Log D analysis using dynamic approach.

Log D the logarithm ( log10 ) of the distribution coefficient ( D ), is one of the important parameters used in Lipinski's rule to assess the druggability of a molecule in pharmaceutical formulations. The distribution of a molecule between a hydrophobic organic phase and an aqueous buffer phase is influenced by the pH of the buffer system. In this work, we used both the conventional algebraic method and the generalized 'dynamic' approach to model the distribution coefficient of amphoteric, diamino-monoprotic molecule and monoprotic acid in the presence of salt or co-solvent. We have shown the equivalence of these methods by analysing the recently reported experimental data of amphoteric molecules such as nalidixic acid, mebendazole, benazepril and telmisartan.

Introduction

Partition coefficient () is defined as the ratio of the concentration of a molecule, whether in ionized or unionized form, distributed between a hydrophobic phase and an aqueous phase [1], [2], [3], [4], [5]. Consider, a weak monoprotic acid, , which can exist in two forms such as, unionized () and ionized () species in an aqueous buffer system. If such an aqueous buffer system is equilibrated with a hydrophobic solvent (e.g. octanol), the unionized species and the ionized species in the aqueous phase will get partitioned into the hydrophobic phase with the partition coefficient defined by, and , respectively. Since, it is less likely for a charged species like , to get partitioned into an octanol phase, prior to partitioning it forms a neutral ion pair with prevalently available cation in the aqueous solution. The distribution coefficient (), on the other-hand is dependent on the partition coefficient () and is defined as, , the ratio of the sum of the concentrations of both ionized and unionized species of a molecule, distributed between the hydrophobic organic phase and the aqueous buffer phase. Since the dissociation of a weak monoprotic acid is dependent on the of the aqueous buffer system, the distribution coefficient also becomes dependent on . In an experiment designed to assess the lipophilicity of a molecule, the distribution coefficient (), is measured at different conditions and the resultant profile of , is fitted to a model, to obtain partition coefficients (), or of all the species present in the system [1], [2], [3], [4], [5].

The mathematical model to predict the profile of simple cases such as monoprotic, diprotic, mono-alkaline and amphoteric can be easily derived using algebraic approach [6]. On the other hand, while studying the effect of salt or co-solvent on the distribution of monoprotic acid, dynamic approach is preferred because of its generality and simplicity in deriving the models [3], [5], [7], [8], [9]. In this article, we explicitly, derive the algebraic and dynamic models for amphoteric, di-amino-monoprotic, and monoprotic in the presence of salt or co-solvent [7], [8], [9]. Further, the profiles of recently reported amphoteric molecules such as nalidixic acid, mebendazole, benazepril and telmisartan, were analysed to show the equivalence of dynamic approach and algebraic method [10].

Theory

A complex dynamic system can be modelled using several analogous kinetic mechanisms. If the experimental data points of the dynamic system is available prior to equilibrium, then the exact kinetic mechanism can be delineated accurately. On the other-hand if the experimental data is available only at equilibrium, then several analogous kinetic mechanisms can be used inter-changeably to determine the equilibrium constants (SI 1 and 2). In analysis, since we deal with systems that are at equilibrium, several analogous kinetic mechanisms are available to model its data. Here we have considered previously reported kinetic mechanisms for amphoteric, monoprotic acid in the presence of salt (KCl) or co-solvent (DMSO) and diamino-monoprotic amphoteric, to model the profile. Additionally, simple cases such as monoprotic acid (SI.3), diprotic acid (SI.4), monoalkaline (SI.5) are detailed in the supplementary information for pedagogic purpose.

