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The Particle Swarm Optimization (PSO) and Support Vector Machines (SVMs) approaches are used for predicting the thermodynamic parameters for the 1:1 inclusion complexation of chiral guests with β-cyclodextrin. A PSO is adopted for descriptor selection in the quantitative structure-property relationships (QSPR) of a dataset of 74 chiral guests due to its simplicity, speed, and consistency. The modified PSO is then combined with SVMs for its good approximating properties, to generate a QSPR model with the selected features. Linear, polynomial, and Gaussian radial basis functions are used as kernels in SVMs. All models have demonstrated an impressive performance with ^{2}

β-cyclodextrin (β-CD) is a cyclic oligosaccharide that naturally contains seven glucose residues linked by (1–4)-glycosidic bonds, with a hydrophilic outer surface and a relative hydrophobic central cavity, which can form complexes with appropriate guest molecules. It has received increasing attention in the pharmaceutical field for modifying drug physicochemical properties, such as solubility, stability and bio-availability, reducing their toxicity and side effects, and suppressing unpleasant taste or smell [

The high interest in the stability constants of CD-host complexes has initiated the search for proper models for predicting these association constants or the related free energies of complexation. The aim is not only to select convenient CDs for the complexation of a particular compound, but also to get some insight into the physico-chemical parameters influencing the affinity between host and guest molecules. The availability of a large amount of experimental data led to several interesting predictive models. The inclusion reactions of a series of benzene derivatives were used for a correlation model [

In this study, two novel approaches, Binary Particle Swarm Optimization (BPSO) [

Several factors such as number of atoms, van der Waals surface area, ionization potential, molecular weight, molar refractivity, atomic connectivity index, molecular flexibility, and angle bend energy, etc., influence thermodynamic properties. Only some of these factors strongly affect these thermodynamic properties and are controlled or set up in advance. The selection of these parameters is traditionally conducted by multiple linear regressions, partial least squares, and principle component analysis methods. Consequently, their assumptions must be verified and validated before the developed model can accurately be used. This results in a predictive model which may compromise the quality of the obtained products and/or efficiency of the modeling process.

With the increasing need for more accurate and practical evaluation QSPR models, techniques in artificial intelligence, particularly Artificial Neural Networks (ANNs), are receiving more attention in industry and academia today because they can be used to learn relationships between thermodynamic properties and their parameters. However, a number of parameters such as network topology, learning rate, and training methods have to be fine-tuned before they are deployed successfully. Furthermore, drawbacks like local optima, overfitting, and long learning time tend to occur.

Theoretically, the aforementioned shortcomings of ANNs have been countered by the development of Support Vector Machines. Unlike ANNs which minimize empirical risk, SVMs are designed to minimize the structural risk, by minimizing an upper bound of the generalization error, rather than the training error. Therefore, the overfitting problem in machine learning is solved successfully. Another outstanding property of SVMs is that the task of training SVMs is mapped to a uniquely solvable linearly constrained quadratic programming problem. This produces a solution that is always unique and globally optimal. They have been extended to solve regression problems as well.

In this paper, Support Vector Regression (SVR), which is based on Support Vector Machines, is investigated as an alternative technique for QSPR prediction. It has shown very good results for function approximation of Quantitative Structure-Activity Relationships (QSAR) [

Since SVM can build a very reliable QSPR model based on the training data, it is incorporated in our feature selection process. The Pearson correlation coefficient (

Therefore, the purpose of this study was to develop a procedure that can determine key features for predicting complexation thermodynamic parameters of β-CD complexes with enantiomeric pairs of a chiral guest.

The proposed methodology consists of two parts: feature selection and QSPR modeling. First, a machine learning technique called Support Vector Machines (SVMs) is used to capture characteristics of QSPR and their factors, because of the SVMs’ superior properties of generalization and global optima. They are next incorporated in an optimization problem so that a relatively new, effective, and efficient optimization algorithm, Particle Swarm Optimization (PSO), is applied to find key parameters. The cooperation between both techniques can produce a very good predictive QSPR model.

