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Protonation Kinetics Compromise Liposomal Fluorescence Assay of Membrane Permeation Deniz Sezer* and Tuğcȩ Oruç Faculty of Engineering and Natural Sciences, Sabancı University, Orhanlı-Tuzla, 34956 Istanbul, Turkey S Supporting Information *

ABSTRACT: The membrane permeation of weak acids and bases couples to the ambient pH and can be studied using pH-sensitive dyes as reporters. Such fluorescence measurements with aliphatic amine drugs have revealed biexponential kinetics of permeation into liposomes (Eyer et al. J. Controlled Release 2014, 173, 102). Permeability coefficients have been obtained using the faster of the two kinetic components. Here, the origin of the biexponential kinetics is studied with a kinetic rate model that in addition to drug permeation accounts for the protonation of the drug and the dye. Surprisingly, the experimental readout is found to strongly depend on the rates of protonation. The analysis demonstrates that fluorescence studies of drug permeation relying on pH-sensitive proxies should be accompanied by comprehensive modeling of the relevant kinetic processes.

I. INTRODUCTION

navigate between the conflicting requirements of solubility in an aqueous environment and in the membrane.10 While AA and CA groups are preferentially ionized in water, their charged forms typically encounter prohibitively large free energy barriers at the hydrophobic core of the membrane and thus have negligibly small permeability coefficients. The neutral forms, on the other hand, encounter substantially smaller free energy barriers and have permeability coefficients that are orders of magnitude larger. Once on the other side of the membrane, the permeating AA (CA) drug takes (releases) a proton to become charged again, inadvertently coupling permeation to pH. As a result, two membrane-separated compartments kept at different pH can support unequal concentrations of a permeating ionizable drug (the pHpartition hypothesis).20,21 This observation has been used to load drugs into vesicles,22 while the reverse effect has been used to measure pH differences across membranes.23 The coupling between permeation and pH can be further exploited to measure the kinetics of permeation through lipid membranes. pH-sensitive fluorescent probes trapped in liposomes have been used extensively to obtain proton permeability coefficients.24−26 More recently, such dyes have been used to measure the kinetics of permeation of AAs and CAs, whose permeation also couples to pH.27,28 In this approach, unilamellar liposomes encapsulating the dye are exposed to a drug solution. The increase of the internal drug concentration modifies the pH inside the vesicle and causes a change in the fluorescence signal. Given the overabundance of AA and CA drugs, the use of pH-sensitive optical reporters should allow for the permeability coefficients of many drug

Fast permeation through lipid membranes is essential for the absorption, distribution, metabolism, and excretion of smallmolecule drugs. The physicochemical factors controlling passive membrane permeation, therefore, have remained at the center stage of pharmaceutical research.1,2 Computational methods to study the permeation of small molecules through lipid bilayers in atomistic detail are currently well developed.3−6 While such computational studies have produced permeability coefficients for many drug molecules,7−10 it has been difficult to assess the accuracy of these quantitative predictions due to the scarcity of experimental measurements under comparable conditions. Fluorescence spectroscopy has been used to quantify the kinetics of drug permeation through model membranes similar to those studied computationally, i.e., lipid bilayers composed of either a single type of lipid (e.g., phospholipid) or a mixture of two lipids (e.g., phospholipid and cholesterol). These approaches, however, require that the permeant itself is fluorescent or exhibits self-quenching fluorescence.11−13 An alternative approach utilizing the photoluminescence of lanthanides has also been developed.14−16 However, in this case, the permeant is required to both absorb in a convenient wavelength window and chelate the lanthanide ion. 14 Furthermore, there is evidence that some of these latter studies are prone to experimental artifact.17 A recent chemogenomic analysis of the ionization constants of more than 3700 human approved drugs has demonstrated that 27% of the examined drugs contained an aliphatic amine (AA) and 20% a carboxylic acid (CA) group.18,19 Subsequently, we have argued that the overabundance of AA and CA moieties among drugs is due to the unique ability of these groups to facilitate permeation through lipid bilayers by allowing drugs to © 2017 American Chemical Society

Received: February 26, 2017 Revised: April 6, 2017 Published: May 4, 2017 5218

DOI: 10.1021/acs.jpcb.7b01881 J. Phys. Chem. B 2017, 121, 5218−5227

Article

The Journal of Physical Chemistry B

comment on the protonation rates deduced by our analysis. The last section summarizes our conclusions. Additional results and detailed examination of the computational findings of ref 29 are presented in the Supporting Information.

molecules that lack special optical properties to be accessed under conditions similar to those in the atomistic simulations. Recent application of this approach to the permeation of seven AA drugs has revealed a biexponential increase of the fluorescence signal.28 Permeability coefficients were deduced from the fluorescence data using only the faster of the two exponential rates.28 The origin of the slower exponential component and its possible relation to the drug permeation is unclear. The permeation of the same AA drugs characterized in the experiments has been subsequently investigated computationally through atomistic molecular dynamics (MD) simulations.29 However, the permeability coefficients calculated according to the well-established computational procedure3 based on the inhomogeneous solubility-diffusion model30 differed by a factor of 4 to a factor of 1230 from the experimental permeabilities. In the case of the drug propranolol, for example, the discrepancy was 11-fold. Passive permeation through lipid bilayers can be viewed as a succession of three steps: insertion into the lipid bilayer, translocation from one of the bilayer leaflets to the other, and dissociation from the membrane.10,16,31 Since each of these subprocesses has a distinct rate, in principle, the increase of the drug concentration in a liposome could be multiexponential. In an effort to reconcile computations and experiments, Dickson et al. formulated and solved numerically a kinetic rate model accounting for the three distinct permeation steps.29 Notably, performing a biexponential fit to the calculated drug concentration and using the faster component of the fit to obtain a permeability coefficient resulted in very good agreement with the experimental values (a factor of 1.1 in the best case and 8 in the worst).29 Again, the slower component of the biexponential fit was not analyzed. Taken together, these recent experiments28 and computations29 suggest that biexponential kinetics may be an essential feature of the permeation of drugs containing AA and, by analogy, CA moieties. Both studies identify the permeability coefficient with the faster of the two kinetic time scales. Singling out only one of the two exponential rates as a proxy of drug permeation, however, calls for a better understanding of the origin of the observed biexponential kinetics. Here, we examine the fluorescence permeation experiment with propranolol, which is one of the studied AA drugs.28 By modeling the protonation (deprotonation) of the drug and the dye, in addition to the permeation of the neutral form of the drug, we are able to reproduce quantitatively the biexponential kinetics of the fluorescence signal. In the Supporting Information, we critically analyze the computational results of ref 29 and show that the three-step permeation process cannot be responsible for the biexponential kinetics observed in the experiments. We conclude that the kinetics of protonation have to be properly accounted for in the analysis of permeation experiments relying on (i) the coupling of permeation and pH and (ii) pH-sensitive reporters. The rest of the paper is organized as follows. The experimental setup of ref 28 is summarized in section II, which also contains background information on how permeability coefficients were deduced from the experimental data. The kinetic rate models that we develop are introduced in section III. These are then used to gain insight into the biexponential permeation kinetics of the drug propranolol (section IV). In the Discussion (section V), we consider other factors that may lead to multiexponential behavior, and

