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The permeation of ions through narrow water-filled channels is essential to life and of rapidly-growing importance in technology. Reaching an understanding of the mechanisms underlying the permeation process requires an interdisciplinary approach, where ideas drawn from physics are of particular importance and have brought encouraging progress in recent years. This Introduction sets into context the several ground-breaking papers presented in the Entropy Special Issue on “The Physics of Ionic Conduction in Narrow Biological and Artificial Channels.

Biological ion channels are fundamental to maintaining life. In this manuscript we apply our recently developed statistical and linear response theory to investigate Na+ conduction through the prokaryotic Na+ channel NaChBac. This work is extended theoretically by the derivation of ionic conductivity and current in an electrochemical gradient, thus enabling us to compare to a range of whole-cell data sets performed on this channel. Furthermore, we also compare the magnitudes of the currents and populations at each binding site to previously published single-channel recordings and molecular dynamics simulations respectively. In doing so, we find excellent agreement between theory and data, with predicted energy barriers at each of the four binding sites of ∼4,2.9,3.6, and 4kT.

We introduce a statistical and linear response theory of selective conduction in biological ion channels with multiple binding sites and possible point mutations. We derive an effective grand-canonical ensemble and generalised Einstein relations for the selectivity filter, assuming strongly coordinated ionic motion, and allowing for ionic Coulomb blockade. The theory agrees well with data from the KcsA K$^+$ channel and a mutant. We show that the Eisenman relations for thermodynamic selectivity follow from the condition for fast conduction and find that maximum conduction requires the binding sites to be nearly identical.

We consider the selective permeation of ions through narrow water-filled channels in the presence of strong interaction between the ions. These interactions lead to highly correlated ionic motion, which can conveniently be described via the concept of a quasiparticle. Here, we connect the quasiparticle’s effective potential and the multi-ion potential of the mean force, found through molecular dynamics simulations, and we validate the method on an analytical toy model of the KcsA channel. Possible future applications of the method to the connection between molecular dynamical calculations and the experimentally measured current-voltage and current-concentration characteristics of the channel are discussed.

We study the highly-concerted motion of ions in anarrow biological ion channel (KcsA) by considering the notionof a quasiparticle, with specific focus on the transition process.Namely, we show that the ion entering or exiting the channelis correlated with the position of the quasiparticle. This resultis of importance in the rate theories of ion conduction throughnarrow channels and artificial nanopores.

Ionic Coulomb blockade (ICB) is an electrostatic phenomenon recently discovered in low-capacitance ion channels/nanopores. Depending on the fixed charge that is present, ICB strongly and selectively influences the ease with which a given type of ion can permeate the pore. The phenomenon arises from the discreteness of the charge-carriers, the dielectric self-energy, an electrostatic exclusion principle, and sequential pore neutralization, and it manifests itself strongly for divalent ions (e.g.\ Ca$^{2+}$). Ionic Coulomb blockade is closely analogous to electronic Coulomb blockade in quantum dots. In addition to the non-local 1D Coulomb interaction considered in the standard Coulomb blockade approach, we now propose a correction to take account of the singular part of the attraction to the binding site (i.e.\ local site binding). We show that this correction leads to a geometry-dependent shift of one of the barrierless resonant conduction points M$_0^{CB}$. We also show that local ion-ion repulsion accounts for a splitting of Ca$^{2+}$ profiles observed earlier in Brownian dynamics simulations., 4 pages, 4 figures, 25 references, ICNF2017 Replacement reasons: Some typo are fixed, notation is optimised)

The conduction and selectivity of calcium/sodium ion channels are described in terms of ionic Coulomb blockade, a phenomenon based on charge discreteness, an electrostatic exclusion principle, and stochastic ion motion through the channel. This novel approach provides a unified explanation of numerous observed and modelled conductance and selectivity phenomena, including the anomalous mole fraction effect and discrete conduction bands. Ionic Coulomb blockade and resonant conduction are similar to electronic Coulomb blockade and resonant tunnelling in quantum dots. The model is equally applicable to other nanopores.