Equivalence of analogous kinetic mechanisms at equilibrium

Considering a simple system with four states/species , , , , ; we show that several analogous kinetic mechanisms can be framed to model it (SI 2). Firstly, we define the ‘analogous kinetic mechanisms’ as a set of kinetic mechanisms whose equilibrium/steady state concentrations are the same for its species across mechanisms. In other words, even though the members of the ‘analogous kinetic mechanisms’ remain distinguishable through their distinct time profiles for , , , and , prior to steady-state or equilibrium, they are indistinguishable at steady state or equilibrium. If the equilibrium constants for one of the members of ‘analogous kinetic mechanisms’ is known then we can easily derive the equilibrium contants for the rest of the members of ‘analogous kinetic mechanisms’, which is stated here as the equivalence of the ‘analogous kinetic mechanisms’ at equilibrium.

If we consider each species as a ‘node’ and the interconnecting equilibrium reactions as bidirectional ‘edges’, then the graph theory suggest a maximum of , edges or equilibrium reactions [11], [12], [13]. For a system with , species, there exist a maximum of , equilibriums. On the other-hand, a minimum of , edges or equilibriums would be required to connect all the four species to obtain a non-disjointed or ‘connected graph’. With a minimum of 3 and a maximum of 6 equilibrums, there exist 38 different analogous kinetic mechanisms for a 4 species system (SI. 2). Out of these 38 possibilities we will consider only two ‘analogous mechanisms’ to show their equivalence. Consider a simple linear mechanism (Fig. 1A) which minimally connects all the four species as shown below (Eq. 1),

Fig. 1

(A) A simple linear kinetic mechanism proposed for a four states/species system, A, B, C, D (). The system is ‘simple’ because, only a minimal of () equilibriums are considered. It is ‘connected’ because, all the species are connected to its neighbour at least once. (B) represents a complex ‘completely connected’ kinetic mechanism for the same system. This system has a maximum number of () equilibriums realizable for a 4 state system. (C) Shows the dynamic profile for [A], [B], [C], [D] using simple model (1 A), (SI. 2.1). (D) shows the dynamic profile for [A], [B], [C], [D] using complex model (1B), (SI. 2.2). Prior to reaching the equilibrium (or pre-steady state phase) for simple and complex mechanisms at 1.5 s, 8×10-4 s, respectively, the time profiles for all 4 species remain distinct and distinguishable between mechanisms; but at equilibrium (steady state), the concentrations of all 4 species are equivalent despite their mechanistic differences. The simulation was carried out using , , , , , , as 7, 1, 72, 1, 21, 1, respectively. The additional kinetic rates seen for (B), , , , , , , were set to 504, 1, 1512, 1, 10584, 1, respectively. The initial concentrations of [A]0, [B]0, [C]0, [D]0 were set to 1,0,0,0, respectively, for both these simulations.

In the above equilibrium, , , , are the forward and , , , are the reverse rate constants for the reactions , , , respectively. The three equilibrium constants , , are defined as , , , respectively. On the other-hand consider a complex mechanism (Fig. 1B) which not only includes Eq. 1, but also three additional equilibriums Eqs. (2), (3), (4),

In the above equations (Eqs. (2), (3), (4)), , , , are the forward and , , , are the reverse rate constants for the reactions , , , respectively, and the corresponding equilibrum constants are defined as , , . If we assume both the mechanisms to be analogous i.e, both lead to an identical ratios of , , , , at equilibrium, then the equilibrium constants , , , are dependent on , , and can be easily derived by comparing a subset of (Eq. 1) and (Eq. 2) to write the following equation (Eq. 5),

The comparison clearly shows that is an abstraction of , hence we can combine the corresponding equilibrium constants and equate . Similarly, based on the comparisons of (Eq. 6) and (Eq. 7), we can write and , repsectively.

