The data of 16 representative PAHs (Figure 1) analyzed by the two ELISA kits were used for modeling. Seven typical physicochemical, electronic and topological descriptors are selected for developing the QSAR models. They are molar solubility (S_m), the logarithm of octanol-water partition coefficient (log K_ow), the gap between the highest occupied molecular orbital energy and the lowest unoccupied molecular orbital energy (E_HOMO − E_LUMO), and four valence molecular connectivity indices (⁰χ^v, ¹χ^v, ²χ^v, ³χ^v). The data of solubility (S) and log K_ow were obtained from Mackay et al. [20]. S_m was calculated by dividing S with the molecular weight. The data of E_HOMO − E_LUMO were from de Lima Ribeiro and Ferreira [21]. The data of ⁰χ^v, ¹χ^v, ²χ^v and ³χ^v were cited from Govers and Aiking [22].

The experimental data of cross-reactivity (CR) for the two commercial PAHs ELISA kits, RaPID and Ris^C, were obtained from Krämer [8]. Since activity data used for QSARs should be in molar dimensions, CR values were converted to molar cross-reactivity (MCR), i.e., the ratio of the molar IC₅₀ of target analyte and the cross-reactant, for QSAR modeling [19]. Then, the predicted MCR was calculated by the QSAR models and converted to CR for comparison with the experimental CR value. We assumed that the CR values “<0.5%” and “<1.6%” were low enough to describe low levels of cross-reactions in the two ELISA kit tests, and reasonably considered the CR value “<” to be “=” for regression modeling [19].

The data of 14 compounds of the 16 PAHs were submitted as training set for regression analysis, and anthracene and benzo[a]pyrene were used as the test set. In order to reduce the colinearity and the number of the molecular descriptors, the analysis of the quantitative relationship between log MCR and the molecular descriptors was performed by stepwise MLR, PCR and PLSR employing SAS 8.1 software. In the stepwise MLR procedure, the data of the seven descriptors of the 16 PAHs were collected in a single data matrix, and the key descriptors were selected by adding descriptors one by one to perform a multivariable regression calculation. The variables significant at the 0.15 level were left in the model. In PCR analysis, the original descriptors were subjected to principal component analysis, and the subset of principal components explaining more than 90% of the variance was extracted. Then, the principal components extracted were subjected to multiple linear regression analysis. The PLSR method reduced large volume of descriptors to several components that were most correlative with the CR. These components were the linear combinations of the descriptors and used as new variables for regression analysis. The optimum number of components for regression analysis was obtained by the leave-one-out cross-validation procedure.

CR is the ratio of the IC₅₀ (the 50% inhibition concentration) of the target analyte and the IC₅₀ of the cross-reactant. Considering the effect of water solubility on the CR value and the immunoassay results, we defined cross-reaction potential (or cross-reaction probability, CRP), i.e., 100-fold the ratio between the solubility (S) of a cross-reactant and the IC₅₀ value [Equation (1)], and used it as a complement of CR to evaluate the extent of cross-reaction. CRP reflects the relative extent of cross-reaction of a non-target cross-reactant compared with the water solubility. The data set of S was from Mackay et al. [20], and IC₅₀ data were from Krämer [8]. We assumed that the IC₅₀ value “>1,000 μg·L⁻¹” was high enough to be considered as “=1,000 μg·L⁻¹” for CRP calculation: (1) CRP ( % ) = S I C 50 × 100

The molecular structures of the 16 PAHs analyzed by the two ELISAs are shown in Figure 1. Since antibodies and antigens in immunoreactions are not mass-equivalent but rather molar-equivalent, molar cross-reactivity (MCR) rather than mass cross-reactivity (CR) is applied to investigate the quantitative structure and cross-reactivity relationships. The obtained stepwise MLR, PCR and PLSR equations and statistical parameters are illustrated in Table 1.

It shows that the regression models for Ris^C kit are all significant (p < 0.005), while the models for RaPID kit are all not significant (p > 0.05). The probable reason is that Ris^C immunoassay employs a monoclonal antibody, while RaPID kit is based on polyclonal antibody. In the stepwise MLR procedure for RaPID, only S_m enters the regression model, and the other six molecular descriptors are excluded from the regression equation. As for Ris^C, S_m and ³χ^v are left in the model. In the PCR procedure, the two most significant principal components (PC1 and PC2) describe respectively 85.7% and 8.3%, and totally 94.0% of the variance. Eigenvectors of the principal components indicate that PC1 demonstrates the integrated character of the seven descriptors, while PC2 mainly represents the character of S_m. The regression equations for RaPID and Ris^C are Equation (2) and Equation (3) respectively. In the PLSR procedure, the models are optimized by leave-one-out cross-validation, and the optimum numbers of components are found to be 4 and 2 for RaPID and Ris^C, respectively. The parameter estimates for centered and scaled data (marked by *) are shown in Equation (4) and Equation (5). The results of stepwise MLR, PCR and PLSR imply that S_m plays an important role in affecting the CR property of the PAHs for the two ELISA kits, and ³χ^v also affects the CR for Ris^C kit to some extent: (2) For RaPID : log MCR = 1.349 + 0.07442 × PC 1 - 0.7234 × PC 2 (3) For Ris C : log MCR = 1.178 + 0.002550 × PC 1 - 0.9075 × PC 2 (4) For RaPID : log MCR ∗ = - 0.6783 × S m ∗ + 1.245 × log K ow ∗ - 0.6106 × ( E HOMO - E LUMO ) ∗ + 0.2556 × χ 0 v ∗ - 0.07527 × χ 1 v ∗ − 0.6231 × χ 2 v ∗ − 1.626 × χ 3 v ∗ (5) For Ris C : log MCR ∗ = - 1.042 × S m ∗ - 0.06846 × log K ow ∗ - 0.1012 × ( E HOMO - E LUMO ) ∗ - 0.1649 × χ 0 v ∗ - 0.1778 × χ 1 v ∗ - 0.2025 × χ 2 v ∗ - 0.2517 × χ 3 v ∗In immunoreactions and immunoassays, the interaction between antigen and antibody is caused by the complementary spatial distribution and the strong affinity between the antigen and the antibody, such as hydrogen bonds, electrostatic interactions, van der Waals forces and hydrophobic interactions. The strong effect of S_m on the CR properties of the two ELISAs reflects the important role of hydrophobic interactions in the PAH-antibody reactions, which confirms the previous result [19]. It is commonly believed that lower order molecular connectivity indices encode mainly the bulk of a molecule, whereas higher order indices encode more subtle features such as the presence of rings and branching patterns. The result that ³χ^v affects CR more than ⁰χ^v, ¹χ^v and ²χ^v implies that molecular shape is more influential than molecular size in the PAH-antibody reactions. It has been reported that E_HOMO and E_LUMO are responsible for the antibody recognition for phenylurea herbicides and organophosphorus pesticides [23–25]. These compounds consist of various functional groups and heteroatoms, while PAHs do not contain substituents and heteroatoms, hence electronic descriptors such as E_HOMO and E_LUMO may have minor effects on the antibody recognition for PAHs. E_HOMO − E_LUMO expresses the necessary energy to excite an electron from the highest occupied molecular orbital to the lowest unoccupied molecular orbital. Since immunoreactions are always not accompanied by a rearrangement of electron density, it is not surprising that E_HOMO − E_LUMO does not exhibit strong effect on the CR in the models.

