Supporting Information.
Synthetic Anionophores for Basic Anion as ‘Presumably OH-/Cl- antiporters’: From the
Synthetic Ion Channels to Multiion Hopping, Modified Anti-Hofmeister Selectivity and
Strongly Positive AMFE.
Sofya Kostina Berezin
sofyaberezin@gmail.com
Table of Contents
SI1. Goldman-Hodgkin-Katz theory versus ‘ion hopping’ over a barrier (or transitionstate theory). ‘One anion’ transporter, ‘multiion hopping’mechanism and AMFE theory.
SI2.
Approximations.
SI3.
Permeability of the ‘unstirred’ layer.
SI4.
Weak vs strong acid: the pH “error” in 10 mM phosphate buffer.
SI5.
Influx of the weak acid HAnion2 OUT into liposomes. A simple model, pH ≥ 7.
SI6.
Influx of the weak acid HAnion2 OUT and H+/OH- transport.
SI7. Anion1 IN / Anion2 OUT exchange, HAnion1 IN - is a strong acid, HAnion2 OUT - is
a weak acid.
SI8.
‘Presumable OH-/Cl- exchange’.
SI 1. GHK theory verus ‘ion hopping’ over a barrier (or transition-state theory).
‘One anion’ transporter, ‘multiion hopping’mechanism and AMFE theory.
GHK voltage equation (or just GHK equation) can be written as following:
Erev 
P [ M  ]  PA [ A ]In
RT
ln M   Out
F
PM  [ M ]In  PA [ A ]Out
, where M+A- is a salt in solution.
GHK flux equation:
J  V
d[C ] IN
zU ([C ] IN  [C ]OUT e  zU ) , where U = Em F/(RT)
 PS
dt
1  e  zU
GHK current equations:
I  V
d [C ] IN zF Pz 2 EF 2 ([C ] IN  [C ]OUT e  zU )

dt S
RT
1  e  zU
(I = J*zF/S, Am-2 = C s-1 m-2= mol s-1C mol-1)
Goldman-Hodgkin-Katz equations is not the Eyring transport model but an approximate
solution of the diffusion equation and Poisson's equation for electrostatic.
Because the ions are subject to the concentration gradients and the electrical potential
gradients, their motion in the one-dimensional system can be described by a combination
of Fick’s law and Poisson's equation as following:
dC
d
J  D
 uzFC
, where D – diffusion coefficient, u – mobility (u = D/RT), C dx
dx
concentration, z – valence of an ion, x – distance and ψ – local potential.
This equation simply expresses the additivity of diffusional and electrophoretic motions.
Let’s assume a constant electric field, i.e. – dψ /dx = constant = -(ψout- ψin)/d = E/d, where
d – is thickness of the membrane, E – membrane potential difference. This implies that ψ
will be a linear function of x as illustrated in Figure S1.
Let’s C' denote the concentration within the membrane (Figure S1). The concentrations
just inside the membrane can be related to corresponding concentrations in the bathing
media by a partition coefficient, ß as CIN = ßC'IN and C'OUT = ßCOUT'. In turn, the
permeability coefficient can defined as P = ßD/d.
Figure S1. Schematic concentration and voltage profiles through the membrane. The
voltage is linear (constant field), while the concentration profile is distorted from linearity
by the electric field. (Adapted from the references 1 and 2.)
Hence, there are only two forces that influence the movement of the charged
particle as it crosses biomembrane. In our case, this charged particle is an (an)ionreceptor complex, which simply drifts down the concentration gradient and the potential
gradient (electric field). The charge, size, diffusion and partitioning coefficients,
concentrations inside and out are to determine the rate of the transmembrane passage
through the regions of matter with different polarity. In turn, the height of the ‘energy
barrier’ (with an exception of its electrical component) is preserved within the
permeability coefficient in a form of partitioning coefficient. This model is known as
solubility-electrodiffusion. At a glance, it is valid as long as the anion-receptor
interactions are relatively weak.
