Kinetic theory of antibiotic ion carrier. 3

Ion selectivity of ionophorous antibiotics has been measured in many equilibrium experiments such as solvent extraction, electric potential and conductance across an artificial bilayer membrane. However in general these measurements cannot be interpreted as the ion selectivity in a non-equilibrium transporting state which is typically the state of a biological system. This point is explained in this paper based on the simple kinetic theories presented in the preceding papers (Huang, 1971a, Huang, 1971b). The exceptional case of the Eyring model and the Eisenman-Ciani-Szabo model is discussed.It is also found that if the no one-ion-exchange assumption in the theory for the valinomycin-type antibiotics (Huang, 1971b) is relaxed such that the antibiotic S reacts with cation I⁺ and anion X^? in the following way I⁺ + S ? IS⁺, X^? + IS⁺ ? ISX, then the previous equation of cation transport remains unchanged while no simple equation can be written for the anion transport.     

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).     

Transcellular Ion Flow in Escherichia coli B and Electrical Sizing of Bacterias

Dielectric breakdown of cell membranes and, in response, transcellular ion flows were measured in Escherichia coli B 163 and B 525 using a Coulter counter as the detector with a hydrodynamic jet focusing close to the orifice of the counter. Plotting the relative pulse height for compensated amplification of a certain size of the cells against increasing detector current, a rather sharp bend within the linear function was found, which did not occur when measuring fixed cells or polystyrene latex. The start current for transcellular ion flow causing the change of the slope is different for the potassium-deficient mutant B 525 in comparison with the wild-type B 163, indicating a change in the membrane structure of B 525 by mutation and demonstrating the sensitivity of the method for studying slight changes in membrane structure in general. The theoretical size distributions for two current values in the range of transcellular ion flow were constructed from the true size distribution at low detector currents, assuming an idealized sharp changeover of the bacterial conductivity from zero to one-third of the electrolyte conductivity.     

A theory of charge separation,ion, electron and proton transport in photosynthetic membranes based on asymmetry of surface charges

We present a model for the light-induced charge separation, proton and ion transport across photosynthetic membranes based on an assumption of the transmembrane surface charge asymmetry. In dark equilibrium, this asymmetry gives rise to an internal membrane electric field whose direction is perpendicular to the membrane surfaces. The role of the field in the light-induced charge separation is similar to the function of the built-in electric field across a solid-state p-n junction. Light-generated free charge carriers in the membrane flow according to its direction and upon recombination on the surface give rise to an electrochemical potential difference for electrons across the membrane. The associated coupled electron-proton transport, and ion diffusion can be viewed as a response of the system to the light-induced redox and electric potential changes.     

Electroporation of Brain Endothelial Cells on Chip toward Permeabilizing the Blood-Brain Barrier

The blood-brain barrier, mainly composed of brain microvascular endothelial cells, poses an obstacle to drug delivery to the brain. Controlled permeabilization of the constituent brain endothelial cells can result in overcoming this barrier and increasing transcellular transport across it. Electroporation is a biophysical phenomenon that has shown potential in permeabilizing and overcoming this barrier. In this study we developed a microengineered in vitro model to characterize the permeabilization of adhered brain endothelial cells to large molecules in response to applied pulsed electric fields. We found the distribution of affected cells by reversible and irreversible electroporation, and quantified the uptaken amount of naturally impermeable molecules into the cells as a result of applied pulse magnitude and number of pulses. We achieved 81 ± 1.7% (N = 6) electroporated cells with 17 ± 8% (N = 5) cell death using an electric-field magnitude of ∼580 V/cm and 10 pulses. Our results provide the proper range for applied electric-field intensity and number of pulses for safe permeabilization without significantly compromising cell viability. Our results demonstrate that it is possible to permeabilize the endothelial cells of the BBB in a controlled manner, therefore lending to the feasibility of using pulsed electric fields to increase drug transport across the BBB through the transcellular pathway.     

Kinetic transport model for cellular regulation of pH and solute concentration in the renal proximal tubule.

