1 Introduction

Detailed knowledge of protonation constants of drug compounds is critical for understanding and predicting mechanisms of action. For example, fat-soluble (lipophilic) drugs diffuse through the cell membrane according to their charges [1, 2]. Added to this, the interaction between solvents with different polarities and certain molecules has been identified as an important factor in drug molecule interactions with DNA or active sites of enzymes, side chains in proteins or generally any binding action [3, 4]. In addition, knowledge of the pK a values in industrial solvents can benefit applications such as the design of liquid pharmaceutical dosages, as well as considerations in drug synthesis and testing such as solubility, purification and drug screening [5,6,7]. Therefore, solvent effects on drug molecule properties, particularly their pK a values, are very important. The dielectric constant (ε) of a solvent, which is related to the solvent polarity (E T) through the Debye equation, is an important physicochemical property mostly analyzed in terms of its effect on drug pK a. This has led to recent analytical initiatives to measure E T within biological microenvironments, in order to better predict the behavior of natural biomolecules and drug molecules [8,9,10,11]. However, the relationship between E T and pK a is not always straightforward, and cannot be used reliably to extrapolate drug molecule behavior in new biochemical environments. Therefore, the multi-parameter model proposed by Kamlet, Abboud, and Taft (KAT) has gained interest among researchers, due to its ability to account for the basic modes of solute–solvent interactions. These include hydrogen-bond acidity (α), hydrogen-bond basicity (β) and dipolarity/polarizability (π*). To facilitate this approach, a general method to modulate easily the solution properties to mimic a range of solvent environments was achieved using aqueous–organic solvent combinations as multipurpose solvents to emulate a wide range of physiochemical properties [12]. Recently, such binary solvents were used to study physiochemical properties such as solubility and acid–base properties of a range of solutes including fluorescent probes, naturally occurring biomolecules and their complexes with foreign species [13,14,15,16,17].

The aim of the present study is to use different binary aqueous/organic solvents as a means to comprehensively study the effect of solvents on the KAT parameters to describe the variation of pK a of mono- and diprotic drug compounds. In addition, computer simulations are used to validate the experimental results and give deeper insight into reaction mechanisms for protonation/deprotonation phenomena and the most probable equilibrium reaction mechanisms between protonated and deprotonated states. We chose the monoprotic and diprotic drug compounds propranolol (PRO) and atenolol (ATN) (Fig. 1), respectively, because they are well-known β-adreno receptor blocking agents, also used to treat hypertension, angina, arrhythmias, anxiety management, migraine prophylaxis, hyperthyroidism and tremors [18]. Propranolol, for example, is considered an essential drug for treating migraine headaches by the World Health Organization [19].

Fig. 1
figure 1

Chemical structures of propranolol (left) and atenolol (right)

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2 Experimental

2.1 Chemicals

PRO and ATN, were obtained from Fluka as analytical reagent grade materials (98 and 95%, respectively) and they were used without further purification. Methanol (99.8%) and perchloric acid (99%) were purchased from Merck (reagent grade materials) and were used as received. Sodium perchlorate (NaClO4) was supplied from Merck and was dried at room temperature for at least 48 h under vacuum before use. NaOH solution was prepared from a titrisol solution (Merck). All dilute solutions were prepared with CO2-free and de-ionized water with a conductance equal to 1.3 ± 0.1 μS·cm−1.

2.2 Apparatus

Electromotive force (emf) measurements of all solutions were made using a pH ion-meter, Metrohm model 781, equipped with a glass-pH electrode (model 6.0258.000) that was modified by replacing its aqueous KCl solution with a solution consisting of 0.01 M NaCl + 0.09 M NaClO4, saturated with AgCl. The electrode was soaked for (15–20) min in the water–organic solvent mixture before the potentiometric measurements. All titrations were carried out in an 80 mL capacity thermostated double-walled glass vessel at 25°C. In this work, the ionic strengths of all solutions were adjusted to 0.1 mol·dm−3 with NaClO4. A peristaltic pump was used to circulate the solutions from the titration vessel to the spectrophotometric cell through Teflon or Tygon tubes, so the whole process was automated with continuous flow and the solution absorbance could be measured simultaneously with the emf. Spectrophotometric measurements were recorded with a UV–Vis Shimadzu 2100 spectrophotometer, controlled with a computer. UV–Vis sampling was accomplished through thermostated 10 mm flow-through quartz cells. To avoid formation of carbonate and carbonic acid by dissolved CO2, which could have affected the pH, we bubbled ultrapure N2 through the reaction solution. Absorbance measurements were made in the spectral window of 200–400 nm with spectral resolution of 0.5 nm. The measured emf was related to solution pH as described in the supplementary information. The spectrophotometric titrations were analyzed with the Squad computer program which is based on titration curves [20, 21].

