Quantitative structure–activity relationship studies of dibenzo[a,d]cycloalkenimine derivatives for non-competitive antagonists of N-methyl-d-aspartate based on density functional theory with electronic and topological descriptors

Peer review under responsibility of Taibah University.

Abstract

To establish a quantitative structure–activity relationship for non-competitive antagonists of the N-methyl-d-aspartate receptor, 48 substituted dibenzo[a,d]cycloalkenimine derivatives were analyzed by principal components, a descendant multiple regression analyses, multiple non-linear regression and an artificial neural network. We propose non-linear and linear quantitative structure–activity models and interpret the activity of the compounds by the multivariate statistical analysis. Density functional theory with Becke's three-parameter hybrid function and Lee–Yang–Parr exchange correlation functional calculations were performed to define the structure, chemical reactivity and properties of the study compounds. The topological and the electronic descriptors were computed with ACD/ChemSketch and Gaussian 03W programs, respectively. The study shows that multiple regression and multiple non-linear regression analyses predict activity; however, predictions made with a 6-2-1 artificial neural network model were more accurate. This model gave statistically significant results and showed good stability to data variation in leave-one-out cross-validation.

Keywords:

• N-Methyl-d-aspartate
• Quantitative structure–activity relation
• Density functional theory
• Dibenzo[a,d]cycloalkenimines
• Artificial neural network
• Cross-validation

1 Introduction

Excitatory amino acids form the mainstay of synaptic transmission in the central nervous system. By the same token, dysfunctional toxic activity of excitatory amino acids can lead to or become instrumental in the progression of a number of neurological and neurodegenerative conditions, such as epilepsy, Huntington disease, Alzheimer disease and schizophrenia. Dementia due to Alzheimer disease is characterized by extracellular plaques containing amyloid (β peptide), which disrupt dendritic morphology and affect glutamate (α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid and N-methyl-d-aspartate) receptor function to alter glutamatergic transmission. Huntington disease manifests as atrophy of the corpus striatum and cortex, with neurons containing the mutant Huntington protein, which are perhaps more susceptible to excitotoxicity from corticostriatal inputs, as reflected by loss of the N-methyl-d-aspartate receptor and interactions with facilitatory group I metabotropic glutamate receptor. In schizophrenia, abnormalities in brain (dendritic) development and synaptic plasticity may precipitate dysfunction of mesolimbic and mesocortical dopaminergic pathways. Here again, aberrations in glutamatergic transmission in the form of N-methyl-d-aspartate receptor hypofunction may be involved [1].

Dizocilpine, a compound originally characterized as an anticonvulsant, is a potent N-methyl-d-aspartate receptor antagonist [2]. It binds selectively and with high affinity to the receptors when they are open [3] and is therefore referred to as a use-dependent N-methyl-d-aspartate receptor open channel blocker with a very slow off-rate. These properties can be exploited to ‘pre-block’ a population of receptors, such as synaptic ones, resulting in selective activation of a different population, such as extra-synaptic receptors. The usefulness of this approach depends on the stability of dizocilpine blockade after washout [4].

Early electrophysiological and ligand binding studies revealed that blockade of N-methyl-d-aspartate receptors by dizocilpine persisted long after the drug had been washed out [5]. This unusual property means that the blockade of receptors is highly stable and can be regarded as ‘irreversible’ over many experiments. It is therefore used to ‘permanently’ block a sub-population of N-methyl-d-aspartate receptors in order to study a separate population of non-blocked receptors. An example of such use is in the study of synaptic and extra-synaptic receptors [6]. Differential signaling by the neuroprotective phasic activation of synaptic N-methyl-d-aspartate receptors as compared with the deleterious tonic activation of extra-synaptic receptors is a topic of much current interest [7–19].

Quantitative structure–activity relations are used to investigate the relations between molecular descriptors of the unique physicochemical properties of a set of compounds and their biological activity or chemical property [20,21]. We attempted to establish a quantitative structure–activity relation for non-competitive antagonists of N-methyl-d-aspartate receptors by studying a series of 48 substituted dibenzo[a,d]cycloalkenimine derivatives.

2 Material and methods

2.1 Experimental data

In order to determine quantitative structure–activity relations for non-competitive antagonists of N-methyl-d-aspartate receptors, we used 48 compounds that have been synthesized and evaluated for their ability to displace dibenzo[a,d]cycloalkenimines from their specific binding sites on rat cortical membranes and for their antagonistic activity against the N-methyl-d-aspartate receptor. As proposed by Thompson et al. [22], 38 molecules were selected for the quantitative model (training set), and 10 were selected randomly to test the performance of the proposed model (test set).

The molecular structures and computational models for the 48 derivatives are shown in Fig. 1 and Table 1 and described by their substituents as R1, R2, R3, R4, R5, R6 and R7. Although Thompson et al. proposed 73 compounds, the structures of the remaining compounds are different from that required for this study.