SVMs represent a relatively new type of learning machine. They are an approximate implementation of the method of structural risk minimization, which attempts to minimize the generalization error, which occurs when the machines are tested with unseen data. The generalization error rate is bounded by the sum of a pair of competing terms, the training error rate and the confidence interval, which depends on the Vapnik-Chervonenkis (VC) dimension. Hence, the VC dimension and the training error (empirical risk) are both minimized at the same time. To realize this in SVMs, a structure is imposed on the set of hyperplanes, by trying to obtain the weight vector

In regression problems, the problem of approximating the following set of data {(_{1}, _{1}),..., (_{l}_{l}^{n}^{n}_{i} is the set of descriptors, and y_{i} is the output, which is the thermodynamic value. The ɛ-insensitive loss function proposed by Vapnik [

min:

Everything above ɛ is captured in slack variables ξ_{i} and everything below -ɛ is captured in slack variables
_{ɛ}

Using the Lagrangian multipliers and the Karush-Kuhn-Tucker (KKT) conditions, the following dual problem is obtained:

Transforming into dual form yields a quadratic programming problem with linear constraints and a positive definite Hessian matrix. This leads to a global optimum. A nonlinear form is usually required to adequately model data. Hence, a nonlinear mapping,

The dimensionality of the intermediate space is thus hidden from the remaining computations. Some of the most widely used kernels, such as linear, polynomial, and Gaussian radial basis functions (RBF), were tested in this study. The kernel function is employed in the optimization models above by replacing 〈.,.〉 with

In summary, the main advantages of SVR are implicit mapping by using kernels in handling nonlinear data, convexity of quadratic optimization, and generalization properties. In addition, distribution of the data is not necessarily assumed in advance, which makes it very promising for real-world problems.

PSO was introduced by Kennedy and Eberhart [

Initialize a population of I particles with random positions and velocities in D dimensions.

Evaluate the desired optimization function in D variables for each particle.

Compare the evaluation with the particle’s previous best value, pbest[i]. If the current value is better than pbest[i], then pbest[i] = current value and the pbest location, pbestx[i][d], is set to the current location in d-dimensional space.

Compare the evaluation with the swarm’s previous best value, (pbest[gbest]). If the current value is better than pbest[gbest]), then gbest = current particle’s array index.

Change the velocity and position of the particle according to the following equations, respectively:

Loop to step 2 until a stopping criterion, a sufficiently good evaluation function value, or a maximum number of iterations, is met.

In feature selection, the input presented to the regression modeling is in the form of a table where the rows represent chemical compounds and the columns are the molecular descriptors. Each compound contains a value for each corresponding factor. How accurately a QSPR model can predict the biological activity of the compounds depends on their values in a subset of the selected features. Hence, the selection of each column or feature is treated as a binary number. A numerical value of zero is used to represent that the corresponding descriptor is not selected for QSPR modeling. Otherwise, a numerical value of one is assigned. This binary problem calls for some modification of the original PSO. Thus presentx[i][d], which represents the value stored by the i^{th} particle in the d^{th} dimension, can only take on a binary value, instead of a real valued number. This indicates whether the d^{th} feature is selected or not. Note that the D dimensions above are equal to the total number of descriptors. After the update step (_{id}, are computed as follows:
_{id} is the fractional coordinates presentx[i][d] after the update step (_{i}_{i}^{th} compound, respectively.

The complex stability constant (ln

The PSO was adopted for major descriptor selection in QSPR of the chiral guest dataset. Swarm parameters are 50 particles and 100 iterations. The iterative PSO attempts to select the key features that maximize the Pearson correlation coefficient (

^{2}^{2}_{Training}^{2}_{Training}

The numbers of descriptors in the QSPR models of these thermodynamic properties are further investigated by using the Gaussian RBF kernel, which gives the best outcome for the chiral guest dataset. The statistics for all PSO-SVM models are given in

Even though PSO-SVM methods are not able to explain the values of descriptors in the models, the maximum outlier from the QSPR models can point out the error of the experimental data. In Δ^{−1}, and ^{−1}, respectively. The experimental results have different values for this enantiomeric pair, whereas the PSO-SVM models have identical results: Δ^{−1}and ^{−1}.

