Ionic Coulomb blockade

Fig. 1. Generic electrostatic and Brownian dynamics model of a channel or nanopore

ICB effects may be derived on the basis of a simplified electrostatics/Brownian dynamics model of a nanopore or of the selectivity filter of an ion channel.[8] The model represents the channel/pore as a charged hole through a water-filled protein hub embedded in the membrane. Its fixed charge ${\displaystyle Q_{\rm {f}}}$ is considered as a uniform, centrally placed, rigid ring (Fig.1). The channel is assumed to have geometrical parameters length ${\displaystyle L\approx 1}$nm and radius ${\displaystyle R\approx 0.3-0.5}$nm, allowing for the single-file movement of partially hydrated ions.

The model represents the water and protein as continuous media with dielectric constants ${\displaystyle \varepsilon _{\rm {w}}=80}$ and ${\displaystyle \varepsilon _{\rm {p}}=2-10}$ respectively. The mobile ions are described as discrete entities with valence ${\displaystyle z}$ and of radius ${\displaystyle R_{\rm {ion}}}$, moving stochastically through the pore, governed by the self-consistently coupled Poisson's electrostatic equation and Langevin stochastic equation.

The model is applicable to both cationic[9] and anionic[14] biological ion channels and to artificial nanopores.[1][3]

The  mobile ion is assumed to be partially hydrated  (typically retaining its first hydration shell [15]) and carrying charge ${\displaystyle q=ze}$ where ${\displaystyle e}$ is the elementary charge (e.g. the ${\displaystyle {\text{Ca}}^{2+}}$ ion with ${\displaystyle z=2}$). The model allows one to derive the pore and ion parameters satisfying the barrier-less permeation conditions, and to do so from basic electrostatics taking account of charge quantisation.

The potential energy ${\displaystyle E_{n}}$ of a channel/pore containing ${\displaystyle n}$ ions can be decomposed into electrostatic energy[1][2][8]${\displaystyle E_{n}^{\rm {ES}}}$ , dehydration energy,[15] ${\displaystyle E_{n}^{\rm {DH}}}$ and ion-ion local interaction energy ${\displaystyle E_{n}^{\rm {INT}}}$:

$${\displaystyle E_{n}=E_{n}^{\rm {ES}}+E_{n}^{\rm {DH}}+E_{n}^{\rm {INT}}...(E_{n}{\text{ Decomposition}})}$$

The basic ICB model makes the simplifying approximation that ${\displaystyle E_{n}=E_{n}^{ES}}$, whence:

$${\displaystyle {\begin{aligned}Q_{n}&=zen+Q_{\rm {f}}&{\text{(Excess charge)}}\\E_{n}&={\dfrac {Q_{n}^{2}}{2C_{s}}}&{\text{(Electrostatic energy)}}\\C_{s}&=4\pi \epsilon _{0}\epsilon _{w}{\dfrac {R^{2}}{L}}&{\text{(Self-capacitance)}}\end{aligned}}}$$

where ${\displaystyle Q_{n}}$ is the net charge of the pore when it contains ${\displaystyle n}$ identical ions of valence ${\displaystyle z}$, the sign of the moving ions being opposite to that of the ${\displaystyle Q_{\rm {f}}}$, ${\displaystyle C_{\rm {s}}}$ represents the electrostatic self-capacitance of the pore, and ${\displaystyle \epsilon _{0}}$ is the electric permittivity of the vacuum.

Fig.2. Resonant barrier-less conduction of ${\displaystyle {\ce {Ca^{2+}}}}$ions, with energies ${\displaystyle E}$ plotted vertically. (a) Plot of ${\displaystyle \mu _{\rm {ex}}}$as a function of fixed charge ${\displaystyle Q_{\rm {f}}/e}$ and position ${\displaystyle x}$ in the channel. At the "resonant" value of ${\displaystyle Q_{\rm {f}}/e=1}$ the transition is almost barrier-less (red cross-section). (b) Plots of ${\displaystyle \Delta E}$ (blue curve) and ${\displaystyle E_{\rm {AFF}}}$ (dashed-green) and their sum ${\displaystyle \mu _{\rm {ex}}}$ (red) against ${\displaystyle x}$ for ${\displaystyle Q_{\rm {f}}/e=1}$, showing that barrier-less conduction originates in a near-cancellation between ${\displaystyle \Delta E}$ and ${\displaystyle E_{\rm {AFF}}}$.

