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.     

General continuum analysis of transport through pores. II. Nonuniform pores.

A general continuum derivation of the nonelectrolyte (Js) and volume (Jv) flux through a pore whose cross section is a function of axial position (nonuniform) is given. In general, the flux equations cannot be reduced to the same form as for a uniform pore and it is not possible to characterize the pore kinetics by three constants as in the uniform pore case. However, it is shown that under certain conditions, the nonuniform pore equations can be approximated by the uniform pore form and can be characterized by three constants (omega, sigma, Lp). The only condition needed to reduce the Jv equation to the uniform form is that the solution be dilute. The deviation of the Js equation from the uniform form is characterized by an asymmetrical function of Jv whose maximum value is estimated. It is shown that the maximum posible fractional deviation of the Js equation from the uniform form is given by the parameter: 0:5sigmaJv/omegaRT. Since this parameter is less then 0.15 for most membrane studies, the nonuniform Js equation can usually be approximated by the uniform pore form. The general results are illustrated by explicit calculations on several models of nonuniform pores. It is shown, for example, that the "equivalent pore radius" defined in the usual way is a function of the experimental parameter that is measured and is not unique.     

The gramicidin a channel: A review of its permeability characteristics with special reference to the single-file aspect of transport

Summary Gramicidin A forms univalent cation-selective channels of 4 Å diameter in phospholipid bilayer membranes. The transport of ions and water throughout most of the channel length is by a singlefile process; that is, cations and water molecules cannot pass each other within the channel. The implications of this single-file mode of transport for ion movement are considered. In particular, we show that there is no significant electrostatic barrier to ion movement between the energy wells at the two ends of the channel. The rate of ion translocation (e.g., Na⁺ or Cs⁺) through the channel between these wells is limited by the necessity for an ion to move six water molecules in single file along with it; this also limits the maximum possible value for channel conductance. At all attainable concentrations of NaCl, the gramicidin A channel never contains more than one sodium ion, whereas even at 0.1M CsCl, some channels contain two cesium ions. There is no necessity to postulate more than two ion-binding sites in the channel or occupancy of the channel by more than two ions at any time.     

Unidirectional fluxes in saturated single-file pores of biological and artificial membranes. II. Asymptotic behavior at high degrees of saturation

A rate-theory analysis of isotope fluxes in filled single-file pores is given. It is assumed that the relation between the activities in solution and the fluxes can be described by the asymptotic laws for high activities. Only pore states containing no more than two vacancies need be taken into account. The flux-ratio exponent n, which is obtained by representing the ratio of unidirectional fluxes as a power of the electrochemical activity ratio, is found to provide information about m, the number of sites per pore. The general relation between n and m, for m ≥ 4, is m ? 3 ≤ n ≤ m ?1. For m = 2 or 3 the minimum value is 1. The theoretical results are discussed with respect to the internal structure of gramicidin pores in artificial bilayer membranes.     

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.     

Brains of Vespertilionids II. Vespertilioninae with special reference to Tylonycteris

As shown in Part I, the Vespertilioninae have on the average the lowest encephalization index (EI) of all the Vespertilionid subfamilies available, and the average size indices (Sis) of most of their brain parts are also lowest. There are, however, clear differences between the genera. The highest indices for the total brain and for many brain parts (OBL, DIE, TEL, PAL, SEP, STR, SCH) were found in Myotis, the highest Sis for NEG and MES in Scotophilus, for CER in Lasiurus, for BOL in Rhogeessa, and for HIP in Cbalinolobus. The lowest values for all brain parts except BOL were found in Tylonycteris (for BOL in Glauconycteris). The average EI of the flat-headed bamboo bats Tylonycteris pachypus and T. robustula was 60, i. e., ²/₅ less than that of the non-Tylonycteris Vespertilionids, which, as the reference group, have an average EI of 100. The brain size reduction may well be related to the adaptation to extreme flat-headedness. The amount of reduction in the various brain parts differs: it is strongest (about ¹/₂) in higher but more dispensable brain parts (STR, HIP, NEO) and distinctly lower (about ¹/₄) in structures closely involved in the fundamental vegetative functions (OBL, MES). Genera with conservative skull characteristics may have derivative characteristics of the brains, and vice versa.     

Mechanisms for induction of drinking with special reference to angiotensin II

The Japanese quail drinks water vigorously after water deprivation, haemorrhage and administration of hypertonic saline solution. Most avian species responded to angiotensin II (AII) by drinking, but carnivorous birds and those originating in arid regions were insensitive. The receptive sites for AII were the subfornical organ (SFO) and the preoptic area (POA) in the Japanese quail. Catecholaminergic fibers proceed from the POA to the SFO. Dipsogenic information generated by AII at the POA is transferred to the SFO through the catecholaminergic nerve fibres. Plasma AII increased following dehydration and haemorrhage and returned to a normal level immediately after rehydration. Following dehydration, arginine vasotocin, aldosterone and corticosterone increased in plasma as well as AII. A single intraperitoneal injection of AII induced increases of arginine vasotocin, aldosterone and corticosterone in plasma. It seems that AII functions as a trigger for release of these other hormones during dehydration.     