II. BACKGROUND II.A. Concentration Change due to Permeation. The change in the numbers of drug molecules inside (N) and outside (N1) a liposome can be described as d N (t ) d N (t ) =− 1 = Aj dt dt

(1)

where A is the area of the membrane interface and j is the density of the net inward drug current. The latter is typically taken to be proportional to the difference of concentrations inside (c) and outside (c1) the liposome, with the constant of proportionality being the permeability coefficient: j = P(c1 − c)

(2)

Using this expression in the right-hand side of (1) and dividing the result by the volume of the liposome (V) or the volume of the outside solution per liposome (V1) leads to the following equations for the drug concentrations: dc(t ) = k[c1(t ) − c(t )], dt

dc1(t ) = −k′[c1(t ) − c(t )] dt (3)

Here, we have defined the rates k = PA /V

and

k′ = PA /V1 = kV /V1

(4)

If no drug is present inside the liposome at the beginning of the experiment and c0 denotes the initial drug concentration outside the liposome, the solution of (3) is c1(t ) = c0 c(t ) = c0

k ⎛ k′ −(k + k ′)t ⎞ ⎜1 + ⎟ e ⎠ k + k′ ⎝ k

k (1 − e−(k + k ′)t ) k + k′

(5)

Both concentrations change monoexponentially with rate A⎛ V⎞ k + k ′ = P ⎜1 + ⎟ V⎝ V1 ⎠

(6)

which is proportional to the permeability coefficient. Note, however, that in addition to the liposome geometry (A and V) this rate also depends on V1. When V1 is much larger than the volume of the liposome, (5) simplifies to c1(t ) ≈ c0 ,

c(t ) ≈ c0(1 − e−kt )

(7)

In this case, the outside drug concentration remains practically constant and the inside concentration changes with a rate essentially equal to k. As a result, from the increase of c(t), the permeability coefficient can be determined using only the geometric parameters of the liposome. If the latter is assumed to be a perfect sphere of radius R, P = kV /A = kR /3

(8)

This relationship was used in ref 28 when analyzing the experimental data. In the case of weak acids and bases, it becomess necessary to consider the permeation of their neutral and ionized forms, 5219


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The Journal of Physical Chemistry B which are typically assumed to take place independently. The discussion below is limited to basic drugs (AAs), but the equations can be easily adapted to acids (CAs). The total drug concentration in a given compartment (c) is the sum of the concentrations of the neutral (c0) and charged (c+) forms. The latter can be written as c + = r +c

c 0 = r 0c ,

(9)

where the fractions r and r = 1 − r depend on the pKa of the drug and the pH of the environment through the Henderson− Hasselbalch equation: 0

+

0

r 0 = [1 + 10(pKa − pH)]−1

Figure 1. Processes taking place in the liposomal permeation assay of ref 28. The gray circle depicts the liposome membrane. B is the aliphatic amine drug and D the pH-sensitive dye HPTS. Their different ionization states interconvert through protonation (deprotonation). The pH is initially 5.9 but increases inside the liposome as a result of drug permeation.

(10)

When the two drug forms permeate the membrane independently, current densities proportional to the differences in concentration exist separately for both [see (2)]: j 0 = P 0(c10 − c 0),

j+ = P+(c1+ − c +)

Table 1. Initial Concentrations and pKa’s of the Species Present outside and inside the Liposome28

(11)

outside

The total current density is the sum of the two independent currents: 0

+

j=j +j =P

0

(c10

0

−c)+P

+

(c1+

+

−c )

pH propranolol (drug, B) HPTS (dye, D) MES (buffer) NaCl

(12)

Using (9) for the inside and outside compartments, the total current density can be related to the total drug concentrations:

9.5a 8.5b 6.1

NaCl, but since we do not model any possible salt-related effects, it is not going to be considered here. In the specific experiment that we analyze, a buffer is present outside the liposomes but not inside (Table 1). Thus, the outside pH should retain its initial value. Because an HPTS molecule is charged both when protonated (−3e) and deprotonated (−4e), the dye should remain in the liposome lumen throughout the experiment (Figure 1, green). As the neutral form of the AA drug penetrates into the liposome and becomes protonated, the pH of the liposomal lumen increases (Figure 1). The change of pH is reflected in the fluorescence signal of the dye, which is measured and analyzed. The relationship between the pH of the solution and the emission intensity of the dye at 520 nm has been determined empirically (Supplementary Figure 1a of ref 28). These data points are shown in Figure 2 with squares. In analogy to other studies employing pH-sensitive dyes,27 the empirical relationship can be described by the functional form

(14)

(15)

Typically, the permeability of the charged species is vanishingly small and the second term on the right-hand side of (15) can be neglected. Then, by measuring the net permeability coefficient, one can deduce the permeability coefficient of the neutral drug form as P 0 ≈ P /r 0 = [1 + 10(pKa − pH)]P

5.9 0M 100 μMb 0M 200 mM

(13)

Thus, when the pH on the two sides of the membrane is the same, the net permeability coefficient can be identified as [see (2)] P = r 0P 0 + (1 − r 0)P+

pKa

b

⇒ [B0out] = 2.5 nM; [B+out] = [Bout] − [B0out] = 9.9975 μM. b⇒ [D4−] = 0.25 μM; [D3−] = [D] − [D4−] = 99.75 μM.