Ion hopping over a barrier (transition-state theory or Eyring rate theory) is more
familiar to a chemist. In the particular case, it becomes the most relevant if there is a
second barrier associated with an anion release from the complex, or if we consider fixed
binding site(s) inside the bilayer as an ion moves along an ion channel. Chemical forces
strongly alter the energy profile along the way and so the movement across is not a
simple electrodiffusion, or using formalism of the partitioning-electrodiffusion model
described above we may say that the diffusion coefficient is no longer a constant.
Let’s consider passage over a single barrier. In passing from one state to another
an activation energy barrier ΔG# must be surmounted. The rate constant, k, is proportional
to the fraction of molecules having the requisite energy, and accordingly is set equal to
koexp(-ΔG#/RT). The exponential can be viewed as the probability of surmounting the
energy barrier at given temperature, and ko as the frequency to attempts per unit time.
ΔG# represents the work done in moving from a local energy minima to the peak. If the
molecule is and ion and there is a potential difference, E ≠ 0, there are two parts of the
energy barrier, chemical, ΔG#, and electrical, ΔE#.
We assume a constant electric field, i.e. – dψ/dx = constant. If x = δ then ψ = E (Figure
S2). As the ion hops from one local minimum to another, along the x axis, the gain (loss)
in the electrical energy can be expressed as zFΔψ, where Δψ - the potential difference, to
be surmounted, and zF - the charge carried by the ion.
Let’s consider a symmetrical case, where the rates for the reversed and forward processes
are exactly the same. For instance, there is a single vacancy in an ionic crystal so as an
ion hops between the two identical sites in the lattice. If this crystal is placed in the
electric field, or charges are applied to its surface, the two sites are no longer the same but
there is zFΔψ difference. In turn, the activation energy of the transition can be expressed
as ΔG# + zFΔψ/2 for the forward and ΔG# - zFΔψ/2 for the reversed processes accordingly
and the rate will be written as following:
-dCIN /dt = koexp(-ΔG#/RT)exp(-zFΔψ/2RT)[C] IN -koexp(-ΔG#/RT)exp(zFΔψ/2RT)[C] OUT=
= k (exp(-zFΔψ/2RT) [C] IN - exp(zFΔψ/2RT) [C] OUT)
In turn, for instance, in the case depicted in Figure S6 the formalism of the
transition-state theory will be written as following:
-dC*/dt=k1[C] IN exp(-zFΔψ/3RT) – k2[C*] exp(-zFΔψ/6RT) – 2k2[C*] exp(zFΔψ/6RT)+
k1[C] OUT exp(zFΔψ/3RT),
-dCIN /dt = - k1[C] IN exp(-zFΔψ/3RT)+ k2[C*] exp(zFΔψ/6RT),
-dCOUT /dt = - k1[C] OUT exp(zFΔψ/3RT)+k2[C*] exp(-zFΔψ/6RT),
where k1 = ko1 exp(-ΔG1#/RT)
k2 = ko2exp(-ΔG2#/RT)
(Adapted from the references 1-3.)
A hypothetical barrier model. The ion fluxes can be calculated from the free-energy
barriers of transitions, which in turn, have two components: ‘chemical’ and electrical, δ –
electrical distance.
Upon desire or when it seems to be necessary, a combination of the two
formalisms can be used to describe the transport phenomena. Thus, the ion transport
across the bilayer can be viewed as alternating steps of the electrodiffusion (GHK theory)
and hopping over barriers (rate theory).
References:
1. Hille, B. Ionic Channels of Excitable Membranes, 3rd ed., Sinauer Associates: Sunderland, MA, 2001.
2. MCB 137 Exercises and Solutions, Lesson 6, Complete Axon chapter.
http://mcb.berkeley.edu/courses/mcb137/exercises/Axon_B104.pdf (accessed Nov. 4,
2013).