CR is one of the most important characteristics of an ELISA test, and influences the extent of cross-reaction and the results of ELISAs significantly. However, due to the difficulty and expense in term of cost and time, not all of the CR data of the cross-reactants are available. Moreover, it is impractical to directly measure the CR of the cross-reactants which are not commercially available, so predicted CR values of the PAHs for the two ELISA kits were calculated using the obtained MLR, PCR and PLSR models, and compared with the experimental data (Table 2, Figure 2). The predicted CR values for Ris^C agree very well with the experimental data, while the predicted and experimental data for RaPID do not agree well with each other. The models were further external validated using the data of three-ringed anthracene and five-ringed benzo[a]pyrene as test set. The range of predicting error for anthracene is from −1.6% to +8.1%, and for benzo[a]pyrene is from −40.8% to −8.3%. It appears that the obtained models can successfully predict the CR for Ris^C kit, but present poor predicting ability for RaPID kit.

Generally speaking, higher CR values imply higher levels of immunoreactions. However, the antigen-antibody reactions in immunoassays are carried out in water or buffer solutions, so if the solubility of a cross-reactant is much lower than the IC₅₀ value, it cannot possibly cause intense cross-reactions in the immunoassays. That is to say, the concentration of this compound in real water samples cannot be high enough to evoke high extent cross-reaction in immunoassays, even though the CR is very high. Some of the 16 PAHs are very hydrophobic compounds, and the solubility is much lower than the tested IC₅₀ value. For example, the IC₅₀ referring to water analysis for benzo[a]pyrene for the RaPID ELISA kit is 6.9 μg·L⁻¹, while the solubility of benzo[a]pyrene is 3.8 μg·L⁻¹ (Table 3), so although benzo[a]pyrene has a high CR of 239% in the RaPID ELISA, the concentration of benzo[a]pyrene in water samples cannot be possibly high enough to evoke a high level of cross-reaction. Hence, considering the important effect of water solubility on immunoreactions and immunoassays, cross-reaction potential (or cross-reaction probability, CRP), i.e., the relative IC₅₀ of a non-target cross-reactant compared with its water solubility, was defined and used as a complement of CR to evaluate the potential and probability that a cross-reaction would occur.

The IC₅₀ and CRP data for the 16 PAHs for RaPID kit are shown in Table 3. The CRP values for Ris^C kit are not calculated because the IC₅₀ values are not available. In addition to the target analyte of phenanthrene, the 15 cross-reactants in RaPID ELISA can be divided into four groups according to CR and CRP (Figure 3): (I) CR > 100%, CRP > 100%; (II) CR > 100%, CRP < 100%; (III) CR < 100%, CRP > 100%; and (IV) CR < 100%, CRP < 100%. The compounds of group (I) might cause intense cross-reactions and affect the determination of phenanthrene, while the group (IV) compounds have little cross-reaction effect on the analysis results. As for the group (II) compounds, the CR is high, while the CPR is low because of the relatively low solubility. The group (III) compounds are two-ringed and three-ringed PAHs, and less cross-reactive but more water soluble. The extent of the cross-reactions of group (II) and (III) compounds depends on both the CR and the CRP properties.

It should be pointed out that the RaPID PAHs ELISA kit is applied not only for water samples [9,10], but more often for soil samples [11,14,15]. In the pretreatment procedure, PAHs were usually extracted from the soil samples by methanol and diluted by buffer. PAHs are very hydrophobic molecules and can be adsorbed to soils at very high concentration. In the immunoassays of PAHs, much attention should be paid to the solubility of the compounds during the procedures of solvent extraction and buffer dilution.

The comparison of RaPID and Ris^C PAHs ELISA kits based on the character and the applicability are illustrated in Table 4. It seems that Ris^C ELISA is more specific, while RaPID ELISA is applied for more kinds of environmental samples. The selection of appropriate ELISAs for PAHs depends on the objective and request of the analysis.