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.     

General continuum analysis of transport through pores. I. Proof of Onsager's reciprocity postulate for uniform pore.

The nonelectrolyte (Js) and volume (Jv) flux across a membrane is usually described in terms of two equations derived from the theory of irreversible thermodynamics: (see article) where delta c and delta P are the concentration and pressure difference; omega and Lp are the diffuse and hydraulic permeability; and sigma s and sigma v are the reflection coefficients. If Onsager's reciprocity postulate is assumed, it can be shown that signa s and sigma v are equal. This is an important assumption because it allows one to apply the continuum theory relationship between sigma s and the pore radius to experimental measurements of sigma v. In this paper, general continuum expressions for both the Jv (a new result) and Js equation will be derived and the equality of sigma s and sigma v proved. The proof uses only general hydrodynamic results and does not require explicit solutions for the drag coefficients or, for example, the assumption that the solute is in the center of the pore. The proof applys to arbitrarily shaped solutes and any pore whose shape is independent of axial position (uniform). In addition, new expressions for the functional dependence of omega and sigma on the pore radius are derived (including the effect of the particle lying off the pore axis). These expressions differ slightly from earlier results and are probably more accurate.     

Two-carrier models for mediated transport I. Theoretical analysis of several two-carrier models

Several possible models of two sequential and two simultaneous carriers of different affinities are theoretically analysed. Following the analysis we suggest for each model an experimental procedure capable of testing and rejecting the model.     

Two-carrier models for mediated transport. I. Theoretical analysis of several two-carrier models.

Several possible models of two sequential and two simultaneous carriers of different affinities are theoretically analysed. Following the analysis we suggest for each model an experimental procedure capable of testing and rejecting the model.     

Semiconductor theory of ion transport in thin lipid membranes: I. Potential and field distributions

In the theory as presented in this paper and the following one, we shall attempt to apply the semiconductor principles and methods to the study of ion transport in thin lipid membranes. Detailed formulations are given on the potential energy barriers at the interfaces, voltage drops in the polar and non-polar regions, and potential and field distributions in the diffuse double layer and within a charged membrane. These results will be used mainly as the boundary conditions for the solution of ion flow as to be given in the following paper. The analysis clearly indicates that the ion transport is interface-limited and is profoundly influenced by the presence of surface charges. An explanation of Na⁺ extrusion in nerve membrane is given based on the field distribution analysis. The theory also suggests that the “membrane potential” depends mainly on surface charges but not necessarily on ion permeation through the membrane.     

A lattice relaxation algorithm for three-dimensional Poisson-Nernst-Planck theory with application to ion transport through the gramicidin A channel.

A lattice relaxation algorithm is developed to solve the Poisson-Nernst-Planck (PNP) equations for ion transport through arbitrary three-dimensional volumes. Calculations of systems characterized by simple parallel plate and cylindrical pore geometries are presented in order to calibrate the accuracy of the method. A study of ion transport through gramicidin A dimer is carried out within this PNP framework. Good agreement with experimental measurements is obtained. Strengths and weaknesses of the PNP approach are discussed.     

Tests of continuum theories as models of ion channels. I. Poisson-Boltzmann theory versus Brownian dynamics

Continuum theories of electrolytes are widely used to describe physical processes in various biological systems. Although these are well-established theories in macroscopic situations, it is not clear from the outset that they should work in small systems whose dimensions are comparable to or smaller than the Debye length. Here, we test the validity of the mean-field approximation in Poisson-Boltzmann theory by comparing its predictions with those of Brownian dynamics simulations. For this purpose we use spherical and cylindrical boundaries and a catenary shape similar to that of the acetylcholine receptor channel. The interior region filled with electrolyte is assumed to have a high dielectric constant, and the exterior region representing protein a low one. Comparisons of the force on a test ion obtained with the two methods show that the shielding effect due to counterions is overestimated in Poisson-Boltzmann theory when the ion is within a Debye length of the boundary. As the ion gets closer to the boundary, the discrepancy in force grows rapidly. The implication for membrane channels, whose radii are typically smaller than the Debye length, is that Poisson-Boltzmann theory cannot be used to obtain reliable estimates of the electrostatic potential energy and force on an ion in the channel environment.     

Structure and dynamics of ion transport through gramicidin A.

Molecular dynamics calculations in which all atoms were allowed to move were performed on a water-filled ion channel of the polypeptide dimer gramicidin A (approximately 600 atoms total) in the head-to-head Urry model conformation. Comparisons were made among nine simulations in which four different ions (lithium, sodium, potassium, and cesium) were each placed at two different locations in the channel as well as a reference simulation with only water present. Each simulation lasted for 5 ps and was carried out at approximately 300 K. The structure and dynamics of the peptide and interior waters were found to depend strongly on the ion tested and upon its location along the pore. Speculations on the solution and diffusion of ions in gramicidin are offered based on the observations in our model that smaller ions tended to lie off axis and to distort the positions of the carbonyl oxygens more to achieve proper solvation and that the monomer-monomer junction was more distortable than the center of the monomer. With the potential energy surface used, the unique properties of the linear chain of interior water molecules were found to be important for optimal solvation of the various ions. Strongly correlated motions persisting over 25 A among the waters in the interior single-file column were observed.     