We analyse the non-equilibrium distribution in dissipative dynamical systems at finite noise intensities. The effect of finite noise is described in terms of topological changes in the pattern of optimal paths. Theoretical predictions are in good agreement with the numerical results.

Fluctuational escape from a multiwell potential is shown to display new features, as compared to the conventional single-well case. The flux J may depend on friction Gamma exponentially strongly, over an exponentially long period; for small enough temperatures, J(Gamma) undergoes marked oscillations in the range of small Gamma, and the time evolution of J changes drastically as Gamma exceeds a critical value.

We investigate, theoretically and by analog experiment, the distribution of paths for large fluctuations away from a stable state. We have found critical broadening of the distribution of the paths coming to a cusp point that represents the simplest generic singularity in the pattern of most probable (optimal) fluctuational paths in nonequilibrium systems. The critical behavior can be described by a Landau-type theory. We predict and observe two-ridged distributions for arrivals on a switching line that separates the areas reached along optimal paths of different types.

A new type of non-linear resonance is reported. It possesses an unusual counterintuitive feature, namely that: strong non-linear response to a weak periodic force can be obtained in the absence of exact resonance between frequency of the force and any eigenfrequency of the system. It is shown that the topology of phase trajectories differs substantially from the case of conventional non-linear resonance. The frequency of a driving force at which the transition between the conventional non-linear resonance and the zero-dispersion one takes place is found. Possible applications are discussed.

The effect of fluctuations on the conductivity of ion channels is investigated. It is shown that modulation of the potential barrier at the selectivity site due to electrostatic amplification of charge fluctuations at the channel mouth exerts a leading-order effect on the channel conductivity. A Brownian dynamical model of ion motion in a channel is derived that takes into account both fluctuations at the channel mouth and vibrational modes of the wall. The charge fluctuations are modeled as a short noise flipping the height of the potential barrier. The wall fluctuations are introduced as a slow vibrational mode of protein motion that modulates ion conductance both stochastically and periodically. The model is used to estimate the contribution of the electrostatic amplification of charge fluctuations to the conductivity of ion channels., Conference, UPON2008, 14 pages

Continuous and pulsed forms of control of a multistable system are compared directly, both theoretically and numerically, taking as an example the switching of a periodically-driven class-B laser between its stable and unstable pulsing regimes. It is shown that continuous control is the more energy-efficient. This result is illuminated by making use of the close correspondence that exists between the problems of energy-optimal control and the stability of a steady state.

How to find the (strongly non‐Boltzmann) distribution in a far‐from‐equilibrium system is a problem of long standing. It appears in many different contexts, with topical examples including stochastic resonance and Brownian ratchets. One of the most promising approaches to the problem is through asymptotic analysis of the Fokker‐Planck equation in the limit of small noise intensity. In simulations and experiments on real systems, however, the noise intensity is necessarily finite. Corrections to allow for finite noise intensity have recently been introduced for the particular case of escape. We are currently investigating the non‐equilibrium distribution over the whole of phase space, for two model systems: the periodically driven, overdamped, Duffing oscillator and the inverted van der Pol oscillator. A modified Monte Carlo technique is being applied to investigate the limit of very small noise intensities. The next‐to‐leading order of approximation of the solution of the Fokker‐Planck equation is used to compare the numerical results with the theory. We show, in particular, how changes in the non‐equilibrium probability distribution induced by finite noise intensity are linked to an observable modification in the pattern of optimal paths. The numerical observations are in good agreement with theory.