Thus, we can conclude that if we have a kinetic mechanism with species, we would require a minimal of equilibriums that uniquely connects these species, so as to determine the additional equilibrium constants existing in other ‘analogous mechanisms’. A comparitive simulation of both these kinetic mechanisms (Fig. 1A, B) using dynamic approach is shown in Fig. 1C & D, to highlight, their differences during pre-steady state phase and their equivalence during the steady state phase. In the following sections, one of the ‘analogous kinetic mechanism’ that best represent the distribution of a molecule between an aqueous buffer and octanol layer will be outlined. Based on the proposed kinetic mechanism, the algebraic and the dynamic models will be derived. The dynamic models proposed here make an assumption that the mass transportation is instantaneously homogenous within each liquid phases for all the species at all instance of time, i.e. perfectly stirred system. The dynamic model for non-stirred systems, which will not be discussed here, would require complex partial differential equations that account for both the spatial and time dependence based on Fick's second law of diffusion.

Amphotheric model for amino acids

Kinetic model for simple amino acids

Consider an amino acid containing a weak mono-protic acid () and a weak basic/alkaline group () distributed between an aqueous buffer and an organic hydrophobic solvent (octanol) (Fig. 2A) [5], [6], [14]. In the aqueous phase, the amino acid, ], exists in an unionized form or , and the ionized forms, or , or , or . The equilibrium among these four states can be written as (Eqs. (8), (9), (10), (11), (12)),, , , , are the forward kinetic rates and , , , , are the reverse kinetic rates for the dissociation of proton from the species, , , , , respectively. , are the forward and reverse kinetic rates for the tautomeric conversion seen between and , respectively. The , is the proton dissociated from or , COOH or (HA), functional groups. As explained in the Section 2.1, the above proposed mechanism is one of the possible “analogous kinetic mechansims”, where the tautomeric equilibrium , with the equilibrium constant, , is explicitly included. Infact, , can be easily derived from other equilibrium constants, by considering its alternate path: , which can be abstracted as, . Subsequently, can be rewritten in terms of and as .

Fig. 2

(A) Shows the kinetic model for a simple amphoteric molecule with four states/species , , , and , in aqueous buffer and when partitioned into octanol phase they exist as , , , and , respectively. (B) shows the simulation of the amphoteric model using algebraic method with equilibrium constants and , partition coefficients, , and .

The four species, , , and , can be partitioned into octanol layer as shown below (Eqs. (13), (14), (15), (16)),, , , , are the forward kinetic rates and , , , are the reverse kinetic rates, respectively, for the partitioning of , ,, , from the aqueous phase into octanol phase, as ,,,, respectively. The partition coefficients for the species , , and are defined as, , , , , respectively. The partition of singly charged species such as , and into the octanol layer is significantly influenced by ion pair formation (salt effect) in the aqueous phase. Whereas, the partitioning of zwitterions , is primarily influenced by its charge neutrality. The concentration of the zwitterions is prevalent at a particular called isoelectric point (), which is defined as the average of the dissociation constant of proton from its acidic group, , (), and its basic group, , ().

Algebraic method for simple amino acids

Since we have eight species () in our proposed kinetic mechanism, we would require only a minimum of ( =7) seven equilibriums (Eqs.,(8), (9), (11), (13), (14), (15), (16)), to define the algebraic model. Given the seven equilibriums, seven algebraic equations can be framed in terms of its species, , , , , , , , . The distribution coefficient (D) for such a system can be defined as,

In the above Eqn. 17, ‘’, is the ratio of the volume of octanol to aqueous buffer (SI. 3.2). By re-expressing seven of the eight species , , , , , , from Eqs., (8), (9), (11), (13), (14), (15), (16), in terms of the eighth species, and substituting the resulting analytical expressions into Eq. 17, we obtain Eq. 18,

Since, and , are concerned with the dissociation of proton () from its base unit (), (i.e. ); and and , are concerned with the dissociation of the proton from the acid unit, , (i.e. ), we can make a valid assumption that , and , in Eq. 18. Further, by substituting , i.e., excluding the neutral species in Eq. 18, we can easily arrive at the expression for diprotic model (SI. 4).