An open circuit kinetic model was developed to calculate the time course of proximal tubule cell pH, solute concentrations, and volume in response to induced perturbations in luminal or peritubular fluid composition. Solute fluxes were calculated from electrokinetic equations containing terms for known carrier saturabilities, allosteric dependences, and ion coupling ratios. Apical and basolateral membrane potentials were determined iteratively from the requirements of cell electroneutrality and equal opposing transcellular and paracellular currents. The model converged to membrane potentials accurate to 0.05% in one to four iterations. Model variables included cell concentrations of Na, K, HCO3, glucose, pH (uniform CO2), volume, and apical and basolateral membrane potentials. The basic model contained passive apical membrane transport of Na/H, Na/glucose, H and K, basolateral transport of Na/3HCO3, K, H, and glucose, and paracellular transport of Na, K, Cl, and HCO3; apical H and basolateral 3Na/2K-ATPases were present. Apical Na/H and basolateral K transport were regulated allosterically by pH. Apical Na/H transport, basolateral Na/3HCO3 transport, and the 3Na/2K-ATPase were saturable. Model parameters were chosen from data in the rat proximal tubule. Model predictions for the magnitude and time course of cell pH, Na, and membrane potential in response to rapid changes in apical and peritubular Na and HCO3 were in excellent agreement with experiment. In addition, the model requires that there exist an apical H-ATPase, basolateral Na/3HCO3 transport saturable with HCO3, and electroneutral basolateral K transport.     

Optimization criteria of the mechanism governing the stability of the membrane potential

For small changes in ion concentration within the physiological range the membrane potential transients can be explained in terms of two linear models both for passive and active transport. Using frog sartorius muscle as a suitable model system the ion pump is considered to work within the steepest range of the flux-concentration characteristic. Further for the small perturbations the equations describing passive ion transport can be safely linearized. The conclusion appears inescapable that for the muscle membrane the intracellular ion concentration adjusts itself in some optimal manner to the level of the extracellular ions. The active ion transport represents a control parameter for the membrane potential. The model structure corresponds to a dynamic system, the control processes of which are optimized with respect to a quadratic integral-criterion function. Here, both the performance index of the control sequence in the membrane processes and the energy consumed by the ion fluxes have been considered for small perturbations of Na⁺, K⁺, and Cl^? in the neighbourhood of the physiological working point. As it is, the control system governing the active and passive ion transport processes is essentially optimized with respect to a minimal energy usage. The amount of energy consumed during the transients predicted by the model has been calculated.     

Modeling epithelial cell homeostasis: Steady-state analysis

Critical to epithelial cell viability is the homeostasis of cell volume and composition during changes in transcellular transport. In this study, two previously developed mathematical models (principal cell of the collecting duct and proximal tubule cell) are approximated by their linearizations about a reference condition. This yields matrices which estimate cell volume, cell composition, and transcellular fluxes in response to perturbations of bath conditions and membrane transporter activity. These approximations are themselves extended with the inclusion of linear dependence of membrane transport coefficients on cell variables (e.g., volume, solute concentrations, or electrical potential). This provides cell models with variable permeabilities, which may be homeostatic, and which can be examined systematically: sequentially testing each membrane permeability and its controlling cell variable. In the proximal tubule approximation, volume-mediated increases in peritubular K—Cl or Na—3HCO₃ cotransport, and volume-mediated decreases in Na,K-ATPase activity are homeostatic; modulation of peritubular K permeability has little impact. In the principal cell model, volume homeostasis is afforded by volume-sensitive peritubular Na/H exchange or Cl⁻ conductance. Predictions from the linear analysis are confirmed in the full models. This approach yields a systematic examination of homeostasis in an epithelial model, and identifies candidate control parameters.     

Water transport across plant tissue: Role of water channels

The contribution of water-filled, selective membrane pores (water channels) is integrated into a general concept of water transport in plant tissue. The concept is based on the composite anatomical structure of tissues which results in a composite transport pattern. Three main pathways of water flow have been distinguished, ie the apoplastic, symplastic and transcellular (vacuolar) paths. Since the symplastic and transcellular components can not be distinguished experimentally, these components are summarized as a cell-to-cell component. Water channel activity may control the overall water flow across tissues provided that the contribution of the apoplastic component is relatively low. The composite transport model has been applied to roots where most of the data are available. Comparison of the hydraulic conductivity at the root cell and organ levels shows that, depending on the species, there may be a dominating cell-to-cell or apoplastic water flow. Most remarkably, there are differences in the hydraulic conductivity of roots which depend on the nature of the force used to drive water flows (osmotic or hydrostatic pressure gradients). This is predicted by the model. The composite transport model explains low reflection coefficients of roots, the variability in root hydraulic resistance and differences between herbaceous and woody species. It is demonstrated that there is also a composite transport of water at the membrane level (water channel arrays vs bilayer arrays). This results in low reflection coefficients of plasma membranes for certain test solutes as derived for isolated internodes of Chara. The titration of water channel activity in this alga with mercurials and its dependence on changes in temperature or external concentration show that water channels do not exclusively transport water. Rather, they are permeable to relatively big uncharged organic solutes. The result indicates that, at least for Chara, the concept of an exclusive transport of water across water channels has to be questioned.     

