The Electrical Conductance of Semipermeable Membranes: II. Unipolar Flow, Symmetric Electrolytes

A general formulation of the problem of stationary ion flow through semipermeable membranes was presented in the first paper of this series. The formalism is applied here to the evaluation of membrane conductance for the special case of unipolar ion flow between symmetric electrolytes. Thus it is assumed that the permeant ions carry one sign of charge only. Furthermore, the valences of all ions in solution on both sides of the membrane are taken to be of equal absolute magnitude. Conductance results obtained by numerical methods are presented for several representative sets of the parameters which characterize the membrane system in equilibrium. These results are discussed qualitatively with emphasis upon the contribution of particular system parameters to the non-linearities observed. Approximate analytic conductance relations, valid for high current levels, are also given.     

STUDIES ON THE PERMEABILITY OF MEMBRANES : IV. VARIATIONS OF TRANSFER NUMBERS WITH THE DRIED COLLODION MEMBRANE PRODUCED BY THE ELECTRIC CURRENT.

The transfer number of Cl in a KCl solution within the pores of a dried collodion membrane is always lower than 0.5. It depends on the concentration of the solution and decreases in general with decreasing concentration. However, the transfer number for any given KCl concentration has the significance of a definite and constant figure only when an infinitely small amount of coulombs is allowed to pass through the system. For finite durations of electric transfer experiments a polarization effect will always change the original transfer number. This polarization consists in an accumulation of the salt at the one boundary and a diminution at the other boundary of the membrane. Again, as the transfer number strongly depends on concentration, this change in concentration will bring about in its turn a gradual change in the transfer number too. It is shown under what conditions the transfer numbers for the anion as obtained by electic transfer experiments are higher or lower than the ones expected without polarization effect. Thus, by changing the character and magnitude of the force driving the ions across the membranes, and according to the history of previous treatment of the membrane, the whole character of what we may call the specific permeablity for ions of the membrane may be varied without any substantial change of the membrane itself concerning its structure, its chemical composition, or its pore size. Contemplation of the results obtained in this series of experiments in the light of the theoretical considerations just outlined has impressed us with the fallacy of speaking of the definite permeability of any type of membrane for electrolytes. The behavior of the membrane toward the passage of electrolytes depends on a variety of conditions. It may be recalled that different investigators have reported widely varying results concerning the permeability of certain physiological membranes for electrolytes. Such experiments as have been described in this paper may lead to an understanding of some of the factors responsible for such variations. We are aware that the collodion membrane in its simplicity is scarcely comparable to the extremely complicated biological membranes. Nevertheless any attempts to understand better the behavior of biological membranes may wisely begin with a study of the simplest prototypes.     

General solution for flow and stress phenomena in elastic vessles (Navier-Stokes)

A method is developed for a full nonlinear evaluation of all velocities and stresses represented in the Navier-Stokes equations and in the general stress tensor. The information required is essentially that for solution of linearized forms. The solution is analytical except for the calculation of the axial velocity, which requires computer assistance to step through time and space. The treatment of the problem, although directed towards solutions involving fluid flow in elastic vessels, is also adaptable to solid deformations (strain vs rate of strain) where the general stress tensor applies. A special case for the distorting ellipse is presented as well as a limited, spatially analytic, solution.     

Foundations of modeling in cryobiology—II: Heat and mass transport in bulk and at cell membrane and ice-liquid interfaces

Modeling coupled heat and mass transport in biological systems is critical to the understanding of cryobiology. In Part I of this series we derived the transport equation and presented a general thermodynamic derivation of the critical components needed to use the transport equation in cryobiology. Here we refine to more cryobiologically relevant instances of a double free-boundary problem with multiple species. In particular, we present the derivation of appropriate mass and heat transport constitutive equations for a system consisting of a cell or tissue with a free external boundary, surrounded by liquid media with an encroaching free solidification front. This model consists of two parts–namely, transport in the “bulk phases” away from boundaries, and interfacial transport. Here we derive the bulk and interfacial mass, energy, and momentum balance equations and present a simplification of transport within membranes to jump conditions across them. We establish the governing equations for this cell/liquid/solid system whose solution in the case of a ternary mixture is explored in Part III of this series.     

