Electrokinetic membrane processes in relation to properties excitable tissues. II. Some theoretical considerations

A quantitative theory is presented for the behavior of a membrane-electrolyte system subject to an electric current flow (the "membrane oscillator"). If the membrane is porous, carries "fixed charges," and separates electrolyte solutions of different conductances, it can be the site of repetitive oscillatory changes in the membrane potential, the membrane resistance, and the hydrostatic pressure difference across the membrane. These events are accompanied by a pulsating transport of bulk solutions. The theory assumes the superposition of electrochemical and hydrostatic gradients and centers round the kinetics of resistance changes within the membrane, as caused by effects from diffusion and electro-osmotic fluid streaming. The results are laid down in a set of five simple, basic expressions, which can be transformed into a pair of non-linear differential equations yielding oscillatory solutions. A graphical integration method is also outlined (Appendix II). The agreement between the theory and previous experimental observations is satisfactory. The applied electrokinetic concepts may have importance in relation to analyses of the behavior of living excitable cells or tissues.     

A theory of membrane permeability: III. The effect of hydrostatic pressure

Using expressions derived in previous papers, the author investigates the behavior of a cell immersed in an infinite medium, under the influence of diffusion of a single solute and flow of water. The effect of hydrostatic pressure on the system is taken into account. It is found that, depending on the values of certain parameters, the cell can collapse, burst, reach a stationary stable state, or execute undamped oscillations; a cell must burst or collapse unless its volume is an increasing function of internal pressure, and it can execute stable oscillations only if its membrane acts as a “potential well” to the molecules of the solute.     

Spontaneous switching of frequency-locking by periodic stimulus in oscillators of plasmodium of the true slime mold

Microfabrication technique was used to construct a model system with a living cell of plasmodium of the true slime mold, Physarum polycephalum, a living coupled oscillator system. Its parameters can be systematically controlled as in computer simulations, so that results are directly comparable to those of general mathematical models. As the first step, we investigated responses in oscillatory cells, the oscillators of the plasmodium, to periodic stimuli by temperature changes to elucidate characteristics of the cells as nonlinear systems whose internal dynamics are unknown because of their complexity. We observed that the forced oscillator of the plasmodium show 1:1, 2:1, 3:1 frequency locking inside so-called Arnold tongues regions as well as in other nonlinear systems such as chemical systems and other biological systems. In addition, we found spontaneous switching behavior from certain frequency locking states to other states, even under certain fixed parameters. This technique can be applied to more complex systems with multiple elements, such as coupled oscillator systems, and would be useful to investigate complicated phenomena in biological systems such as information processing.     

Mechanical Equilibrium of Biological Membranes

The nature of mechanical and electrical forces on biological membranes in relation to mechanical equilibrium is examined. The presence of a double layer of electric charge is shown to give rise to an effective pressure drop across a curved membrane of finite thickness. For certain geometric shapes of a membrane, the magnitude of the pressure drop due to electrostatic forces may set a limit on the hydrostatic pressure drop that the membrane can support without buckling. The results are applied to the equilibrium shape of the red blood cell.     

Are axoplasmic microtubules necessary for membrane excitation?

Summary The excitability of the squid giant axon was studied as a function of transmembrane hydrostatic pressure differences, the latter being altered by the technique of intracellular perfusion. When a KF solution was used as the internal medium, a pressure difference of about 15 cm water had very little effect on either the membrane potential or excitability. However, within a few minutes after introducing either a KCl-containing, a KBr-containing, or a colchicine-containing solution as the internal medium, with the same pressure difference across the membrane, the axon excitability was suppressed. In these cases, removal of the pressure difference restored the excitability, indicating that the structure of membrane was not irreversibly damaged. Electron-microscopic observations of these axons revealed that the perfusion with a KF solution or colchicine-containing solution preserves the submembranous cytoskeletal layer, whereas perfusion with a KCl or KBr solution dissolves it. These results suggest that the submembranous cytoskeletons including microtubules provide an important mechanical support to the excitable membrane but are not essential elements in channel activities.     