Table 1 The structure and the observed for the 48 MK801 derivatives (training and test set).

No.	R1	R2	R3	R4	R5	R6	R7	Obs.
Test set
1	H	CH3	CH3	H	H	H	H	−0.215
2	H	H	CH3	H	H	H	H	−1.252
3	CH3	CH3	CH3	H	H	H	H	−0.149
4	H	OH	CH3	H	H	H	H	1.279
5	H	H	CH2CH3	H	H	H	H	−1.347
6	H	CH2CH3	H	H	H	H	H	1.380
7	OH	OH	CH2CO2ET	H	H	H	H	3.653
8	OH	H	CH2CO2ET	H	H	H	H	0.556
9	H	H	CH2CH2OH	H	H	H	H	−0.585
10	H	H	CH2CO2ET	H	H	H	H	−0.260
Training set
11	H	H	CH2OH	H	H	H	H	−0.456
12	H	H	CH2F	H	H	H	H	−0.796
13	H	H	CH2CH2F	H	H	H	H	−0.759
14	H	H	CH2SC6H5	H	H	H	H	1.863
15	H	H	CH2S(O)C6H5	H	H	H	H	2.204
16	OH	H	CH3	H	H	H	H	−1.114
17	F	H	CH3	H	H	H	H	−0.032
18	H	H	CH=CH2	H	H	H	H	−1.060
19	OH	H	CH2CO2H	H	H	H	H	2.447
20	OH	H	CH2CONH2	H	H	H	H	0.806
21	Cl	H	CH2CO2H	H	H	H	H	3.653
22	Cl	H	CH2CONH2	H	H	H	H	1.954
23	H	H	CH(CO)CO2H	H	H	H	H	3.000
24	Cl	H	CH2CH2Cl	H	H	H	H	1.724
25	H	H	CH3	H	H	Cl	H	−1.076
26	H	H	CH3	H	Cl	H	H	−1.959
27	H	H	CH3	H	H	Br	H	−0.745
28	H	H	CH3	H	H	OCH3	H	−1.444
29	H	H	CH3	H	H	OH	H	−1.638
30	H	H	CH3	H	NH2	H	H	−1.569
31	H	H	CH3	H	Br	H	H	−1.097
32	H	H	CH3	H	OCH3	H	H	−1.337
33	H	H	CH3	H	OH	H	H	−1.745
34	H	H	CH3	H	CH2OH	H	H	−0.863
35	H	H	CH3	H	CH3	H	H	−1.469
36	H	H	CH3	H	(CH2)3CH3	H	H	0.097
37	H	H	CH3	H	C6H5	H	H	−1.495
38	H	H	CH3	OCH3	H	H	H	−0.215
39	H	H	CH3	H	H	H	OCH3	−1.481
40	H	H	CH3	OH	H	H	H	−0.558
41	H	H	CH3	H	H	H	OH	−1.310
42	H	H	CH3	H	F	F	H	−1.509
43	H	CH3	H	H	H	H	H	1.081
44	H	(CH2)2OH	H	H	H	H	H	1.854
45	H	H	(CH2)2CH3	H	H	H	H	0.921
46	OH	H	(CH2)2OH	H	H	H	H	−1.328
47	H	H	CH(OH)CO2Et	H	H	H	H	−0.408
48	Cl	H	(CH2)2OH	H	H	H	H	−0.553

Fig. 1 Schematic diagram of MK801 skeleton.

2.2 Computational methods

The activity of these compounds was correlated with various physicochemical parameters by density functional theory. The three-dimensional structures were generated with Gauss View 3.0, and all calculations were performed with Gaussian 03W programs. Geometrical optimization of 48 compounds was carried out by Lee–Yang–Parr exchange correlation functional with the 6-31G (d) basic set [23–27]. The geometry of the compounds was determined by optimizing all geometrical variables with no symmetry constraints [28]. ChemSketch [29] was used to calculate the other molecular descriptors.

The quantum chemistry descriptors were obtained for the model from the density functional theory calculations as follows: total energy (E), highest occupied molecular orbital (HOMO) energy (E_HOMO), lowest unoccupied molecular orbital (LOMO) energy (E_LUMO), the difference between LUMO and HOMO energy (Gap), the total dipole moment of the compound (μ, Debye), absolute hardness (η), absolute electronegativity (χ) and the reactivity index (ω) [30].

η, χ and ω were determined from:$\eta =\frac{E_{\text{LUMO}}-E_{\text{HOMO}}}{2},\chi =\frac{E_{\text{LUMO}}-E_{\text{HOMO}}}{2}\text{and}\omega =\frac{{\chi }^2}{2\eta }\text{.}$

The Advanced Chemistry Development ChemSketch program was used to calculate formula weight, molar volume (cm³), molar refractivity (cm³), parachor (cm³), density (g/cm³), refractive index (n), surface tension (γ (dyne/cm)) and polarizability (α_e (cm³)) [31].