Some Results on the Tukey-Mclaughlin and Yuen Methods for Trimmed Means when Distributions are Skewed

In applied work, distributions are often highly skewed with heavy tails, and this can have disastrous consequences in terms of power when comparing groups based on means. One solution to this problem in the one-sample case is to use the TUKEY and MCLAUGHLIN (1963) method for trimmed means, while in the two-group case YUEN's (1974) method can be used. Published simulations indicate that they yield accurate confidence intervals when distributions are symmetric. Using a Cornish-Fisher expansion, this paper extends these results by describing general circumstances under which methods based on trimmed means can be expected to give more accurate confidence intervals than those based on means. The results cover both symmetric and asymmetric distributions. Simulations are also used to illustrate the accuracy of confidence intervals using trimmed means versus means.     

Steady-state electrodiffusion. Scaling, exact solution for ions of one charge, and the phase plane.

This is the first of two papers dealing with electrodiffusion theory (the Nernst-Planck equation coupled with Gauss's law) and its application to the current-voltage behavior of squid axon. New developments in the exact analysis of the steady-state electrodiffusion problem presented here include (a) a scale transformation that connects a given solution to an infinity of other solutions, suggesting the po-sibility of direct comparison of electrical data for membranes with different thicknesses and other properties; (b) a first-integral relation between the electric field and ion densities more general than analogous relations previously reported, and (c) an exact solution for the homovalent system, i.e., a membrane system permeated by various ion species of the same charge. The latter is a generalization of the known one-ion solution. The properties of the homovalent solution are investigated analytically and graphically. In particular we study the phase-plane curves, which reduce to the parabolas discussed by K. S. Cole in the special case in which the current-density parameter (a linear combination of the ionic current densities) is zero.     

A Physical Interpretation of the Phenomenological Coefficients of Membrane Permeability

A "translation" of the phenomenological permeability coefficients into friction and distribution coefficients amenable to physical interpretation is presented. Expressions are obtained for the solute permeability coefficient ω and the reflection coefficient σ for both non-electrolytic and electrolytic permeants. An analysis of the coefficients is given for loose membranes as well as for dense natural membranes where transport may go through capillaries or by solution in the lipoid parts of the membrane. Water diffusion and filtration and the relation between these and capillary pore radius of the membrane are discussed. For the permeation of ions through the charged membranes equations are developed for the case of zero electrical current in the membrane. The correlation of σ with ω and L_p for electrolytes resembles that for non-electrolytes. In this case ω and σ depend markedly on ion concentration and on the charge density of the membrane. The reflection coefficient may assume negative values indicating anomalous osmosis. An analysis of the phenomena of anomalous osmosis was carried out for the model of Teorell and Meyer and Sievers and the results agree with the experimental data of Loeb and of Grim and Sollner. A set of equations and reference curves are presented for the evaluation of ω and σ in the transport of polyvalent ions through charged membranes.     

Solving the minsum product rate variation problem as an assignment problem

In this paper, we consider the Minimum Product Rate Variation Problem (PRVP), which consists in sequencing parts of different types so that the sum of discrepancy functions between actual and ideal productions is minimal. Such a problem can be reduced to an Assignment problem (AP) with a matrix of a special structure. Following this approach, the efficiency of different algorithms for an AP applied to the minsum PRVP case was compared. As a result, a new algorithm that exploits specific PRVP matrix properties is presented. Computational experiments are presented, including both types of objective functions, symmetric, as described in the literature, and asymmetric. The proposed approach allows, for the first time, obtaining optimal solutions for instances of dimensions up to 10,000 parts of different types to be produced.     