2.3 Computational Details

All calculations of neutral and cationic forms of the drug molecules and water cluster molecules were carried out using the GAMESS-US version May 1, 2013 (R1) [22]. Geometry optimizations and frequency calculations were performed at the B3LYP/6-311++G(d,p) level of theory. To include the solvent effect in our calculation, solvation model density (SMD) was used. The SMD solvation model was used to calculate solute–solvent interactions in different binary mixtures of water and methanol. The SMD model calculates short-range interaction energies between solvent and solute in different binary mixtures of water and methanol by using a modified solvent-accessible surface area that incorporates parameters for atomic and molecular surface tensions and hydrogen-bond acidity and basicity [23, 24].

For building the structure and visualizing all species involved in the reactions, Gabedit v2.4.0 [25], Jmolv13.0.16 [26] and wxMacMolPltv7.4.4 [27] programs were used. At first, all proposed geometries were fully optimized without any geometrical constraint and then vibrational tests were conducted to make sure that there are no imaginary frequencies for the optimized structures. A cluster continuum method which takes advantage of both implicit and explicit solvent molecules were used to simulate different binary mixtures of methanol–water for calculating pK a values [28]. In this explicit approach only water molecules are accounted for. On the other hand, in this implicit approach water is considered as a main solvent in SMD model with the effect of methanol in the binary mixtures added to water solvent considered by changing the dielectric constant of water to the experimental values reported in Table 1. For calculating pK a values, several reactions were tested; some of the reactions were not considered further due to large deviations from the experimental results.

Table 1 The comparison of B3LYP-SMD/6-311++G (d,p) calculated and experimental pK a values of PRO and ATN (at 25 °C)
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2.3.1 Fitting

pK a values were correlated with the KAT solvent properties by means of multiple regression analysis using the Gauss–Newton non-linear least-squares method by a suitable computer program (Microsoft Excel Solver and Linest) [29]. Further information is given in the supporting information file. We refined the pK a values by minimizing the error squares sum from Eq. 1:

$$ U = \sum {\left( {{\text{p}}K_{ \exp } {-}{\text{p}}K_{\text{fitting}} } \right)^{ 2} }. $$
(1)

3 Results and Discussion

3.1 Titrations

The speciation of drugs in the aqueous solution can be readily illustrated by their distribution curves as a function of pcH. In this work, the fractions of various forms of representative drugs, such as H2ATN2+, HATN+, ATN (Fig. 2a) and HPRO+, PRO (Fig. 2b) are depicted in Fig. 2. For ATN, the results indicate that in the pH range of 3–4, the major species are H2ATN2+, HATN+, in the 8–10 pH range the major species are HATN+, ATN and PRO (Fig. 2a, b), while at pHs > 10 the dominant species are neutral ATN and PRO. The speciation of these drugs in binary solutions significantly impacts biomicro environment-related processes such as metal complexation, decomposition, drug–solvents, proteins and cell interactions. The pK a values were measured from titration curves, by drop wise addition of NaOH to the drug solution in the concentration range 1.0–3.0 mmol·dm−3. Both titrate and titrant had the same percentages of methanol for each titration run.

