Theory of single-file noise

A general theoretical approach to the analysis of electric fluctuations generated by the so-called single-file diffusion through narrow channels is presented. The formalism is a slight extension of an approach to electric fluctuations in discrete transport systems with negligible interactions between the particles recently developed by one of the authors. In the single-file transport mechanism interactions between the particles must be taken into account. Three main results of principal interest are: (a) the electric fluctuations around stationary states (at equilibrium and non-equilibrium) are determined by the time-dependent solutions of the macroscopic single-file transport equations, (b) as a direct consequence of the interactions between the ions in the single-file transport the macroscopic time-dependent current and the autocorrelation function of the microscopic current fluctuations can exhibit damped oscillatory behavior, and the current noise spectrum can show peaking, (c) the number of binding sites for the ions within the pores seems to have a strong influence on the oscillatory behavior: with increasing number of binding sites the damping of the oscillations decreases and the peaking of the spectrum becomes stronger.     

Rate theory models for ion transport through rigid pores. II. Time-dependent analysis of the single-file model with special reference to oscillatory behavior

This article deals with the time-dependent evolution of the single-file movement of ions through channels of both biological and artificial membranes. The single-file transport process may exhibit not only the usual relaxation behaviour but also oscillatory behaviour as a steady state is approached after an initial perturbation. A necessary condition for the occurrence of oscillations is that the system acts sufficiently far from equilibrium. The occurrence of oscillations is due to the interactions within the transport system which are taken into account by the single-file model; these are the electrostatic repulsion between the ions being transported, and the competition of the ions for the free binding sites within the pore. Information about the strength of the interactions can be obtained by measuring the damping of the transport observables (e.g. the electric current): The stronger the inter-ionic repulsion, the more apparent the oscillatory behaviour will become. Furthermore, the damping is influenced by the microscopic structure of the transport system (i.e. the energy profile of the pores). With an increasing degree of microscopicity, i.e. with a decreasing number of binding sites and an increasingly irregular pore profile, the oscillations become more damped. However, a considerable oscillatory behaviour can only be predicted for pores with both a sufficiently regular structure and a sufficiently large number of binding sites. For this class of pores, however, the measurement of the damping represents an appropriate method of gaining information which could exceed that obtainable from the usual methods of measuring stationary quantities (e.g. stationary conductance). Moreover, our goal is to explain theoretically how the oscillatory behaviour can be interpreted in terms of the order inherent in the ionic movement, which is determined by both the external and internal forces and the microscopic properties of the system.     

Current noise around steady states in discrete transport systems

Subject of this paper is the transport noise in discrete systems. The transport systems are given by a number (n) of binding sites separated by energy barriers. These binding sites may be in contact with constant outer reservoirs. The state of the system is characterized by the occupation numbers of particles (current carriers) at these binding sites. The change in time of the occupation numbers is generated by individual “jumps” of particles over the energy barriers, building up the flux matter (for charged particles: the electric current). In the limit n → ∞ continuum processes as e.g. usual diffusion are included in the transport model. The fluctuations in occupation numbers and other quantities linearly coupled to the occupation numbers may be treated with the usual master equation approach. The treatment of the fluctuations in fluxes (current) makes necessary a different theoretical approach which is presented in this paper under the assumption of vanishing interactions between the particles. This approach may be applied to a number of different transport systems in biology and physics (ion transport through porous channels in membranes, carrier mediated ion transport through membranes, jump diffusion e.g. in superionic conductors). As in the master equation approach the calculation of correlations and noise spectra may be reduced to the solution of the macroscopic equations for the occupation numbers. This result may be regarded as a generalization to non-equilibrium current fluctuations of the usual Nyquist theorem relating the current (voltage) noise spectrum in thermal equilibrium to the macroscopic frequency dependent admittance.The validity of the general approach is demonstrated by the calculation of the autocorrelation function and spectrum of current noise for a number of special examples (e.g, pores in membrances, carrier mediated ion transport).     