In general, the fractions of the neutral drug inside and outside the liposome need not be equal (r01 ≠ r0), unless the two compartments have equal pH. In this latter case, dropping the subscripts of r0 and r+, we have j = (r 0P 0 + r +P+)(c1 − c)

5.9 10 μMa 0M 10 mM 190 mM

inside

a

j = P 0(r10c1 − r 0c) + P+(r1+c1 − r +c) = (P 0r10 + P+r1+)c1 − (P 0r 0 + P+r +)c

a

(16)

This expression was used in the analysis of ref 28. II.B. Fluorescence Experiments. From the experiments reported in ref 28, we only consider the ones in which liposomes containing the pH-sensitive dye HPTS are free in solution. In fact, only the measurement for the drug propranolol is going to be analyzed. The experimental situation is depicted schematically in Figure 1. Initially, the pH inside and outside the liposomes is 5.9, the dye molecules (D) are inside the liposomes at a concentration of 100 μM and the drug molecules (B) are outside the liposomes at a concentration of 10 μM (Table 1). From these and the pKa’s of the drug and the dye (Table 1, last column), the concentrations of the species with different protonation states are readily obtained using (10) (Table 1, footnotes). The inside and outside solutions also contained

Figure 2. Emission intensity of the dye HPTS at 520 nm changing with the ambient pH. Measurements of ref 28 (symbols) and a fit with the function in eq 17 (black line). 5220


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The Journal of Physical Chemistry B F(pH) = 1 −

0.995 1 + e−2(7.2 − pH)

III. METHODS III.A. Kinetic Model of the Relevant Processes. Rather than analyzing the experimental fluorescence signal using eqs 8 and 16, whose validity and limitations were discussed in the previous section, we explicitly model the processes that modify the concentrations inside the liposome (Figure 1). The species that are considered are the neutral and charged forms of the drug (B0, B+), protons (H+), and the two ionization forms of the dye (D4−, D3−). Because the total dye concentration in the liposome remains constant, i.e., [D4−(t)] + [D3−(t)] = [D], only [D4−(t)] will be explicitly monitored. The kinetic rate equations describing how these concentrations change in time are

(17)

which is drawn with a solid line in Figure 2. Because F(pH) is not linear in the pH range of the experiment (from 5.9 to 6.6), we transform the calculated pH time traces using (17) before comparing with experiment (section IV.C). In ref 28, the fluorescence signal was fitted with the biexponential function F(t ) = A e−kat + Be−kbt + F0

(ka > k b)

(18)

The fit fixes the parameters A, B, ka, and kb, while F0 is determined by the starting value of the signal. Only the ratio of A and B is of physical significance because the absolute intensity of the signal is arbitrary. The best-fitting parameters ka, kb, and A/B for the drug propranolol (from Supplementary Table 1 of ref 28) are given in Table 2. The time scales corresponding to the biexponential fit are ka−1 = 0.015 s and kb−1 = 0.06 s.

d[B0(t )] = J 0 (t ) − JBH (t ), dt d[H+(t )] = −JBH (t ) − JDH (t ), dt

Table 2. Biexponential Fit Parameters (ka, kb, A/B), Liposome Diameter (2R), and Permeability of Neutral Propranolol (P0) as Reported in ref 28a

a

kb (s−1)

A/B

2R (nm)

1/r0

P0 (cm s−1)

66.7

17.5

3.56

176

3982

1.55

where the J’s on the right-hand sides of the equalities account for the separate processes that modify the concentrations. These are described below. The inward current of B0 due to its permeation through the membrane is [see first equation in (3)]

The fraction of neutral propranolol, r0, corresponds to pH 5.9.

0 J 0 (t ) = k0([Bout ] − [B0(t )])

The permeability coefficient of the drug has been obtained from the rate constant of the faster exponential component according to (8).28 Using the average hydrodynamic diameter of the liposomes, which is also given in Table 2 (from footnote c of Table 1 of ref 28), we find

k0 = P 0A /V = 3P 0/R

(23)

Although it has not been specified whether in the experiments the outside volume per liposome was much larger than the liposome volume,28 when writing (22), we assumed that the outside drug concentration remains unchanged. (If necessary, this assumption can be relaxed by additionally following the change of the drug concentrations outside the liposome.) Inside the liposome, the neutral form of the drug is subject to the process (Figure 1)

Finally, the permeability coefficient of the neutral form of the drug has been determined from (16).28 The inverse of the fraction of neutral propranolol at pH 5.9 is given in Table 2. Using this value, we find 0

(22)

with [from (4)]

(19)

P 0 = 3982 × 1957 × 10−7 cm s−1 = 0.779 cm s−1

d[D4 −(t )] = −JDH (t ) dt (21)

ka (s−1)