Fig. 2
figure 2

Equilibrium distribution of different protonation states of ATN (a) and PRO (b) as a function of pcH in water (0% methanol). Blue, green, and red circles represent doubly, singly and non-protonated molecules, respectively (Color figure online)

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For minimizing the sums of the squares of the deviations of the experimental absorbances from the calculated ones, the measured absorbance and pcH data from the titration in the computer program were fitted with Eqs. 2–4:

$$ {\text{H}}_{ 2} \left( {\text{ATN}} \right)^{ 2+ } \rightleftharpoons {\text{H}}\left( {\text{ATN}} \right)^{ + } {\text{ + H}}^{ + } K_{ 1} = [{\text{H}}\left( {\text{ATN}} \right)^{ + } ][{\text{H}}^{ + } ]/[{\text{H}}_{ 2} \left( {\text{ATN}} \right)^{ 2+ } ] $$
(2)
$$ {\text{H}}\left( {\text{ATN}} \right)^{ + } \rightleftharpoons {\text{ATN }} + {\text{ H}}^{ + } K_{ 2} = {\text{ [ATN] }}[{\text{H}}^{ + } ]/[{\text{H}}\left( {\text{ATN}} \right)^{ + } ] $$
(3)
$$ {\text{H}}\left( {\text{PRO}} \right)^{ + } \rightleftharpoons {\text{PRO }} + {\text{ H}}^{ + } K = [{\text{PRO}}]{\text{ [H}}^{ + } ] / [ {\text{H}}\left( {\text{PRO}} \right)^{ + } ] $$
(4)

Since PRO is monoprotic and ATN is diprotic, one pK a value was extracted from the titration of PRO (pK aPRO), whereas two pK a values were obtained from the titration of ATN (pK aATN1, pK aATN2) by fitting the calculated UV–Vis adsorption values to experimental ones (the chosen wavelengths were around 250 nm). pK aPRO, pK aATN1, pK aATN2 are listed in Table 1 together with some values reported in the literature [30,31,32]. Titrations were performed for ATN and PRO solutions in binary solvents with different water–methanol mixing ratios to explore the solvent effect on the pK a values. The UV–Vis spectra of propranolol and atenolol in 30% water–methanol are shown in Fig. 3.

Fig. 3
figure 3

Experimental absorption spectra recorded in the course pH-metric titrations of (a) ATN and (c) PRO. Spectral data sets were acquired from their respective analytes in the concentration range 1.0 × 10−3 mol·dm−3–3.0 × 10−3 mol·dm−3 in a 30% (v/v) MeOH–water binary solvent mixture, with the pH ranging from 1.8 to 12. In all cases temperature and ionic strength were held at 25 °C and 0.1 mol·dm−3 of NaClO4. (b) and (d) are average values of absorbance at different pH values in a range of wavelength 250–260 nm for ATN and PRO, respectively. Sampling was accomplished in flow-through quartz cells with 10 mm optical path length

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3.2 Solvent Effect

In general, the standard Gibbs energy of acid–base equilibria consists of an electrostatic term, which can be estimated by the Born equation (S5), and a non-electrostatic term, which includes specific solute–solvent interaction [33, 34]. When the electrostatic effects predominate (as in case of pK aATN2), then the plot of pK a versus ε −1 of the media should be linear. As seen in Fig. 4, the correlation between pK aATN2 and ε −1 of the water–methanol mixtures is linear (r2 > 0.99). A linear trend for pK aATN1 or pK aPRO could be achieved; however it yielded a far worse r2 fit value. Therefore, we tested two more complex approaches (required to better understand solvent effects) that accounted for solvent–solvent and solute–solvent interactions. First, we tested two approaches to evaluate the solute–solvent interaction. In the first approach, the correlation between pK a and solvent polarity (E T) was investigated. As seen from the plots of pK a versus E T, Fig. 5, the relationship is nonlinear. We attributed this to the fact that water preferentially dissolves methanol compared to the solute [35, 36]. As the percentage of methanol was increased in the mixtures, the self-associated structure of water gradually breaks down and at the methanol-rich region, preferential solvation by water appears mainly because of solute–solvent hydrogen bonding. In the second approach we used a multi-parametric approach (KAT) developed by Kamlet et al. [37,38,39] to predict pK a. The solvatochromic parameters used in this approach were the hydrogen-bond acidity (α), hydrogen-bond basicity (β) and dipolarity/polarizability (π*) of the water–methanol solvent mixtures. The latter term, π*, is the index of the solvent polarizability, which is a measure of the ability of a solvent to stabilize a charge or a dipole by its own dielectric effects. The contributions of these parameters to the measured pK a values were determined from the plots of each property versus the mole fraction of the organic solvent (Table 2). Together, these parameters constitute more comprehensive measures of solvent character than E T or any other single physical parameter.