Theory of transport noise in membrane channels with open-closed kinetics

A theoretical approach to transport noise in kinetic systems, which has recently been developed, is applied to electric fluctuations around steady-states in membrane channels with different conductance states. The channel kinetics may be simple two state (open-closed) kinetics or more complicated. The membrane channel is considered as a sequence of binding sites separated by energy barriers over which the ions have to jump according to the usual single-file diffusion model. For simplicity the channels are assumed to act independently. In the special case of ionic movement fast compared with the channel open-closed kinetics the results agree with those derived from the usual Master equation approach to electric fluctuations in nerve membrane channels.For the simple model of channels with one binding site and two energy barries the coupling between the fluctuations coming from the open-closed kinetics and from the jump diffusion is investigated.     

Computer simulation of single-file transport

A stochastic model of single-file transport was developed as the Markov process in continuous time technique. The model was constructed using an EC-1060 computer. Unidirectional fluxes were investigated and populations of channels were correlated with flux fluctuations. The profiles of channel populations were shown to have nonlinear shapes even with the transport of nonelectrolyte (the classical diffusion approach gives linear profiles). The relationship between the paired correlation function F(AB) and the concentration of transported particles was examined. The F(AB) profile was shown to become flattened (or exponential for asymmetrical cases) at high concentrations. The concentration dependence jA/jA0 ratio were analyzed, where jA is a single-file unidirectional flux, jA0 is unidirectional flux for the case of free diffusion. An interesting "stack" phenomenon was observed for abnormal time correlations of single-file fluxes.     

Nonequilibrium voltage fluctuations in biological membranes. I. General framework of charge transport in discrete systems and related voltage noise

This paper continues our work on the theory of nonequilibrium voltage noise generated by electric transport processes in membranes. Introducing the membrane voltage as a further variable, a system of kinetic equations linearized in voltage is derived by which generally the time-dependent behaviour of charge-transport processes under varying voltage can be discussed. Using these equations, the treatment of voltage noise can be based on the usual master equation approach to steady-state fluctuations of scalar quantities. Thus, a general theoretical approach to nonequilibrium voltage noise is presented, completing our approach to current fluctuations which had been developed some years ago. It is explicitly shown that at equilibrium the approach yields agreement with the Nyquist relation, while at nonequilibrium this relation is not valid. A further general property of voltage noise is the reduction of low-frequency noise with increasing number of transport units as a consequence of the interactions via the electric field. In a second paper, the approach will be applied for a number of special transport mechanisms, such as ionic channels, carriers or electrogenic pumps.     

A multi-substrate single-file model for ion-coupled transporters.

Ion-coupled transporters are simulated by a model that differs from contemporary alternating-access schemes. Beginning with concepts derived from multi-ion pores, the model assumes that substrates (both inorganic ions and small organic molecules) hop a) between the solutions and binding sites and b) between binding sites within a single-file pore. No two substrates can simultaneously occupy the same site. Rate constants for hopping can be increased both a) when substrates in two sites attract each other into a vacant site between them and b) when substrates in adjacent sites repel each other. Hopping rate constants for charged substrates are also modified by the membrane field. For a three-site model, simulated annealing yields parameters to fit steady-state measurements of flux coupling, transport-associated currents, and charge movements for the GABA transporter GAT1. The model then accounts for some GAT1 kinetic data as well. The model also yields parameters that describe the available data for the rat 5-HT transporter and for the rabbit Na(+)-glucose transporter. The simulations show that coupled fluxes and other aspects of ion transport can be explained by a model that includes local substrate-substrate interactions but no explicit global conformational changes.     