MECHANICAL BEHAVIOR OF PLANT TISSUES AS INFERRED FROM THE THEORY OF PRESSURIZED CELLULAR SOLIDS

The mechanical behavior of plant tissues and its dependency on tissue geometry and turgor pressure are analytically dealt with in terms of the theory of cellular solids. A cellular solid is any material whose matter is distributed in the form of beamlike struts or complete “cell” walls. Therefore, its relative density is less than one and typically less than 0.3. Relative density is the ratio of the density of the cellular solid to the density of its constitutive (“cell wall”) material. Relative density depends upon cell shape and the density of cell wall material. It largely influences the mechanical behavior of cellular solids. Additional important parameters to mechanical behavior are the elastic modulus of “cell walls” and the magnitude of internal “cell” pressure. Analyses indicate that two “stiffening” agents operate in natural cellular solids (plant tissues): 1) cell wall infrastructure and 2) the hydrostatic influence of the protoplasm within each cellular compartment. The elastic modulus measured from a living tissue sample is the consequence of both agents. Therefore, the mechanical properties of living tissues are dependent upon the magnitude of turgor pressure. High turgor pressure places cell walls into axial tension, reduces the magnitude of cell wall deformations under an applied stress, and hence increases the apparent elastic modulus of the tissue. In the absence of turgid protoplasts or in the case of dead tissues, the cell wall infrastructure will respond as a linear elastic, nonlinear elastic, or “densifying” material (under compression) dependent upon the magnitude of externally applied stress. Accordingly, it is proposed that no single tangent (elastic) modulus from a stress-strain curve of a plant tissue is sufficient to characterize the material properties of a sample. It is also suggested that when a modulus is calculated that it be referred to as the tissue composite modulus to distinguish it from the elastic modulus of a noncellular solid material.     

Stabilizing role of calcium store-dependent plasma membrane calcium channels in action-potential firing and intracellular calcium oscillations

In many biological systems, cells display spontaneous calcium oscillations (CaOs) and repetitive action-potential firing. These phenomena have been described separately by models for intracellular inositol trisphosphate (IP3)-mediated CaOs and for plasma membrane excitability. In this study, we present an integrated model that combines an excitable membrane with an IP3-mediated intracellular calcium oscillator. The IP3 receptor is described as an endoplasmic reticulum (ER) calcium channel with open and close probabilities that depend on the cytoplasmic concentration of IP3 and Ca2+. We show that simply combining this ER model for intracellular CaOs with a model for membrane excitability of normal rat kidney (NRK) fibroblasts leads to instability of intracellular calcium dynamics. To ensure stable long-term periodic firing of action potentials and CaOs, it is essential to incorporate calcium transporters controlled by feedback of the ER store filling, for example, store-operated calcium channels in the plasma membrane. For low IP3 concentrations, our integrated NRK cell model is at rest at -70 mV. For higher IP3 concentrations, the CaOs become activated and trigger repetitive firing of action potentials. At high IP3 concentrations, the basal intracellular calcium concentration becomes elevated and the cell is depolarized near -20 mV. These predictions are in agreement with the different proliferative states of cultures of NRK fibroblasts. We postulate that the stabilizing role of calcium channels and/or other calcium transporters controlled by feedback from the ER store is essential for any cell in which calcium signaling by intracellular CaOs involves both ER and plasma membrane calcium fluxes.     

Coupled multi-component systems: A simple membrane model

We present initial results regarding the existence, stability and interactionof linear and nonlinear vibrational modes in a system of two coupled, onedimensional lattices with unequal numbers of masses. The effects on thesenonlinear modes of coupling a near continuum system to a discrete systemusing a nonlinear coupling are examined. This numerical model is a firststep towards investigating the dynamical behavior of a flexible sheetcoupled nonlinearly to a semi-rigid support, a system which couldconceivably represent a biological cell membrane with a supporting proteinnetwork. General implications for the dynamical behavior of continuumsystems coupled nonlinearly to discrete systems are introduced.     

A minimal,compartmental model for a dendritic origin of bistability of motoneuron firing patterns

Various nonlinear regenerative responses, including plateau potentials and bistable repetitive firing modes, have been observed in motoneurons under certain conditions. Our simulation results support the hypothesis that these responses are due to plateau-generating currents in the dendrites, consistent with a major role for a noninactivating calcium L-type current as suggested by experiments. Bistability as observed in the soma of low- and higher-frequency spiking or, under TTX, of near resting and depolarized plateau potentials, occurs because the dendrites can be in a near resting or depolarized stable steady state. We formulate and study a two-compartment minimal model of a motoneuron that segregates currents for fast spiking into a soma-like compartment and currents responsible for plateau potentials into a dendrite-like compartment. Current flows between compartments through a coupling conductance, mimicking electrotonic spread. We use bifurcation techniques to illuminate how the coupling strength affects somatic behavior. We look closely at the case of weak coupling strength to gain insight into the development of bistable patterns. Robust somatic bistability depends on the electrical separation since it occurs only for weak to moderate coupling conductance. We also illustrate that hysteresis of the two spiking states is a natural consequence of the plateau behavior in the dendrite compartment.     