P = kaR/3 = 1957 × 10−7 cm s−1

d[B+(t )] = JBH (t ), dt

(20)

kon

B0 + H+ HooI B+

−1

Interestingly, ref 28 reports P = 1.55 cm s (Table 2), which is 2 times larger than what we obtained in (20). Thus, either the radius of the liposome (rather than the diameter) is equal to 176 nm or in ref 28 the diameter (rather than the radius) was used in eq 19. Nevertheless, to be consistent with ref 28, we use the value in Table 2 in the numerical analysis of section IV. The conclusions, however, will be shown to be independent of this choice (Figure 5). The analysis of ref 28 summarized above raises the following questions: Since the fluorescence signal does not originate directly from the drug molecules, why should its rate of change be related to the permeability coefficient of the drug through (8)? Even if it were, should it be proportional to the net permeability (P) [eq 19] or to the permeability of the neutral drug form (P0), which is the only one assumed to permeate the membrane? Furthermore, the derivation of (15) required the assumption of equal pH in the donor and acceptor compartments. However, in the experiment, the pH inside the liposome changes with time. It is therefore not clear why the initial pH and not the final pH, for exampleshould be used in (16). Finally, what is the origin of the slower exponential rate kb? Does it also contain information about the membrane permeation of the drug?

(24)

koff

where kon and koff are the rates of protonation and deprotonation. These are related through the equilibrium constant of the deprotonation reaction, Ka = 10−pKa, as koff = konK a

(25)

The net concentration current associated with the rightward (protonation) direction of (24) is JBH (t ) = kon([B0(t )][H+(t )] − K a[B+(t )])

(26)

When positive, this current decreases [B0] and [H+] and increases [B+], thus the choice of signs on the right-hand sides of the first three equations in (21). Similarly, the dye inside the liposome undergoes protonation/deprotonation (Figure 1) D kon

D4 − + H+ HooDI D3 −

(27)

koff

with net rightward (protonation) current D JDH (t ) = kon ([D4 −(t )][H+(t )] − K aD[D3 −(t )])

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(28)


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The Journal of Physical Chemistry B The rate equations (21) together with the expressions for the separate concentration currents J constitute our full kinetic model. III.B. The Limit of Instantaneous Protonation. When analyzing fluorescence experiments that probe the membrane permeation of AA drugs28 or small CAs,27 the kinetics of protonation have not been considered. Instead, the protonation (deprotonation) of the permeant has been treated on the level of its pKa. In ref 27, for example, the concentration ratio of charged and uncharged species was deduced from the pH and pKa using the Hederson−Hasselbalch equation. Similarly, in ref 28, the net permeability coefficient was rescaled to obtain the permeability of the neutral drug (eq 16). Assuming that the processes of protonation (deprotonation) are fast, the rates of protonation (kon and kDon) can be eliminated from our kinetic model by requiring that the reactions (24) and (27) are at quasi-equilibrium at all times, i.e., [B0(t )][H+(t )] = Ka , [B+(t )]

[D4 −(t )][H+(t )] = K aD [D3 −(t )]

c3 = K aD/K a − 1 c 2 = K a + h0 + b − d0 4 − (d0 3 + h0 + 2b + K a)K aD/K a c1 = b(d0 4 − h0) + [2b(d0 3 + h0) + b2 + K ab]K aD/K a c0 = −b2(d0 3 + h0)K aD/K a (36)

This makes x(t) a unique function of b(t). As a result, when protonation (deprotonation) is practically instantaneous, the full model (21) can be replaced by the following simpler model db(t ) 0 ] − b(t ) + x(t )} = k0{[Bout dt

which explicitly keeps track of the kinetics of the concentration b(t) only. The concentrations of all molecular species of interest are recovered from b(t) [and x(t), which is also obtained from b(t)] using (30), (34), (31), and (32). III.C. The Limit of Instantaneous Permeation. The opposite limit of permeation being the fastest process can also be considered. In this case, the inside concentration of the permeating species becomes instantaneously equal to the outside concentration and remains constant afterward:

(29)

To impose these conditions, we denote the concentration (in units of molar, M) of the neutral drug molecules that have entered into the liposome from outside up to time t by b(t). Note that this is not the actual concentration of B0 at time t, since some fraction of these molecules have captured a proton and become B+. Let x(t) denote the concentration (in M) of the neutral molecules that have been protonated. Because initially there were no drug molecules inside the liposome, at any given time t, we have [B0] = b − x ,

[B+] = x

0 [B0(t )] = [Bout ]

[D3 −] = d0 3 + y

This makes the rate equation of [B ] in the full kinetic model (21) redundant. In addition, the drug protonation current JBH (eq 26) becomes inst 0 JBH (t ) = kon([Bout ][H+(t )] − K a[B+(t )])

(30)

d[B+(t )] d[D4 −(t )] inst (t ), = JBH = −JDH (t ), dt dt d[H+(t )] inst (t ) − JDH (t ) = −JBH dt

(31)

IV. RESULTS IV.A. Instantaneous Protonation. Concentrations were first calculated with the model (37) in which protonation is treated as instantaneous. The time traces obtained by using the experimental parameters (Table 1) and the experimental value of the propranolol permeability coefficient (Table 2) are shown in Figure 3. Note that, while the concentration of the neutral drug (dashed, blue line) is in units of nM, those of the protonated drug (solid, blue line) and deprotonated dye (green) are in μM. The pH (black) is plotted rather than the concentration of H+. We emphasize that this calculation does not contain any free fitting parameters. All of the simulated processes in our model are driven by the initial gradient of the drug concentration. Thus, a quasiequilibrium should be reached when the gradient of the neutral

(b − x)(h0 − x − y) = K ax (33)

Using the first equality in (33), y is readily expressed in terms of x: y = h0 − x − K ax /(b − x)

(34)

Substituting this result into the second equality, we deduce that x should be the physically meaningful root (i.e., 0 < x < b) of the cubic equation c3x 3 + c 2x 2 + c1x + c0 = 0

(40)

Although not immediately clear form the rate equations, the permeability coefficient has disappeared entirely from the kinetic model (40), since P0 was present only in the current J0 (eq 22). Thus, in this limit, the kinetics of the concentrations do not contain any information about the permeability coefficient.