3. Yaroslavtsev AB Properties of solids through the eyes of a chemist. M. MUCTR.
Mendeleev, 1992. Ярославцев А.Б. Свойства твердых тел глазами химика. М.: РХТУ
им. Д.И.Менделеева, 1992. (Russian language)
For instance, we may consider a ‘one anion’ transporter using analogy of a
classical ‘one ion’ channel. In contrast to Goldman-Hodgkin-Katz (GHK) theory that at a
glance, describes permeation as passage over a single barrier, a ‘one anion’ transporter
includes two symmetric barriers with one energy minimum in between (Figure S2, A and
B). Therefore, it more accurately describes the behavior of the K+-valinomycin complex
or a hypothetical Cl- - ‘valinomycin for anions’ complex inside the bilayer. (It is relevant
to note that our model yet differs from the one of valinomycin, where the permeation of
the free receptor and not of the anion-receptor complex is thought to be rate-limiting.)
Being particularly interested in supramolecular chemistry of anions inside biomembrane,
we wish to study the exact opposite so permeation of anion-receptor complex and not of
the ‘unloaded’ carrier determines the rate of ion passage. We may also note that GHK
theory may yet adequately describe the system depicted in Figure S2A (weak
interactions) but not the case depicted in Figure S2B (strong interactions.)
In turn, the ‘one anion’ transporter inevitably evolves to add one, two or more
binding sites. Such systems correspond to numerous examples of the synthetic
compounds, cation- or anion-selective, where a (multi)ion hopping mechanism has been
proposed, implied, or could have been proposed. Anion-π slides, synthetic lipids that use
relay mechanism for anion translocation (S. Matile), guanosine-based Na+ transporter
(J.T. Davis), and helical peptides bearing two, three or more crown-ether units (N.
Voyer), are just a few such systems to mention.
A.
B.
C.
Figure S2. Simple barrier models. The ion fluxes can be calculated from the free-energy
barriers of transitions, which in turn, have two components: ‘chemical’ and electrical, δ –
electrical distance. Even if ‘chemical’ energy profile is symmetric as depicted the
total free energy profile would be asymmetric. A. A ‘one anion’ transporter, weak
binding. B. A ‘one anion’ transporter, strong binding. C. A (multi)ion hopping
mechanism, strong binding - two ions in the adjacent binding sites efficiently destabilize
each other, lowering the kinetic barriers.
SI2. Approximations.
Table S1. Hydration energies and permeabilities of the selected inorganic anions.
∆GHydr (Anion) kJ mol-1
-363
-328
-311
-278
-228
PAnion *109 cm s-1
2.2
7.3
19
60
116
Anion
ClBrNO3IClO4-
The literature data in Table S1 have been used to construct the dependence of PAnion- on
∆GHydr as depicted in Figure S3. This approximate relation y = 6020.5*exp (-0.0302*x)
has been used to obtain the permeability values for anions of the weak acids, N3-, SCN-,
F- and CH3COO-, tabulated in Table 1 in the article.
Figure S3. Dependence of PAnion- on ∆GHydr
Table S2. Experimental and theoretical values of the partition coefficient
(hydrophobicity) for the weak acids HAnion, Poctanol/water (reported by Walter et al)
and LogPHAnion (ACD Lab); experimental values of the permeability PHAnion, cm s-1
reported by Walter et. al.
HAnion LogPHAnion
(ACD Lab)
HF
Poct/water
(experimental)
4.2*10-6
PHAnion, cm s-1
3.1*10-4
0.429
3.2*10-1
2.6
C2H4O2 -0.322
5.3*10-4
5.0 * 10-3
6.0*10-2
2.9
6.9*10-5
9.2*10-4
HSCN
HN3
-0.068
HCl
HNO3
-0.773
The data in Table S2 have been used to construct the dependence of log PHAnion on
LogPHAnion depicted in Figure S4.