A simple network thermodynamic method for series-parallel coupled flows: II. The non-linear theory, with applications to coupled solute and volume flow in a series membrane

Building on the linear network thermodynamic model presented in the first paper in this series (Mikulecky, Wiegand &; Shiner, 1977), a method of non-linear analysis is developed based mainly on the methods of Chua (1969). Using coupled salt and volume flow through membranes as an example, it is shown that the Kedem &; Katchalsky (1963a,b,c) linear model of series combinations of membranes behaves in a physically unrealistic manner (negative and infinite concentrations occur in the compartment between the membranes). Using the model introduced by Patlak, Goldstein &; Hoffman (1963) to obtain characteristics for a non-linear 2-port, the series network is well behaved and self-regulated. In the coordinate system of the linear reciprocal 2-port, the non-linear 2-port is shown to be a non-reciprocal element. Another co-ordinate system, which utilizes unidirectional fluxes (as measured by isotope fluxes, for example) and their conjugate driving forces, shows the non-linear 2-port to be separable in the sense of Li (1962) and Rosen (1968). A set of three pseudo-independent force-flow pairs completely characterizes the system and is compatible with a very simple non-linear network analysis which utilizes experimentally accessible and practical parameters such as the unidirectional solute fluxes, concentrations in the baths, volume flow and the hydrostatic minus effective osmotic pressure. As constitutive relations, the forward and backward permeabilities, which are dependent on volume flow and the filtration coefficient are all that are needed. A reflection coefficient appears in the driving force conjugate to volume flow in that the effective osmotic pressure is δΔπ and thus four independent coefficients, two of which depend on volume flow, are needed to determine the system. From the network analysis, the global properties of the series and/or parallel combinations of non-linear 2-ports are readily obtained. Although applied to a particular example, these non-linear methods are perfectly general. They are essentially the graphical form of the algebraic analysis in the linear theory, and are essentially based on the generalized version of Kirchhoff's laws used in graph and network theory. In the particular case of coupled solute and volume flow through membranes the non-linear element has a natural piecewise linearization which allows the linear theory to be applied using different values for the elements in each region.In an initial analysis, a linear network is drawn using the I-equivalent network of 1-port elements to model the linear 2-port in a series system. From this over-simplication a great deal of the qualitative behavior of the system can be visualized. For example, it is an easy way to see that solute will be accumulated or depleted in the central compartment between the membranes in an asymmetric series system. After this the non-linear analysis is used to further quantitatively describe the system's behavior and such phenomena as rectification of volume flow in the series membrane system. An appendix introduces a general theory for a class of non-linear transport phenomena.     

DONNAN EQUILIBRIUM AND THE PHYSICAL PROPERTIES OF PROTEINS : IV. VISCOSITY—Continued.

1. The proof is completed that the influence of electrolytes on the viscosity of suspensions of powdered particles of gelatin in water is similar to the influence of electrolytes on the viscosity of solutions of gelatin in water. 2. It has been suggested that the high viscosity of proteins is due to the existence of a different type of viscosity from that existing in crystalloids. It is shown that such an assumption is unnecessary and that the high viscosity of solutions of isoelectric gelatin can be accounted for quantitatively on the assumption that the relative volume of the gelatin in solution is comparatively high. 3. Since isoelectric gelatin is not ionized, the large volume cannot be due to a hydration of gelatin ions. It is suggested that this high volume of gelatin solutions is caused by the existence in the gelatin solution of submicroscopic pieces of solid gelatin occluding water, the relative quantity of which is regulated by the Donnan equilibrium. This would also explain why the influence of electrolytes on the viscosity of gelatin solutions is similar to the influence of electrolytes on the viscosity of suspensions of particles of gelatin. 4. This idea is supported by experiments on solutions and suspensions of casein chloride in which it is shown that their viscosity is chiefly due to the swelling of solid particles of casein, occluding quantities of water regulated by the Donnan equilibrium; and that the breaking up of these solid particles into smaller particles, no longer capable of swelling, diminishes the viscosity. 5. This leads to the idea that proteins form true solutions in water which in certain cases, however, contain, side by side with isolated ions and molecules, submicroscopic solid particles capable of occluding water whereby the relative volume and the viscosity of the solution is considerably increased. This accounts not only for the high order of magnitude of the viscosity of such protein solutions but also for the fact that the viscosity is influenced by electrolytes in a similar way as is the swelling of protein particles. 6. We therefore reach the conclusion that there are two sources for the viscosity of protein solutions; one due to the isolated protein ions and molecules, and the other to the submicroscopic solid particles contained in the solution. The viscosity due to the isolated molecules and ions of proteins we will call the general viscosity since it is of a similar low order of magnitude as that of crystalloids in solution; while the high viscosity due to the submicroscopic solid protein particles capable of occluding water and of swelling we will call the special viscosity of protein solutions. Under ordinary conditions of hydrogen ion concentration and temperature (and in not too high a concentration of the protein in solution) the general viscosity due to isolated ions and molecules prevails in solutions of crystalline egg albumin and in solutions of metal caseinates (where the metal is monovalent) while under the same conditions the second type of viscosity prevails in solutions of gelatin and in solutions of acid-salts of casein; and also in solutions of crystalline egg albumin at a pH below 1.0 and at higher temperatures. The special viscosity is higher in solutions of gelatin than of casein salts for the probable reason that the amount of water occluded by the submicroscopic solid gel particles in a gelatin solution is, as a rule, considerably higher than that occluded by the corresponding particles of casein.     