Fig. 4
figure 4

a Plot of pK aATN2 versus ε −1 of different mixed solvents performed in stable conditions of temperature and ionic strength. b Plots of pK aATN1 (red line) and pK aPRO (blue line) versus ε −1 of different mixed solvents performed in stable conditions of temperature and ionic strength [at 25 °C and an ionic strength of 0.1 mol·dm−3 (NaClO4)] (Color figure online)

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Fig. 5
figure 5

a Plot of pK aATN2 versus the E T of different mixed solvents performed in stable conditions of temperature and ionic strength. b Plots of pK aATN1 (red line) and pK aPRO (blue line) versus the E T of different mixed solvents at 25 °C and an ionic strength of 0.1 mol·dm−3 (NaClO4) (Color figure online)

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Table 2 Polarity and KAT solvatochromic parameters of different water–methanol mixtures [40, 41]
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A general multi-parametric equation, Eq. 5, was used to predict pK a values

$$ { \log }_{ 10} K = A_{0} + a\alpha + b\beta + {p\pi} ^{*} $$
(5)

where A 0 represents the regression value and a, b and p are the regression coefficients, which measure the relative susceptibilities of the solvent-dependence of pK a to α, β, and π* parameters, respectively. The solvatochromic parameters (α, π*, and β) of Kamlet–Taft’s scales vary from 0.0 to about 1 for these solvents.

Although E T is identified as the main contributor to variations of pK a values in binary water–organic solvents, the results show single-parameter correlations did not give good results based on their low r2 values, 0.91–0.94, while the correlation analysis with dual-parameter equations shown in Eq. 6 indicates significant improvement:

$$ {\text{p}}K_{\text{aPRO}} = 6. 9 9 { }( \pm 0. 2 4) + 0. 9 2 { }( \pm 0. 3 5)\alpha + 1. 1 4 { }( \pm 0. 5 6)\pi^{*} $$
(6a)
$$ (N = 10,\;\rm{r}^{2} = 0.99, \,\rm ose = 1.64 \times 10^{ - 2} , \, \rm rss = 1.87\times 10^{ - 3} ,\;\rm f{ - }test = 253.8) $$
$$ {\text{p}}K_{\text{aATN2}} = 1. 2 3( \pm 0. 8 2){-}0.0 3( \pm 0. 50)\beta + 2.0 7( \pm 0. 50)\pi^{*} $$
(6b)
$$ (N = 10,\;\rm{r}^{2} = 0. 9 9,\;{\text{ose}} = 1.09\times 10^{- 2} ,\;{\text{rss}} = 8. 46 \times 10^{- 4},\;{\text{f-test}} = 368.1) $$
$$ {\text{p}}K_{\text{aATN1}} = 6.48 { }\,( \pm 0. 1 8) + 1.0 4 { }\,( \pm 0. 2 7)\,\alpha + 1. 5 1 { }( \pm 0. 4 2)\,\pi^{*} $$
(6c)
$$ (N = 10,\rm{r}^{ 2} = 0. 9 9,{\text{ ose}} = 1. 2 5\times 10^{ - 2} ,{\text{ rss}} = 1.0 9\times 10^{ - 3} ,{\text{ f-}} {\text{test}} = 643.8) $$

where N, rss, ose and r2 represent the number of the samples (mixed solvents), the residual sum of squares, the overall error and regression coefficient, respectively.

It can be seen from Table 1 that the dielectric constants of the solvent mixtures decrease as the solutions are enriched in organic solvent. The values of pK aPRO and pK aATN2 and pK aATN1 all decrease with decreasing dielectric constant of the medium. Nevertheless, this property is often inadequate to explain this variation as a single parameter. Therefore, a dual-parametric approach according to the KAT equation was applied to find out which parameter is responsible for this behavior. The positive π* coefficients in the correlation analysis of pK aPRO, pK aATN2 and pK aATN2 by the KAT equation imply that a decrease in the polarity of the mixed solvents, decreases the protonation constant. This indicates the polarity parameter, π*, is the most important (with a large difference of affinity with the donor or acceptor acidity or basicity parameters) in the correlation analysis of the pK a values of PRO and ATN. Furthermore, the positive coefficient of α or negative coefficient of β in the correlation of pK a suggests that increasing basicity of the solvent mixtures decreases the pK a values of PRO and ATN as can be seen in Eq. 6.