2.3 Statistical analysis

To explain the structure–activity relations, we used principal component analysis, multiple linear and non-linear regression in XLSTAT software [32]. The artificial neural network and leave-one-out cross-validation were performed with a program written in C language in Matlab 7.

Principal component analysis is a statistical technique used for summarizing information encoded in the structures of compounds and for understanding the distribution. It is essentially a descriptive statistical method for presenting the maximum information contained in Tables 2 and 3 in graphical form.

Multiple linear regression is used to study the relation between one dependent and several independent variables. It minimizes differences between actual and predicted values and was used to select the descriptors to be used as inputs into multiple non-linear regression and the artificial neural network. Multiple linear and non-linear regression were used to predict effects on the activity of dibenzo[a,d]cycloalkenimines (K_i). Equations were justified by the correlation coefficient (r), the mean squared error, Fisher's F statistic and the significance level (F value) [33].

Artificial neural networks are artificial systems that simulate the function of the human brain. A neural network has three components: the processing elements or nodes, the topology of the connections between the nodes, and the rule by which new information is encoded in the network. While there are a number of models, the most frequently used artificial neural network in quantitative structure–activity relation studies is a three-layered feed-forward network [34]. In this type of network, the neurons are arranged in layers, with an input layer, one hidden layer and an output layer. Each neuron in each layer is fully connected to the neurons of a succeeding layer, and there are no connections between neurons in the same layer. According to the supervised learning model, the networks are taught by giving them examples of input patterns and the corresponding target outputs. Through an iterative process, the connection weights are modified until the network gives the desired results for the training set of data. A back-propagation algorithm is used to minimize the error function. This algorithm has been described previously, with a simple example of application [35]. A detailed description of this algorithm is given elsewhere [36].

Cross-validation is used to explore the reliability of statistical models. In this technique, a number of modified data sets are created by deleting one or a small group of molecules, known respectively as “leave-one-out” and “leave-some-out” [37–39]. For each data set, an input–output model is prepared, and its accuracy in predicting the responses of the remaining data (those that were not used in the model) is evaluated. In this study, we used the leave-one-out procedure.

3 Results and discussion

3.1 Data set for analysis

The quantitative structure–activity relation analysis was performed using the Log K_i of the 48 selected molecules that have been synthesized and evaluated for their ability to displace dibenzo[a,d]cycloalkenimines (experimental values) as reported by Thompson et al. [22], The values of the 16 chemical descriptors as shown in Table 2.

Table 2 The values of the 16 chemical descriptors.

