ABSTRACT
The biological activities of a congeneric series of pyropheophorbides used as sensitizers in photodynamic therapy have been predicted on the basis of their molecular structures, using multiple linear regression and artificial neural network (ANN) computations. Theoretical descriptors (a total of 81) were calculated by the 3DNET program based on the three-dimensional structure (3D) of the geometry-optimized molecules. These input descriptors were tested as independent variables and used for model building. Systematic descriptor selections yielded models with one, two or three descriptors with good cross-validation results. The predictive abilities of the best fitting models were checked by shuffling and cross-validation procedures. ANN was suitable for building models for both linear and nonlinear relationships. Lipophilicity was sufficient to predict the accumulation of the sensitizers in the target tissue. Weighted holistic invariant molecular descriptors weighted by atomic mass, Van der Waals volume or electronegativity were also needed to predict photodynamic activity properly. Our models were able to predict the biological activities of 13 pyropheophorbide derivatives solely on the basis of their 3D molecular structures. Moreover, linear and nonlinear variable selection methods were compared in models built linearly and nonlinearly. It is expedient to use the same method (linear or nonlinear) for variable selection as for parameter estimation.
Abbreviations:
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- ANN
-
- artificial neural network
-
- 2D
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- two-dimensional
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- 3D
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- three-dimensional
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- MLR
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- multiple linear regression
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- PCA
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- principal component analysis
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- PDT
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- photodynamic therapy
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- QSAR
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- quantitative structure–activity relationship.
INTRODUCTION
Photodynamic therapy (PDT) is one of the promising methods in cancer treatment. It uses a combination of a photosensitizing agent and visible light for therapy of many kinds of solid tumors. The light-activated sensitizer usually produces active oxygen species (e.g. singlet oxygen), which results in tumor destruction (1).
Photofrin II, a complex mixture of porphyrins based on a hematoporphyrin derivative, is a photosensitizer widely used in clinical trials (2). It is not an ideal sensitizer because (1) it is unstable during storage or after administration; (2) it has a relatively low optical extinction coefficient in the 600–900 nm range, where light exhibits optimal penetration through tissue; and (3) its excretion is quite slow. Therefore, development of new photosensitizers, which are pure materials and are activated strongly by red light above 630 nm wavelength, is an important goal. A wide range of tetrapyrrole derivatives such as porphyrins, chlorins, benzochlorins, bacteriochlorins, pheophorbides, phthalocyanines and naphthalocyanines are being studied as possible second-generation photosensitizers. Hundreds of compounds designed as potential PDT agents have been synthesized so far in different laboratories. The most promising compounds are usually tested in cell cultures or on live animals. These tests in biological systems require a long time and a large number of animals, making the selection process cumbersome and expensive. Therefore, a preselection of promising molecules on the basis of their structures would be preferable. Moreover, the large number of tested compounds makes possible a rational design of new, more effective molecules. Similarly, good models are needed to understand the mechanistic action of PDT. Prediction of the biological activity of compounds with known structures becomes possible only after understanding the main factors influencing PDT effects. An idea about the substructures necessary to achieve high PDT activity can also be obtained after modeling these effects.
Nowadays, quantitative structure–activity relationships (QSAR) are widely used in rational drug design. Quantitative relationships are frequently found between the differences in biological activity and the changes in physicochemical properties. A linear relationship was observed, for example, between the oil–water partition coefficient and the narcotic activities of a series of compounds. Therefore, lipophilicity can be used to describe nonspecific biological activities for certain groups of compounds. The combination of different physicochemical parameters in a linear additive manner gave QSAR a head start. Later, a parabolic equation was formulated for the quantitative description of nonlinear lipophilicity–activity relationships, which are very common (3). Nowadays, we can use several other methods for studying such relationships.