Rate theory models for ion transport through rigid pores. I. Time-dependent analysis in the case of vanishing interactions

In this first of a series of papers concerning the theoretical analysis of rate theory models for ion transport through rigid pores, the case of vanishing interactions is investigated. "Rigidity" means that ions crossing membranes through pores see a fixed structure of the pores, not changing in time. A single pore is considered to be a sequence of (n + 1) activation barriers separated by n energy minima. The explicit analytical treatment is restricted to pores with regular internal barrier structure, including the nonequilibrium situation of an applied electric field. In this case the connection with continuum diffusion models is demonstrated by performing in the limit n leads to infinity (n = number of binding sites within the pores) the transition to continuum. Thus, from diffusion equations describing a discrete number of jumps, the corresponding diffusion-like partial differential equations and boundary conditions are generated. For regular pores, from the time dependent solutions of the discrete equations, the corresponding solutions of the continuum equations are explicitly generated. The time-dependent relaxation behaviour of the discrete model is in good agreement with the continuum model if one assumes more than two binding sites in the pores.     

Determinants of time-dependent membrane conductance. The nonrole of classical ion-membrane molecule interactions

We have examined the steady-state and time-dependent electrical properties of a model membrane system. The model assumes that the directed velocity and energy of ions moving through the membrane are determined by the applied electric field, ionic diffusion forces, and central elastic collisions between ions and membrane molecules. A simple analysis of the steady-state electrical properties of the model yields results identical with ones obtained previously using a more complex analysis procedure. The time-dependent conductance changes of the model in response to a step change in electric field strength when there is solution symmetry display three qualitative patterns dependent on the nature of the ion-membrane molecule interaction. One of the patterns of conductance change is quite similar to that observed in the sodium conductance system of a number of excitable tissues: an initial conductance rise to a maximum (activation) followed by a decay to a final steady-state value (inactivation). However, the correspondence between the time-dependent model behavior and known experimental behavior of excitable systems is only qualitative. We conclude that the classical ion-membrane molecule interactions we consider are not involved in determining time-dependent conductance processes in the excitable systems for which comparison is possible.     

Light scattering spectroscopy of the squid axon membrane

Light scattering studies on the giant squid axon were done using the technique of optical mixing spectroscopy. This experimental approach is based on the use of laser light to detect the fluctuations of membrane macromolecules which are associated with conductance fluctuations. The light scattering spectra were similar to the Lorentzian-like behavior of conductance fluctuations, possibly reflecting an underlying conformational change in the specific membrane sites responsible for the potassium ion conductance. The amplitude of the spectra measured, increased when the membrane was depolarized and decreased on hyperpolarization. The spectra were fit to the sum of two terms, a (1/fcomponent and a simple Lorentzian term. Spectra from deteriorating axons did not show sensitivity to membrane potential changes. It is shown theoretically that fluctuations due to the voltage-dependent variable, n, of the Hodgkin-Huxley formalism are identical to the voltage fluctuations. The derived power spectrum is that of a second order system, capable of showing resonance peaking only if the voltage dependence of the potassium rate constants is included in the analysis. The lack of resonance peaking in the observed light scattering spectra, indicates that the data are best described by a damped second order system.     

Anomalous mole fraction effect, electrostatics, and binding in ionic channels.

Ionic channels bathed in mixed solutions of two permeant electrolytes often conduct less current than channels bathed in pure solutions of either. For many years, this anomalous mole fraction effect (AMFE) has been thought to occur only in single-file pores containing two or more ions at a time. Most thinking about channels incorporates this view. We show here that the AMFE arises naturally, as an electrostatic consequence of localized ion specific binding, if the average current through a channel is described by a theory (Poisson-Nernst-Planck, PNP) that computes the average electric field from the average concentration of charges in and near the channel. The theory contains only those ion-ion interactions mediated by the mean field, and it does not enforce single filing. The AMFE is predicted by PNP over a wide range of mean concentrations of ions in the channel; for example, it is predicted when (on the average) less, or much less, than one ion is found in the channel's pore. In this treatment, the AMFE arises, in large measure, from a depletion layer produced near a region of ion-specific binding. The small excess concentration of ions in the binding region repels all nearby ions of like charge, thereby creating a depletion layer. The overall conductance of the channel arises in effect from resistors in series, one from the binding region, one from the depletion zone, and one from the unbinding region. The highest value resistor (which occurs in the depletion zone) limits the overall series conductance. Here the AMFE is not the result of single filing or multiple occupancy, and so previous views of permeation need to be revised: the presence of an AMFE does not imply that ions permeate single file through a multiply occupied pore.     