(32)

where h0 is the initial proton concentration (equal to 10−5.9 M in the experiments in ref 28). In terms of these variables, the quasi-equilibrium conditions (29) can be written as

(d0 4 − y)(h0 − x − y) = K aD(d0 3 + y)

(39)

In this case, the following simpler kinetic model, in which only three internal concentrations change in time, needs to be integrated:

Finally, the proton concentration at time t is [H+] = h0 − x − y

(38) 0

Similarly, let d03 and d04 be the initial concentrations of D3− and D4−, respectively (their numerical values were given in footnote b of Table 1), and y(t) be the concentration (in M) of the protons captured by D4− up to time t. Then, [D4 −] = d0 4 − y ,

(37)

(35)

with coefficients 5222


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Figure 3. Concentrations according to the model with instantaneous protonation (deprotonation). The two drug forms are blue, the deprotonated dye is green, and the pH is black. An integration time step of 1 μs was used to resolve the fast permeation kinetics (P0 = 1.55 cm s−1, R = 88 nm).

Figure 4. Concentrations according to the model with protonation kinetics. Protonation rates kon = 1010.67 s−1 M−1 (propranolol) and kDon = 108.2 s−1 M−1 (dye) were used in addition to previous model parameters. A monoexponential function with rate k0 = 3P0/R = 0.5 × 106 s−1 (red line) matches the rapid increases of [B0].

form of the drug is dissipated. Indeed, [B0] in Figure 3 reaches a plateau when it becomes equal to [B0out] = 2.5 nM (Table 1, footnote). In contrast, the charged drug form, which is not allowed to permeate in the model, does not equilibrate with its concentration outside the liposome ([B+out] ≈ 10 μM). As a result, at steady state, the total drug concentration inside the liposome is about 2 μM, which is 5 times less than the outside drug concentration, in accordance with the pH-partition hypothesis.20,21 Regarding the kinetics of relaxation to quasi-equilibrium, the increase of [B0] is observed to be sigmoidal. Thus, it would be impossible to describe it as a sum of two exponential functions. Therefore, performing a biexponential fit and deducing the permeability coefficient from the faster exponential rate would not work. In the case of ionizable drugs, where the drug permeation couples to other processes involving protonation, the concentrations of the species change with complex kinetics. For properly inferring the permeability coefficient of the neutral drug, therefore, all processes that couple to the pH have to be modeled explicitly. All concentrations in Figure 3 are seen to reach their steady state values in a few ms. In contrast, the fluorescence signal in the propranolol experiments of ref 28 changes on time scales of 15 ms (ka−1) and 60 ms (kb−1), which are an order of magnitude longer. This discrepancy between the model with instantaneous protonation and experiment prompted us to go back to the full kinetic model (21). IV.B. Complete Kinetics and Instantaneous Permeation. In addition to the permeability coefficient of the drug, which is known from experiment, the numerical integration of the full model requires knowledge of the protonation rates of the drug (kon) and the dye (kDon). Treating these as fitting parameters, we achieved reasonable agreement with experiment for kon = 1010.67 s−1 M−1 and kDon = 108.2 s−1 M−1. Before comparing the fit with the experimental fluorescence (Figures 5 and 6), however, let us examine the calculated concentrations, which are plotted in Figure 4. Compared to the model with instantaneous protonation, the full kinetic model contains only two additional parameters (kon and kDon). Being rates, these should leave the quasi-equilibrium state unaffected. Indeed, all concentrations in Figure 4 reach the same steady-state values as in Figure 3. However, the time scales to reach this steady state are different in the two models.

In Figure 4, the pH (black) and the concentration of the ionized form of the drug (solid, blue line) increase on time scales of tens of ms, similar to the fluorescence signal in the experiment. Perhaps surprisingly, the concentration of the permeating neutral form of the drug (dashed, blue line) increases on a much faster time scale in Figure 4 than in Figure 3. Numerical comparison with a monoexponential function of the form (7) with c0 = [B0out] (Figure 4, red line) reveals that [B0] increases with a rate exactly equal to k0 = P0A/V. For P0 = 1.55 cm s−1 and R = 88 nm, k0 = 0.5 × 106 s−1 corresponds to a time scale of 2 μs. In other words, according to the full kinetic model (21), the equilibration of the inside and outside concentrations of the permeating neutral drug is practically instantaneous on the time scale of the experiment. The same concentrations calculated according to the model with instantaneous permeation (40) are shown in the Supporting Information (Figure S1). The kinetics of the concentrations on a ms time scale are visually indistinguishable from those in Figure 4. Thus, for the experimental permeability coefficient, the permeation of propranolol appears to be practically instantaneous compared to the time scales of protonation. IV.C. Sensitivity to Permeation and Protonation and Uniqueness of Fit. For direct comparison with experiment, the pH time traces were converted to fluorescence using the function F(pH) in (17). The results for the full kinetic model are plotted with solid lines in Figure 5. Notably, taking the experimental permeability coefficient at face value (i.e., P0 = 1.55 cm s−1), we have been able to reproduce the biexponential increase of the experimental signal (green dashed line) by fitting the protonation rates of the drug and the dye (black line). Thus, a quantitative description that uses the experimental permeability coefficient and explains the biexponential increase of fluorescence has emerged. Taking the achieved description of the experiment as a reference, we now examine how sensitive the calculated fluorescence is to the value of the permeability coefficient. To this end, simulations were performed with P0 that was 2 times smaller (eq 20) and 2 times larger (Figure 5, blue and orange lines) than the experimental value. The fluorescence signal for infinitely fast permeability coefficient was also calculated from the pH trace of the model (40) (dashed, red line). Strikingly, all of these predictions are basically indistinguishable. Further5223