This work demonstrated that the combination of PSO and SVMs can be applied to effectively and efficiently select major features in QSPR modeling of the thermodynamic parameters of 1:1 inclusion complexation of enantiomeric pairs of chiral guests with β-CD. This responds to the needs of drug designers for prediction of the thermodynamic parameters of new compounds in complexation with β-CD. The method was based on a discrete binary modification of PSO. The fitness function was the Pearson correlation which was curve fitted by SVMs. The modified PSO appeared to be an effective and efficient algorithm, which robustly finds near-optimal and consistent results with short computer code and simple mathematical operators, while converging rather quickly. The SVMs showed excellent performance in predicting ln

This work is financially supported by Thailand Research Fund (grant number RSA5080001). We are appreciative to Paul V. Neilson for carefully reading through the manuscript.

Plots of calculated thermodynamic parameters by PSO-SVM with Gaussian RBF kernel models with 4 features,

Experimental Thermodynamic parameters: ln ^{−1}), Δ^{−1}), Δ^{−1}) and ^{−1}) of 74 chiral compounds in 1:1 inclusion complexation with β-CD taken from Ref. [

^{b} | |||||||||
---|---|---|---|---|---|---|---|---|---|

| |||||||||

1 | 4.11 | −10.18 | −8.14 | 2.04 | 4.43 | −11.61 | −8.82 | 0.78 | |

2 | 4.21 | −10.44 | −8.17 | 2.27 | 4.43 | −11.61 | −8.82 | 0.78 | |

3 | 2.54 | −6.30 | −25.50 | −19.20 | 2.54 | −6.94 | −24.09 | −17.72 | |

4^{a} |
2.84 | −7.04 | −23.80 | −16.80 | 2.54 | −6.94 | −24.09 | −17.72 | |

5 | 4.83 | −11.97 | −16.70 | −4.70 | 4.65 | −11.48 | −15.77 | −3.92 | |

6 | 4.87 | −12.07 | −17.10 | −5.00 | 4.65 | −11.48 | −15.77 | −3.92 | |

7 | (1 |
4.01 | −9.90 | −10.00 | −0.10 | 3.84 | −9.35 | −10.11 | 0.65 |

8^{a} |
(1 |
3.83 | −9.50 | −10.00 | −0.50 | 3.84 | −9.35 | −10.11 | 0.65 |

9 | ( |
5.46 | −13.52 | −9.20 | 4.30 | 5.21 | −13.45 | −9.58 | 4.22 |

10 | ( |
5.43 | −13.50 | −9.30 | 4.20 | 5.21 | −13.45 | −9.58 | 4.22 |

11 | 2,3- |
4.76 | −11.81 | −7.56 | 4.25 | 4.70 | −12.05 | −7.98 | 3.30 |

12 | 2,3- |
4.74 | −11.76 | −7.49 | 4.27 | 4.70 | −12.05 | −7.98 | 3.30 |

13^{a} |
(2 |
4.42 | −10.95 | −8.07 | 2.90 | 4.79 | −10.23 | −7.23 | 2.90 |

14 | (2 |
4.44 | −11.01 | −7.79 | 3.20 | 4.79 | −10.23 | −7.23 | 2.90 |