Der Einfluß von Elektrolyten auf Chloroplasten beim Gefrieren und Trocknen

Summary The effect of freezing, desiccation and various electrolytes on photophosphorylation, electron transport and some enzyme reactions of isolated spinach chloroplasts has been investigated. Freezing of broken chloroplasts took place at-25°C for 3 hrs; desiccation was performed at +2°C in vacuo over CaCl₂ for 3 hrs. The influence of various electrolytes during freezing or drying or during incubation of thylakoids or stroma enzymes for 3 hrs at +2°C in electrolyte solutions was determined. After treatment, the activities of a number of enzymes and enzyme systems were measured under normal conditions, e. g. in the absence of elevated electrolyte levels in a reaction medium which contained only the substrates and cofactors which are necessary for the respective enzyme reactions.Only photophosphorylation and electron transport were affected by freezing, desiccation and high concentrations of electrolytes; various soluble enzymes investigated here were not inactivated under the same conditions. In general, mild dehydration and lower concentrations of electrolytes resulted in an irreversible inactivation of ATP synthesis but did not impair ferricyanide reduction. With increasing dehydration or at higher concentrations of electrolytes the Hill reaction was also inhibited. In a certain range of dehydration and electrolyte concentration uncoupling of photophosphorylation from electron transport took place. Sugar protects the sensitive structures against the deleterious effect of both dehydration and high concentration of electrolytes.Various electrolytes affected thylakoid membranes differently. Inactivation of the membranes increased with increasing ion radius and decreasing hydration envelope of univalent or divalent cations. Divalent cations were more destructive than univalent cations. Anions did not follow these rules. Within a group of similar anions (halides or organic anions) effectivity decreased with increasing hydration envelope. On a molar basis, polyvalent anions were less effective than univalent anions. Inactivation by anions followed Hofmeister's series in seversed order. However, exceptions were observed and it appears that various ions affect the membrane in a specific manner.Inactivation of photophosphorylation and electron transport due to freezing or desiccation is identical to that due to high concentrations of electrolytes. This suggests that during dehydration due to freezing or drying the concentration of electrolytes in the remaining solution is responsible for the inactivation of the sensitive membranes.     

A simple network thermodynamic method for series-parallel coupled flows III. Application to coupled solute and volume flows through epithelial membranes

Using the network thermodynamic methods presented in the first two papers in this series (Mikulecky, Wiegand & Shiner, 1977; Mikulecky, 1977), an application to the coupled flows of salt and volume through epithelial membranes is developed. In another paper, the kidney proximal tubule is treated as a specific example (Thomas & Mikulecky,1978; Thomas, 1977). Two different experimental situations are examined in detail. The first is the case of transport across an epithelium with no external driving forces and a pump; the second is the case with transepithelial driving forces for both volume flow and solute flow but with no pump. For the linear network, the result for the most general case with a pump and transepithelial driving forces is obtained by superposition.The results obtained for the linear model lead to a number of interesting conclusions. The composition of transported fluid is independent of pump rate but does depend on the characteristics of the various membranes in the epithelium. The link between the genetic determination of epithelial membranes and the tonicity of blood is discussed. The absence of a need for a “standing gradient” in the interstitial space is clearly demonstrated, although the actual tonicity of that space can play a significant role, as is shown in detail in the work on the kidney proximal tubule. The topological aspects of the experimental design affect the manifestation of Onsager reciprocity in different parts of the system. Also, much of the epithelial function can be seen in terms of simple network concepts, such as the current divider properties of parallel pathways. The changes in topology brought about by different experimental design can be seen to alter these current-division patterns.     