A Test of the Theory of the Steady-State Properties of an Ion Exchange Membrane with Mobile Sites and Dissociated Counterions

An experimental model system, formally equivalent to a liquid ion exchange membrane having completely dissociated sites and counterions, has been devised in order to test the steady-state properties recently deduced theoretically for such a membrane by Conti and Eisenman, (1966). In this system we have obtained quantitative experimental confirmation of the following theoretical expectations. (a) The current-voltage relationship is nonlinear and exhibits finite limiting currents with strong applied fields. (b) The mobile sites rearrange within the “membrane” under applied electric field to give a linear concentration profile and a logarithmic electric potential profile in the steady state. We have also extended the theory to consider the instantaneous conductance in the steady state. Theory and experiment indicate that in a mobile site membrane the instantaneous conductance in the steady state is not given by the chord conductance of the steady-state current-voltage relationship, in contrast to the situation in a fixed site membrane. This finding suggests a way of testing whether ions permeate across an unknown membrane by a fixed site or a dissociated mobile site mechanism.     

Dynamics of cellular homeostasis: Recovery time for a perturbation from equilibrium

In the collecting ductin vivo, the principal cell encounters a wide range in luminal flow rate and luminal concentration of NaCl. As a consequence, there are substantial variations in the transcellular fluxes of Na⁺ and Cl⁻, conditions which would be expected to perturb cell volume and cytosolic concentrations. Several control mechanisms have been identified which can potentially blunt these perturbations, and these entail cellular regulation of the luminal membrane Na⁺ channel and peritubular membrane K⁺ and Cl⁻ channels. To illustrate the impact of these regulated channels, a mathematical model of the principal cell of the rat cortical collecting duct has been developed, in which ion channel permeabilities are either constant or regulated. In comparison to the model with fixed permeabilities, the model with regulated channels demonstrates enhanced cellular homeostasis following steady-state variation in luminal NaCl. However, in the transient response to a cytosolic perturbation, the difference in recovery time between the models is small. An approximate analysis is presented which casts these models as dynamical systems with constant coefficients. Despite the presence of regulated ion channels, concordance of each model with its linear approximation is verified for experimentally meaningful perturbations from the reference condition. Solution of a Lyapunov equation for each linear system yields a matrix whose application to a perturbation permits explicit estimation of the time to recovery. Comparison of these solution matrices for regulated and non-regulated cells confirms the similarity of the dynamic response of the two models. These calculations suggest that enhanced homeostasis by regulated channels may be protective, without necessarily hastening recovery from cellular perturbations.     

Insight into the mechanisms of neuronal processing from electric fish

Weakly electric fish use their electric fields to locate objects and communicate with each other. Their electric discharges vary with species, gender, and social status. This variation is mediated by steroid and peptide hormones that influence ion currents through changes in gene expression or phosphorylation state. Understanding how electric fish decode the perturbations of their electric fields that result from interactions with the discharges of other fish or prey is illuminating general mechanisms of neuronal processing. Their central sensory circuits are specialized to process amplitude modulated signals, to detect microsecond variations in spike timing, and are dynamically reconfigured depending on the stimulus parameters.     

On the admittance of membranes associated with channel conduction. Application to channels at non-equilibrium steady state

The small-signal admittance of membranes associated with channel conduction is derived for a general channel model. A general channel model is represented by a set of chemical reactions with each species of the reactions representing a channel state. The membrane admittance is shown to be related to the phenomenological relaxation matrix of the reactions. If the kinetic reactions are at a non-equilibrium steady state, the relaxation matrix may have complex eigenvalues and the equivalent circuit of the membrane admittance may contain RLC or RLC-like branches. For equilibrium kinetic systems, on the other hand, the equivalent circuit contains only RL or RC branches. Thus, the membrane admittance of equilibrium channels is quite different from that of non-equilibrium channels. In particular, we show that the low frequency feature in the admittance of squid axons as observed by Fishman, Poussart, Moore &; Siebenga (1977) can be obtained easily from a non-equilibrium cycling steady-state model.     

Periodic band pattern as a dissipative structure in ion transport systems with cylindrical shape

A theory is presented for appearance of periodic band patterns of ion concentration and electric potential associated with electric current surrounding a unicellular or multicellular system of a cylindrical shape. A flux continuity at the membrane (or the surface) is reduced to a nonlinear equation expressing passive and active fluxes across the membrane and intracellular diffusion flux. It is shown that, when an external parameter is varied from the sub-critical region, i.e. the homogeneous flux state, a symmetry breaking along a longitudinal axis usually appears prior to the one along a circumferential direction. The spectrum analysis shows that the correlation length is longer in the longitudinal direction. Growth of the band pattern from a patch-shaped pattern is demonstrated by the use of numerical calculations of proton concentration on the two-dimensional space of cylindrical surface. An experimental example of formative process of H⁺ banding is given for the internodal cell ofChara. It is shown that small patches on the surface decline or are sometimes gathered to the band surrounding the circle. The resulting pattern is suggested as a kind of dissipative structure appearing far from equilibrium.     