Phase-State Dependent Current Fluctuations in Pure Lipid Membranes

Current fluctuations in pure lipid membranes have been shown to occur under the influence of transmembrane electric fields (electroporation) as well as a result from structural rearrangements of the lipid bilayer during phase transition (soft perforation). We demonstrate that the ion permeability during lipid phase transition exhibits the same qualitative temperature dependence as the macroscopic heat capacity of a D15PC/DOPC vesicle suspension. Microscopic current fluctuations show distinct characteristics for each individual phase state. Although current fluctuations in the fluid phase show spikelike behavior of short timescales (∼2 ms) with a narrow amplitude distribution, the current fluctuations during lipid phase transition appear in distinct steps with timescales of ∼20 ms. We propose a theoretical explanation for the origin of timescales and permeability based on a linear relationship between lipid membrane susceptibilities and relaxation times near the phase transition.     

Dynamic ion-ion and water-ion interactions in ion channels.

The dynamic interactions among ions and water molecules in ion channels are treated based on an assumption that ions at binding sites can be knocked off by both transient entering ions and local water molecules. The theory, when applied to a single-site model K+ channel, provides solutions for super- and subsaturations, flux-ratio exponent (n') greater than 1, osmotic streaming current, activity-dependent reversal potentials, and anomalous mole-fraction behavior. The analysis predicts that: (a) the saturation may but, in general, does not follow the Michaelis-Menten relation; (b) streaming current results from imbalanced water-ion knock-off interactions; (c) n' greater than 1 even for single-site channels, but it is unlikely to exceed 1.4 unless the pore is occupied by one or more ion(s); (d) in the calculation involving two permeant ion species with similar radii, the heavier ions show higher affinity; the ion-ion knock-off dissociation from the site is more effective when two interacting ions are identical. Therefore, the "multi-ion behaviors" found in most ion channels are the consequences of dynamic ion-ion and water-ion interactions. The presence of these interactions does not require two or more binding sites in channels.     

ZnO nanoparticles modulate the ionic transport and voltage regulation of lysenin nanochannels

Background

The insufficient understanding of unintended biological impacts from nanomaterials (NMs) represents a serious impediment to their use for scientific, technological, and medical applications. While previous studies have focused on understanding nanotoxicity effects mostly resulting from cellular internalization, recent work indicates that NMs may interfere with transmembrane transport mechanisms, hence enabling contributions to nanotoxicity by affecting key biological activities dependent on transmembrane transport. In this line of inquiry, we investigated the effects of charged nanoparticles (NPs) on the transport properties of lysenin, a pore-forming toxin that shares fundamental features with ion channels such as regulation and high transport rate.

Results

The macroscopic conductance of lysenin channels greatly diminished in the presence of cationic ZnO NPs. The inhibitory effects were asymmetrical relative to the direction of the electric field and addition site, suggesting electrostatic interactions between ZnO NPs and a binding site. Similar changes in the macroscopic conductance were observed when lysenin channels were reconstituted in neutral lipid membranes, implicating protein-NP interactions as the major contributor to the reduced transport capabilities. In contrast, no inhibitory effects were observed in the presence of anionic SnO₂ NPs. Additionally, we demonstrate that inhibition of ion transport is not due to the dissolution of ZnO NPs and subsequent interactions of zinc ions with lysenin channels.