In conjunction with the experimental determination of pK a values, quantum mechanical DFT calculations were also conducted. As discussed in the supporting information, solvation Gibbs energies (G) play an important role in prediction of molecular structures and pK a constants due to its relation to the tendency of the system to react or change. Since ε is the most sensitive to change for different water–methanol solvent compositions, we changed its value according to Table 1 for each binary solvent, in order to accurately evaluate G for each drug molecule. The general form of the proton transfer reaction shown in the supporting information (Eq. S10) gives all possible n-hydrated species (n = 0–3) for PRO and ATN (Fig. 6). According to the results, the following reactions are the best match to those of experimental results.

Fig. 6
figure 6

Selected optimized structures of bare and cluster ATN and PRO molecules at the DFT-B3LYP/6-311++G (d,p)/SMD level

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$$ {\text{ATN}}:{\text{H}}_{ 2} {\text{L}}^{ 2+ } + {\text{OH}}^{-} \left( {{\text{H}}_{ 2} {\text{O}}} \right)_{ 2} \rightleftharpoons {\text{HL}}^{ + } \left( {{\text{H}}_{ 2} {\text{O}}} \right)_{ 3} $$
(7)
$$ {\text{ATN}}:{\text{HL}}^{ + } \left( {{\text{H}}_{ 2} {\text{O}}} \right)_{ 2} + {\text{OH}}^{-} \left( {{\text{H}}_{ 2} {\text{O}}} \right) \rightleftharpoons {\text{L}}\left( {{\text{H}}_{ 2} {\text{O}}} \right)_{ 3} + {\text{H}}_{ 2} {\text{O}} $$
(8)
$$ {\text{PRO}}:{\text{HL}}^{ + } + {\text{OH}}^{ - } \left( {{\text{H}}_{ 2} {\text{O}}} \right) \rightleftharpoons {\text{L}}\left( {{\text{H}}_{ 2} {\text{O}}} \right) + {\text{H}}_{ 2} {\text{O}} $$
(9)

In Eqs. 8 and 9, L(H2O) n and HL+(H2O) n represent the neutral and the first protonated forms of ATN and PRO solvated with n molecules of water, respectively, and in Eq. 7, H2L2+ represents the second protonated form of ATN. Isolated H2O molecules are shown as products in Eqs. 8 and 9.

$$ {\text{RD}} = \frac{{\left| {{\text{p}}K_{\text{cal}} - {\text{p}}K_{ \exp } } \right|}}{{{\text{p}}K_{ \exp } }} $$
(10)
$$ {\text{\% MRD}} = \mathop \sum \nolimits \frac{\text{RD}}{n} *100 $$
(11)

The computational pK a values are listed in Table 1 and were compared with experimental pK a values by relative deviation (RD) and percentage of mean relative deviation (%MRD), which are defined in Eq. 10 and 11, respectively, with n being the number of the samples (mixed solvents). The mean relative differences of 0.19, 0.09 and 0.41% between experiment and calculation for pK aPRO, pK aATN1 and pK aATN2 strongly supported our model. Moreover, both methods capture the tendency of pK a to decrease with increasing percentage of methanol.

4 Conclusions

In this work pK a values of two important anti-hypertension drugs, ATN and PRO, were determined using a potentiometric method in various aqueous methanol mixtures. The solvent dependence of their acid–base equilibria were determined by UV-spectroscopy/potentiometric titration and correlations with the KAT solvatochromic parameters. Using linear relationships between these solvatochromic parameters and protonation constants, the pK a values were obtained with good accuracy. The experimental results are supplemented with DFT calculations, which provide a critical comparative evaluation of both the experimental determination of pK a and the computational prediction of pK a on ATN and PRO drugs. The cluster continuum method with implicit-explicit solvent molecules was used to calculate the pK a values in various binary mixtures of water and methanol, the total Gibbs energies were obtained for the ATN and PRO clusters at the DFT-B3LYP/6-311++G(d,p) level of theory. For applying the solvent effects on the species involved in the selected proton transfer reactions, the SMD solvation model was used. It is shown that there is a good correlation between experimental and theoretical results.