Ion transport through pores: transient phenomena

A theoretical analysis of non-stationary states in membrane pores is given in this paper, which is based on the rate-theory treatment of transport processes. The principal aim of this study is to give a basis for the interpretation of relaxation experiments in which an external parameter, such as the voltage across the membrane, is suddenly displaced. From the time course of the membrane current information about the microscopic properties of the pore may be obtained. The pore is considered as a sequence of binding sites, separated by energy barriers over which the ion has to jump. It is found that under certain conditions damped oscillations occur after the initial perturbations of the membrane. In all other cases the approach towards the steady state may be described by a discrete spectrum of n relaxation times, where n is the number of binding sites within the pore. In the case of a pore with regular energy profile (internal barriers of identical height) the relaxation times may be obtained as the roots of Tchebycheff polynomials for arbitrary n. It is shown that the present treatment becomes identical with the continuum analysis of transport processes in the limit of large n.     

Opto-Mechanical Coupling in Interfaces under Static and Propagative Conditions and Its Biological Implications

Fluorescent dyes are vital for studying static and dynamic patterns and pattern formation in cell biology. Emission properties of the dyes incorporated in a biological interface are known to be sensitive to their local environment. We report that the fluorescence intensity of dye molecules embedded in lipid interfaces is indeed a thermodynamic observable of the system. Opto-mechanical coupling of lipid-dye system was measured as a function of the thermodynamic state of the interface. The corresponding state diagrams quantify the thermodynamic coupling between intensity I and lateral pressure π. We further demonstrate that the coupling is conserved upon varying the temperature T. Notably, the observed opto-mechanical coupling is not limited to equilibrium conditions, but also holds for propagating pressure pulses. The non-equilibrium data show, that fluorescence is especially sensitive to dynamic changes in state such as the LE-LC phase transition. We conclude that variations in the thermodynamic state (here π and T, in general pH, membrane potential V, etc also) of lipid membranes are capable of controlling fluorescence intensity. Therefore, interfacial thermodynamic state diagrams of I should be obtained for a proper interpretation of intensity data.     

A chemical instability mechanism for asymmetric cell differentiation

We propose a mechanism of asymmetric mitosis applicable for cells even without inherent architectural asymmetry and in the presence of symmetric conditions such as a homogeneous environment. The theory is based on the instability of the symmetric development in time of the prospective daughters of a cell in mitosis. The macroscopic equations of chemical reaction, diffusion, and permeation of the various chemical species in the cell are given formal expression, and are then linearized about the symmetric development in order to test the stability to asymmetric perturbations. Instability to such perturbations indicates the deterministic onset of asymmetric division (differentiation). Only small external gradients of concentration, temperature, light, etc. are necessary to polarize the asymmetry for the purpose of a particular morphology. The theory is compared qualitatively with experiments on melanocytogenesis, is used to suggest new experiments, and is proposed as a possible alternative to other mechanisms. Possible application of the theory to the experiments on the first division in the egg of Fucus is considered. Finally, a simple model of a product-enhanced reaction mechanism is developed in detail which shows that the history of the initial preparation of the cell prior to mitosis may play a role in determining the possibility of asymmetric division.     

Thermodynamics and bioenergetics

Bioenergetics is concerned with the energy conservation and conversion processes in a living cell, particularly in the inner membrane of the mitochondrion. This review summarizes the role of thermodynamics in understanding the coupling between the chemical reactions and the transport of substances in bioenergetics. Thermodynamics has the advantages of identifying possible pathways, providing a measure of the efficiency of energy conversion, and of the coupling between various processes without requiring a detailed knowledge of the underlying mechanisms. In the last five decades, various new approaches in thermodynamics, non-equilibrium thermodynamics and network thermodynamics have been developed to understand the transport and rate processes in physical and biological systems. For systems not far from equilibrium the theory of linear non-equilibrium thermodynamics is used, while extended non-equilibrium thermodynamics is used for systems far away from equilibrium. All these approaches are based on the irreversible character of flows and forces of an open system. Here, linear non-equilibrium thermodynamics is mostly discussed as it is the most advanced. We also review attempts to incorporate the mechanisms of a process into some formulations of non-equilibrium thermodynamics. The formulation of linear non-equilibrium thermodynamics for facilitated transport and active transport, which represent the key processes of coupled phenomena of transport and chemical reactions, is also presented. The purpose of this review is to present an overview of the application of non-equilibrium thermodynamics to bioenergetics, and introduce the basic methods and equations that are used. However, the reader will have to consult the literature reference to see the details of the specific applications.