Dynamic method for simple amino acids

Based on the kinetic mechanism (Eqs. (8), (9), (10), (11), (12), (13), (14), (15), (16)), the rate equation (SI.1) can be written for eight species, , , , , , , , and , as follows,

By numerically integrating the above set of coupled differential equations, Eqs. 19–26, we obtain the concentration of eight species, , , , , , , , and at different time points. The resultant concentrations at equilibrium, can be substituted into the Eq. 17, to obtain the distribution coefficient. In the above model, if we were to account for the dynamics of , then we could write,

The “Buffer terms” as mentioned in the Eq. 27, are the terms that arise from the dissociation of the weak acid and weak base moieties that are specific to the buffer system [15]. In this work, we prefer to assume or , to be constant with respect to time, i.e. , due to fact that experiments are usually carried out in a controlled buffer system. To simulate the profile in Fig. 2B, the parameters such as , , , , , , , , , , , , , , , , , , were set to , 1.0, , 1.0, , 1.0, ,1.0, ,1.0, , 1.0, , 1.0, , 1.0, , 1.0, respectively. The initial concentrations of all the eight species , , , , , , and , were set to 1.0,0,0,0,0,0, 0,0, respectively and the , was varied linearly between 1 and 10.

Monoprotic acid with salt (KCl)

Consider the distribution of a weak mono-protic acid () between an aqueous solvent and an organic hydrophobic solvent (octanol) in the presence of a salt such as potassium chloride (KCl) (Fig. 3A) [3], [5], [16], [17]. In this mechanism, we first write the equilibriums existing in the aqueous phase, then the interface (aqueous-octanol partition) and finally the octanol phase.

Fig. 3

(A) Shows the kinetic model for a monoprotic acid in the presence of a salt. In this mechanism, there exist a reversible equilibrium between , and in aqueous phase and a reversible equilibrium between and in the octanol phase. The formation of in the octanol layer is mediated through the partitioning of the ion pairs , , and from the aqueous to octanol layer. (B) Shows the profiles of the monoprotic acid when the ratio of monoprotic acid to salt is varied between 1:0 to 1:1000. (C) shows how the , , of the monoprotic acid varies as the salt concentration increases, from 0 to 1000, plotted here in scale. It is clear that only increases linearly with increase in salt concentration, whereas, and , remains constant. (Though not shown, the plot of non-logarithmic form of vs is also linear). In the simulation, the , and of the monoprotic acid were set to 103, 10-3 and 10-3 respectively, the concentration of monoprotic acid was set to 1, and the concentration of salt was set to 0, 1, 10, 100, 1000 to simulate different , profiles. The resultant profiles were fitted to the monoprotic model, , to obtain the apparent , and , the logarithmic form of which are plotted against the logarithm of salt concentration in (C).

In aqueous phase, the weak acid, ], exists in the unionized form and ionized form . The equilibrium between these two states can be written as (Eq. 28),

, are the forward and reverse kinetic rates for the dissociation of the weak acid, ], respectively. Since the dissociation of the salt KCl into and (in aqueous buffer) is complete and irreversible we can write (Eq. 29) with, , as its kinetic forward rate constant.

The unionized species, gets partitioned into octanol layer with the forward and reverse kinetic rates, and , respectively, and the corresponding partition equilibrium, (Eq. 30),

The formation of the ionic complex, , in the aqueous phase is highly probably at high salt concentration, whereas at low concentrations (« saturation limit), it remain dissociated as and . We represent the partitioning of and into octanol layer as Eq. 31, with the forward and reverse kinetic rates as and , respectively; and the corresponding partition coefficient as, .

Even though the above Eq. 31, could have been written in a more explicit manner, , with the inclusion of an intermediate ionic complex () in aqueous phase, the experimental determination of the association constant for such an ionic complex is practically difficult. Hence, we prefer to use an abstracted mechanism as proposed by Scherrer [18] for all the partition equilibriums concerned with the ionic species.

The ionized species , would require neutralization of its negative charge through the prevalent cation, from KCl, before partitioning into octanol layer. The forward and reverse kinetic rates of the partition of are , , respectively, with its partition equilibrium constants defined as, (Eq. 32),

The ions (from , (Eq. 28)) and (from ) gets partitioned into octanol with the forward and reverse kinetic rates as , , respectively, with its partition equilibrium constants defined as, (Eq. 33).