15 | 4.26 | −10.57 | −8.90 | 1.70 | 4.25 | −9.67 | −9.46 | 2.74 | |

16^{a} |
4.23 | −10.50 | −9.20 | 1.30 | 4.25 | −9.67 | −9.46 | 2.74 | |

17 | 5.97 | −14.80 | −9.70 | 5.10 | 6.11 | −14.00 | −9.94 | 5.16 | |

18 | 5.91 | −14.64 | −9.80 | 4.80 | 6.11 | −14.00 | −9.94 | 5.16 | |

19 | 6.49 | −16.09 | −13.82 | 2.30 | 6.40 | −14.95 | −12.37 | 2.43 | |

20 | 6.36 | −15.77 | −12.80 | 3.00 | 6.40 | −14.95 | −12.37 | 2.43 | |

21^{a} |
5.72 | −14.19 | −11.00 | 3.20 | 5.33 | −13.67 | −11.20 | 2.95 | |

22 | 5.65 | −14.01 | −10.60 | 3.40 | 5.33 | −13.67 | −11.20 | 2.95 | |

23 | ( |
8.23 | −20.41 | −30.10 | −9.70 | 8.03 | −19.49 | −29.63 | −9.67 |

24^{a} |
8.20 | −20.32 | −29.60 | −9.30 | 8.03 | −19.49 | −29.63 | −9.67 | |

25 | ( |
4.96 | −12.29 | −9.30 | 3.00 | 4.97 | −12.47 | −10.07 | 2.65 |

26 | ( |
4.94 | −12.25 | −10.10 | 2.20 | 4.97 | −12.47 | −10.07 | 2.65 |

27 | ( |
5.58 | −13.80 | −12.05 | 1.80 | 5.50 | −13.90 | −12.45 | 2.40 |

28 | ( |
5.60 | −13.90 | −12.40 | 1.50 | 5.50 | −13.90 | −12.45 | 2.40 |

29^{a} |
( |
5.18 | −12.85 | −17.80 | −5.00 | 5.23 | −15.01 | −19.07 | −6.17 |

30 | ( |
5.33 | −13.22 | −17.70 | −4.50 | 5.23 | −15.01 | −19.07 | −6.17 |

31 | (1 |
2.94 | −7.30 | −15.50 | −8.20 | 3.02 | −8.52 | −13.02 | −6.63 |

32^{a} |
(1 |
3.18 | −7.90 | −8.30 | −0.40 | 3.02 | −8.52 | −13.02 | −6.63 |

33 | ( |
7.87 | −19.50 | −27.10 | −7.60 | 7.62 | −19.33 | −27.04 | −6.91 |

34 | ( |
7.80 | −19.34 | −27.20 | −7.90 | 7.62 | −19.33 | −27.04 | −6.91 |

35 | ( |
6.34 | −15.70 | −20.70 | −5.00 | 6.46 | −15.44 | −20.53 | −5.74 |

36 | ( |
6.19 | −15.35 | −19.50 | −4.20 | 6.46 | −15.44 | −20.53 | −5.74 |

37^{a} |
5.00 | −12.40 | −8.90 | 3.50 | 4.81 | −12.64 | −8.69 | 2.17 | |

38 | 4.99 | −12.37 | −10.00 | 2.40 | 4.81 | −12.64 | −8.69 | 2.17 | |

39 | (1 |
4.44 | −11.01 | −3.98 | 7.03 | 4.66 | −12.04 | −4.54 | 7.43 |

40^{a} |
(1 |
4.45 | −11.04 | −4.21 | 6.83 | 4.66 | −12.04 | −4.54 | 7.43 |

41 | ( |
5.80 | −14.37 | −7.85 | 6.50 | 5.59 | −14.24 | −7.94 | 6.15 |

42 | ( |
5.79 | −14.36 | −7.87 | 6.50 | 5.59 | −14.24 | −7.94 | 6.15 |

43 | 3.47 | −8.60 | −7.00 | 1.60 | 3.39 | −7.95 | −6.32 | 2.02 | |

44 | 3.00 | −7.40 | −4.90 | 2.50 | 3.39 | −7.95 | −6.32 | 2.02 | |

45^{a} |
Gly-D-Phe | 3.85 | −9.54 | −7.93 | 1.60 | 3.96 | −10.44 | −11.79 | 1.74 |