	PM	MR	MV	Pc	n	γ	D	αe	Log(−E)	EHOMO	ELUMO	Gap	μ	χ	η	ω
1	235.324	73.95	208.3	535.7	1.628	43.7	1.129	29.31	4.288	−5.767	−0.157	5.610	0.805	2.962	2.805	1.564
2	221.297	69.02	193.3	510.8	1.632	48.7	1.144	27.36	4.263	−5.948	−0.184	5.764	1.079	3.066	2.882	1.631
3	249.350	78.56	223.1	574.1	1.621	43.8	1.117	31.14	4.311	−5.698	−0.133	5.564	0.850	2.915	2.782	1.527
4	237.296	70.83	185.3	512.7	1.689	58.5	1.280	28.08	4.309	−6.047	−0.190	5.857	0.578	3.119	2.928	1.661
5	235.324	73.64	210.9	555.6	1.615	48.1	1.115	29.19	4.288	−5.941	−0.195	5.745	1.055	3.068	2.873	1.638
6	235.324	73.96	211.2	537.4	1.617	41.9	1.114	29.32	4.288	−5.817	−0.176	5.641	0.795	2.997	2.820	1.592
7	325.358	87.98	231.1	673.8	1.686	72.1	1.407	34.88	4.473	−5.911	−0.226	5.684	1.043	3.068	2.842	1.656
8	309.359	86.17	239.1	669.7	1.640	61.4	1.293	34.16	4.442	−6.009	−0.153	5.856	1.987	3.081	2.928	1.621
9	251.323	75.18	208.4	572.3	1.640	56.8	1.205	29.80	4.331	−5.982	−0.251	5.731	2.315	3.116	2.866	1.695
10	293.360	84.68	247.4	659.9	1.600	50.5	1.185	33.57	4.408	−5.910	−0.130	5.780	2.357	3.020	2.890	1.578
11	237.296	70.54	191.9	532.5	1.656	59.2	1.236	27.96	4.309	−5.873	−0.137	5.736	1.140	3.005	2.868	1.574
12	239.287	69.17	199.8	525.3	1.608	47.7	1.197	27.42	4.323	−6.068	−0.307	5.761	1.379	3.188	2.880	1.764
13	253.314	73.81	216.3	216.3	1.597	46.5	1.170	29.26	4.344	−6.105	−0.369	5.736	2.442	3.237	2.868	1.827
14	329.458	102.25	259.4	723.9	1.717	60.5	1.260	40.53	4.550	−5.959	−0.582	5.377	1.954	3.271	2.689	1.989
15	345.457	103.26	276.7	758.4	1.668	56.3	1.248	40.93	4.574	−6.105	−0.698	5.407	1.549	3.402	2.704	2.140
16	237.296	70.51	185.0	520.6	1.687	62.5	1.282	27.95	4.309	−6.057	−0.231	5.826	0.526	3.144	2.913	1.697
17	239.287	69.67	191.1	505.1	1.649	48.7	1.250	27.62	4.323	−6.179	−0.370	5.810	2.012	3.274	2.905	1.845
18	233.308	75.10	194.2	526.2	1.700	53.9	1.201	29.77	4.287	−5.991	−0.234	5.756	1.113	3.112	2.878	1.683
19	281.306	76.70	197.3	587.2	1.705	78.4	1.425	30.40	4.407	−6.074	−0.257	5.816	1.150	3.165	2.908	1.723
20	280.321	78.70	203.4	599.5	1.700	75.3	1.377	31.20	4.397	−6.027	−0.241	5.786	2.947	3.134	2.893	1.698
21	299.752	80.67	208.1	597.1	1.702	67.7	1.440	31.98	4.556	−6.253	−0.500	5.753	2.145	3.376	2.877	1.982
22	298.767	82.75	214.3	609.9	1.699	65.5	1.390	32.80	4.549	−6.128	−0.341	5.787	3.421	3.234	2.894	1.808
23	281.306	76.65	201.0	574.8	1.687	66.7	1.398	30.38	4.407	−5.873	−0.524	5.349	2.147	3.199	2.674	1.913
24	304.214	83.85	225.1	613.9	1.667	55.2	1.350	33.24	4.647	−6.445	−0.727	5.718	3.110	3.586	2.859	2.249
25	255.742	73.92	205.3	546.6	1.639	50.2	1.245	29.30	4.489	−6.088	−0.427	5.661	3.107	3.258	2.831	1.874
26	255.742	73.92	205.3	546.6	1.639	50.2	1.245	29.30	4.489	−6.124	−0.488	5.636	1.569	3.306	2.818	1.939
27	300.193	76.71	209.5	561.3	1.653	51.4	1.432	30.41	4.946	−6.060	−0.437	5.623	3.026	3.249	2.812	1.877
28	251.323	75.70	217.3	567.4	1.613	46.4	1.156	30.01	4.331	−5.543	−0.118	5.425	1.235	2.830	2.712	1.477
29	237.296	70.91	191.7	525.8	1.661	56.5	1.237	28.11	4.309	−5.600	−0.146	5.455	1.498	2.873	2.727	1.513
30	236.312	73.82	204.1	546.0	1.642	51.1	1.150	29.26	4.297	−5.237	−0.082	5.154	2.809	2.659	2.577	1.372
31	300.193	76.71	209.5	561.3	1.653	51.4	1.432	30.41	4.946	−6.104	−0.488	5.616	1.479	3.296	2.808	1.934
32	251.323	76.27	215.5	46.4	1.625	46.4	1.160	30.23	4.331	−5.626	−0.168	5.459	2.299	2.897	2.729	1.538
33	237.296	71.73	197.9	533.3	1.644	52.6	1.190	28.43	4.309	−5.678	−0.221	5.458	2.176	2.949	2.729	1.594
34	251.323	75.47	207.1	564.3	1.649	55.0	1.213	29.92	4.331	−5.854	−0.145	5.709	1.931	2.999	2.854	1.576
35	235.324	73.85	209.6	548.4	1.622	46.8	1.122	29.27	4.288	−5.869	−0.158	5.711	1.341	3.014	2.856	1.590
36	277.403	87.84	259.1	666.9	1.593	43.8	1.070	34.82	4.354	−5.860	−0.149	5.711	1.395	3.004	2.855	1.581
37	297.393	93.62	258.6	684.1	1.643	48.9	1.149	37.11	4.391	−5.794	−0.684	5.109	0.978	3.239	2.555	2.053
38	251.323	76.27	215.5	562.6	1.625	46.4	1.160	30.23	4.331	−5.557	−0.121	5.436	2.180	2.839	2.718	1.483
39	251.323	75.70	217.3	567.4	1.613	46.4	1.156	30.01	4.331	−5.528	−0.165	5.363	0.961	2.847	2.681	1.511
40	237.296	71.73	197.9	533.3	1.644	52.6	1.190	28.43	4.309	−5.612	−0.164	5.448	1.980	2.888	2.724	1.531
41	237.296	70.91	191.7	525.8	1.661	56.5	1.237	28.11	4.309	−5.587	−0.203	5.383	1.154	2.895	2.692	1.557
42	257.278	69.01	201.7	525.0	1.599	45.8	1.275	27.36	4.375	−6.089	−0.501	5.588	2.061	3.295	2.794	1.943
43	221.297	69.34	193.6	497.4	1.635	43.5	1.142	27.49	4.263	−5.837	−0.185	5.652	0.839	3.011	2.826	1.604
44	251.323	75.49	208.7	554.4	1.643	49.8	1.204	29.93	4.331	−5.757	−0.162	5.595	1.968	2.960	2.798	1.565
45	249.350	78.28	227.4	595.3	1.604	46.9	1.096	31.03	4.311	−5.938	−0.196	5.742	1.079	3.067	2.871	1.638
46	267.322	76.67	200.1	582.1	1.691	71.5	1.335	30.39	4.371	−6.092	−0.295	5.797	1.587	3.194	2.899	1.759
47	309.359	86.12	242.9	658.3	1.627	53.9	1.273	34.14	4.442	−5.794	−0.340	5.454	3.325	3.067	2.727	1.725
48	285.768	80.55	211.1	592.0	1.688	61.7	1.350	31.93	4.531	−6.277	−0.559	5.718	3.765	3.418	2.859	2.043