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for agreement with experiment, in addition to the rates of the two exponential components (ka and kb in Table 2), their relative weights (A/B in Table 2) also need to be reproduced correctly. While not appreciably affected by permeability coefficients larger than about 0.5 cm s−1, the rise in fluorescence is predicted to slow down for smaller values (Figure S4). Since the initial slope of the fluorescence response decreases when P0 is reduced (Figure S4) but increases when kon is increased (Figure S2), the experimental rise can be matched by many different combinations of these parameters. One example is provided in Figure S5. Clearly, if the three parameters P0, kon, and kDon in our kinetic model are treated as free fitting parameters, the fit to the biexponential experimental signal is not unique. This prevents us from deducing the exact permeability coefficient of propranolol (or its exact protonation rate, for that matter) from the available experimental data. Additional permeation experiments, perhaps at different initial pH, would need to be performed. Nevertheless, we emphasize that reading out the permeability coefficient from the rate of increase of the experimental fluorescence using (19) is unjustified, given the sensitivity of this increase to the rate of protonation of the drug. IV.D. Implications for Other Similar Experiments. The finding that rates of protonation may need to be explicitly considered when relying on pH-sensitive reporter dyes to measure the membrane permeation of weak bases and acids appears to be new. It, therefore, becomes necessary to comment on why protonation could legitimately be treated as instantaneous in other similar experiments, e.g., those reported in ref 27. Without going into the details of the experiments in ref 27, which monitored the membrane permeation of small CAs using a pH-sensitive dye as a proxy, it suffices to say that there the liposomes were much larger (R ≈ 20 μm) than what we have considered so far (R ≈ 0.1 μm). Note that larger R translates into slower permeation rate (eq 23). For sufficiently large R, permeation should slow down enough for protonation to be (almost) instantaneous in comparison. Using the propranolol parameters that were employed so far (as in Figure 4, for example), pH profiles were calculated for liposomes with increasing radii. The predictions of the full kinetic model (21) (solid lines) and the model (37) treating protonation as instantaneous (dashed lines) are compared in Figure 7. Clearly, for the permeation kinetics of propranolol to become slower than the protonation kinetics, thus justifying analysis on the level of the equilibrium dissociation constants (or pKa’s), the liposome radius must be more than 100 times larger than in the experiments of ref 28. Since R ≈ 20 μm in ref 27 (and the permeation of the studied CAs was significantly slower compared to propranolol), treating the protonation as instantaneous appears to be justified in this case.

Figure 5. Comparison of fluorescence intensities for various permeability coefficients (solid lines) and instantaneous permeation (dashed, red line) with the biexponential fit to experiment28 (dashed, green line). Vertical lines are the time scales of the biexponential fit (ka−1 and kb−1 from Table 2).

more, they all agree with the biexponential fit to the experimental fluorescence parametrized by ka, kb, and A/B from Table 2 (dashed, green line). The two time scales of the biexponential fit (vertical lines) are clearly present in the calculated traces. Thus, according to the quantitative description that we have constructed, any value of the permeability coefficient that is larger than about 0.5 cm s−1 would lead to fluorescence that exhibits the biexponential increase observed in the experiment. Apparently, all of these permeability coefficients, including the experimental one, fall in a regime of fast permeation where the experiment cannot meaningfully discriminate between them. In particular, this holds true for the value P0 = 17.4 cm s−1 that was obtained in ref 29 from atomistic MD simulations of neutral propranolol using the standard computational approach.5 In contrast to its poor sensitivity to P0, the calculated fluorescence is very sensitive to the rates of protonation, as demonstrated in Figure 6 where kon is decreased by a small factor of 1.3 = 100.1 (red line) and kDon is increased by a factor of 1.6 = 100.2 (blue line). (Additional plots demonstrating the exquisite sensitivity of the fluorescence signal to the protonation rates are given in Figures S2 and S3.) Note that

V. DISCUSSION V.A. Biexponential Kinetics. The membrane permeation of aliphatic amine (AA) drugs has been reported to exhibit kinetics on two different time scales.28 Only the faster time scale was identified with the permeability of the drug in the original analysis.28 Above, we provided evidence that the two exponential rates revealed by the experiment depend strongly on the protonation rates of the drug and the dye (Figure 6) and are almost insensitive to the rate of drug permeation for the permeability coefficients deduced by Eyer et al.28 (Figure 5).

Figure 6. Comparison of fluorescence intensities for various protonation rates (solid lines) with the biexponential fit to experiment (dashed, green line). Modifying kon = 1010.67 s−1 M−1 by a factor of 100.1 (red line) or kDon = 108.2 s−1 M−1 by a factor of 100.2 (blue line) results in clear deviations from experiment. (Numbers in the legend are base-10 logarithms of kon and kDon. Two vertical lines as in Figure 5.) 5224