Conclusion

We conclude that electrostatic interactions between positively charged ZnO NPs and negative charges within the lysenin channels are responsible for the inhibitory effects on the transport of ions. These interactions point to a potential mechanism of cytotoxicity, which may not require NP internalization.

     

Nonequilibrium ion transport through pores

In this paper is presented an investigation of the influence of the internal structure of pores in membranes on a) the time dependent macroscopic relaxation current after a voltage jump, b) the macroscopic frequency dependent admittance and c) the microscopic current fluctuations around stationary (nonequilibrium) states. All these quantities are determined by the time dependent transport equations, which are calculated with the use of the eigenvectors and eigenvalues of the matrix of coefficients, occurring in the transport equations. Numerical calculations for channels with up to 31 barriers are presented. The treatment of the fluctuations is done with the use of a general approach to nonequilibrium transport noise recently developed by one of the authors. It is shown that the influence of the internal barrier structure as, e.g., the height of central or decentral barriers in the pores is of great complexity. Nevertheless we hope that the calculations lead to a better understanding especially of the microscopic nonequilibrium transport fluctuations in complex systems.This work has been supported by the Deutsche Forschungsgemeinschaft     

K(+) versus Na(+) ions in a K channel selectivity filter: a simulation study

Molecular dynamics simulations of a bacterial potassium channel (KcsA) embedded in a phospholipid bilayer reveal significant differences in interactions of the selectivity filter with K(+) compared with Na(+) ions. K(+) ions and water molecules within the filter undergo concerted single-file motion in which they translocate between adjacent sites within the filter on a nanosecond timescale. In contrast, Na(+) ions remain bound to sites within the filter and do not exhibit translocation on a nanosecond timescale. Furthermore, entry of a K(+) ion into the filter from the extracellular mouth is observed, whereas this does not occur for a Na(+) ion. Whereas K(+) ions prefer to sit within a cage of eight oxygen atoms of the filter, Na(+) ions prefer to interact with a ring of four oxygen atoms plus two water molecules. These differences in interactions in the selectivity filter may contribute to the selectivity of KcsA for K(+) ions (in addition to the differences in dehydration energy between K(+) and Na(+)) and the block of KcsA by internal Na(+) ions. In our simulations the selectivity filter exhibits significant flexibility in response to changes in ion/protein interactions, with a somewhat greater distortion induced by Na(+) than by K(+) ions.     

Predicting nonspecific ion binding using DelPhi

Ions are an important component of the cell and affect the corresponding biological macromolecules either via direct binding or as a screening ion cloud. Although some ion binding is highly specific and frequently associated with the function of the macromolecule, other ions bind to the protein surface nonspecifically, presumably because the electrostatic attraction is strong enough to immobilize them. Here, we test such a scenario and demonstrate that experimentally identified surface-bound ions are located at a potential that facilitates binding, which indicates that the major driving force is the electrostatics. Without taking into consideration geometrical factors and structural fluctuations, we show that ions tend to be bound onto the protein surface at positions with strong potential but with polarity opposite to that of the ion. This observation is used to develop a method that uses a DelPhi-calculated potential map in conjunction with an in-house-developed clustering algorithm to predict nonspecific ion-binding sites. Although this approach distinguishes only the polarity of the ions, and not their chemical nature, it can predict nonspecific binding of positively or negatively charged ions with acceptable accuracy. One can use the predictions in the Poisson-Boltzmann approach by placing explicit ions in the predicted positions, which in turn will reduce the magnitude of the local potential and extend the limits of the Poisson-Boltzmann equation. In addition, one can use this approach to place the desired number of ions before conducting molecular-dynamics simulations to neutralize the net charge of the protein, because it was shown to perform better than standard screened Coulomb canned routines, or to predict ion-binding sites in proteins. This latter is especially true for proteins that are involved in ion transport, because such ions are loosely bound and very difficult to detect experimentally.     

Open channel noise. V. Fluctuating barriers to ion entry in gramicidin A channels.