The principle is to perform in the first time, a main component analysis (PCA), which allows us to eliminate descriptors that are highly correlated (dependent), then perform a decreasing study of multiple linear regression based on the elimination of descriptors aberrant until a valid model (including the critical probability: p-value <0.05 for all descriptors and the model complete).

3.2 Principal component analysis

All 16 descriptors (variables) coding the 48 molecules were submitted to principal components analysis, and 17 components were obtained (Fig. 2). The first three axes, F1, F2 and F3, contributed 41.5%, 21.6% and 11.1%, respectively, to the total variance, and the total information was estimated to be 74.2%. The Pearson correlation coefficients are summarized in Table 3; the matrix provides information on the negative and positive correlations between variables. Correlations among the 16 descriptors are shown in Table 3 as a correlation matrix; in Fig. 3, these descriptors are represented in a correlation circle.

Table 3 The correlation matrix (Pearson (n)) between different obtained descriptors.

	PM	MR	MV	Pc	n	γ	D	αe	Log(−E)	EHOMO	ELUMO	Gap	μ	χ	η	ω	Log Ki
PM	1
MR	0.862	1
MV	0.720	0.911	1
Pc	0.536	0.572	0.473	1
n	0.368	0.243	−0.175	0.262	1
γ	0.457	0.233	−0.107	0.297	0.834	1
D	0.587	0.164	−0.136	0.219	0.722	0.782	1
αe	0.862	1	0.911	0.572	0.243	0.233	0.164	1
Log(−E)	0.710	0.406	0.281	0.271	0.302	0.235	0.687	0.406	1
EHOMO	−0.398	−0.169	−0.026	−0.156	−0.351	−0.354	−0.551	−0.169	−0.484	1
ELUMO	−0.619	−0.486	−0.339	−0.263	−0.339	−0.199	−0.491	−0.486	−0.629	0.632	1
Gap	−0.083	−0.249	−0.290	−0.053	0.124	0.260	0.235	−0.249	0.017	−0.673	0.148	1
μ	0.395	0.186	0.130	−0.003	0.135	0.203	0.396	0.186	0.434	−0.232	−0.350	−0.037	1
χ	0.544	0.337	0.177	0.223	0.382	0.317	0.580	0.337	0.603	−0.930	−0.872	0.355	0.312	1
η	−0.082	−0.248	−0.289	−0.052	0.125	0.261	0.236	−0.248	0.019	−0.674	0.147	1	−0.036	0.356	1
ω	0.610	0.446	0.289	0.261	0.367	0.255	0.539	0.446	0.635	−0.783	−0.976	0.068	0.339	0.957	0.069	1
Log Ki	0.520	0.437	0.245	0.345	0.473	0.489	0.455	0.437	0.219	−0.335	−0.243	0.196	0.044	0.327	0.196	0.291	1

Fig. 2 The principal components and there variances.

Fig. 3 Correlation circles.

As molar refractivity (α_e) and (gap, η) are perfectly correlated (r = 1), both variables are redundant. Molar refractivity, molar volume and polarizability are highly correlated (r (molar volume, molar refractivity) = 0.911, r (molar volume, polarizability) = 0.911. E_HOMO and absolute electronegativity are strongly negatively correlated (r = −0.930), and E_LUMO, absolute electronegativity and reactivity index are strongly negatively correlated: r (E_LUMO, reactivity index)) = −0.976, (absolute electronegativity, reactivity index) = −0.957. The variables polarizability, gap, molar volume and absolute electronegativity were therefore removed.

