**2.1 Determination of octanol-water partition coefficients (K** **ow** **) of azide ion / hydrazoic acid for different pH values using a reversed-phase liquid chromatography and UV detection.**

In our study we considered experimental n-octanol water partition coefficients for azidic acid and azide ion, respectively. In principle one can apply one of the developed constant pH simulation algorithms [24, 25], but it is worth to stress that HN3 pKa value at the interface is highly dependent on the coordinate and is computationally as demanding as pKa values in protein interior [26, 27] and in contrast to proteins inaccessible to experiment [28].

Since hydrazoic acid is an ionizable species its octanol-water partition coefficient (Kow) depends on pH. We have decided to measure Kow for two pH values and apply extrapolation approach for all the pH values [29]. Determination of 1-octanol-water (octanol-water) partition coefficients (Kow) of AHA was carried out using two different pH values of azide solutions (pH 2.0 and 8.0) and a reversed-phase liquid chromatography with UV detection. pH-dependent partition coefficients Kow(pH) were calculated from the experimental data using equation:

$$\begin{array}{c}{K}_{ow}\left(pH\right)=\frac{{\left[A\right]}_{oct}}{{\left[A\right]}_{aq}}\#\left(1\right)\end{array}$$

where [A]oct represents total azide concentration (sum of neutral HN3 and deprotonated N3− species) in organic phase of 1-octanol and [A]aq represents total azide concentration in aqueous phase.

Since HPLC method does not distinguish between neutral and ionizable species of hydrazoic acid and azide anion, partition coefficient Kow(pH) below can be expressed as distribution coefficient Q(pH). Azide exists in its neutral or protonated form both in aqueous solution and in 1-octanol. The ratio of concentrations of azide in 1-octanol (organic phase) and in water regardless of its protonation state at a certain pH value of the environment is described by distribution coefficient Q. Q(pH) is an equilibrium constant and a measure of free energy. We define Q(pH) as:

$$\begin{array}{c}Q\left(pH\right)=\frac{{\left[{A}^{0}\right]}_{org} + {\left[{A}^{-}\right]}_{org} }{{\left[{A}^{0}\right]}_{aq}+ {\left[{A}^{-}\right]}_{aq}}\#\left(2\right)\end{array}$$

where [A0]org stands for concentration of neutral form of azide (HN3, hydrazoic acid) in the organic phase (1-octanol), [A0]aq for the concentration in the aqueous solution, [A¯]org for concentration of deprotonated form of azide (N3¯) in the organic phase and [A¯]aq for the concentration of azide in the aqueous solution.

By expanding the numerator and denominator of the right-hand side of the Eq. (2) by factor 1/[A−]aq one obtains

$$\begin{array}{c}Q=\frac{{\left[{A}^{0}\right]}_{org}/{\left[{A}^{-}\right]}_{aq}+ {\left[{A}^{-}\right]}_{org}/{\left[{A}^{-}\right]}_{aq} }{{\left[{A}^{0}\right]}_{aq}/{\left[{A}^{-}\right]}_{aq} + 1}\#\left(3\right)\end{array}$$

If a new variable P0 is introduced as the partition coefficient of neutral species; P0 = [A0]org/[A0]aq, and P¯ as the partition coefficient of deprotonated species P¯ = [A¯]org/[A¯]aq, equation simplifies to a more compact form:

$$\begin{array}{c}Q=\frac{{P}^{0}{\left[{A}^{0}\right]}_{aq}/{\left[{A}^{-}\right]}_{aq} + {P}^{-} }{ {\left[{A}^{0}\right]}_{aq}/{\left[{A}^{-}\right]}_{aq}+ 1}\#\left(4\right)\end{array}$$

[A0]aq/[A−]aq is the equilibrium constant of neutral and deprotonated species in organic and in aqueous solution that depends both on pKa value of azide and the aqueous phase pH value.