Additionally, as seen with the monoprotic acid (SI.3), it is also possible for a fraction of the ionized species , to get directly partitioned into the octanol layer without ion-pair formation as , whose forward and reverse kinetic rates are given by , , respectively, with its partition equilibrium constants defined as, (Eq. 34),

Finally, all the four species , , , , that got partitioned into the octanol layer undergo a dynamic equilibrium whose forward and reverse kinetic rates are given as , , respectively, with its equilibrium constants defined as, (Eq. 35).

Based on the above kinetic mechanism Eqs., (28), (29), (30), (31), (32), (33), (34), (35), the algebraic model and the dynamic models were derived and used for this analysis (SI 7).

Monoprotic acid in the presence of co-solvent

Consider the distribution of a weak mono-protic acid () between an aqueous buffer and an organic hydrophobic solvent (octanol) in the presence of co-solvent () such as DMSO (Fig. 4A) [3], [4].

Fig. 4

(A) The kinetic model for a monoprotic acid in the presence of a co-solvent ([S]). In this mechanism, there exists a reversible equilibrium between , and in the aqueous phase, which also partitions into octanol as , and , respectively. All four species, , , , get co-solvated by the solvent, present in aqueous(), or octanol phase () with the stoichiometry of , , , , to form ,, ,, respectively. (B) & (D) Shows the profiles of the monoprotic acid when the ratio of monoprotic to solvent is varied between 1:0 and 1:1000, with the stoichiometry of , , , , to be 1:0:0:0 for (B) and 1:1:1:1 for (D), respectively. (C) & (E) shows how the , , of the monoprotic acid varies as the solvent concentration increases, from 0 to 1000 (shown here in scale). It is clear that the modulation of , and is strongly dependent on the degree of the co-solvation of different species. In the simulation, the , and of the monoprotic acid were set to 103,10-3 and 10-2 respectively, the concentration of monoprotic acid was set to 1 and the solvent concentration was set to 0, , , , , , , , , , , to simulate 12 different , profiles. Only five of the solvent profiles corresponding to 0, 1, 10, 100, 1000 were plotted in (B) and (D). The simulated profiles were fitted to the monoprotic model, , to obtain the apparent , and , the logarithmic form of which are plotted against the logarithm of solvent concentrations.

In aqueous phase, the weak acid, ], exists in unionized form and ionized form . The equilibrium between these two states can be written as (Eq. 36),

The equilibrium constant, , is defined based on its forward and reverse kinetic rates, , , respectively. The distribution of a co-solvent between an aqueous buffer and octanol is given by (Eq. 37),

, are the forward and reverse kinetic rates for the partition of the co-solvent ] between the aqueous and octanol phase (), respectively. The co-solvation of unionized form and in aqueous buffer can be written as (Eqs. (38), (39))

The equilibrium constant, , is defined based on the forward and reverse kinetic rates, , , respectively, for the co-solvation of in the aqueous phase. , are the forward and reverse kinetic rates for the co-solvation of with its equilibrium constant defined as . , , are the number of co-solvent molecules required to co-solvate , , respectively.

, and , gets partitioned into octanol layer as shown below (Eqs. (40), (41)),

The partition coefficient, , is defined based on the forward and reverse kinetic rates, , , respectively. Similarly, , is defined based on the forward and reverse kinetic rates, , , respectively. The co-solvation of and in octanol phase can be written as (Eqs. (42), (43)),

, are the forward and reverse kinetic rates for the co-solvation of in the octanol phase, respectively, the corresponding equilibrium constant is defined as, . Similarly, , , are the forward and reverse kinetic rates for the co-solvation of in the octanol phase, respectively, the corresponding equilibrium constant is defined as, . , , are the number of co-solvent molecules required to co-solvate , , respectively.