In the projection of the compounds in the plane of the three first axes, F1, F2 and F3 (Fig. 4), they are distributed in three regions. Region 1 contains those with log (−E) values between 4.263 (−18,323.14 eV) and 4.354 (−22,594.36 eV), region 2 contains compounds with values between 4.371 (−23,496.33 eV) and 4.489 (−30,831.88 eV), and region 3 contains compounds with values between 4.531 (−33,962.53 eV) and 4.946 (−88,307.99 eV).

Fig. 4 Cartesian diagram showing the separation between the tree regions and the dispersal of different molecules by groups.

3.3 Multiple linear regression

In order to propose a mathematical model and to evaluate physicochemical effects on the activity of the entire set of 48 compounds quantitatively, we submitted the data matrix constituted from the 12 variables corresponding to the training set to descendent multiple regression analysis. The correlation coefficient, r, the coefficient of determination, r², mean squared error and F values were used to select the best regression performance.

Multiple linear regression allowed connection of the structural descriptors for the activity of each of the 38 compounds in order to evaluate the effect of the substituent quantitatively. The descriptors selected were molar refractivity, surface tension (γ), density, log (−E), E_LUMO and reactivity index (ω).

The quantitative structure–activity relation model built with multiple linear regression is represented by:(1) $\text{log}{K}_i=-9.783+0.192\text{molar refractivity}-0.134\text{surface tension}+24.864\text{density}-9.411\text{log}(-E)+13.416E_{\text{LUMO}}+9.651\text{reactivity index}$(1) $N=38r=0.731r^2=0.535F=5.934\text{Mean}\text{squared}\text{error}=1.276$

A high correlation coefficient and a low mean squared error indicate that the model is reliable. As the F value is less than 0.05, we would be taking a less than 0.01% risk in assuming that the null hypothesis is wrong. Therefore, we can conclude that the model provides a significant amount of information.

The quantitative structure–activity relation model showed that the N-methyl-d-aspartate antagonist activity can be explained by a number of electronic and topological factors. The negative correlation between surface tension and the total energy and the ability to displace the activity of dibenzo[a,d]cycloalkenimines results in a decrease in log (K_i), while the positive correlation between the topological descriptors density and molar refractivity and electronic descriptors E_HOMO and E_LUMO indicates the ability to displace dibenzo[a,d]cycloalkenimine activity, with an increase in log (K_i).

The predicted log (K_i) activities calculated from equation 1 in the optimal multiple linear regression model and the observed values are given in Table 4. The correlations between the predicted and observed activities and the residue values are illustrated in Fig. 5. The descriptors proposed in Eq. (1)(1) $\text{log}{K}_i=-9.783+0.192\text{molar refractivity}-0.134\text{surface tension}+24.864\text{density}-9.411\text{log}(-E)+13.416E_{\text{LUMO}}+9.651\text{reactivity index}$(1) were therefore used as the input parameters for multiple non-linear regression and the artificial neural network.

Table 4 The observed, the predicted activities (Log Ki), and residue according to different methods for the 38 MK801 derivatives (training set).