We have measured the fluctuations in the current through gramicidin A (GA) channels in symmetrical solutions of monovalent cations of various concentrations, and compared the spectral density values with those computed using E. Frehland's theory for noise in discrete transport systems (Frehland, E. 1978. Biophys. Chem. 8:255-265). The noise for the transport of NH4+ and Na+ ions in glycerol-monooleate/squalene membranes could be accounted for entirely by "shot noise" in the process of transport through a single-filing pore with two ion binding sites. However, in confirmation of results in a previous paper (Sigworth, F. J., D. W. Urry, and K. U. Prasad. 1987. Biophys. J. 52:1055-1064) currents of Cs+ showed a substantial excess noise at low ion concentrations, as did currents of K+ and Rb+. The excess noise was increased in thicker membranes. The observations are accounted for by a theory that postulates fluctuations of the entry rates of ions into the channel on a time scale of approximately 1 microsecond. These fluctuations occur preferentially when the channel is empty; the presence of bound ions stabilizes the "high conductance" conformation of the channel. The fluctuations are sensed to different degrees by the various ion species, and their kinetics depend on membrane thickness.     

Current fluctuations in discrete transport systems far from equilibrium. Breakdown of the fluctuation dissipation theorem

A recently developed theoretical approach to transport fluctuations around stable steady states in discrete biological transport systems is used in order to investigate general fluctuation properties at nonequilibrium. An expression for the complex frequency dependent admittance at nonequilibrium is derived by calculation of the linear current response of the transport systems to small disturbances in the applied external voltage. It is shown that the Nyquist or fluctuation dissipation theorem, by which at equilibrium the macroscopic admittance or linear response can be expressed in terms of fluctuation properties of the system, breaks down at nonequilibrium. The spectral density of current fluctuations is decomposed into one term containing the macroscopic admittance and a second term which is bilinear in current. This second term is generated by microscopic disturbances, which cannot be excited by external macroscopic perturbations. At special examples it is demonstrated that this second term is decisive for the occurrence of excess noise e.g. the 1/f(2)-Iorentzian noise generated by the opening and closing of nerve channels in biological membranes.     

Sodium flux ratio in Na/K pump-channels opened by palytoxin

Palytoxin binds to Na(+)/K(+) pumps in the plasma membrane of animal cells and opens an electrodiffusive cation pathway through the pumps. We investigated properties of the palytoxin-opened channels by recording macroscopic and microscopic currents in cell bodies of neurons from the giant fiber lobe, and by simultaneously measuring net current and (22)Na(+) efflux in voltage-clamped, internally dialyzed giant axons of the squid Loligo pealei. The conductance of single palytoxin-bound "pump-channels" in outside-out patches was approximately 7 pS in symmetrical 500 mM [Na(+)], comparable to findings in other cells. In these high-[Na(+)], K(+)-free solutions, with 5 mM cytoplasmic [ATP], the K(0.5) for palytoxin action was approximately 70 pM. The pump-channels were approximately 40-50 times less permeable to N-methyl-d-glucamine (NMG(+)) than to Na(+). The reversal potential of palytoxin-elicited current under biionic conditions, with the same concentration of a different permeant cation on each side of the membrane, was independent of the concentration of those ions over the range 55-550 mM. In giant axons, the Ussing flux ratio exponent (n') for Na(+) movements through palytoxin-bound pump-channels, over a 100-400 mM range of external [Na(+)] and 0 to -40 mV range of membrane potentials, averaged 1.05 +/- 0.02 (n = 28). These findings are consistent with occupancy of palytoxin-bound Na(+)/K(+) pump-channels either by a single Na(+) ion or by two Na(+) ions as might be anticipated from other work; idiosyncratic constraints are needed if the two Na(+) ions occupy a single-file pore, but not if they occupy side-by-side binding sites, as observed in related structures, and if only one of the sites is readily accessible from both sides of the membrane.