In the above mechanism, we could have included additional partition equilibriums such as, and , but as explained in Section 2.1, these two paths can be achieved through already existing paths such as, and . Hence, we can easily derive and write the expression for and as and , respectively. Based on the above kinetic mechanism Eqs.(36), (37), (38), (39), (40), (41), (42), (43), the algebraic model and the dynamic models were derived and used for this analysis (SI 8).

Results

The experimental data of nalidixic acid, mebendazole, benazepril, telmisartan obtained from a recent study were used in our current analysis [10]. The data of nalidixic acid and mebendazole could be well represented through a monoprotic (SI 3, Eqn. S21) and monalkaline model (SI 5, Eqn. S65), respectively (Fig. 5A & B). The profile of benazepril could be explained through a simple amino acid model (monoprotic-monalkaline) (Eq. 18, SI 6, Fig. 5C). Telmisartan, on the other-hand, required a complex monoprotic-dialkaline model to fit its experimental data (SI 9, Eqn. S168, Fig. 5D). The optimized , , and values were consistent with the previous studies and are summarised in Table 1 [10].

Fig. 5

Experimental analysis of (A) nalidixic acid, (B) mebendazole, (C) benazepril and (D) telmisartan using monoprotic, monoalkaline, simple amphoteric and diamino-monoprotic amphoteric models, respectively. The data were fitted using an in-house written matlab code [31].

Table 1

Fit parameters for nalidixic acid, mebendazole, benazepril and telmisartan based on monoprotic, monoalkaline, simple amphoteric and diamino-monoprotic acid models, respectively. The parameters such as ,,, are equivalent to the conventional which is related to the dissociation of the ion from the acid moieties. The ,, are not the conventional that is concerned with the dissociation of ion from the base moieties.

	Nalidixic acid	Mebendazole	Benazepril	Telmisartan
Model	monoprotic	monoalkaline	simple amphoteric	monoprotic-diamino amphoteric
Parameters
	-1.36 ± 0.07 (P_HA)	-3.06 ± 0.05 (P_BOH)	0.46 ± 0.11 (P_HAB,P_A−BH^+)	4.16 ± 0.27 (P_HABB, P_A−BBH^+,P_A−BH^+B)
Partition coefficients (P)	1.71 ± 0.17 (P_A^−)	-1.51 ± 0.26 (P_B^+)	-1.31 ± 0.05 (P_HABH^+,P_A−B)	1.46 ± 0.09 (P_HABBH^+,P_HABH^+B, P_A−BH^+BH^+,P_A−BB)
				-71825692.17 ± 0.00 (P_HABH^+BH^+)
pK_a/pK_b	6.35 ± 0.13 (pK_a)	10.25 ± 0.17 (pK_b)	2.88 ± 0.10 (pK_A)	5.6 ± 0.41 (pK_A)
			5.48 ± 0.11 (pK_B)	4.44 ± 0.63 (pK_B_1)
				3.61 ± 0.36 (pK_B_2)

The quantitative relationship between the salt concentration and the distribution parameters such as , , were derived for monoprotic acid (SI 7) and the simulation was carried out to assess the effect of salt on profile (Fig. 3B, C). The results show that the salt affects the value in a linear manner (directly proportional), whereas, the, and remained unaffected [5]. Similar to salt, the effect of co-solvent (e.g. DMSO) on the profile of a monoprotic acid was assessed through another set of simulations (SI 8). The profile was primarily influenced by the stoichiometry (, , , ) and the binding affinities (, , , ) of the co-solvent towards the four species , , and , present in the system. Keeping the binding affinities to be constant at 1:1:1:1, we studied the effect of different stoichiometry on profiles. In the first case, we assumed only to interacts with the co-solvent () with a stoichiometric ratio of to be 1:1 or higher (2:1 or 3:1). This could be realized by setting the overall stoichiometric ratio of : : : to be 1:0:0:0, respectively. In this case, the , remained constant (Fig. 4B, C), whereas, was decreasing non-linearly and , was increasing non-linearly with the addition of co-solvent. On the other-hand, if we assume, , , , to be 1,1,1,1, all the parameters varied non-linearly, except, which remained constant (Fig. 4D, E) [3], [5]. Finally, if we assume, a higher stoichiometric ratios, such as 2,1,1,1, or 3,1,1,1, all the parameters varied non-linearly with increase in co-solvent concentration (). Thus, we observe that the effect of co-solvent on profile is non-linear, and is significantly depended on the stoichiometry ratios that determine the degree of co-solvation of different monoprotic species.