No.	Obs.	MLR		MNLR		ANN		CV
		Pred	Resid	Pred	Resid	Pred	Resid	Pred	Resid
11	−0.456	−0.635	0.179	−0.527	0.071	−0.818	−0.362	−0.161	0.295
12	−0.796	−0.911	0.115	−0.771	−0.025	−0.818	−0.022	−0.819	−0.023
13	−0.759	−0.954	0.195	−1.429	0.670	−0.818	−0.059	−0.820	−0.061
14	1.863	1.633	0.230	1.284	0.579	1.786	−0.077	−0.224	−2.087
15	2.204	1.764	0.440	2.148	0.056	2.348	0.144	1.531	−0.673
16	−1.114	−0.012	−1.102	−0.256	−0.858	−0.818	0.296	−0.807	0.307
17	−0.032	0.306	−0.338	0.060	−0.092	−0.818	−0.786	−0.172	−0.140
18	−1.060	0.034	−1.094	−0.976	−0.084	−0.818	0.242	−1.098	−0.038
19	2.447	1.587	0.860	2.282	0.165	2.825	0.378	3.327	0.880
20	0.806	1.259	−0.453	0.611	0.195	0.300	−0.506	1.038	0.232
21	3.653	1.987	1.666	3.905	−0.252	3.480	−0.173	2.998	−0.655
22	1.954	1.956	−0.002	2.096	−0.142	2.033	0.079	0.443	−1.511
23	3.000	0.720	2.280	1.636	1.364	2.874	−0.126	2.685	−0.315
24	1.724	0.702	1.022	0.964	0.760	1.394	−0.330	−0.220	−1.944
25	−1.076	−1.251	0.175	−1.782	0.706	−0.818	0.258	−0.809	0.267
26	−1.959	−1.443	−0.516	−1.900	−0.059	−0.818	1.141	−0.777	1.182
27	−0.745	−0.634	−0.111	−0.792	0.047	−0.793	−0.048	−0.796	−0.051
28	−1.444	−0.814	−0.630	−0.685	−0.759	−0.818	0.626	−0.795	0.649
29	−1.638	−0.888	−0.750	−0.770	−0.868	−0.818	0.820	−0.852	0.786
30	−1.569	−2.162	0.593	−1.331	−0.238	−0.818	0.751	−1.285	0.284
31	−1.097	−0.768	−0.329	−0.861	−0.236	−0.780	0.317	−0.760	0.337
32	−1.337	−0.688	−0.649	−0.961	−0.376	−0.818	0.519	−0.799	0.538
33	−1.745	−1.604	−0.141	−1.649	−0.096	−0.818	0.927	−0.784	0.961
34	−0.863	0.004	−0.867	−0.571	−0.292	−0.818	0.045	−0.817	0.046
35	−1.469	−1.109	−0.360	−0.462	−1.007	−0.818	0.651	−0.794	0.675
36	0.097	0.093	0.004	1.248	−1.151	−0.818	−0.915	−0.176	−0.273
37	−1.495	−0.487	−1.008	−0.752	−0.743	−0.817	0.678	−0.133	1.362
38	−0.215	−0.588	0.373	−0.635	0.420	−0.818	−0.603	−0.168	0.048
39	−1.481	−1.117	−0.364	−1.187	−0.294	−0.818	0.663	−0.794	0.687
40	−0.558	−1.447	0.889	−1.273	0.715	−0.818	−0.260	−0.836	−0.278
41	−1.310	−1.228	−0.082	−1.325	0.015	−0.818	0.492	−1.760	−0.450
42	−1.509	−0.113	−1.396	0.109	−1.618	−0.818	0.691	−0.792	0.717
43	1.081	−1.028	2.109	−0.333	1.414	−0.818	−1.899	0.990	−0.091
44	1.854	0.144	1.710	−0.471	2.325	−0.818	−2.672	−0.223	−2.077
45	0.921	−1.183	2.104	−0.504	1.425	−0.818	−1.739	−0.198	−1.119
46	−1.328	0.441	−1.769	−0.680	−0.648	−0.749	0.580	−0.697	0.631
47	−0.408	1.461	−1.869	−0.234	−0.174	−0.782	−0.374	−0.162	0.246
48	−0.553	0.560	−1.113	0.361	−0.914	0.100	0.653	−0.158	0.395

Table 5 The observed, the predicted activities (Log (K_i)), and residue according to MLR and MNLR for the 10 tested compounds (test set).

No.	Obs.	MLR		MNLR
		Pred-test	Resid	Pred-test	Resid
1	−0.215	−0.740	−0.525	−0.400	−0.185
2	−1.252	−1.460	−0.208	−0.517	0.735
3	−0.149	−0.419	−0.270	−0.202	−0.053
4	1.279	0.737	−0.542	0.647	−0.632
5	−1.347	−1.530	−0.183	−0.766	0.581
6	1.380	−0.855	−2.235	−0.347	−1.727
7	3.653	3.291	−0.362	3.478	−0.175
8	0.556	2.472	1.916	1.220	0.664
9	−0.585	−0.765	−0.180	−1.611	−1.026
10	−0.260	1.170	1.430	0.242	0.502

Fig. 5 Correlations of observed and predicted activities and the residues values calculated using MLR.

3.4 Multiple non-linear regression

We used also a non-linear regression model to improve the structure–activity relation and evaluate the effect of substituents quantitatively. We applied the descriptors proposed by multiple linear regression for the 38 molecules in the training set and used the coefficients r, r² and F to select the best regression performance. We used a pre-programmed function of XLSTAT as follows:$Y=a+(b{X}_1+c{X}_2+d{X}_3+e{X}_4\dots )+(f{X}_1^2+g{X}_2^2+h{X}_3^2+i{X}_4^2\dots )$where a, b, c, d,… represent the parameters and X₁, X₂, X₃, X₄,… represent the variables.

The resulting equation was:(2) $\text{log}{K}_i=-97.482-0.893\text{molar refractivity}+0.154\text{surface tension}-220.100\text{density}+117.475\text{log}(-E)+18.508E_{\text{LUMO}}+2.298\text{reactivity index}+0.006{(\text{molar refractivity})}^2-0.003{(\text{surface tension})}^2+98.188{(\text{density})}^2-14.042{(\text{log}(-E))}^2+6.731{(E_{\text{LUMO}})}^2+1.963{(\text{reactivity index})}^2$(2) $N=38r=0.852r^2=0.726\text{Mean}\text{squared}\text{error}=0.933$

The predicted activities calculated from Eq. (2)(2) $\text{log}{K}_i=-97.482-0.893\text{molar refractivity}+0.154\text{surface tension}-220.100\text{density}+117.475\text{log}(-E)+18.508E_{\text{LUMO}}+2.298\text{reactivity index}+0.006{(\text{molar refractivity})}^2-0.003{(\text{surface tension})}^2+98.188{(\text{density})}^2-14.042{(\text{log}(-E))}^2+6.731{(E_{\text{LUMO}})}^2+1.963{(\text{reactivity index})}^2$(2) and the observed values are given in Table 4. The correlations of the predicted and observed activities and the residues values are illustrated in Fig. 6.