In the development of second-generation photosensitizers, the relationship between the structure and the properties affecting tumoricidal effect of compounds has been studied (4). In 1990, a comparison between the tumor localizing properties and hydrophilicity, as well as dimerization abilities of 28 porphyrins and pheophorbides, was published (5). It was found that the tumoricidal activity depends on a delicate balance between the hydrophilic and hydrophobic characters of the compounds. Another study examined the relationship between the photophysical properties and the photodynamic activity of five tetrapyrroles (6). A good correlation between generation of singlet oxygen and PDT effect was observed. An in vitro structure–activity relationship of a set of silicon phthalocyanine sensitizers was published in 1997 (7). However, none of these studies was quantitative.
Henderson and co-workers (8,9) reported that PDT activity is a nonlinear function of lipophilicity. The study used a semiempirical, nonlinear activity–lipophilicity relationship model and found that lipophilicity is highly predictive for photodynamic activity, whereas accumulation of photosensitizers in the cancer tissue is not sufficient for good tumoricidal effect. In a QSAR study (10), where comparative molecular field analysis was used, 21 porphyrins were examined as anti–human immunodeficiency virus agents. To our knowledge, this is the only study that applied a three-dimensional (3D) QSAR model to photosensitizers of tetrapyrrole structure. The photoinduced toxicity of polycyclic aromatic hydrocarbons has already been studied by QSAR methodology, using highest occupied molecular orbital–lowest unoccupied molecular orbital energies as descriptors (11).
3D QSAR has been developed from the beginning to map receptor surfaces for noncovalent interactions of ligands in different positions of substitution.
One of the most promising methods for 3D QSAR modeling is the use of artificial neural networks (ANN). This method mimics a biological neuron stimulated by one or more inputs and generates an output that is sent to other neurons. The output depends on each of the inputs and on the nature of each input connection. This connection can increase, decrease or delay the output. The individual neuron can be modeled by a simple, weighted sum of inputs or by complicated differential equations. A fully connected three-layer, feed-forward computational neural network with back-propagation training is widely used in chemistry (12,13).
We applied the 3D neural network method described earlier by Kövesdi et al. (14) to study the PDT activity of a pyropheophorbide series analyzed by Henderson et al. (8). The purpose was to find better fits and highly predictive models for the synthesis of effective photosensitizers with potential use in cancer therapy. Moreover, we wanted to extend a variable selection and model-building method suitable for other kinds of sensitizers. Our approach was to use a linear modeling first, which is a well known and the most accepted method, and then apply ANN to demonstrate the superiority of the latter.
MATERIALS AND METHODS
Chemical structures Structures of pyropheophorbide derivatives are given in Fig. 1. The conformational search module of ChemPlus™ extension for HyperChem program (15) was used to find the global minimum conformer of the studied sensitizers. During this procedure, all the dihedral angles around the single bonds of the substituents were randomly varied to generate new structures. The geometry was optimized using MM+ molecular mechanics method, which uses harmonic oscillator function to calculate potentials for bonds and angles. To find the local minimum of energy, 256 iterations were completed for each structure with 0.1 kcal/Å convergence criterion for the gradient. Both geometric and energetic criteria were applied to decide when to accept the resulting conformer. The last accepted structure was employed as the initial structure for the next optimization. Energies of 1024 conformers for every compound were calculated, minimized and compared.
. Chemical structures of pyropheophorbide derivatives. M1, R1= methyl; M2, R1= 1-propyl; M3, R1= 1-pentyl; M4, R1= 1-hexyl; M5, R1=cis-3-hexenyl; M6, R1=trans-3-hexenyl; M7, R1= cyclohexenyl; M8, R1= 2-hexyl; M9, R1= 1-heptyl; M10, R1= 1-octyl; M11, R1= 1-nonyl; M12, R1= 1-decyl; and M13, R1= 1-dodecyl
In the case of minimum conformers, the alkyl ether chain was partially (black balls in Fig. 2A) or completely (black balls in Fig. 2B) bent over the main plane of the pheophorbide ring. This suggests that there are strong Van der Waals interactions between the side chain and the pheophorbide ring. The propionic acid chain (light-gray balls in Fig. 2A,B) was also bent over the base structure when the ether substituent (R1) varied between methyl (M1) and 2-hexyl (M8) groups (Fig. 2A). The propionic acid chain was perpendicular to the main plane of the molecule when the R1 substituent varied between 1-heptyl (M9) and 1-dodecyl (M13) chains (Fig. 2B).
