This technique is able to explore a broad range of solutions, searching for the best ones, by maximizing or minimizing a selected fitness function. This is done mimicking the natural selection, where the best solutions replace the less performing. In biological terms, one would say that the best genes in the population displace the less fitting. In our case, every descriptor represents a gene, and a set of descriptors represents a chromosome. The fitness of a chromosome is related to the matching model performances. Starting with a pool of chromosomes, small subsets of chromosomes are picked randomly, and the best become parents. Couples of parent chromosomes are then crossed at a random position (crossing-over), thus obtaining the offspring, whose chromosomes are a combination of the parent ones. If among the new chromosomes one or more of them outperform the less fitting in the parent population, these chromosomes will replace the less performing. Repeating the aforesaid procedure many times, and introducing also random mutations (descriptor substitution) in the chromosomes, the result at the end of the procedure is a population of models with better performances than the models introduced at the beginning. In order to prevent a completely random beginning of the GA, in QSARINS, the best set of descriptors extracted from the all subset process is used as the core of the chromosomes of the initial population. In QSARINS, the tuning of the GA can be done changing the population size, the mutation rate, and the number of generations. A fundamental option is the selection of the fitness function to be used by GA. In the work, leave-one-out cross-validation (Q2 LOO) was used as fitness function throughout the GA process. When increasing the model size does not improve the Q2 value significantly, the GA selection will be stopped. Q2 LOO used as fitness function, is useable to select models with high fitting with the minimum number of descriptors. However, it is essential to note that they are fitting criteria, so they provide no information on the predictive ability of the models. For this reason, it is here proposed to use Q2 LOO as fitness function for the selection of predictive models33. The important parameters used in the GA process were set as below: population size 100, maximum allowed descriptors in a model 10 and reproduction/mutation trade-off 0.5. Finally, we obtained a 10-descriptor subset, which keeps most interpretive information for logPo/w. Four descriptors were calculated for each compound in the data set. The selected descriptors are: SKMostHydrophobic Area, SAHydrophobic Area, SKAverage, XKAverage Hydrophobicity, PSA, Average Potential, Polar Surface Area Excluding P & S, 4Path Count, ChiV6chain and AlphaR.
The datasets used in QSPR analysis are, as previously mentioned, composed of descriptors that should be correlated with the corresponding experimental responses. At this step it is necessary to apply a quantitative method able to find the existing relationship between a limited number of structural descriptors and the modeled response. In QSARINS, the used method is the MLR approach that can be demonstrated by the following formula:
where a linear relationship is computed between the studied responses (yi) and the selected values of the descriptors (xij); ei is the random error (called also model residual). The intercept (b0) and the coefficients (bj) are thus to be evaluated. The equation (2) can be rewritten in a more compact form using the matrix notation:
where y is the responses vector, b the vector of the coefficients and e is the vector of the errors. X is the matrix of the model, where the columns are the descriptors. In this software, to estimate the vector of the coefficients, the OLS technique is used:
where H is the leverage (or hat) matrix that relates the calculated and the experimental responses. The diagonal elements of the hat matrix h ii are useable to determine the distance of the i object from the centre of the chemical space of the model34, 35, thus, for checking the structural applicability domain (AD) of the model.
Evalution of QSPR model is a very important aspect. It is acknowledged that the goodness-of-fit is very important for QSPR models. The quality of goodness-of-fit of the models is quantified by the R2 squared correlation coefficient, R2 adj is adjusted squared correlation coefficient, s is the standard error of the regression and F is the Fisher ratio for regression. R2 is a statistic that will give some information about the goodness of fit of a model. R2 is defined as:
where RSS is the residual sum of squares and TSS is the total sum of squares. Adjusted R2 detects the possible overfitting of a model so, used as fitness functions, are useful to select models with high fitting with the minimum number of descriptors. Adjusted R2 is defined as:
where n is the number of members of the training set and m is the number of descriptors included in the model. The Adjusted R2 is a better measure of the proportion of variance in the data explained by the correlation than R2. The standard error indicates dispersion degree of random error. F-ratio test in regression is defined as the ratio between the variance explained by the model to the residual variance. The larger R2, R2 adj and F, the smaller s, and the model will have more fitting ability.
Model calculation and evaluation are the basic steps in QSPR analysis, but are not sufficient to guarantee the model validity. Validation is fundamental to ensure the reliability of data predicted by the models. Validation of QSPR model is very important aspect, thus internal and external validation is considered to be necessary for model validation35.
Internal validation is obtained from analyzing of each one of individual objects that configure the final equation. This procedure is leave-one-out (LOO) cross-validation. This process was done in training set and Q2 LOO is calculated.
where TSS is the total sum of squares that is the sum of squared deviations from the data set mean and PRESS is the sum of squares of the prediction errors. The larger Q2 LOO and the model will have more predictive ability. However, a perturbation of only one compound at a time is very weak to demonstrate real model robustness. In QSARINS, the stronger Leave-More (or many)-Out (LMO) technique is also included. This technique studies the behavior of the model when a larger number of compounds are eliminated. LMO is used to counteract the slight overoptimism of LOO-cross-validation. The model under analysis can be considered stable if the R2 and Q2 values calculated in every LMO iteration and their averages (R2 LMO and Q2 LMO), are close to R2 LOO and Q2 LOO values of the model36.
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