1
0.5
0
-1
-0.8
-0.6
-0.4
-0.2
-0.5
0
0.2
0.4
0.6
-1
y = 2.9471x - 0.9865
R2 = 0.969
-1.5
-2
-2.5
-3
-3.5
Figure S4. Dependence of log PHAnion on LogPHAnion
This approximate relation y = 2.9471*x - 0.9865 has been used to obtain/approximate the
permeability values for hydrazoic acids HN3, PHN3 = 6.5 *10-2 cm s-1, depicted in Table 1.
SI 3. Permeability of the ‘unstirred’ layer.
P unstirred layer can be estimated as following:
P unstirred layer = D/δ = 10-5 cm2 s-1/ 1000 nm = 10-9 m2 s-1 / 10-6 m = 10-5 cm s-1 where D =
10-5 cm2 s-1 - typical value for the diffusion coefficient in water for small molecule, δ =
1000 nm - thickness of the unstirred layer.
Therefore, if speed at which the molecule moves across the unstirred layer is much higher
than the speed at which it moves across the membrane (P unstirred layer >> P m) than
unstirred layer effect is negligible.
For instance, P m (F-) = 0.83 *10 -9 cm s-1 >> P unstirred layer
However, if speed at which the molecule moves across the unstirred layer and the speed
at which it moves across the membrane are comparable than apparent permeability
should be expected as a combination of the two phenomena.
For instance, P m (HF) = 3.1 * 10-4 > P unstirred layer
SI4. Weak vs strong acid: the pH “error” in 10 mM phosphate buffer.
Let’s imagine that we replaced all Anion1- in a 100 mM solution, 10 mM phosphate
buffer at pH = 7 with CH3COO-. (Anion1- corresponds to a strong acid.) The
concentration of CH3COOH in the medium can be obtained as following:
pH = pKa + Log[Anion2-]/[HAnion2],
pH = pKa(CH3COOH) + Log( 0.1 - [CH3COOH]/[ CH3COOH]),
7 = 4.8 + Log(0.1 – [CH3COOH]/[ CH3COOH]), which gives [CH3COOH] = 0.000627.
But if the anion of the weak acid undergoes hydrolysis, the pH of the medium should also
rise. There is 0.006131 mM of H2PO4- at pH = 7 in 10 mM buffer in Anion1- solution,
which drops to [H2PO4-]’ = 0.006131 - 0.000627 = 0.005504 mM in CH3COO-, which in
turn, corresponds to pH’ = 7.1. Therefore, we can say that the actual concentration of the
acetic acid in the solution is less than 0.015846 mM and the actual pH change is less than
0.1.
HAnion2
%
0.099999984
0.099984154
1.58489E-08
1.58464E-05
1.58E-05
0.015846
10
7
6.800000806
3.800000834
[CH3COO-]
0.1
0.099999369
0.1
0.099373
6.30952E-07
0.000627
0.000631
0.627
10
7
5.200000938
2.20000086
[F- ]
0.1
0.1
pH
Log (Anion2/HAnion2)
Anion2-
pHpKa
[H2PO4-]
[H2PO4]'
pH'
6.8
3.8
1.582E-05
0.006131
1.58E-05
0.006115
10.0004
7.00297
5.2
2.2
1.582E-05
0.006131
1.52E-05
0.005504
10.0177
7.11215
SI5. Influx of the weak acid HAnion2 OUT into liposomes. A simple model, pH ≥ 7.
Let’s consider Anion1-IN / Anion2-OUT system in the absence of an ionophore, 10 mM
phosphate buffer, HAnion2 is a weak acid and HAnion1 is a strong acid.
We have a liposomal suspension, Anion1-IN/Anion1-OUT, 10 mM phosphate buffer, pHIN =
pHOUT = 6.4 and an isoosmotic solution containing F-, 10 mM phosphate buffer, pH = 6.4.
As soon as a small aliquot of this liposomal suspension is diluted into the isoosmotic
medium containing the anion of a weak acid, the intravesicular pH drops due to the influx
of HF. Because HF is the only permeant specie, the influx of HF terminates as soon as
concentrations of it inside and out equalize. We note that 1) it takes several ms (or s) to
establish the equilibrium, 2) Em = 0 during this process because charged species do not
permeate across the bilayer on a relevant time scale.