Free energy for azide deprotonation (ΔG) with a certain value of pKa in the solution with certain pH value reads:

$$\begin{array}{c}\varDelta G={ln}10\bullet {k}_{b}T\bullet \left(pH-{pK}_{a}\right)\#\left(5\right)\end{array}$$

where *k**B* is Boltzmann's constant and equals 1.987 kcal (mol K)−1 and T is the absolute temperature in Kelvins (310 K at physiological conditions). Eq. 5 implies that ionizable group with a certain pKa value is in contact and equilibrium with the aqueous solution with a certain pH value. There is one to one correspondence between the equilibrium constant and free energy [30] and is expressed as

$$\begin{array}{c}\varDelta G=-{k}_{b}T\bullet {ln}\left(\frac{{\left[{A}^{0}\right]}_{aq}}{{\left[{A}^{-}\right]}_{aq}}\right)\#\left(6\right)\end{array}$$

By combining Equations 5 and 6 and by considering the identity \(\text{ln}x=\text{ln}10\bullet \text{log}x\), and introducing the relation \(\text{log}\beta =pH-{pK}_{a}\) or \(\beta ={10}^{{\text{p}\text{K}}_{\text{a}}-\text{p}\text{H}}\), it is possible to write an equation for azide distribution coefficient as a function of pH and pKa.

$$\begin{array}{c}Q=\frac{{P}^{0} + \beta {P}^{-} }{1+ \beta }=\frac{{P}^{0}{10}^{{pK}_{a}-pH} + {P}^{-} }{1+ {10}^{{pK}_{a}-pH}}\#\left(7\right)\end{array}$$

Distribution coefficient Q is the equilibrium constant for azide distribution between the organic phase and the aqueous phase regardless of the azide protonation state. The corresponding free energy ΔG, associated with azide transfer from aqueous solution with certain pH value to the membrane is:

$$\begin{array}{c}\varDelta G=-{k}_{b}T\bullet {ln}Q=-{k}_{b}T\bullet {ln}\frac{{P}^{0} + \beta {P}^{-} }{1+ \beta }=-{k}_{b}T\bullet {ln}\left(\frac{{P}^{0} {10}^{{pK}_{a}-pH}+ {P}^{-} }{1+ {10}^{{pK}_{a}-pH}}\right)\#\left(8\right)\end{array}$$

Since extracellular fluid (ECF) and cytoplasm (CYT) have different pH values of 7.4 and 7.2, free energy for transfer of AHA is not zero. AHA tends to concentrate in the compartment with more favourable solvation free energy. To calculate relative population of AHA in the cell membrane, the appropriate Q for given pH value is used:

$$\begin{array}{c}{\left[A\right]}_{ memb\left({pH}_{1}\right)}={Q}_{\left({pH}_{1}\right)}\bullet {\left[A\right]}_{ ECF\left({pH}_{1}\right)}={Q}_{\left({pH}_{1}\right)}\#\left(9\right)\end{array}$$

The same relation can be used for equilibrium between the cytoplasm and the membrane. By combining these two relative populations of AHA in the cytoplasm and ECF can be found.

$$\begin{array}{c}{\left[A\right]}_{ CYT\left({pH}_{2}\right)}=\frac{1}{{Q}_{\left({pH}_{2}\right)}}\bullet {\left[A\right]}_{ memb\left({pH}_{1}\right)}=\frac{{Q}_{\left({pH}_{1}\right)}}{{Q}_{\left({pH}_{2}\right)}}\#\left(10\right)\end{array}$$

The relative AHA population in cytoplasm is therefore simply the ration of both Q.

By inserting Eq. 7, we can derive a simplified formula:

$$\begin{array}{c}{\left[A\right]}_{ CYT\left({pH}_{2}\right)}=\frac{{Q}_{\left({pH}_{1}\right)}}{{Q}_{\left({pH}_{2}\right)} }=\frac{{10}^{{pH}_{1}-{pK}_{a}} }{{10}^{{pH}_{2}-{pK}_{a}}}={10}^{{pH}_{1}-{pH}_{2}}\#\left(11\right)\end{array}$$

As seen in Eq. 11, the overall AHA concentration in the cytoplasm is pKa independent and depends only on the pH value difference of the cytoplasm and ECF.

## 2.2. Determination of effective permeability, *P**e* *(pH*) using the method PAMPA

Simulation of azide partition between extracellular fluid and biological membrane was performed using the method PAMPA (Parallel Artificial Membrane Permeability Assay), the schematics of which is seen in Fig. 1. PAMPA is an *in vitro* tool for high-throughput prediction of *in vivo* drug permeability and is also useful for the assessment of passive transport mechanisms [31]. Our experiments were performed using Corning BioCoat Pre-Coated PAMPA Plate System with 96-well insert system with 0.45 µm PVDF (polyvinyliden fluoride) filter plate which has been pre-coated with structured layers of phospholipids and a matched receiver microplate. The thickness *d* of phospholipid membrane was 125 µm = 1 250 000 Å.

PAMPA experimental conditions are described in Table S3 in Supplementary materials. Prior to use, the pre-coated PAMPA plate system was warmed to room temperature for 60 minutes. The compound solutions of NaN3 with concentrations of 0.241 and 0.120 mg mL− 1 were prepared by dilution of stock solutions (1.2 mg mL− 1) into a PBS buffer adjusted to two pH values of 7.4 and 8.0 with 1M NaOH. 300 µL of solvent per well was pipetted into acceptor plate and 200 µL of sample per well was pipetted into upper donor plate. Upper donor plate with sample was sealed with the Sealing Tape for 96-Well Plates to prevent hydrazoic acid to evaporate and contaminate other wells. We placed the filter plate on the receiver plate and incubated the assembly at room temperature for 5 hours, without stirring. After that we separated plates and determined the compound concentrations in both plates using HPLC with UV detection; chromatographic conditions are listed in Table S4 in Supplementary materials.

Determination of effective permeability, *P**e**(pH*) for azide (cm s− 1) at pH values of 7.4 and 8.0 was calculated using as follows:

$$\begin{array}{c}{P}_{e}=\frac{-ln\left[1-{C}_{A}\left(t\right)/{C}_{equilibrium}\right]}{S*\left(\frac{1}{{V}_{D}}+\frac{1}{{V}_{A}}\right)*t}\#\left(12\right)\end{array}$$

We calculated Mass Retention (%) using Eq. 13:

$$\begin{array}{c}1-\frac{\left[{C}_{D}\left(t\right)\bullet {V}_{D}+{C}_{A}\left(t\right)\bullet {V}_{A}\right]}{{C}_{0}\bullet {V}_{0}}\#\left(13\right)\end{array}$$

where *C**0* is initial azide concentration in donor well (mM), *C**D**(t)* is azide concentration in donor well at time *t* (mM), *C**A**(t)* is azide concentration in acceptor well at time t (mM), *V**D* is donor well volume of 0.2 mL (including compound in buffer), *V**A* is acceptor well volume of 0.3 mL (including buffer), *S* is filter area of 0.3 cm2 and *t* is incubation time (5 h = 18000 s). Equilibrium concentration, *C**equilibrium* (µM), was calculated as

$$\begin{array}{c}{C}_{equilibrium}\frac{\left[{C}_{D}\left(t\right)\bullet {V}_{D}+{C}_{A}\left(t\right)\bullet {V}_{A}\right]}{{V}_{D}+{V}_{A}}\#\left(14\right)\end{array}$$

## 2.3. Theoretical prediction of permeability (Pt)

Herman J.C. Berendsen was among the first who performed molecular simulation of membranes to address their structure, stability, and transport properties [32–37], followed by his disciples that included simulation of membrane embedded transporters [38–42]