Electrical instability in cardiac muscle: phase singularities and rotors

A dynamical system is "excitable" at some stage in its behavior (e.g. at a rest state or while it is nearly at rest prior to a spontaneous event) if a small, but not too small, stimulus of the right kind elicits an immediate big reaction that eventually leads back to the original state. During this return to excitability a typical system is not excitable. An excitable system need not have an attracting rest state; a spontaneous oscillator can be excitable, too, as is common in biological and in chemical excitable kinetics. In a medium characterized by such excitable dynamics at every point, the excitation can propagate as a travelling pulse. Undamaged cardiac muscle shares with other excitable media certain features of such pulse propagation in two and three dimensions. Among the new electrophysiological phenomena thus anticipated are paired mirror-image vortices ("rotors") organized around phase singularities. These should arise in the myocardium near the intersection of a moving critical contour of phase in the normal cycle of excitation and recovery with a momentary critical contour of local stimulus strength. Such intersections, and the corresponding aftermath of paired rotors, should only occur following certain combinations of stimulus size and stimulus timing. Plotting those combinations on a "vulnerability diagram", one delineates a domain for creation of rotors (corresponding to tachycardia) surrounded on all sides by a halo of combinations at which just a few repetitive responses follow stimulation. The experiments called for to check these implications have now been carried out in the special case of electrically-induced tachycardia in healthy canine ventricle. They support the two-dimensional theory, so a new experiment is suggested to demonstrate wholly intramural three-dimensional vortex filaments.     

Asymmetric electrotonic coupling between the soma and dendrites alters the bistable firing behaviour of reduced models

The goal of the study was to investigate the influence of asymmetric coupling, between the soma and dendrites, on the nonlinear dynamic behaviour of a two-compartment model. We used a recently published method for generating reduced two-compartment models that retain the asymmetric coupling of anatomically reconstructed motor neurons. The passive input-output relationship of the asymmetrically coupled model was analytically compared to the symmetrically coupled case. Predictions based on the analytic comparison were tested using numerical simulations. The simulations evaluated the nonlinear dynamics of the models as a function of coupling parameters. Analytical results showed that the input resistance at the dendrite of the asymmetric model was directly related to the degree of coupling asymmetry. In contrast, a comparable symmetric model had identical input resistances at both the soma and dendrite regardless of coupling strength. These findings lead to predictions that variations in dendritic excitability, subsequent to changes in input resistance, might change the current threshold and onset timing of the plateau potential generated in the dendrite. Since the plateau potential underlies bistable firing, these results further predicted that asymmetric coupling might alter nonlinear (i.e. bistable) firing patterns. The numerical simulations supported analytical predictions, showing that the fully bistable firing pattern of the asymmetric model depended on the degree of coupling asymmetry and its correlated dendritic excitability. The physiological property of asymmetric coupling plays an important role in generating and stabilizing the bistability of motor neurons by interacting with the excitability of dendritic branches.     

Nonlinear material and ionic transport through membrane nanotubes

Membrane nanotubes (NTs) and their networks play an important role in intracellular membrane transport and intercellular communications. The transport characteristics of the NT lumen resemble those of conventional solid-state nanopores. However, unlike the rigid pores, the soft membrane wall of the NT can be deformed by forces driving the transport through the NT lumen. This intrinsic coupling between the NT geometry and transport properties remains poorly explored. Using synchronized fluorescence microscopy and conductance measurements, we revealed that the NT shape was changed by both electric and hydrostatic forces driving the ionic and solute fluxes through the NT lumen. Far from the shape instability, the strength of the force effect is determined by the lateral membrane tension and is scaled with membrane elasticity so that the NT can be operated as a linear elastic sensor. Near shape instabilities, the transport forces triggered large-scale shape transformations, both stochastic and periodic. The periodic oscillations were coupled to a vesicle passage along the NT axis, resembling peristaltic transport. The oscillations were parametrically controlled by the electric field, making NT a highly nonlinear nanofluidic circuitry element with biological and technological implications.     

Behavior of plant plasma membranes under hydrostatic pressure as monitored by fluorescent environment-sensitive probes

We monitored the behavior of plasma membrane (PM) isolated from tobacco cells (BY-2) under hydrostatic pressures up to 3.5 kbar at 30 °C, by steady-state fluorescence spectroscopy using the newly introduced environment-sensitive probe F2N12S and also Laurdan and di-4-ANEPPDHQ. The consequences of sterol depletion by methyl-β-cyclodextrin were also studied. We found that application of hydrostatic pressure led to a marked decrease of hydration as probed by F2N12S and to an increase of the generalized polarization excitation (GPex) of Laurdan. We observed that the hydration effect of sterol depletion was maximal between 1 and 1.5 kbar but was much less important at higher pressures (above 2 kbar) where both parameters reached a plateau value. The presence of a highly dehydrated gel state, insensitive to the sterol content, was thus proposed above 2.5 kbar. However, the F2N12S polarity parameter and the di-4-ANEPPDHQ intensity ratio showed strong effect on sterol depletion, even at very high pressures (2.5-3.5 kbar), and supported the ability of sterols to modify the electrostatic properties of membrane, notably its dipole potential, in a highly dehydrated gel phase. We thus suggested that BY-2 PM undergoes a complex phase behavior in response to the hydrostatic pressure and we also emphasized the role of phytosterols to regulate the effects of high hydrostatic pressure on plant PM.     