Figure S5. The equilibrium that establishes upon dilution of liposomal suspension, 100
mM NaClIN and 100 mM NaClOUT, pH = 10 into the isoosmotic medium containing NaF
at pH = 10.
Flowchart S1.
Equations.
{Reservoirs}
d/dt (R1) = + J1
INIT R1 = 0
d/dt (R2) = - J1
INIT R2 = 0.1*0.002
d/dt (R3) = + J2
INIT R3 = 0.000005*4.3*10^(-19)*7.1*10^12
d/dt (R4) = - J2
INIT R4 = 0.000005*0.002
{Flows}
J1 = F2*2.8*10^(-14)*7.1*10^12*(F3-F1) * 10^3
J2 = J1
{Functions}
F1 = F6 * 10^(-F4)/10^(-3.2)
F2 = 0.31*10^(-5)
F3 = F7 * 10^(-F5)/10^(-3.2)
F4 = 7.2+log10((0.01-F8)/(F8))
F5 = 7.2+log10((0.01-F9)/(F9))
F6 = R1/(4.3*10^(-19)*7.1*10^12)
F7 = R2/(0.002)
F8 = R3/(4.3*10^(-19)*7.1*10^12)
F9 = R4/0.002
Equations + Explanation.
d/dt (R1) = J1
J1, mol s-1
R1 – N(F-),
– HF flow
mol – amount of the fluoride anion in liposomes
INIT R1 = 0
d/dt (R2) = - J1
R2 –
N(F-), mol – amount of the fluoride anion in the extravesicular solution
INIT R2 = 0.1*0.002
INIT R4 = [F-]OUT Initial,
M * V, L
d/dt (R3) = J2
INIT R3 = 0.000005*4.3*10^(-19)*7.1*10^12
J2, mol s-1 – ‘flow’ of [H2PO4-] into liposomes.
R3 – N(H2PO4-), mol – amount of H2PO4- anions
in liposomes
At pH = 10.5 in 10 mM phosphate buffer we have [NaH2PO4] = 0.000005, M
INIT R3 = [H2PO4-]IN initial, M
* Vliposome, L * Nliposomes
d/dt (R4) = - J2
INIT R4 = 0.000005*0.002
R4 –
N(H2PO4-), mol - amount of H2PO4- anions in the extravesicular solution
INIT R4 = [H2PO4-]OUT initial, M * Vliposome, L * Nliposomes
J1 = F2*2.8*10^(-14)*7.1*10^12*(F3-F1) * 10^3
J1 = PHF,
m s-1 * Aliposome, m2 * Nliposomes * ([HF]OUT - [HF]IN), mol L * 103
J2 = J1
J2([H2PO4-]) = J1(HF)
F2 – PHF, m s-1
F1 - [HF]IN, M
F3 – [HF]OUT, M
F2 = 0.31*10^(-5)
F1 = F6 * 10^(-F4)/10^(-3.2)
F3 = F7 * 10^(-F5)/10^(-3.2)
[HF] IN = [F-]IN * [H+]IN /Ka(HF)
[HF]OUT = [F-]OUT * [H+]OUT / Ka(HF)
F4 – pHIN
F5 – pHOUT
F4 = 7.2+log10((0.01-F8)/(F8))
F5 = 7.2+log10((0.01-F9)/(F9))
F6 – [F-]IN, M
F7 – [F-]OUT, M
F6 = R1/(4.3*10^(-19)*7.1*10^12)
F7 = R2/(0.002)
[F-]IN = N(F-)IN, M / (Vliposome, L * Nliposomes)
[F-]OUT = N(F-)OUT , mol / VOUT, L
F8 - [H2PO4-]IN, M
F9 - [H2PO4-]OUT, M
F8 = R3/(4.3*10^(-19)*7.1*10^12)
F9 = R4/0.002
[H2PO4-]IN = N(H2PO4-)IN, M / (Vliposome, L * Nliposomes)
[H2PO4-]OUT = N(H2PO4-)OUT, M / V, L
SI6. Influx of the weak acid HAnion2 OUT and H+/OH- transport.