Fig. 6 Correlations of observed and predicted activities and the residues values calculated using MNLR.

The true predictive power of a quantitative structure–activity relation model is its ability to predict accurately the activities of compounds in an external test set (compounds not used in the model development). The activities of the remained 10 compounds are deduced from the training set by multiple linear and non-linear regression. Their structures and the observed and calculated log (K_i) values are given in Tables 1 and 5.

Comparison of the values of log (K_i-test) and log (K_i-obs.) shows that good predictions for the 10 compounds:

• Multiple linear regression:$N=10r_{\text{test}}=0.750r_{\text{test}}^2=0.563$

• Multiple non-linear regression:$N=10r_{\text{test}}=0.845r_{\text{test}}^2=0.715$

3.5 Artificial neural networks

In order to increase the probability of characterizing the compounds well, artificial neural networks can be used to generate predictive models of quantitative structure–activity relations between a set of molecular descriptors obtained from multiple linear regression and the observed activities. The model was prepared with the properties of several of the compounds. A parameter, ρ, has been proposed for determination of the number of hidden neurons, which play a major role in determining the best artificial neural network architecture [40,41], defined as follows:$\rho =\frac{\text{Number of data points in the training set}}{\text{Sum of the number of connections in the artificial neural network}}$

In order to avoid over-fitting or under-fitting, it is recommended that 1.8 < ρ < 2.3 [42]. The output layer represents the calculated activity values (log (K_i)). The architecture of the artificial neural network used in this work was 6-2-1, with ρ = 2.11.

The correlation between the calculated and experimental artificial neural network and the residue values were highly significant, as illustrated in Fig. 7 and as indicated by the r and r² values. The predicted activities calculated with the artificial neural network and the observed values are given in Table 4.$N=38r=0.849r^2=0.721$

Fig. 7 Correlations of observed and predicted activities and residues values calculated using ANN.

The r² value confirms that the results of the artificial neural network were the best for building quantitative structure–activity models. ‘Leave-one-out’ is an approach particularly well adapted for estimating the predictive ability of these models. The correlations between the observed activities, the cross-validation, calculated values and the residues are illustrated in Fig. 8 and Table 4.$N=38r_{\text{cv}}=0.836r_{\text{cv}}^2=0.699$

Fig. 8 Correlations of observed and predicted activities and residues values calculated using CV.

The good results obtained with cross-validation and for the prediction of the activities of the 10-compound test set show that the model proposed in this paper can accurately predict activity and that the selected descriptors are pertinent. The results obtained by multiple linear and non-linear regression are sufficient to conclude that the model performs well. Although the good predictive ability might be due to chance, we consider it a positive result. This model could therefore be used for all the dibenzo[a,d]cycloalkenimine derivatives (Table 1) and could add to the search for non-competitive antagonists of N-methyl-d-aspartate receptors and their interaction with the receptor.

Our tests with multiple linear and non-linear regression and artificial neural network models showed a substantially better predictive capability of the artificial neural network model. It showed a satisfactory relation between the molecular descriptors and the activity of the compounds and a good correlation with cross-validation (r_cv = 0.836).

4 Conclusion

Multiple linear and non-linear regression and an artificial neural network were used to construct a quantitative structure–activity relation model for antagonists of N-methyl-d-aspartate receptors and compared. The artificial neural network had substantially better predictive capability than the other two models, with greater predictive power. We established satisfactory relations between several descriptors and antagonist activity to the N-methyl-d-aspartate receptor, with cross-validation. The results show that the model proposed in this paper can predict activity accurately and that the selected descriptors are pertinent.

The accuracy and predictability of the proposed models were illustrated by comparison of the key statistical terms r or r² for the different models (Table 4). The proposed methods will reduce the time and cost of synthesis and determination of the activity of N-methyl-d-aspartate receptor antagonists based on dibenzo[a,d]cycloalkenimine derivatives. Furthermore, the descriptors are sufficiently rich in chemical, electronic and topological information to encode structural features that could be used with other descriptors in the development of predictive, quantitative structure–activity models.

Acknowledgements

We are grateful to the Association marocaine des Chimistes théoriciens for help with the programs.

Notes

Peer review under responsibility of Taibah University.