Diffusion is a passive process of the net movement of a substance down driven by a concentration gradient. Diffusion is described by diffusion equation in n-dimensions:

$$\begin{array}{c}\frac{\partial u}{\partial t}=D{\nabla }^{2}u\#\left(15\right)\end{array}$$

where *u* is the coordinate and time-dependent concentration, *D* stands for diffusion coefficient and \({\nabla }^{2}\) is Laplace operator. It must be noted, that in our calculation *D* is coordinate and time independent. Furthermore, in complex environments, such as in a nerve cell, the concentration difference is not the only force driving diffusion. Therefore, we include the coordinate-dependent chemical potential (\(\mu\)) to the Eq. 15, which alters the time dependent concentration profiles. This is a special case of a more general Fokker-Planck equation, called the Smoluchowski equation [43], which reads:

$$\begin{array}{c}\frac{\partial u}{\partial t}=\nabla \left[D\left(\nabla u+\frac{1}{{k}_{b}T}\left(u\bullet \nabla \mu \right)\right)\right]\#\left(16\right)\end{array}$$

where \(\mu\) is the coordinate-dependent chemical potential, *k**b* is Boltzmann constant, *T* is absolute temperature and \(\nabla\) is gradient of a function.

Since the diffusion equation with coordinate-dependant chemical potential described above does not allow for analytical solution, we proceeded with numerical solving on a grid. Algorithms for numerical solving of such diffusion equation appeared only recently [44]. Especially illustrative is a carefully performed computational study of oxygen diffusion through the membranes by Ghysels et al. [45]. We have used finite differences numerical method. Two numerical simulations were made, once with dimensions of a realistic cell membrane with thickness of 50 Å and one with real PAMPA membrane thickness of 125 µm according to the manufacturer. First experiment used grid spacing of 1 Å and time step of 0.1 ns, while second experiment used grid spacing of 1 µm and time step of 1 µs. In both simulations the diffusion coefficient D from Huang and co-workers of 820 µm2 s− 1 [46] was used. Moreover, both simulations used the free energy profiles of the transition to and from the membrane, which were experimentally determined earlier in this study.

Secondly, the permeability across a membrane is described by the following equation:

$$\begin{array}{c}j=-P\varDelta c\#\left(17\right)\end{array}$$

where *j* is the flux, *P* is the membrane permeability and \(\varDelta c\) is the difference of concentration of both sides of the membrane. Flux was calculated numerically from diffusion equation with coordinate-dependent chemical potential, while concentration difference was initially 1.

## 2.4. Preparation of solutions

Solvents of sample solution were prepared by mixing acetonitrile and water in ratio 1:10 (V/V). Solvents were adjusted to pH 2.0 with 20% H3PO4 and to pH 8.0 with 0.01 M NaOH before filling to the calibration mark.

First, calibration curves were prepared at 5 concentration levels; for pH value of 2.0 from 0.12 to 1.5 mg mL− 1 of NaN3 (that corresponds 0.08 to 1.0 mg mL− 1 of HN3) and 0.0019 to 0.12 mg mL− 1 of NaN3 (that corresponds 0.0013 to 0.08 mg mL− 1 of HN3) for pH value 8.0.

Aqueous sample solutions of NaN3 with pH values of 2.0 and 8.0 were prepared also; concentration of sample solution with pH value of 2.0 was 1.5 mg mL− 1 of NaN3 (equal to 1.0 mg mL− 1 of HN3) and sample solutions with pH value of 8.0 was 12 mg mL− 1 (equal to 8.0 mg mL− 1 of HN3). Then, equal volumes, 15 mL of solution of NaN3 and 15 mL of 1-octanol, was pipetted into a 50 mL separatory funnel and mixed well every 10 minutes in 1 hour to equilibrate. Finally, upper organic and lower aqueous phase was sampled. Aqueous phase with pH value of 8.0 was diluted to factor 0.0008. All solutions for calibration curves and final solutions of organic and aqueous phases were injected into a HPLC system.