Adaptive bolus-chasing computed tomography angiography in the cases of symmetric and asymmetric arterial flows in peripheral arteries

Synchronization of the contrast bolus peak and CT imaging aperture is a crucial issue for computed tomography angiography (CTA). It affects the CTA image quality and the amount of contrast dose. A whole-body CTA procedure means to scan from the abdominal aorta to pedal arteries. In this context, the synchronization is much more difficult with the asymmetric arterial flow in lower extremities than in the case of symmetric arterial flow. In this paper, we propose an adaptive optimal controller to chase the contrast bolus peak while it propagates in the aorta and lower extremities with symmetric flow. In the case of asymmetric flow after the contrast bolus splitting into two lower limbs, we propose a dynamic programming approach to cover the lower limbs optimally. Simulation and experimental results show that the proposed methods outperform the current constant-speed method substantially.     

Impurity radiation from a tokamak plasma

In tokamak operating modes, energy balance is often governed by impurity radiation. This is the case near the divertor plates, during impurity pellet injection, during controlled discharge disruptions, etc. The calculation of impurity radiation is a fairly involved task (it is sometimes the most difficult part of the general problem) because the radiation power is determined by the distribution of ions over the excited states and by the rate constants of elementary processes of radiation and absorption. The objective of this paper is to summarize in one place all the approximate formulas that would help investigators to describe radiation from the most often encountered impurities in a fairly simple way in their calculations accounting for plasma radiation, without reference to special literature. Simple approximating formulas describing ionization, recombination, and charge-exchange processes, as well as radiative losses from ions with a given charge, are presented for five impurity species: beryllium, carbon, oxygen, neon, and argon. Estimating formulas that allow one to take into account plasma opacity for resonant photons in line impurity radiation are also presented.     

Voltage-dependent block of anthrax toxin channels in planar phospholipid bilayer membranes by symmetric tetraalkylammonium ions. Effects on macroscopic conductance

In a recent paper (Blaustein, R. O., T. M. Koehler, R. J. Collier, and A. Finkelstein, 1989. Proc. Natl. Acad. Sci. USA. 86:2209-2213) we described the general channel-forming properties of the PA65 fragment of anthrax toxin in planar phospholipid bilayer membranes. In the present paper we extend our previous studies of the permeability properties of this channel, using a series of symmetric tetraalkylammonium (TAA) ions. Our main finding is that at micromolar concentrations on either the cis (toxin-containing) or trans side of a membrane containing many (greater than 1,000) channels, these ions, ranging in size from tetramethylammonium to tetrahexylammonium, induce a voltage-dependent reduction of membrane conductance. (We attribute a similar voltage-dependent reduction of membrane conductance by millimolar concentrations of HEPES to a cationic form of this buffer present at micromolar concentrations.) In going from large negative to large positive voltages (on the TAA side) one sees that the conductance first decreases from its value in the absence of TAA, reaches a minimum, and then rises back at larger positive voltages toward the level in the absence of TAA. Our interpretation of this behavior is that these symmetric TAA ions block the cation-selective PA65 channel in a voltage-dependent manner. We postulate that there is a single site within the channel to which TAA ions can bind and thereby block the passage of the major current-carrying ion (potassium). A blocking ion is driven into the site by modest positive voltages, but is driven off the site and through the channel by larger positive voltages, thus explaining the relief of block. (In the accompanying paper [Blaustein, R. O., E. J. A. Lea, and A. Finkelstein. 1990. J. Gen. Physiol. 96:921-942] we confirm this interpretation of the data by analysis at the single-channel level.) This means that these blocking ions can pass through the channel; the permeability to tetrahexylammonium, the largest ion studied, implies that the narrowest part of the channel has a diameter of at least 11 A.     