Discussion

Among all the molecules considered in this study, only telmisartan required a complex di-alkaline-monoprotic model to explain its profile. The kinetic mechanism of telmisartan consisted of sixteen species with eight species in aqueous phase and another eight species in the octanol phase. Ignoring the eight species in the octanol phase, if we were to propose a kinetic mechanism for the rest of the eight species in the aqueous phase, the graph theory predicts a total of ~2.7×108 possibilities (SI. 2) [11]. Based on the fact that the equilibrium constants are equivalent for ‘analogous kinetic mechanisms’, we choose a minimal of seven (i.e. ) equilibriums out of the maximum 28 equilibriums () to explain an eight species system (). With the inclusion of the law of conservation of mass for telmisartan (in both aqueous and octanol phase) and considering the partition equilibriums for the eight species (between aqueous and octanol layer), we arrive at 16 algebraic equations to solve for the equilibrium concentrations of 16 species. In contrast to algebraic method, in dynamic approach, the rate equations for sixteen species depends significantly on the connectivity seen among species as proposed in the kinetic mechanism. Though the concentrations of all the 16 species will vary significantly during the pre-steady state phase, the concentrations of all the species will remains invariably the same for ‘analogous kinetic mechanisms’ at equilibrium or steady state phase. The model-fit for benazepril and telmisartan (Fig. 5C & D) was carried out using algebraic method (Eq. 18 & Eqn. S168) and the resultant optimized parameters were used to simulate profiles through dynamic approach (Eqs. 19–26; Eqns. S169-S184) The values obtained through algebraic and dynamic methods were comparable and equivalent to a degree of two decimal points for most of the data points (Table 2).

Table 2

Comparison of the log D values back calculated using algebraic and dynamic approach for Benazepril and Telmisartan, using simple amphoteric and diamino-monoprotic acid models, respectively. Accuracy to a degree of two decimal points was observed for most of the data points analysed through algebraic and dynamic approach.

Benazepril				Telmisartan
		Calculated (log_10D)				Calculated (log_10D)
pH	Experimental (log_10D)	Algebraic	Dynamic	pH	Experimental (log_10D)	Algebraic	Dynamic
1.95	0.38	0.39	0.39	1.95	0.24	0.25	0.25
2.42	0.74	0.74	0.74	2.42	1.07	1.04	1.02
3.00	1.06	1.06	1.07	3.00	1.99	2.06	2.03
3.75	1.29	1.24	1.25	3.75	3.32	3.19	3.16
4.19	1.28	1.26	1.27	4.19	3.65	3.63	3.62
4.75	1.25	1.23	1.23	4.75	3.63	3.91	3.96
5.30	0.99	1.09	1.09	5.75	3.96	3.77	3.79
6.04	0.60	0.68	0.67	6.04	3.84	3.60	3.61
6.65	0.34	0.22	0.21	6.89	2.85	2.90	2.90
7.22	-0.12	-0.15	-0.15	7.60	2.08	2.27	2.28
7.65	-0.36	-0.32	-0.32	8.00	1.91	1.96	2.00
				8.90	1.53	1.56	1.57
				9.72	1.57	1.47	1.48
				10.52	1.38	1.46	1.46
				10.91	1.52	1.46	1.44