46 | Gly-L-Phe | 3.99 | −9.89 | −8.59 | 1.30 | 3.96 | −10.44 | −11.79 | 1.74 |

47 | ( |
6.47 | −16.05 | −5.61 | 10.44 | 6.11 | −14.54 | −5.92 | 9.37 |

48^{a} |
( |
6.40 | −15.87 | −5.36 | 10.51 | 6.11 | −14.54 | −5.92 | 9.37 |

49 | (1 |
7.77 | −19.30 | −19.50 | −0.20 | 8.05 | −19.47 | −19.79 | −1.12 |

50 | (1 |
7.75 | −19.20 | −20.00 | −0.80 | 8.05 | −19.47 | −19.79 | −1.12 |

51 | ( |
2.40 | −5.90 | −4.90 | 1.00 | 2.20 | −5.63 | −5.17 | 1.11 |

52 | ( |
2.20 | −5.40 | −4.60 | 0.80 | 2.20 | −5.63 | −5.17 | 1.11 |

53^{a} |
( |
4.20 | −10.42 | −7.80 | 2.60 | 4.13 | −10.17 | −6.94 | −0.86 |

54 | ( |
4.28 | −10.60 | −8.20 | −2.40 | 4.13 | −10.17 | −6.94 | −0.86 |

55 | ( |
2.40 | −5.90 | −4.40 | 1.50 | 2.79 | −7.68 | −6.63 | 1.75 |

56^{a} |
( |
2.30 | −5.70 | −5.10 | 0.60 | 2.79 | −7.68 | −6.63 | 1.75 |

57 | ( |
5.16 | −12.80 | −17.48 | −4.70 | 5.45 | −13.14 | −16.92 | −4.19 |

58 | ( |
4.95 | −12.27 | −16.35 | −4.10 | 5.45 | −13.14 | −16.92 | −4.19 |

59 | D-phenylalanine amide | 4.62 | −11.44 | −10.00 | 1.40 | 4.66 | −11.71 | −10.01 | 0.87 |

60 | L-phenylalanine amide | 4.69 | −11.63 | −10.60 | 1.00 | 4.66 | −11.71 | −10.01 | 0.87 |

61^{a} |
D-phenylalanine methyl ester | 2.40 | −5.90 | −5.60 | 0.30 | 3.16 | −7.07 | −3.56 | 0.58 |

62 | L-phenylalanine methyl ester | 2.48 | −6.20 | −5.00 | 1.20 | 3.16 | −7.07 | −3.56 | 0.58 |

63 | ( |
4.54 | −11.26 | −9.79 | 1.50 | 4.77 | −12.20 | −9.15 | 1.81 |

64^{a} |
( |
4.55 | −11.29 | −9.91 | 1.40 | 4.77 | −12.20 | −9.15 | 1.81 |

65 | ( |
6.00 | −14.86 | −8.62 | 6.24 | 5.22 | −14.41 | −8.72 | 5.34 |

66 | ( |
6.06 | −15.03 | −8.68 | 6.35 | 5.22 | −14.41 | −8.72 | 5.34 |

67 | ( |
4.13 | −10.23 | −7.54 | 2.69 | 3.85 | −9.42 | −6.96 | 2.24 |

68 | ( |
4.14 | −10.26 | −7.30 | 2.96 | 3.85 | −9.42 | −6.96 | 2.24 |

69^{a} |
( |
4.48 | −11.10 | −9.34 | 1.80 | 5.06 | −10.83 | −8.09 | 2.99 |

70 | ( |
4.42 | −10.95 | −8.65 | 2.30 | 5.06 | −10.83 | −8.09 | 2.99 |

71 | ( |
3.53 | −8.74 | −8.81 | −0.10 | 4.06 | −8.72 | −8.84 | 1.18 |

72^{a} |
( |
3.58 | −8.88 | −8.69 | 0.20 | 4.06 | −8.72 | −8.84 | 1.18 |

73 | (1 |
8.77 | −21.74 | −20.40 | 1.30 | 8.67 | −21.18 | −20.19 | 2.03 |

74 | (1 |
8.76 | −21.71 | −20.30 | 1.40 | 8.67 | −21.18 | −20.19 | 2.03 |

Compounds in test set;

The descriptors in the QSPR models are provided in

The average predictive ability of PSO-SVMs QSPR models with 8 descriptors.

| ||||||||
---|---|---|---|---|---|---|---|---|

^{2}_{Training} |
^{2}_{Testing} |
^{2}_{Training} |
^{2}_{Testing} |
^{2}_{Training} |
^{2}_{Testing} |
^{2}_{Training} |
^{2}_{Testing} | |

0.8201 | 0.6666 | 0.8239 | 0.6349 | 0.9048 | 0.8455 | 0.8220 | 0.8257 | |

0.9993 | 0.7358 | 0.9994 | 0.8213 | 0.9992 | 0.8432 | 0.9991 | 0.8251 | |

0.9983 | 0.9762 | 0.9987 | 0.9713 | 0.9983 | 0.9350 | 0.9986 | 0.8853 |

The average predictive ability of PSO-SVM with Gaussian RBF kernel models.