Thermodynamics and statistical mechanics describe systems that have variable numbers of particles via the chemical potential ${\displaystyle \mu }$, defined as Gibbs free energy ${\displaystyle G}$ per particle:[16][17]

$${\displaystyle {\begin{aligned}G_{n}&=E_{n}-TS_{n}&{\text{(Gibbs free energy)}}\\\mu _{n}&=G_{n+1}-G_{n}&{\text{(Chemical potential)}}\end{aligned}}}$$

, where ${\displaystyle G_{n}}$ is the Gibbs free energy for the system of ${\displaystyle n}$ particles. In thermal and particle equilibrium with bulk reservoirs, the entire system has a common value of chemical potential ${\displaystyle \mu =\mu _{F}}$ (the Fermi level in other contexts).[16] The free energy needed for the entry of a new ion to the channel is defined by the excess chemical potential ${\displaystyle \mu _{\rm {ex}}=\mu _{n}-\mu _{F}}$[16] which (ignoring an entropy term ) can be written as

$${\displaystyle {\begin{aligned}\mu _{\rm {ex}}&=E_{n+1}-E_{n}=\Delta E+E_{\rm {AFF}}&{\text{(Coulomb gap)}}\\\Delta E&={\frac {z^{2}e^{2}}{2C_{s}}};&{\text{(Charging energy)}}\\E_{\rm {AFF}}&={\frac {ze}{C_{s}}}(zen+Q_{\rm {f}})&{\text{(Affinity energy)}}\end{aligned}}}$$

where ${\displaystyle \Delta E}$ is the charging energy (self-energy barrier) of an incoming ion and ${\displaystyle E_{\rm {AFF}}}$is its affinity (i.e. energy of attraction to the binding site ${\displaystyle Q_{\rm {f}}}$). The difference in energy between ${\displaystyle \Delta E}$ and ${\displaystyle \Delta E_{\rm {AFF}}}$ (Fig.2.) defines the ionic energy level separation (Coulomb gap) and gives rise to most of the observed ICB effects.

In selective ion channels, the favoured ionic species passes through the channel almost at the rate of free diffusion, despite the strong affinity to the binding site. This conductivity-selectivty paradox has been explained as being a consequence of selective barrier-less conduction.[6][10][17][18] In the ICB model, this occurs when ${\displaystyle \Delta E}$ is almost exactly balanced by ${\displaystyle E_{\rm {AFF}}}$ (${\displaystyle \mu _{\rm {ex}}\approx 0}$), which happens for a particular value of ${\displaystyle Q_{\rm {f}}}$ (Fig.2.).[12] This resonant value of ${\displaystyle Q_{\rm {f}}}$ depends on the ionic properties ${\displaystyle z}$ and ${\displaystyle R_{\rm {ion}}}$ (implicitly, via the ${\displaystyle R_{\rm {ion}}}$-dependent dehydration energy [6][15]), thereby providing a basis for selectivity.

Fig.3. Ionic Coulomb blockade illustrated by BD-simulations of Ca${\displaystyle ^{2+}}$ conduction, as the fixed charge ${\displaystyle Q_{\rm {f}}}$ is varied: (a) Ca${\displaystyle ^{2+}}$ conduction bands; (b) Ca${\displaystyle ^{2+}}$ occupancy, forming a Coulomb staircase; and (c) Ground state energy (red)

The ICB model explicitly predicts an oscillatory dependence of conduction on ${\displaystyle Q_{\rm {f}}}$, with two interlaced sets of singularities associated with a sequentially increasing number of ions ${\displaystyle n=1,2,3,...}$ in the channel (Fig.3A). Electrostatic blockade points ${\displaystyle Z_{n}}$ correspond to minima in the ground state energy of the pore (Fig.3C).

$${\displaystyle E_{\rm {G}}(Q_{\rm {f}})=\min _{n}{E_{n}(Q_{\rm {f}})}\quad \quad {\text{(Ground state}})}$$

The ${\displaystyle Z_{n}}$ points (${\displaystyle \partial E_{n}/\partial Q_{\rm {f}}=0}$) are equivalent to neutralisation points[12] where ${\displaystyle Q_{n}=0}$.

Resonant conduction points ${\displaystyle M_{n}}$ correspond to the barrier-less condition: ${\textstyle \mu _{\rm {ex}}=0}$, or ${\displaystyle \Delta E\approx -E_{\rm {AFF}}}$.

The values of ${\displaystyle Z_{n}{\text{ and }}M_{n}}$[2] are given by the simple formulae

$${\displaystyle {\begin{aligned}Z_{n}&=-zen&{\text{(Electrostatic blockade)}}\\M_{n}&=-ze(n+1/2)&{\text{(Resonant conduction)}},\end{aligned}}}$$

i.e. the period of conductance oscillations in ${\displaystyle Q_{\rm {f}}}$, ${\displaystyle \Delta =|M_{n+1}-M_{n}|=|Z_{n+1}-Z_{n}|=|ze|}$.