The effect of salt on the profile of monoprotic acid as assessed through simulation, suggests an apparent increase in value with increase in salt concentration () (SI 7, Fig. 3B & C). To explain this effect quantitatively, we compare Eqn. S186 with Eqn. S185 (i.e. the simple monoprotic model), and obtain the expression for apparent as . The equation clearly shows that, and are linearly related to each other, with the slope and intercept, and , respectively (SI.10). When a salt (KCl) is added to a monoprotic acid, it tend to increase the formation of ion pair, , in the aqueous phase. The neutral ion pair easily partitions into octanol in the form of , thereby, increasing the total concentration of in the organic layer. Since, is defined as , where, = , an increase in due to addition of salt will proportionally increase the value of too [3], [5].

The effect of co-solvent on a monoprotic acid is more complex compared to the effect of salt (SI 8, Fig. 4C, D). We compare Eqn. S191 with Eqn. S185 to obtain the relationship between the apparent , , and and the co-solvent concentration ( and ) as follows, , and (SI.10) [3]. In the above expressions, and are the concentrations of the co-solvent in the octanol and the aqueous phase, respectively. The degree of non-linearity in the above expressions are introduced by the stoichiometric ratios, , , and . Previous studies have shown that the addition of co-solvent like DMSO will increase the , of the monoprotic acid in the aqueous solutions (distribution into octanol was not considered) [19]. This can be easily realised by assuming, , , , to be 1,0,1,0, respectively, and taking logarithm on both sides in the expression for , which yields, . This expression clearly shows that, when the co-solvent concentration is less (, the apparent or for a monoprotic acid is constant and is equal to . On the other-hand, if the co-solvent concentration is high (, then the apparent becomes linearly proportional to logarithm of co-solvent concentration in aqueous phase ().

At high concentrations, the concentration of the analytes can differ significantly from its thermodynamic ‘activity’ (), under such circumstances, it is necessary to make appropriate corrections in the equilibrium constants (dissociation or partition) [20], [21], [22]. Consider a simple monoprotic case, where, expression for distribution coefficient can be written as . As the concentrations of the species , , , or tend to increase significantly, then, we have to redefine, distribution coefficient as , by replacing the concentrations of the species with its corresponding activities. For example, the activity of can be defined as , where, is called the ‘activity coefficient’. Further, activity coefficient itself is a variable that is dependent on ‘ionic strength’ (I), which, in turn is a function of the concentration (molarity or molality) of the species and its charge. Several theories (Debye-Huckel, Pitzer, etc.) are available to calculate the activity coefficient of a given analyte at a given molarity or molality value [22], [23]. Experimentally, , can be measured through direct methods such as electrochemistry at the interface of immiscible liquids (ITIES) [24], [25], [26], [27], or indirect methods such as potentiometric, chromatography (HPLC, HPTLC, LCMS) [5]. The direct method has the advantage of innately taking into account the correction factors for the temperature and ionic concentrations, and also yields instantaneous time profile data suitable for dynamic analysis.

The data analysis of multiple species system, becomes increasingly complex because of the inclusion of large number of parameters in the model. For such complex models it is often recommended to carry out ‘sensitivity analysis’ to identify the parameters of significance in order to reduce the complexity of the models [28], [29], [30]. A simple sensitivity analysis, could be based on Jacobian matrix obtained at global minimum of the model fit, which can be normalized to assess the significance of each parameter present in the model. On the other-hand sophisticated, Monte-Carlo based approaches are available which additionally provide insights on the degree of interaction present among parameters in the model [30].

Supporting Information

Supporting Information contains explicit derivation of log10 D for mono-protic, di-protic acid, mono-alkaline, mono-protic acid with salt, mono-protic acid with solvent and monoprotic-dialkaline molecule. The expressions for apparent , and for monoprotic acid in the presence of salt and co-solvent are provided. Matlab codes to derive algebraic model (monoprotic, diprotic, monoalkaline, diamino monoprotic) and simulation of dynamic approach are provided (monoprotic with co-solvent).