Electrostatic modeling of ion pores. II. Effects attributable to the membrane dipole potential.

This paper presents calculations of the shielded dipole potential in the interior of a pore piercing a lipid membrane that is at a potential V0 with respect to the aqueous solution. Except in the case of long narrow pores, there is substantial shielding of the membrane dipole potential. The associated dipole field never extends a significant distance into the aqueous region. The fact that the single-channel conductance of gramicidin B is only twice as large in glyceryl monooleate membranes as in phosphatidyl choline (PC) membranes, even though PC is approximately 120 mV more positive with respect to water, is interpreted in terms of the potential energy profile calculated for a gramicidin-like channel. It is demonstrated that the membrane dipole potential can significantly affect channel conductance only if the pore is narrow and if the peak in the potential energy profile occurs in the pore interior.     

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.     

A model proposed for the process of evolution with special reference to plants

Summary A spiral running along the surface of a cone standing on top is proposed as a model for evolution, the progress of which is considered as composed of a linear progression following the direction of time and representing the linear increase in the number of taxa, while a circular component stands for the ever recurring functional types of organisms.     

Physiological analysis on oscillatory behavior of glucose-insulin regulation by model with delays

Despite temporally forced transmission driving many infectious diseases, analytical insight into its role when combined with stochastic disease processes and non-linear transmission has received little attention. During disease outbreaks, however, the absence of saturation effects early on in well-mixed populations mean that epidemic models may be linearised and we can calculate outbreak properties, including the effects of temporal forcing on fade-out, disease emergence and system dynamics, via analysis of the associated master equations. The approach is illustrated for the unforced and forced SIR and SEIR epidemic models. We demonstrate that in unforced models, initial conditions (and any uncertainty therein) play a stronger role in driving outbreak properties than the basic reproduction number R₀, while the same properties are highly sensitive to small amplitude temporal forcing, particularly when R₀ is small. Although illustrated for the SIR and SEIR models, the master equation framework may be applied to more realistic models, although analytical intractability scales rapidly with increasing system dimensionality. One application of these methods is obtaining a better understanding of the rate at which vector-borne and waterborne infectious diseases invade new regions given variability in environmental drivers, a particularly important question when addressing potential shifts in the global distribution and intensity of infectious diseases under climate change.     

Ultrastructure of the anal organ of Drosophila larva with reference to ion transport

The ultrastructure of the anal organ of the full-grown larva of Drosophila melanogaster is described. The thin cuticle is characterized by epicuticular depressions which contain particulate material. In AgNO(3)-treated larvae, silver grains tend to penetrate the cuticle at the epicuticular depressions. At the basal surface, the epithelial cells exhibit narrow, parallel membrane infoldings which bear a particulate coat on the cytoplasmic surface. The infoldings are also attached around the cytoplasmic surface of endocuticular tubercles, thereby greatly increasing the absorptive surface area. At the apical surface, the membrane invaginations, which are closely associated with mitochondria, anastomose freely and extend deeply into the cytoplasm. The lateral membranes are linked by desmosomes and septate junctions. They are highly folded, are closely associated with mitochondria, and enclose intercellular channels and spaces. The epithelial cells are rich in mitochondria, glycogen particles and tracheoles. Numerous vesicles, multivesicular bodies, lysosome-like dense bodies and sparse endoplasmic reticulum are found in the cytoplasm. In concentrated medium, the epithelial cells show complete absence of the membrane infoldings and invaginations and reduction in the number of mitochondria. The ultrastructural features of the anal organ are consistent with its function in ion transport.     

Semiconductor theory of ion transport in thin lipid membranes: II. Surface recombination

Following the theory of surface recombination in semiconductors, we have derived an expression for the rate of ion recombination at the membrane surface. The surface recombination rate is used in the boundary conditions of current flows at the interfaces. Expressions for the ion fluxes are then derived as functions of environmental variables and membrane parameters. Our analysis strongly suggests that the ion flow through a thin lipid membrane consists of two major components: the surface barrier jumping current and the surface recombination current that are controlled decisively by surface barrier height, surface trap density and surface recombination rates. These general formulations are useful not only for the calculation but also for the understanding of ion transport in thin lipid membranes under a variety of experimental conditions. The implications of this theory to biological membranes and its possible extensions are discussed.