. 3D structures of compounds optimized by conformational analysis. A: M3 is an example of a molecule with shorter alkyl ether chains. B: M11 is a compound with longer side chain. White balls represent the pheophorbide base, black balls represent alkyl ether side chain and light-gray balls represent other side chains
We used these conformers to represent the 3D structures of the molecules. The two-dimensional (2D) and 3D molecular structures, as well as the biological activity data, were stored in MDL IsisBase format (16).
Calculation of descriptors (3DNET software package) Theoretical 2D and 3D QSAR descriptors were calculated from molecular data sets (MDL SDF format) using the 3DNET program (17), which has been designed for fast and effective 3D QSAR calculations. All the systematically checked descriptors, a total of 81 (18–34), are included in Table 1.
Multiple linear regression computation In this comparative study, multiple linear regression (MLR) illustrates the best description method by a linear model. In our calculation, the number of descriptors (81) is much higher than the number of compounds. Hence, only forward selection could be applied. All descriptors that exceeded the 5% significance level were retained in the model. Calculations were carried out using the Statistica™ software package (Statsoft, Tulsa, OK).
ANN computation We also used the 3DNET program (17) for the neural network computations and calculation of descriptors. The program contains a fully connected, three-layer, feed-forward computational neural network with back-propagation training. The network architecture is illustrated in Fig. 3.
. Schematic architecture of a typical three-layer neural network. For details, see Materials and Methods
The back-propagation algorithm uses a gradient method with continuously decreasing learning rate during the learning epochs. In this net, the input and output layers are linear, and the hidden layer has neurons with a hyperbolic tangent (tanh) transfer function. The leave-one-out cross-validation procedure is automatic.
The 3DNET program applies a quick and effective algorithm that calculates the relative significance of the input descriptors. It consists of adding a surplus input layer and pushing the input values toward zero in a stepwise manner. The back-propagation algorithm tries to decrease the growing error of the calculated outcomes by restoring those inputs that are relevant for the calculation of the outcomes by increasing their weights. The extra network weights determined for each input were sorted, and the largest one was taken as having 100% of relative significance on a linear scale. Descriptors with less than 5% relative importance were left out. Then, the calculation was done again without these descriptors. The remaining descriptors were selected according to their importance one by one until six of them remained. The last six descriptors were tested in every combination. The promising models (from one to six descriptors) were selected by leave-one-out cross-validation procedure.
Validation of predictive ability The following methods have been elaborated in chemometrics to prove the reliability of models in cases when not enough data are available: (1) shuffling of values for dependent variable (activity data); (2) leave-one-out cross-validation; (3) leave-n-out cross-validation; and (4) external validation.
During the shuffling procedure, biological activity values are reallocated (mixed together), whereas the other descriptor values remain in the same position. If the statistical properties do not change significantly, then the model before shuffling is not better than the one obtained using random numbers as descriptors.
In the leave-one-out cross-validation, the first sensitizer is left out, and then a model is built using the remaining compounds. After this, the model is used to verify the calculated biological activity of the sensitizer that was left out. Each molecule is left out once. Based on the predicted PDT activities of all 13 compounds, the goodness of prediction is calculated and characterized by Q2.
where i= number of tested molecules, Yexp= experimental activity, Ypred= calculated activity without using the left-out compound in the model building and Yi exp= average of experimental activities.
The leave-n-out cross-validation was carried out with six randomly left-out compounds by a process similar to the leave-one-out cross-validation.
A real external validation was not possible because extra measured data were not available. However, we can test the present data set formally by splitting it into training and validation sets. Five compounds were selected for the external validation set. These compounds were not used in the model building process. So, the model was built from the data matrix of the other eight molecules. In this process, the model building is independent of the validation set. The calculated activity values were used to demonstrate the predictive ability of the selected descriptor combinations.