Figure S6. The equilibrium that establishes upon dilution of liposomal suspension, 100
mM NaClIN and 100 mM NaClOUT, pH = 10 into the isoosmotic medium containing NaF
at pH = 10; H+ and OH- ions permeate on a relevant time scale.
Let’s consider the same Anion1-IN / Anion2-OUT system in the absence of an ionophore, 10
mM phosphate buffer, HAnion2 is a weak acid and HAnion1 is a strong acid. However,
here HAnion2 is not the only permeant specie but H+ and OH- ions may also permeate
across on a relevant time scale. If H+/OH- permeate fast enough, the new equilibrium is
established. In turn, if H+/OH- do not permeate fast enough, we are looking at the kinetics
as the liposome slowly gets positively charged on the outside. The Flowchart S2 that
describes this process is partly analogous to Flowchart S1. Only the difference between
the two is explained in details.
HAnion2 = HF
Flowchart S2.
Equations + Explanation.
{Reservoirs}
d/dt (R1) = + J1
INIT R1 = 0
d/dt (R2) = - J1
INIT R2 = 0.1*0.002
d/dt (R3) = + J2
INIT R3 = 0.000005*4.3*10^(-19)*7.1*10^12
d/dt (R4) = - J2
INIT R4 = 0.000005*0.002
d/dt (R6) = + J3
INIT R6 = 0
J3, mol s-1 – H+
R6 – N(H+)Excess,
flow
mol – amount of the H+ that accumulates in the extravesicular medium
due to passage of H+ ions across the bilayer
d/dt (R5) = - J4
INIT R5 = 0
J4, mol s-1 – OH- flow
R5 – N(OH-)Excess, mol –
amount of the OH- anion that accumulates in the extravesicular
medium due to passage of OH- ions across the bilayer
{Flows}
J1 = F2*2.8*10^(-14)*7.1*10^12*(F3-F1) * 10^3
J2 = J1 - J3 - J4
-
+
J2([H2PO4 ]) = J1(HF) - J3(H ) - J4(OH-)
J3 = F13*3*10^(-14)*7.1*10^12 *(10^(-F4)*F11-10^(- F5)*F12) * 10^3
J4 = F13*3*10^(-14)*7.1*10^12 *(10^(F5-14)*F11-10^(F4-14)*F12) * 10^3
J3 = PH+, m s-1 * Aliposome, m2 * Nliposomes * (10-pH OUT *F11 – 10-pH IN *F12)* 103
J4 = POH-, m s-1 * Aliposome, m2 * Nliposomes * (10(pH OUT -14) * F11– 10(pH IN -14)*F12)*
103
{Functions}
F1 = F6 * 10^(-F4)/10^(-3.2)
F2 = 0.31*10^(-5)
F3 = F7 * 10^(-F5)/10^(-3.2)
F4 = 7.2+log10((0.01-F8)/(F8))
F5 = 7.2+log10((0.01-F9)/(F9))
F6 = R1/(4.3*10^(-19)*7.1*10^12)
F7 = R2/(0.002)
F8 = R3/(4.3*10^(-19)*7.1*10^12)
F9 = R4/0.002
F10 = (F16-F14) *96480/(8.314*298)
F10 – U –
reduced membrane potential
U = Em, V * F, Coulomb mol -1 /(R, J mol-1 K-1 * T, K), where Em, V = F16 – F 14
F11 = IF F10 >0.01 OR F10 <-0.01 then F10/(1-2.72^(-F10)) else 1/(1+F10/2+F10^2/6)
F12 = F11*2.72^(-F10)
F13 = 10^(-8)
F17 = 4*10^(-11)
F13 – PH+ = 10-8, m s-1
F17 – POH- = 4*10-11, m s-1
F15 = F16 – F 14
F15 –
Em, V = EH+, V – EOH-, V
F14 = R6/(7.1*10^12) *96480/(10^(-2)*3*10^(-14))
F14 - EH+,
V = N(H+)Excess/Nliposomes * F, Coulomb mol -1/(C, Coulombs V-1 m-2 * A, m2)
F16 = R5/(7.1*10^12)*96480/(10^(-2)*3*10^(-14))
F16 - EOH-,
V = N(OH-)Excess/Nliposomes * F, Coulomb mol -1/(C, Coulombs V-1 m-2 * A, m2)
SI7. Anion1 IN / Anion2 OUT exchange, HAnion1 IN - is a strong acid, HAnion2 OUT is a weak acid.
Anion1 IN / F-OUT exchange, HAnion1 - is a strong acid.
Flowchart S3.
Flowchart S3 is largely analogous to Flowchart S2. Only the difference between the two
is explained in details.
Equations + Explanation.
{Reservoirs}
d/dt (R1) = + J1
INIT R1 = 0
d/dt (R2) = - J1
INIT R2 = 0.1*0.002
d/dt (R3) = + J2
INIT R3 = 0.006815*4.3*10^(-19)*7.1*10^12
d/dt (R4) = - J2
INIT R4 = 0.0008182*0.00204
INIT R4(3) = [H2PO4-]OUT(IN) initial,
M * Vliposome, L * Nliposomes
At pH = 6.87 in 10 mM phosphate buffer we have [NaH2PO4] = 0.006815, M
At pH = 8.25 in 10 mM phosphate buffer we have [NaH2PO4] = 0.0008182, M
d/dt (R6) = + J3
INIT R6 = 0
d/dt (R5) = - J4
INIT R5 = 0
d/dt (R7) = - J5
INIT R7 = 0.1*4.3*10^(-19)*7.1*10^12
R7- N(Anion1-),
mol – amount of Anion1 in liposomes.
d/dt (R8) = + J5
INIT R8 = 0.1*0.04*10^(-3)
N(Anion1-), mol – amount of Anion1 in the extravesicular solution (40 μL of the
liposomal suspension 100 mM NaAnion IN/NaAnion OUT is injected into 2 ml solution)
R8 -
d/dt (R9) = + J6
INIT R9 = 0
N(F-)anionophoretic, mol – amount of the F- anion in liposomes that accumulates using
aniophoretic pathway
R9 -
{Flows}
J1 = F2*2.8*10^(-14)*7.1*10^12*(F3-F1) * 10^3+J6
mol s-1 – total flow of F- anion into liposome as a combination of the
electronetutral (HF) and electrogenic (J6 (F-)) pathways.
J1 (HF +F-),
J2 = (J1-J3-J4-J6)
J2([H2PO4-]) = J1(HF+F-) - J3(H+) - J4(OH-) – J6(F-)
J3 = F13*3*10^(-14)*7.1*10^12 *(10^(-F4)*F11-10^(- F5)*F12) * 10^3
J4 = F17*3*10^(-14)*7.1*10^12 *(10^(F5-14)*F11-10^(F4-14)*F12) * 10^3
J5 = F18*3*10^(-14)*7.1*10^12*(F19*F12-F20*F11) * 10^3
J6 = F21*3*10^(-14)*7.1*10^12*(F7*F11-F6*F12) * 10^3
J5 = PAnion-,
m s-1 * Aliposome, m2 * Nliposomes * ([Anion1-]IN, M*F12 - [Anion1-]OUT, M
*F11)*103
J6 = PF-,
m s-1 * Aliposome, m2 * Nliposomes * ([F-]OUT, M*F11 – [F-]IN, M *F12)*103
F1 = F6 * 10^(-F4)/10^(-3.2)
F2 = 0.31*10^(-5)
F3 = F7 * 10^(-F5)/10^(-3.2)
F4 = 7.2+log10((0.01-F8)/(F8))
F5 = 7.2+log10((0.01-F9)/(F9))
F6 = R1/(4.3*10^(-19)*7.1*10^12)
F7 = R2/(0.00204)
F8 = R3/(4.3*10^(-19)*7.1*10^12)
F9 = R4/0.00204
F10 = (F16-F14+F23-F22)*96480 /(8.314*298)
Em, V = F16 - F14 + F23 - F22 = EH+ – EOH- + EAnion- – EFF11 = IF F10 >0.01 OR F10 <-0.01 then F10/(1-2.72^(-F10)) else 1/(1+F10/2+F10^2/6)
F12 = F11*2.72^(-F10)