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form of the protonophore FCCP (254 Da, logP 3.7), whose size and lipophilicity are very similar to those of propranolol (259 Da, logP 3.48), has been estimated to be equal to 50 cm s−1 at room temperature from kinetic modeling of experimental data.33 Admittedly, in this case, the lipid bilayer was composed of DPhPC lipids, instead of POPC used in the experiments of ref 28. For salicylic acid (138 Da, logP 2.26), which is less lipophilic than propranolol and whose protonated carboxyl group is rather polar, a lower limit of 0.2 cm s−1 has been estimated for the permeability coefficient through a lipid bilayer of DPhPC lipids.17 Thus, the permeability coefficient of neutral propranolol is likely larger than 0.5 cm s−1. According to our analysis, this puts the permeation of propranolol into the fast regime where the fluorescence signal is predicted to be rather insensitive to the exact numerical value of P0 (Figure 5). If so, we expect the rates kon and kDon obtained by fitting to experiment to be reliable. By matching the biexponential fluorescence signal quantitatively (Figures 5 and 6), the kinetic rate model formulated here produced numerical estimates for the protonation rates of propranolol and the dye HPTS. Although these estimates cannot be treated as definitive, it is necessary to comment on their magnitudes. The on rate of protonation reflects the frequency of productive encounters between hydronium ions and the protonation site of the molecule. Leaving chemistry aside, an encounter rate for diffusion in three dimensions can be estimated from dimensional analysis as kenc ∼ aDc,34 where a is the linear dimension of the target (i.e., protonation site), D is the diffusion constant of protons,35 and c is their number concentration. The concentration of protons at pH 6 is c ≈ 10−7 protons per nm3 (number concentration, not molar). With D ≈ 10 nm2 ns−1,36 and taking a ∼ 0.1 nm as the size of the protonation site, we deduce kenc ∼ 102 s−1. The corresponding time scale of encounter, k−1 enc ∼ 10 ms, is comparable to the experimental time scales of 15 and 60 ms. Dividing the estimated encounter rate kenc by the molar concentration of protons (10−6 M at pH 6) allows us to estimate an on rate of protonation: kon ∼ 108 s−1 M−1. This estimate is similar to the protonation rate of the dye HPTS obtained in the previous section (kDon = 108.2 s−1 M−1). The deduced protonation rate of propranolol was larger by more than two orders of magnitude (kon = 1010.67 s−1 M−1). The 300-fold difference between the on rates of propranolol and HPTS (534 Da) must be due to factors that influence the frequency of their productive encounters with hydronium ions. In the case of HPTS, the favorable electrostatic interactions between its three negatively charged sulfate groups and a positively charged hydronium may be reducing the effective proton diffusion constant in the vicinity of the dye, and thus reducing the encounter rate of a proton with the protonation site. Atomistic simulations of proton hopping kinetics at lipid bilayer surfaces have shown substantially slower rates of proton transfer when hydronium ions were engaged in favorable electrostatic contacts with the phosphate groups of the lipids.37 On the other hand, the large protonation rate of propranolol may indicate that encounters with protons take place in two rather than in three dimensions. Indeed, protonation rates larger than what would be expected from diffusive encounters in 3D have been reported for membrane-bound proton acceptors, suggesting that the membrane acts as a “protoncollecting antenna”.38,39 Notably, the propranolol protonation rate that we estimated from a fit to the experimental

Figure 7. Calculated pH time traces assuming finite (solid lines) and infinitely fast (dashed lines) protonation rates of the drug and the dye. Significant differences remain for liposome radii up to a factor of 300 larger than in experiment.28 (P0 = 1.55 cm s−1, kon = 1010.67 s−1 M−1, and kDon = 108.2 s−1 M−1.)

However, other factorslike diffusion inside the vesicle and diffusion inside the membranecould also contribute to the biexponential kinetics observed in the experiments. A time scale for diffusion inside a vesicle can be estimated from dimensional considerations:32 tdiff ∼ L2/D. Here D is the diffusion coefficient of the molecule in water and L is the linear dimension of the liposome (either radius or diameter). Using D ≈ 1 nm2 ns−1 and L ≈ 100 nm, we find tdiff ∼ 10 μs, which is 3 orders of magnitude smaller than the experimental time scales. Thus, for the experiments in ref 28, diffusion inside the vesicle can be safely treated as instantaneous. (Note, however, that the time scale of diffusion increases quadratically with liposome size. For the liposomes employed in ref 27, for example, we have tdiff ∼ 0.4 s. Appropriately, simultaneously with permeation, the authors solved the diffusion equation to calculate how the concentration profile along the radial direction of a liposome changed with time.27) Recently, it was argued that the biexponential permeation kinetics observed in ref 28 were due to the heterogeneous diffusion of the drug through the membrane.29 A kinetic model accounting for the separate permeation steps of insertion, translocation, and dissociation was formulated and solved numerically using the experimental parameters.29 When applied to propranolol, the novel computational procedure of ref 29 produced a permeability coefficient of the neutral drug that was only a factor of 2 larger than the experimental estimate. In Figure 5, we already showed that increasing the experimental permeability coefficient by a factor of 2 has a marginal effect on the fluorescence. Thus, the two exponential time scales observed in the permeation experiments of propranolol cannot be due to the heterogeneity of drug diffusion in the membrane. (A detailed examination of the novel analysis of ref 29 is provided in section II of the Supporting Information.) V.B. Permeability Coefficient and Rates of Protonation. When the permeability coefficient was treated as a free parameter and allowed to attain values below 0.5 cm s−1, the fit to the experimental fluorescence provided by our kinetic model was not unique (e.g., Figure S5). However, there are reasons to believe that the permeability coefficient of neutral propranolol should be larger than 0.5 cm s−1. As already mentioned, Dickson et al.29 have calculated P0 = 17.4 cm s−1 from atomistic simulations of neutral propranolol using the standard computational procedure. The permeability coefficient of the neutral 5225


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The Journal of Physical Chemistry B fluorescence signal falls directly in the range of the on rates reported in ref 39 (1010−1012 s−1 M−1). Thus, the majority of the permeating, neutral propranolol molecules may be getting protonated before dissociating from the membrane surface.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support by the Turkish Academy of Sciences (TUBA) through an Outstanding Young Scientist (GEBIP) award to D.S. is gratefully acknowledged.