For ${\displaystyle z=2}$, in a typical ion channel geometry, ${\displaystyle \Delta E/(k_{\rm {B}}T)\approx 20\gg 1}$, and ICB becomes strong. Consequently, plots of the BD-simulated ${\displaystyle {\ce {Ca^2+}}}$current ${\displaystyle J}$ vs ${\displaystyle Q_{\rm {f}}}$ exhibit multi-ion conduction bands - strong Coulomb blockade oscillations between minima ${\displaystyle Z_{n}}$and maxima ${\displaystyle M_{n}}$(Fig.3A)).[12]

The point ${\displaystyle Z_{0}=0}$ corresponds to an uncharged pore with ${\displaystyle Q_{\rm {f}}=0}$. Such pores are blockaded for ions of either sign.

The ICB oscillations in conductance correspond to a Coulomb staircase in the pore occupancy ${\displaystyle P_{\rm {c}}}$, with transition regions corresponding to ${\displaystyle M_{n}}$ and saturation regions corresponding to ${\displaystyle Z_{n}}$ (Fig.3B) . The shape of the staircase is described by the Fermi-Dirac (FD) distribution,[2] similarly to the Coulomb staircases of quantum dots.[5] Thus, for the ${\displaystyle 0\rightarrow 1}$ transition, the FD function is:

$${\displaystyle {\begin{aligned}P_{\rm {c}}&=\left[1+{\dfrac {1}{P_{\rm {b}}}}\exp \left({\dfrac {\mu _{\rm {ex}}}{k_{\rm {B}}T}}\right)\right]^{-1};&{\text{(Fermi-Dirac distribution)}}\\\mu _{\rm {ex}}&={\frac {ze}{C_{s}}}\left(Q_{\rm {f}}-M_{0}\right).\end{aligned}}}$$

Here ${\displaystyle \mu _{\rm {ex}}}$ is the excess chemical potential for the particular ion and ${\displaystyle P_{\rm {b}}}$ is an equivalent bulk occupancy related to pore volume. The saturated FD statistics of occupancy is equivalent to the Langmuir isotherm[19] or to Michaelis-Menten kinetics.[20]

It is the factor ${\displaystyle 1/P_{\rm {b}}}$ that gives rise to the concentration-related shift in the staircase seen in Fig.3B.

Addition of the partial excess chemical potentials ${\displaystyle \mu _{\rm {ex}}^{Y}}$ coming from different sources ${\displaystyle Y}$(including dehydration,[15] local binding,[21] volume exclusion etc.[9][17]) leads to the ICB barrier-less condition ${\displaystyle \mu _{\rm {ex}}=0}$ leads to a proper shift in the ICB resonant points ${\displaystyle M_{n}}$, described by a "shift equation" :[22][21]

$${\displaystyle \Delta M_{n}=-{\dfrac {C_{s}}{ze}}\sum _{Y}{\mu _{\rm {ex}}^{Y}}\quad \quad {\text{(Shift equation)}}}$$

i.e. the additional energy contributions ${\textstyle \mu _{\rm {ex}}^{Y}}$ lead to shifts in the resonant barrier-less point ${\displaystyle M_{0}}$.

The more important of these shifts (excess potentials) are:

•  A concentration-related shift ${\displaystyle \mu _{\rm {ex}}^{\rm {ES}}=-k_{\rm {B}}T\log(P_{\rm {b}})}$ arising from the bulk entropy[17]
•  A dehydration-related shift ${\displaystyle \mu _{\rm {ex}}^{\rm {DH}}}$, arising from partial dehydration penalty [15]
• A local binding-related shift ${\displaystyle \mu _{\rm {ex}}^{\rm {INT}}}$, coming from energy of local binding [21] and surface effects.[23]

Following its prediction based on analytic theory[1][2] and molecular dynamics simulations, experimental evidence for ICB emerged from experiments[3] on monolayer ${\displaystyle {\ce {MoS2}}}$ pierced by a single ${\displaystyle 0.6}$nm nanopore. Highly non-Ohmic conduction was observed between aqueous ionic solutions on either side of the membrane. In particular, for low voltages across the membrane, the current remained close to zero, but it rose abruptly when a threshold of about ${\displaystyle 400}$mV was exceeded. This was interpreted as complete ionic Coulomb blockade of current in the (uncharged) nanopore due to the large potential barrier at low voltages. But the application of larger voltages pulled the barrier down, producing accessible states into which transitions could occur, thus leading to conduction.