A numerical technique is introduced that reduces exponentially the time required for Monte Carlo simulations of non-equilibrium systems. Results for the quasi-stationary probability distribution in two model systems are compared with the asymptotically exact theory in the limit of extremely small noise intensity. Singularities of the non-equilibrium distributions are revealed by the simulations., Comment: 4 pages, 4 figures

We discuss activated escape from a metastable state of a system driven by a time-periodic force. We show that the escape probabilities can be changed very strongly even by a comparatively weak force. In a broad parameter range, the activation energy of escape depends linearly on the force amplitude. This dependence is described by the logarithmic susceptibility, which is analyzed theoretically and through analog and digital simulations. A closed-form explicit expression for the escape rate of an overdamped Brownian particle is presented and shown to be in quantitative agreement with the simulations. We also describe experiments on a Brownian particle optically trapped in a double-well potential. A suitable periodic modulation of the optical intensity breaks the spatio-temporal symmetry of an otherwise spatially symmetric system. This has allowed us to localize a particle in one of the symmetric wells. (c) 2001 American Institute of Physics.

It is demonstrated that the signal-to-noise ratio for a weak periodic signal in a superconductive loop with a Josephson junction (a superconducting quantum interference device, or SQUID) can be substantially enhanced, over a wide range of frequencies, by the addition of noise. This manifestation of zero-dispersion stochastic resonance (ZDSR) is shown to occur for a wide variety of loop parameters and signal frequencies. Unlike most earlier examples of stochastic resonance, ZDSR does not depend on fluctuational transitions between coexisting stable states. Rather, it exploits the noise-enhanced susceptibility that arises in underdamped nonlinear oscillators for which the oscillation eigenfrequency possesses one or more extrema as a function of energy. The phenomenon is investigated theoretically, and by means of analog and digital simulations. It is suggested that ZDSR could be used to enhance the sensitivity of radio-frequency SQUIDs and other SQUID-based devices. In the course of the work, two additional useful results were obtained: (a) an asymptotic expression describing ZDSR for the general case in the limit of weak dissipation; (b) a method for the numerical calculation of fluctuation spectra in bistable or multistable underdamped systems.

Kim and Lee (Phys. Rev. E 52, 473; 1995) report relativity-induced resonances in periodically driven oscillators. We comment that zero-dispersion nonlinear resonance (ZDNR) will occur in some of the systems considered, outline the physical origins of the ZDNR, and propose an explanation of a discrepancy noted by Kim and Lee between their theoretical and numerical values of the energy at the stationary stable points of Poincare sections.

The response of an overdamped system to quasi-monochromatic (high-frequency narrow band) noise (QMN) has been investigated by means of analogue electronic experiments. Measurements of the stationary and prehistory probability distributions demonstrate that large occasional, fluctuations are markedly different in character from those that occur when the system is driven by white or exponentially correlated noise. In particular, we report the first experimental observation of a switching point, a critical value of the coordinatex that separates regions which, with an overwhelming probability, are accessed by the fluctuating system along topologically different paths. The results are discussed in terms of, and shown to be consistent with, an existing theoretical description of the QMN-driven system.

The susceptibility of an overdamped Markov system fluctuating in a bistable potential of general form is obtained by analytic solution of the Fokker-Planck equation (FPE) for low noise intensities. The results are discussed in the context of the LRT theory of stochastic resonance. They go over into recent results (Gang Hu et al {\em Phys. Lett. A} {\bf 172}, 21, 1992) obtained from the FPE for the case of a symmetrical potential, and they coincide with the LRT results (Dykman et al, {\em Phys. Rev. Lett.} {\bf 65}, 2606, 1990; {\em JETP Lett} {\bf 52}, 144, 1990; {\em Phys. Rev. Lett.} {\bf 68}, 2985, 1992) obtained for the general case of bistable systems., Comment: 9 pages. No special macros. No figures

A Hamiltonian approach is introduced for the reconstruction of trajectories and models of complex stochastic dynamics from noisy measurements. The method converges even when entire trajectory components are unobservable and the parameters are unknown. It is applied to reconstruct nonlinear models of rodent–predator oscillations in Finnish Lapland and high-Arctic tundra. The projected character of noisy incomplete measurements is revealed and shown to result in a degeneracy of the likelihood function within certain null-spaces. The performance of the method is compared with that of the conventional Markov chain Monte Carlo (MCMC) technique.