Experimental biological data Experimental data on sensitizer concentration in tumor tissue and tumor growth delay have been reported by Henderson et al. (8). An accumulation model was built based on the logarithm of reciprocal tumor photosensitizer levels in nanomoles per gram of tumor tissue measured 24 h after injection of the sensitizer. The PDT activity was expressed as the logarithm of the median time required for a radiation-induced fibrosarcoma tumor to grow to a size of 400 mm3 after treatment. For determination of the PDT activity, the treatment was started when the tumors had reached a surface diameter of 4–6 mm and a thickness of 2–4 mm (meaning a tumor volume of about 50 mm3) inoculated in female C3H mice. Tumors were exposed to a light dose of 135 J/cm2 24 h after the injection of photosensitizers. The tumor diameter measurement was started 24 h after the PDT treatment, and it was repeated every other day. The tumor volume was calculated using the formula V= (l×w2), where l is the longest axis of the tumor, and w is the axis perpendicular to l in the plane of the animal surface. The time for growth of the tumor to a size of 400 mm3 was estimated by interpolation of the times just before and after 400 mm3 was reached (8).
RESULTS
Relationship between photosensitizer structure and accumulation
The relationship between the structures and the biological activities of a congeneric series of pyropheophorbide photosensitizers (Fig. 1) in PDT has been analyzed. First, we looked at the correlation between the octanol–water partition coefficient (logP) or lipophilicity and the accumulation of the sensitizers in the tumor tissue using MLR method.
The relationship between the level of sensitizers and logP changed in a nearly linear manner (Fig. 4), suggesting that even linear regression can provide acceptable results.
. Experimental accumulation (8) versus calculated logP data for photosensitizers. For details, see Materials and Methods
The best MLR models listed in part A of Table 2 were built in two ways: (1) finding the best two-descriptor model (model I, using logP and atomic lipophilicity weighted by atomic volume, VLIPO), and (2) using principal component analysis (PCA) to improve the variable selection process (model II). Using PCA model building, the whole input data matrix (i.e. all descriptors and the accumulation together) has been subjected to analysis. Two principal components were retained in the model. We have selected descriptors that were in proximity to the points of accumulation (proximity was evaluated by visual inspection in the 2D factor space). Four descriptors (Hildebrand solubility parameter, HILI; WHIM descriptor for atomic mass, KMASS; QN-Bodor descriptor, QN; and calculated dipole moment, μ) have been chosen by the standard forward stepwise procedure from among the PCA-preselected descriptors (HILI, μ, QN, KMASS, KPOS, KVDW, KETPI, KLIPO, KEN, KPOL and HDSA1 [for an explanation of the descriptors, see Table 1]).
The squared value of the correlation coefficient (R2) indicates the goodness of fit, which was close to 1 in both cases, showing that the descriptors in each model explain the accumulation with good efficiency. The low Q2 value in model I indicates an unacceptable predictive ability.
We have also listed, in part A of Table 2, the best ANN models for predicting sensitizer enrichment in the tumor tissue. For each of these models, the number of hidden neurons was two. Both goodness of fit and predictive ability of all three models gave high values. To compare linear and nonlinear variable selection methods, the best MLR model (model II) was tested using ANN. At the same time, the ANN-selected descriptors (model III) were used to build a linear model by MLR. The results in part A of Table 3 show that changing the variable selection method worsened the predictive ability of the models.
Relationship between photosensitizer structure and PDT activity
PDT activity was measured as the effect of photosensitizers on tumor growth delay by Henderson et al. (8). We found the same effect that they did, i.e. the relationship between PDT activity and lipophilicity was nonlinear (Fig. 5).