Extended Kalman Filter for Estimation of Parameters in Nonlinear State-Space Models of Biochemical Networks

It is system dynamics that determines the function of cells, tissues and organisms. To develop mathematical models and estimate their parameters are an essential issue for studying dynamic behaviors of biological systems which include metabolic networks, genetic regulatory networks and signal transduction pathways, under perturbation of external stimuli. In general, biological dynamic systems are partially observed. Therefore, a natural way to model dynamic biological systems is to employ nonlinear state-space equations. Although statistical methods for parameter estimation of linear models in biological dynamic systems have been developed intensively in the recent years, the estimation of both states and parameters of nonlinear dynamic systems remains a challenging task. In this report, we apply extended Kalman Filter (EKF) to the estimation of both states and parameters of nonlinear state-space models. To evaluate the performance of the EKF for parameter estimation, we apply the EKF to a simulation dataset and two real datasets: JAK-STAT signal transduction pathway and Ras/Raf/MEK/ERK signaling transduction pathways datasets. The preliminary results show that EKF can accurately estimate the parameters and predict states in nonlinear state-space equations for modeling dynamic biochemical networks.     

The measurement and mechanism of lipid peroxidation in biological systems

The basic chemistry of the propagation of lipid peroxidation reactions has been known for years, but the mechanism of initiation of this process in biological membrane systems is still uncertain. Currently available assays for measuring peroxidation are reviewed--the more specific the assay used, the less peroxide is found in healthy human tissues and body fluids. Lipid peroxidation can arise as a consequence of tissue injury in many disease states and may sometimes contribute significantly to worsening the tissue injury.     

Lens intracellular hydrostatic pressure is generated by the circulation of sodium and modulated by gap junction coupling

We recently modeled fluid flow through gap junction channels coupling the pigmented and nonpigmented layers of the ciliary body. The model suggested the channels could transport the secretion of aqueous humor, but flow would be driven by hydrostatic pressure rather than osmosis. The pressure required to drive fluid through a single layer of gap junctions might be just a few mmHg and difficult to measure. In the lens, however, there is a circulation of Na(+) that may be coupled to intracellular fluid flow. Based on this hypothesis, the fluid would cross hundreds of layers of gap junctions, and this might require a large hydrostatic gradient. Therefore, we measured hydrostatic pressure as a function of distance from the center of the lens using an intracellular microelectrode-based pressure-sensing system. In wild-type mouse lenses, intracellular pressure varied from ～330 mmHg at the center to zero at the surface. We have several knockout/knock-in mouse models with differing levels of expression of gap junction channels coupling lens fiber cells. Intracellular hydrostatic pressure in lenses from these mouse models varied inversely with the number of channels. When the lens' circulation of Na(+) was either blocked or reduced, intracellular hydrostatic pressure in central fiber cells was either eliminated or reduced proportionally. These data are consistent with our hypotheses: fluid circulates through the lens; the intracellular leg of fluid circulation is through gap junction channels and is driven by hydrostatic pressure; and the fluid flow is generated by membrane transport of sodium.     

Water Stress and Protein Synthesis: II. Interaction between Water Stress, Hydrostatic Pressure, and Abscisic Acid on the Pattern of Protein Synthesis in Avena Coleoptiles

Water stress causes a reduction in hydrostatic pressure and can cause an increase in abscisic acid in plant tissues. To assess the possible role of abscisic acid and hydrostatic pressure in water stress effects, we have compared the effects of water stress, abscisic acid, and an imposed hydrostatic pressure on the rate and pattern of protein synthesis in Avena coleoptiles. Water stress reduces the rate and changes the pattern of protein synthesis as judged by a double labeling ratio technique, Abscisic acid reduces the rate but does not alter the pattern of protein synthesis. Gibberellic acid reverses the abscisic acid-induced but not the stress-induced inhibition of protein synthesis. The effect of hydrostatic pressure depends on the gas used. With a 19: 1 N₂-air mixture, the rate of protein synthesis is increased in stressed but not in turgid tissues. An imposed hydrostatic pressure alters the pattern of synthesis in stressed tissues, but does not restore the pattern to that found in turgid tissues. Because of the differences in response, we conclude that water stress does not affect protein synthesis via abscisic acid or reduced hydrostatic pressure.