F13 = 10^(-3)
F14 = R6/(7.1*10^12)*96480/(10^(-2)*3*10^(-14))
F15 = (F16-F14)
F16 = R5/(7.1*10^12)*96480/(10^(-2)*3*10^(-14))
F17 = 4*10^(-11)
F18 =
F18 - PAnion1- = , m s-1
F19 = R7/(4.3*10^(-19)*7.1*10^12)
F20 = R8/(2.04*10^(-3))
F19 – [Anion1-]IN, M
F20 – [Anion1-]OUT, M
F21 = 0.83*10^(-11)
F21 - PF- = 0.83
*10-11, m s-1
F22 = R9/(7.1*10^12)*96480/(10^(-2)*3*10^(-14))
F23 = (0.1*4.3*10^(-19) - R7/(7.1*10^12))*96480/(10^(-2)*3*10^(-14))
F22 = EF-,V
= N(F-)anionophoretic/Nliposomes * F, Coulomb mol -1/(C,Coulombs V-1 m-2 *A, m2)
= ([Anion1-]IN initial * VLiposome, L - (N(Anion1-)IN/Nliposomes)) * F, Coulomb
mol -1/(C, Coulombs V-1 m-2 * A, m2)
F23 = EAnion1-,V
SI8. ‘Presumable OH-/Cl- exchange’.
Figure S7. Presumable OH-/Cl- exchange.
Here we consider a classical experiment (‘presumable Cl-/OH- exchange’) where the
anion gradient as well as the pH gradient are imposed. We ignore the electrogenic nature
of the phenomena, and the goal is to consider and compare just the initial fluxes of the
anions in solution.
If PAnion- is in a range about (2.2-116)*10-11 m s-1, the ion transport will proceed on a
relevant time scale. Because hydration energies for Cl- and OH- are similar we could have
assigned POH- ≈ PCl- = 2.2*10-11 m s-1. However, because reported values POH- for pure
phospholipid bilayer (without anionophore) equal 10-9 – 4*10-11 m s-1 we use them
instead.
J(Anion-)
= PAnion-, m s-1 * Aliposome, m2 * Nliposomes * ([Anion-]OUT, mol L-1) * 103 =
= 116 * 10-11 * 3 *10-14 * 7.1*1012 * 0.1 *103 = 2.47*10-8 mol s-1
J(Cl-) = PCl- *
Aliposome * Nliposomes * ([Cl-]IN, mol L-1) * 103 =
= 2.2 * 10-11 * 3 *10-14 * 7.1*1012 * 0.1 *103 = 4.69*10-10 mol s-1
J(OH-) = POH- *
Aliposome * Nliposomes * ([OH-]OUT , mol L-1) * 103 =
= 4 * 10-11 * 3 *10-14 * 7.1*1012 * 10-4 *103 = 8.5 * 10-13 mol s-1
or = 10-9 * 3 *10-14 * 7.1*1012 * 10-4 *103 = 2 * 10-11 mol s-1
Therefore, even if we assume the pHOUT = 10 and use the largest reported value POH- =109
m s-1, the flux of OH- ion is way below the flux rate for the least permeant inorganic
anions of the salts.
An interested researcher can also take a look at the phosphate buffer, knowing that
concentration of H2PO4- + HPO42- is only 10 mM and hydration energy follows an order
Cl- < CH3COO- < H2PO4-/ HPO42- .