Valence selectivity is the channel's ability to discriminate between ions of different valence ${\displaystyle z}$, wherein e.g. a calcium channel favours ${\displaystyle {\text{Ca}}^{2+}}$ ions over ${\displaystyle {\text{Na}}^{+}}$ ions by a factor of up to 1000×.[24] Valence selectivity has been attributed variously to pure electrostatics,[11] or to a charge space competition mechanism,[25] or to a snug fit of the ion to ligands,[26] or to quantised dehydration.[27] In the ICB model, valence selectivity arises from electrostatics, namely from ${\displaystyle z}$-dependence of the value of ${\displaystyle Q_{\rm {f}}=M_{n}=-ze(n+1/2)}$ needed to provide for barrier-less conduction.

Correspondingly, the ICB model provides explanations of why site-directed mutations that alter ${\displaystyle Q_{\rm {f}}}$ can destroy the channel by blockading it, or can alter its selectivity from favouring ${\displaystyle {\text{Ca}}^{2+}}$ ions to favouring ${\displaystyle {\text{Na}}^{+}}$ ions, or vice versa [28].

Divalent (e.g. ${\displaystyle {\text{Ca}}^{2+}}$) blockade of monovalent (e.g. ${\displaystyle {\text{Na}}^{+}}$) currents is observed in some types of ion channels. Namely,[24] ${\displaystyle {\text{Na}}^{+}}$ ions in a pure sodium solution pass unimpeded through a calcium channel, but are blocked by tiny (nM) extracellular concentrations of ${\displaystyle {\text{Ca}}^{2+}}$ ions.[24] ICB provides a transparent explanation of both the phenomenon itself and of the Langmuir-isotherm-shape of the current vs. ${\displaystyle \log[{\text{Ca}}^{2+}]}$ attenuation curve, deriving them from the strong affinity and an FD distribution of ${\displaystyle {\ce {Ca^2+}}}$ions.[2][13] Vice versa, appearance divalent blockade presents strong evidence in favour of ICB

Similarly, ICB can account for the divalent (Iodide ${\displaystyle {\ce {I^2-}}}$) blockade that has been observed in biological chloride (${\displaystyle {\ce {Cl-}}}$)-selective channels.[14]

ICB and ECB should be considered as two versions of the same fundamental electrostatic phenomenon. Both ICB and ECB are based on charge quantisation and on the finite single-particle charging energy ${\displaystyle \Delta E}$, resulting in close similarity of the governing equations and manifestations of these closely related phenomena. Nonetheless, there are important distinctions between ICB and ECB: their similarities and differences are summarised in Table 1.

Table 1. Comparison between ICB and ECB

Property	ICB	ECB
Mobile charge carriers	cations (${\displaystyle {\text{Na}}^{+},{\text{K}}^{+},{\text{Ca}}^{2+},}$ etc...), anions (${\displaystyle {\text{Cl}}^{-},{\text{I}}^{2-},}$ etc.)	electrons (${\displaystyle e^{-}}$)
Valence of mobile charge carriers, ${\displaystyle z}$	positive (+1, +2, +3,...), negative (-1, -2...)	${\displaystyle z=-1}$
Transport engine	Classical diffusion	QM tunneling
Conductance oscillations	Yes, valence dependent	Yes
Coulomb staircase for occupancy, ${\displaystyle P_{c}}$	Yes, FD-shaped	Yes, FD-shaped

Coulomb blockade can also appear in superconductors; in such a case the free charge carriers are Cooper pairs (${\displaystyle z=-2}$) [29]

In addition, Pauli spin blockade [30] represents a special kind of Coulomb blockade, connected with Pauli exclusion principle.

Despite appearing in completely classical systems, ICB exhibits some phenomena reminiscent of quantum-mechanics (QM). They arise because the charge/entity discreteness of the ions leads to quantisation of the energy ${\displaystyle \Delta E}$ spectrum and hence to the QM-analogies:[31]

• Noise-driven diffusive motion provides for escape over barriers, comparable to QM-tunnelling in ECB.
• The particular FD shape[2] of the ${\displaystyle {\text{Ca}}^{2+}}$ occupancy vs ${\displaystyle \log {[{\text{Ca}}^{2+}]}}$ plays a significant role in the ICB explanation of the divalent blockade phenomenon.[13] The appearance of an FD distribution in the diffusion of classical particles obeying an exclusion principle, has been demonstrated rigorously.[19][32][33]