. Nonlinear relationship between PDT activity (8) and calculated log P data for photosensitizers
To find the best computational method to describe PDT activity, first we tried linear calculation techniques using all 81 variables. Again, preselection of variables was done by PCA as described before. Standard forward selection technique, as seen in part B of Table 2, has selected two variables for the best linear model (model VI) at 5% significance level: the maximum of molecular lipophilicity potential on the Van der Waals surface, MMLP and the electrostatic total hydrogen bond basicity, ESTB. The three best nonlinear models (built using two hidden neurons) resulting from a systematic descriptor selection by ANN containing one to three hidden neurons are also listed in part B of Table 2. For comparison, the table contains data obtained by two one-descriptor models (numbers of hidden neurons are five) using calculated and experimental lipophilicity data (models X and XI, respectively). The latter corresponds to the variable used in the structure–activity report (8) published on the relationship between lipophilicity and tumor growth delay. As it can be seen from the Q2 data, log P was not appropriate for describing the PDT activity. Inclusion of another variable (the Wiener index, WINI, in model IX) improved the predictive ability of the ANN model, whereas the best models (VII and VIII) were built using three variables. The ratio between the number of molecules and network parameters was 1.3 for these models.
Descriptors of the best MLR model at 5% significance level (model VI) were tested by ANN. ANN-selected descriptors (model VII) were also used to build linear models by MLR. The correlation between the experimental and calculated tumor growth delay data (R2) and the predictive ability (Q2) of the best MLR and ANN models are summarized in part B of Table 3. As in the case of the photosensitizer accumulation data, changing the variable selection method did not improve the predictive ability of the models for PDT activity.
DISCUSSION
Prediction of accumulation
The results of prediction of accumulation by MLR calculation (Table 2, part A) show that model I, with good R2 value, has low predictive ability represented by the leave-one-out cross-validated Q2 value. This means that the model is slightly overfitted.
The best nonlinear one-descriptor model (model V) uses the calculated lipophilicity as a descriptor variable. We obtained slightly better prediction when using the best two- or three-descriptor models (models IV and III, respectively). Plotting the experimental versus calculated accumulation data of photosensitizers based on the results of a three-descriptor (Fig. 6A) and a one-descriptor (Fig. 6B) ANN model shows that there is virtually no difference between them for prediction of accumulation.
. Prediction of photosensitizer accumulation using nonlinear (ANN) models. A: The best three-descriptor model (model III). B: The best one-descriptor model (model V)
When comparing linear and nonlinear calculation methods, the most useful variable for describing the structure–accumulation relationship is, without any doubt, logP. This is not surprising because lipophilicity is expected to describe the accumulation of sensitizers because it depends to a large extent on drug transport through the lipophilic cell membrane. The best predictive model is built by ANN (Q2= 0.903), but MLR (Q2= 0.836) seems to be almost as useful as the nonlinear method in describing this relationship.
Prediction of PDT activity
The statistical characteristics in part B of Table 2 show that the fit made for PDT activity based on MLR calculations is not particularly good. The curvature seen in Fig. 5 shows that the linear model is not adequate because there is a considerable nonlinearity in the biological activity as a function of logP.
On the other hand, ANN gave a good description of PDT activity, especially when using two- or three-descriptor models (part B of Table 2).
The predictive ability of the best one-descriptor model (model X), which solely uses calculated lipophilicity data as a descriptor variable, is Q2= 0.365 and is far from being satisfactory (Fig. 7).
. Experimental versus predicted PDT activity for photosensitizers calculated by the one-descriptor ANN model (model X)
ANN calculation using logP alone corresponds to the only nonlinear quasi-mechanistic model described in the literature so far (8). The network architecture performs much better when using three descriptors as shown, for example, in Fig. 8 (model VII). The 0.879 value for Q2 denotes that this model can explain 87.9% of the changes in the activity.