VI. CONCLUSION The origin of the biexponential kinetics of drug permeation reported in ref 28 was investigated by formulating and numerically integrating a kinetic rate model that, in addition to permeation, accounted for the protonation (deprotonation) of the drug and the dye (eq 21). For the purposes of comparison, two limiting cases of the full kinetic model were also considered: one in which protonation (deprotonation) was treated as occurring instantaneously (eq 37) and another in which the permeation of the neutral drug was instantaneous (eq 40). In permeation experiments with pH-sensitive fluorophores, it is commonly assumed that protonation reactions are at quasiequilibrium at all times,27,28 which amounts to treating protonation (deprotonation) as instantaneous compared to the other dynamic processes. Here, the kinetics of protonation were shown to be essential for the quantitative analysis of the experimental data of ref 28. Furthermore, when the permeability of the neutral form of the drug was relatively fast, as in the case of propranolol, the kinetics of protonation were estimated to remain important for liposomes as large as R ≈ 20 μm (Figure 7). In contrast to the demonstrated sensitivity to the rates of protonation (Figure 6), changing the permeation rate of propranolol by a factor of 2 or increasing it all the way up to infinity (i.e., instantaneous permeation) had little effect on the predicted fluorescence (Figure 5). We therefore conclude that the biexponential kinetics observed in the experiment with propranolol reflect the rates of protonation of the drug and the dye, rather than the rate of permeation of the neutral drug. Whether the protonation kinetics similarly contaminate the fluorescence signal in the case of the other drugs examined in ref 28 can only be ascertained through a comprehensive analysis of the experimental data along the lines illustrated here for propranolol. Such an effort would likely require additional experiments to independently measure the rates of protonation and is beyond the goals of the present paper. Nevertheless, the presented analysis clearly demonstrates that using eqs 8 and 16 to deduce a permeability coefficient from the rate of increase of the fluorescence signal is not justified when the permeation of a weak base is monitored through a pH-sensitive proxy. It is therefore likely that the numerical values of the other permeability coefficients reported in ref 28 are also flawed.

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(1) Krämer, S. D.; Aschmann, H. E.; Hatibovic, M.; Hermann, K. F.; Neuhaus, C. S.; Brunner, C.; Belli, S. When barriers ignore the “ruleof-five”. Adv. Drug Delivery Rev. 2016, 101, 62−74. (2) Lopes, D.; Jakobtorweihen, S.; Nunes, C.; Sarmento, B.; Reis, S. Shedding light on the puzzle of drug-membrane interactions: Experimental techniques and molecular dynamics simulations. Prog. Lipid Res. 2017, 65, 24−44. (3) Marrink, S.-J.; Berendsen, H. J. C. Simulation of water transport through a lipid membrane. J. Phys. Chem. 1994, 98, 4155−4168. (4) Loverde, S. M. Molecular simulation of the transport of drugs across model membranes. J. Phys. Chem. Lett. 2014, 5, 1659−1665. (5) Lee, C. T.; Comer, J.; Herndon, C.; Leung, N.; Pavlova, A.; Swift, R. V.; Tung, C.; Rowley, C. N.; Amaro, R. E.; Chipot, C.; Wang, Y.; Gumbart, J. C. Simulation-based approaches for determining membrane permeability of small compounds. J. Chem. Inf. Model. 2016, 56, 721−733. (6) Di Meo, F.; Fabre, G.; Berka, K.; Ossman, T.; Chantemargue, B.; Paloncýová, M.; Marquet, P.; Otyepka, M.; Trouillas, P. In silico pharmacology: Drug membrane partitioning and crossing. Pharmacol. Res. 2016, 111, 471−486. (7) Ulander, J.; Haymet, A. D. J. Permeation across hydrated DPPC lipid bilayers: Simulation of the titrable amphiphilic drug valproic acid. Biophys. J. 2003, 85, 3475−3484. (8) Liu, W.; Zhang, S.; Meng, F.; Tang, L. Molecular simulation of ibuprofen passing across POPC membrane. J. Theor. Comput. Chem. 2014, 13, 1450033. (9) Carpenter, T. S.; Kirshner, D. A.; Lau, E. Y.; Wong, S. E.; Nilmeier, J. P.; Lightstone, F. C. A method to predict blood-brain barrier permeability of drug-like compounds using molecular dynamics simulations. Biophys. J. 2014, 107, 630−641. (10) Oruç, T.; Kücu̧ ̈k, S. E.; Sezer, D. Lipid bilayer permeation of aliphatic amine and carboxylic acid drugs: rates of insertion, translocation and dissociation from MD simulations. Phys. Chem. Chem. Phys. 2016, 18, 24511−24525. (11) Shimanouchi, T.; Ishii, H.; Yoshimoto, N.; Umakoshi, H.; Kuboi, R. Calcein permeation across phosphatidylcholine bilayer membrane: Effects of membrane fluidity, liposome size, and immobilization. Colloids Surf., B 2009, 73, 156−160. (12) Jespersen, H.; Andersen, J. H.; Ditzel, H. J.; Mouritsen, O. G. Lipids, curvature stress, and the action of lipid prodrugs: Free fatty acids and lysolipid enhancement of drug transport across liposomal membranes. Biochimie 2012, 94, 2−10. (13) Arouri, A.; Lauritsen, K. E.; Nielsen, H. L.; Mouritsen, O. G. Effect of fatty acids on the permeability barrier of model and biological membranes. Chem. Phys. Lipids 2016, 200, 139−146. (14) Thomae, A. V.; Wunderli-Allenspach, H.; Krämer, S. D. Permeation of aromatic carboxylic acids across lipid bilayers: the pHpartition hypothesis revisited. Biophys. J. 2005, 89, 1802−1811. (15) Gardiner, J.; Thomae, A. V.; Mathad, R. I.; Seebach, D.; Krämer, S. D. Comparison of permeation through phosphatidylcholine bilayers of N-dipicolinyl- alpha- and -beta-oligopeptides. Chem. Biodiversity 2006, 3, 1181−1201. (16) Thomae, A. V.; Koch, T.; Panse, C.; Wunderli-Allenspach, H.; Krämer, S. D. Comparing the lipid membrane affinity and permeation of drug-like acids: The intriguing effects of cholesterol and charged lipids. Pharm. Res. 2007, 24, 1457−1472. (17) Saparov, S. M.; Antonenko, Y. N.; Pohl, P. A new model of weak acid permeation through membranes revisited: Does Overton still rule? Biophys. J. 2006, 90, L86−L88.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.7b01881. Additional simulation results and information on the four-state model of permeation (PDF)

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Deniz Sezer: 0000-0002-9635-4369 5226


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