| ||||||||
---|---|---|---|---|---|---|---|---|

^{2}_{Training} |
^{2}_{Testing} |
^{2}_{Training} |
^{2}_{Testing} |
^{2}_{Training} |
^{2}_{Testing} |
^{2}_{Training} |
^{2}_{Testing} | |

8 | 0.9983 | 0.9762 | 0.9987 | 0.9713 | 0.9983 | 0.9350 | 0.9986 | 0.8853 |

7 | 0.9977 | 0.9534 | 0.9981 | 0.9778 | 0.9978 | 0.9325 | 0.9981 | 0.8936 |

6 | 0.9963 | 0.9629 | 0.9967 | 0.9039 | 0.9966 | 0.9271 | 0.9967 | 0.8868 |

5 | 0.9869 | 0.9292 | 0.9872 | 0.9507 | 0.9932 | 0.9142 | 0.9919 | 0.8812 |

4 | 0.9534 | 0.9020 | 0.9496 | 0.8498 | 0.9820 | 0.8563 | 0.9572 | 0.8707 |

Descriptors in the best predictive ability of PSO-SVM with Gaussian RBF kernel models.

^{2}_{Training} |
^{2}_{Testing} |
||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

| |||||||||||

8 | 0.9982 | 0.9903 | 2 | 20 | 31 | 76 | 80 | 143 | 164 | 185 | |

7 | 0.9982 | 0.9922 | 2 | 12 | 90 | 94 | 122 | 144 | 164 | ||

6 | 0.9968 | 0.9879 | 3 | 20 | 30 | 94 | 140 | 167 | |||

5 | 0.9929 | 0.9829 | 3 | 20 | 27 | 59 | 134 | ||||

4 | 0.9641 | 0.9530 | 12 | 79 | 114 | 134 | |||||

8 | 0.9985 | 0.9924 | 3 | 11 | 28 | 79 | 94 | 112 | 136 | 144 | |

7 | 0.9978 | 0.9928 | 2 | 9 | 111 | 123 | 129 | 133 | 140 | ||

6 | 0.9969 | 0.9894 | 3 | 12 | 94 | 124 | 133 | 140 | |||

5 | 0.9935 | 0.9849 | 3 | 20 | 27 | 59 | 134 | ||||

4 | 0.9688 | 0.9281 | 20 | 72 | 94 | 122 | |||||

8 | 0.9987 | 0.9510 | 28 | 54 | 76 | 83 | 124 | 173 | 181 | 186 | |

7 | 0.9977 | 0.9385 | 20 | 79 | 91 | 140 | 143 | 153 | 187 | ||

6 | 0.9965 | 0.9417 | 2 | 21 | 27 | 112 | 154 | 176 | |||

5 | 0.9943 | 0.9374 | 16 | 36 | 79 | 91 | 122 | ||||

4 | 0.9794 | 0.9408 | 20 | 21 | 30 | 36 | |||||

8 | 0.9986 | 0.8949 | 3 | 23 | 36 | 44 | 94 | 112 | 114 | 159 | |

7 | 0.9982 | 0.9371 | 9 | 20 | 33 | 100 | 122 | 158 | 199 | ||

6 | 0.9952 | 0.8991 | 9 | 12 | 93 | 94 | 154 | 164 | |||

5 | 0.9868 | 0.9079 | 6 | 30 | 39 | 76 | 90 | ||||

4 | 0.9754 | 0.9113 | 8 | 76 | 114 | 129 |

The selected descriptors in four feature QSPR models.

8 | 2D | Weiner polarity number |

12 | 2D | PEOE Charge BCUT (3/3) |

20 | 2D | Molar Refractivity BCUT (3/3) |

21 | 2D | PEOE Charge GCUT (0/3) |

30 | 2D | Molar Refractivity GCUT (1/3) |

36 | 2D | Atom information content (mean) |

50 | 2D | Number of chiral centers |

72 | 2D | Total positive partial charge |

76 | 2D | Total positive 0 van der Waals surface area |

79 | 2D | Total positive 3 van der Waals surface area |

94 | 2D | Fractional positive van der Waals surface area |

114 | 2D | Third alpha modified shape index |

122 | 2D | Number of H-bond donor atoms |

129 | 2D | van der Waals polar surface area |

134 | 2D | Bin 3 SlogP_(0.00, 0.10] |