. Predictive ability of the best ANN model (model VII) for PDT activity of photosensitizers
The best descriptor combinations (from one to six descriptors) were selected by leave-one-out cross-validation procedure. These combinations (models VII and VIII) include, in addition to logP, the weighted holistic invariant molecular (WHIM) descriptors, i.e. related to quadratic contribution to the total molecular dimensions, which are weighted by scalar field values as atomic mass (AMASS), atomic Van der Waals volume (AVDW) or atomic electronegativity (AEN). The importance of these molecular size descriptors is acceptable because the side chains of the studied compounds are bent over the main plane of the molecule. Lipophilicity (logP) could explain the accumulation or the transport of the sensitizers through the cell membranes, but it does not seem to be enough to explain PDT activity. Our results indicate that the transport of the sensitizers, their binding or other physicochemical properties within the cell related to the atomic mass, Van der Waals volume or electronegativity should be important in explaining the effect of photosensitizing drugs on tumor cell killing.
Validation of the predictive ability: (1) Shuffling of the structures and activity values three times yields an average cross-validated Q2 as bad as −0.610 (data not shown). This proves that there is a definite role of the selected descriptors in describing the biological activity studied. In our case, the fit after shuffling became unacceptable. The statistical properties (Q2 and residual error) changed significantly. The extremely bad fit indicates that the model before shuffling is much better than the one obtained using random numbers as descriptors. (2, 3) The leave-n-out procedure supports the result of good predictive ability of the models obtained by leave-one-out cross-validation. Both validation processes selected model VII as the best variable combination. (4) The external validation process on eight training compounds using the descriptors of model VII resulted in Q2= 0.834, with a standard error of 0.080 for the five selected molecules. The predicted activity value (Ypred) for M1 molecule is 2.042, whereas the experimental value (Yexp) is unknown. The other molecules tested were M4 (Ypred= 2.524, Yexp= 2.46), M5 (Ypred= 2.144, Yexp= 2.22), M11 (Ypred= 2.355, Yexp= 2.36) and M13 (Ypred= 2.227, Yexp= 2.18). Calculation of the activity data of molecules M1 and M13 can be done by extrapolation.
Comparing the linear and nonlinear models, it is interesting to note that variables selected by forward stepwise manner cannot provide either reasonable description or prediction even when using ANN (Table 3, part B). In our case, the descriptor combination selected by nonlinear way did not give satisfactory results by linear calculations either. Because of the strong nonlinear relationship between the PDT activity and logP, ANN is expected to build a better model for predicting the tumor growth suppressing activity of photosensitizers. The R2 and Q2 values show that ANN is a much better tool in building promising models for description and prediction of this kind of biological activities.
In summary, it can be established that mixing linear and nonlinear calculation methods is not a good policy in predicting either the accumulation or the PDT activity of photosensitizing molecules. It is expedient to remain with the same variable selection and model building methods for both MLR and ANN.
CONCLUSIONS
It is relatively easy to calculate the aforementioned descriptors using computational methodologies for any new molecule to be tested. The biological activity and accumulation of new pyropheophorbide molecules can then be predicted on the basis of suitable models presented in this study, which is important from the point of view of chemical synthesis.
We recommend using ANN models for predicting PDT activity because it is highly nonlinear with respect to the QSAR parameters like logP.
For prediction of linear relationships such as accumulation, MLR can successfully compete with ANN models.
It is worth noting that the pyropheophorbide molecules analyzed in this study differ from each other only in their alkyl ether side chains R1. Our results, most probably, do not shed light onto the main molecular features responsible for their biological activity. In fact, we only investigated how the effect of varying the side chain of a congeneric series of pyropheophorbides can be explained and modeled in QSAR terms.
Nevertheless, from a robust pool of QSAR descriptors, the neural network itself can select the information-rich ones even for a limited, although structurally closely related number of molecules. Our model, which was calculated solely from the 3D molecular structure of a series of compounds, is able to predict the tumoricidal activity of some phorbide derivatives. In addition to lipophilicity parameters, the most important descriptors (WHIM descriptors related to quadratic contributions to the total molecular dimensions weighted by atomic mass, Van der Waals volume or electronegativity) tell us how the substituents modify the molecular size.
In our case, ANN was superior to other methods like principal component regression, partial least squares and locally weighted regression (35) in predicting PDT activity.
Acknowledgements
The authors express their thanks to the National Scientific Research Foundation of Hungary for financial help (OTKA T034986).