Optimising Body Layout Design of Limbed Machines

This paper presents our efforts to explain why mammals have large thigh muscles while insects have small ones. After a discussion of this observation a definition of body foot ratio is defined which describes how animals stand and how their legs are arranged. To investigate the mechanics, we present a closed optimum solution of the body foot ratio for a 2D two-leg walking machine. A multi-walker is used as a case for 3D general analysis, and the numerical simulation is presented. Both 2D and 3D case studies can explain the above observations of mammals and insects. These findings can also be used as a guide for the design of man-made limbed machines.     

Properties of microsolvated ions: from the microenvironment of chromophore and alkali metal ions in proteins to negative ions in water clusters

Here we discuss the fascinating chemistry and physics of microsolvated ions that bridge the transition from bare ions in gas phase to ions in solution. Such ions occur in many situations in biochemistry and are crucial for several functions; metal ions, for example, must remove their water shell to pass through ion pumps in membranes. Furthermore, only a few water molecules are buried in the hydrophobic pockets of proteins where they are bound to charged amino acid residues or ionic chromophores. Another aspect is the reactivity of microsolvated ions and the importance in atmospheric, organic and inorganic chemistry. We close by a discussion of the stability of molecular dianions, and how hydration affects the electronic binding energy. There is a vast literature on microsolvated ions, and in this review we are far from being comprehensive, rather we mainly bring examples of our own work.     

Advances in blind source separation (BSS) and independent component analysis (ICA) for nonlinear mixtures

In this paper, we review recent advances in blind source separation (BSS) and independent component analysis (ICA) for nonlinear mixing models. After a general introduction to BSS and ICA, we discuss in more detail uniqueness and separability issues, presenting some new results. A fundamental difficulty in the nonlinear BSS problem and even more so in the nonlinear ICA problem is that they provide non-unique solutions without extra constraints, which are often implemented by using a suitable regularization. In this paper, we explore two possible approaches. The first one is based on structural constraints. Especially, post-nonlinear mixtures are an important special case, where a nonlinearity is applied to linear mixtures. For such mixtures, the ambiguities are essentially the same as for the linear ICA or BSS problems. The second approach uses Bayesian inference methods for estimating the best statistical parameters, under almost unconstrained models in which priors can be easily added. In the later part of this paper, various separation techniques proposed for post-nonlinear mixtures and general nonlinear mixtures are reviewed.     

Nonequilibrium voltage fluctuations in biological membranes. II. Voltage and current noise generated by ion carriers, channels and electrogenic pumps

As applications of the general theoretical framework of charge transport in biological membranes and related voltage and current noise, a number of model calculations are presented for ion carriers, rigid channels, channels with conformational substates and electrogenic pumps. The results are discussed with special reference to the problem of threshold values for sensory transduction processes and their limitations by voltage fluctuations. Furthermore, starting from the special results of model calculations, an attempt is made to determine more general aspects of electric fluctuations generated by charge-transport processes in biological membranes: different frequency dependences of voltage and current noise, and dependence of noise intensities with increasing distance from the equilibrium state.     

Real-time technique for conversion of skin temperature into skin blood flow: human skin as a low-pass filter for thermal waves

Monitoring of skin blood flow oscillations related with mechanical activity of vessels is a very useful modality during diagnosis of peripheral hemodynamic disorders. In this study, we developed a new model and technique for real-time conversion of skin temperature into skin blood flow oscillations, and vice versa. The technique is based on the analogy between the thermal properties of the human skin and electrical properties of the special low-pass filter. Analytical and approximated impulse response functions for the low- and high-pass filters are presented. The general algorithm for the reversible conversion of temperature into blood flow is described. The proposed technique was verified using simulated or experimental data of cold stress, deep inspiratory gasp, and post-occlusive reactive hyperaemia tests. The implementation of the described technique will enable to turn a temperature sensor into a blood flow sensor.