Low Reynolds number steady state flow through a branching network of rigid vessels: I. A mixture theory

A mixture theory has been used to formulate a theory of blood perfusion. By means of a formal averaging procedure the discrete network of microvessels is transformed into a continuum. During this procedure, the distinction between arterioles, capillaries and venules is preserved by means of an arteriovenous parameter. In this paper, two equations are derived for the case of low Reynolds number steady-state flow through a rigid vessel network: the extended Darcy equation and the continuity equation. A verification of the theory is presented, on the basis of a network analysis.     

Ion currents through pores. The roles of diffusion and external access steps in determining the currents through narrow pores.

External access steps, which may include restricted aqueous diffusion, are introduced into a kinetic model for ion transport through narrow pores. The conductance-concentration relation and the concentration dependence of the biionic permeability are calculated using two alternative assumptions: (a) access to the mouth of the pore is allowed only when no ion is within the lumen or at either mouth; (b) ions remain at the mouth only very transiently. With either assumption the concentration dependence of the fluxes is the same as in previous treatments in which all steps in access were lumped into a single process. Also as before, the biionic permeability ratio is independent of concentration so long as the lumen is never doubly occupied. For narrow pores, such as those formed by gramicidin A, the slowest external portion of the access process must occur close to the pore's mouth, and thus the region an ion must occupy to gain access is small. As a consequence, the probability of finding an ion within this region is also small. On this basis, it is argued that the second assumption is appropriate for these pores. The kinetic equations that result are identical to those used by Urban, B., S.B. Hladky, and D.A. Haydon (1980, Biochim. Biophys. Acta. 602:331-354).     

Brownian dynamics simulations of ion transport through the VDAC

It is important to gain a physical understanding of ion transport through the voltage-dependent anion channel (VDAC) because this channel provides primary permeation pathways for metabolites and electrolytes between the cytosol and mitochondria. We performed grand canonical Monte Carlo/Brownian dynamics (GCMC/BD) simulations to explore the ion transport properties of human VDAC isoform 1 (hVDAC1; PDB:2K4T) embedded in an implicit membrane. When the MD-derived, space-dependent diffusion constant was used in the GCMC/BD simulations, the current-voltage characteristics and ion number profiles inside the pore showed excellent agreement with those calculated from all-atom molecular-dynamics (MD) simulations, thereby validating the GCMC/BD approach. Of the 20 NMR models of hVDAC1 currently available, the third one (NMR03) best reproduces both experimental single-channel conductance and ion selectivity (i.e., the reversal potential). In addition, detailed analyses of the ion trajectories, one-dimensional multi-ion potential of mean force, and protein charge distribution reveal that electrostatic interactions play an important role in the channel structure and ion transport relationship. Finally, the GCMC/BD simulations of various mutants based on NMR03 show good agreement with experimental ion selectivity. The difference in ion selectivity between the wild-type and the mutants is the result of altered potential of mean force profiles that are dominated by the electrostatic interactions.     

A case of apparent active transport. I

When an experimenter determines the “internal concentration” of a substance in a cell (or cell suspension) it is in general the average concentration (quantity of substance divided by cell volume or volume of cell water) which is measured. When this concentration is less than that in the ambient medium but there is either no flow into the cell or flow from the cell into the medium, then (under the usually tacit assumption of spatial uniformity in the cell) the possibility of active transport is considered. The possibility that lack of spatial uniformity could lead to apparent active transport was early proposed by A. Bierman and later examined quantitatively by N. Rashevsky for a special case. In this paper spherical cells are treated but under quite general conditions regarding the metabolic aspects of the problem. It is shown that apparent active transport can result for a metabolite which is a reactant in one set of reactions and a product in another provided the sites of these sets of reactions are spatially separated in the cell.     

A case of apparent active transport. I

When an experimenter determines the “internal concentration” of a substance in a cell (or cell suspension) it is in general the average concentration (quantity of substance divided by cell volume or volume of cell water) which is measured. When this concentration is less than that in the ambient medium but there is either no flow into the cell or flow from the cell into the medium, then (under the usually tacit assumption of spatial uniformity in the cell) the possibility of active transport is considered. The possibility that lack of spatial uniformity could lead to apparent active transport was early proposed by A. Bierman and later examined quantitatively by N. Rashevsky for a special case. In this paper spherical cells are treated but under quite general conditions regarding the metabolic aspects of the problem. It is shown that apparent active transport can result for a metabolite which is a reactant in one set of reactions and a product in another provided the sites of these sets of reactions are spatially separated in the cell.     

Development of models of active ion transport for whole-cell modelling: cardiac sodium-potassium pump as a case study

This study presents a method for the reduction of biophysically-based kinetic models for the active transport of ions. A lumping scheme is presented which exploits the differences in timescales associated with fast and slow transitions between model states, while maintaining the thermodynamic properties of the model. The goal of this approach is to contribute to modelling of the effects of disturbances to metabolism, associated with ischaemic heart disease, on cardiac cell function.

The approach is illustrated for the sodium-potassium pump in the myocyte. The lumping scheme is applied to produce a 4-state representation from the detailed 15-state model of Läuger and Apell, Eur. Biophys. J. 13 (1986) 309, for which the principles of free energy transduction are used to link the free energy released from ATP hydrolysis (ΔG_ATP) to the transition rates between states of the model. An iterative minimisation algorithm is implemented to determine the transition rate parameters based on the model fit to experimental data. Finally, the relationship between ΔG_ATP and pump cycling direction is investigated and compared with recent experimental findings.