Ionic Coulomb blockade


Ionic Coulomb blockade is an electrostatic phenomenon that appears in ionic transport through mesoscopic electro-diffusive systems and manifests itself as oscillatory dependences of the conductance on the fixed charge in the pore.
ICB represents an ion-related counterpart of the better-known electronic Coulomb blockade that is observed in quantum dots. Both ICB and ECB arise from quantisation of the electric charge and from an electrostatic exclusion principle and they share in common a number of effects and underlying physical mechanisms. ICB provides some specific effects related to the existence of ions of different charge where integer is ion valence and is the elementary charge, in contrast to the single-valence electrons of ECB.
ICB effects appear in tiny pores whose self-capacitance is so small that the charging energy of a single ion becomes large compared to the thermal energy per particle. In such cases there is strong quantisation of the energy spectrum inside the pore, and the system may either be “blockaded” against the transportation of ions or, in the opposite extreme, it may show resonant barrier-less conduction, depending on the free energy bias coming from ,, or.
The ICB model claims that is a primary determinant of conduction and selectivity for particular ions, and the predicted oscillations in conductance and an associated Coulomb staircase of channel occupancy vs are expected to be strong effects in the cases of divalent ions or trivalent ions.
Some effects, now recognised as belonging to ICB, were discovered and considered earlier in precursor papers on electrostatics-governed conduction mechanisms in channels and nanopores.
The manifestations of ICB have been observed in water-filled sub-nanometre pores through a 2D MoS2 monolayer, revealed by Brownian dynamics simulations of calcium conductance bands in narrow channels, and account for a diversity of effects seen in biological ion channels. ICB predictions have also been confirmed by a mutation study of divalent blockade in the NaChBac bacterial channel.

Model

Generic electrostatic model of channel/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. The model represents the channel/pore as a charged hole through a water-filled protein hub embedded in the membrane. Its fixed charge is considered as a uniform, centrally placed, rigid ring. The channel is assumed to have geometrical parameters length nm and radius nm, allowing for the single-file movement of partially hydrated ions.
The model represents the water and protein as continuous media with dielectric constants and respectively. The mobile ions are described as discrete entities with valence and of radius, 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 and anionic biological ion channels and to artificial nanopores.

Electrostatics

The mobile ion is assumed to be partially hydrated and carrying charge where is the elementary charge. 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 of a channel/pore containing ions can be decomposed into electrostatic energy, dehydration energy, and ion-ion local interaction energy :
The basic ICB model makes the simplifying approximation that, whence:where is the net charge of the pore when it contains identical ions of valence, the sign of the moving ions being opposite to that of the, represents the electrostatic self-capacitance of the pore, and is the electric permittivity of the vacuum.

Resonant barrier-less conduction

Thermodynamics and statistical mechanics describe systems that have variable numbers of particles via the chemical potential, defined as Gibbs free energy per particle:, where is the Gibbs free energy for the system of particles. In thermal and particle equilibrium with bulk reservoirs, the entire system has a common value of chemical potential . The free energy needed for the entry of a new ion to the channel is defined by the excess chemical potential which can be written as where is the charging energy of an incoming ion and is its affinity. The difference in energy between and defines the ionic energy level separation 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 consequaence of selective barrier-less conduction. In the ICB model, this occurs when is almost exactly balanced by , which happens for a particular value of . This resonant value of depends on the ionic properties and , thereby providing a basis for selectivity.

Oscillations of conductance

The ICB model explicitly predicts an oscillatory dependence of conduction on, with two interlaced sets of singularities associated with a sequentially increasing number of ions in the channel.
Electrostatic blockade points correspond to minima in the ground state energy of the pore. The points are equivalent to neutralisation points where.
Resonant conduction points correspond to the barrier-less condition:, or.
The values of are given by the simple formulaei.e. the period of conductance oscillations in,.
For, in a typical ion channel geometry,, and ICB becomes strong. Consequently, plots of the BD-simulated Ca^2+current vs exhibit multi-ion conduction bands - strong Coulomb blockade oscillations between minima and maxima ).
The point corresponds to an uncharged pore with. Such pores are blockaded for ions of either sign.

Coulomb staircase

The ICB oscillations in conductance correspond to a Coulomb staircase in the pore occupancy, with transition regions corresponding to and saturation regions corresponding to . The shape of the staircase is described by the Fermi-Dirac distribution, similarly to the Coulomb staircases of quantum dots. Thus, for the transition, the FD function is: Here is the excess chemical potential for the particular ion and is an equivalent bulk occupancy related to pore volume. The saturated FD statistics of occupancy is equivalent to the Langmuir isotherm or to Michaelis-Menten kinetics.
It is the factor that gives rise to the concentration-related shift in the staircase seen in Fig.3B.

Shift of singular points

Addition of the partial excess chemical potentials coming from different sources leads to the ICB barrier-less condition leads to a proper shift in the ICB resonant points, described by a "shift equation" : i.e. the additional energy contributions lead to shifts in the resonant barrier-less point.
The more important of these shifts are:

Sub-nm MoS2pores

Following its prediction based on analytic theory and molecular dynamics simulations, experimental evidence for ICB emerged from experiments on monolayer MoS2 pierced by a single 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 mV was exceeded. This was interpreted as complete ionic Coulomb blockade of current in the 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.

In biological ion channels

The realisation that ICB could occur in biological ion channels accounted for several experimentally observed features of selectivity, including:

Valence selectivity

Valence selectivity is the channel's ability to discriminate between ions of different valence, wherein e.g. a calcium channel favours ions over ions by a factor of up to 1000×. Valence selectivity has been attributed variously to pure electrostatics,
or to a charge space competition mechanism,
or to a snug fit of the ion to ligands,
or to quantised dehydration.
In the ICB model, valence selectivity arises from electrostatics, namely from -dependence of the value of needed to provide for barrier-less conduction.
Correspondingly, the ICB model provides explanations of why site-directed mutations that alter can destroy the channel by blockading it, or can alter its selectivity from favouring ions to favouring ions, or vice versa .

Divalent blockade

Divalent blockade of monovalent currents is observed in some types of ion channels. Namely, ions in a pure sodium solution pass unimpeded through a calcium channel, but are blocked by tiny extracellular concentrations of ions. ICB provides a transparent explanation of both the phenomenon itself and of the Langmuir-isotherm-shape of the current vs. attenuation curve, deriving them from the strong affinity and an FD distribution of Ca^2+ions. Vice versa, appearance divalent blockade presents strong evidence in favour of ICB
Similarly, ICB can account for the divalent blockade that has been observed in biological chloride -selective channels.

Special features

Comparisons between ICB and ECB

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, 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.
PropertyICBECB
Mobile charge carrierscations,
anions 		electrons
Valence of mobile charge carriers,positive,
negative
Transport engineClassical diffusionQM tunneling
Conductance oscillationsYes, valence dependentYes
Coulomb staircase for occupancy,Yes, FD-shapedYes, FD-shaped

Particular cases

Coulomb blockade can also appear in superconductors; in such a case the free charge carriers are Cooper pairs
In addition, Pauli spin blockade represents a special kind of Coulomb blockade, connected with Pauli exclusion principle.

Quantum analogies

Despite appearing in completely classical systems, ICB exhibits some phenomena reminiscent of quantum-mechanics. They arise because the charge/entity discreteness of the ions leads to quantisation of the energy spectrum and hence to the QM-analogies: