Self-sustained oscillations of electric potential in a model membrane

A model membrane constructed from a Millipore filter, whose pores are filled with dioleyl phosphate molecules, exhibits a self-oscillation of the electric potential with a period of about a few seconds in the presence of a salt-concentration difference, pressure difference and/or electric current across the filter. In this paper, the effects of chemicals such as KCl, CaCl2, pH and sucrose on the self-oscillation are investigated experimentally. These chemical substances are shown to alter the characteristic properties as the frequency of oscillation. Theoretical consideration of electrochemical interaction between these substances and DOPH molecules gives a fairly good explanation of the observed results.     

Calcium signalling in non-excitable cells: notes on oscillations and store refilling

Technical advances in studying cellular calcium concentrations, and discoveries about many aspects of signal transduction have transformed this field of biology since this Journal was launched a decade ago. At that time monitoring of the key variable, cytoplasmic free Ca2+ concentration [Ca2+]i, was possible with aequorin or arsenazo ill mainly in large invertebrate cells, though pioneering work with aequorin micro-injection into cardiac and smooth muscle had just started. At that time there was also intense activity by a few groups aiming to make Ca-selective micro-electrodes selective and sharp enough to measure [Ca2+]i in small cells. Also the use of electropermeabilized cells had begun to allow the defining of the concentrations of Ca2+ required to activate secretion in mammalian cells. Nearly all this work and all the relevant electrophysiology relating to calcium signalling had been done in excitable cells, basically muscle and nerve, and was aimed at understanding contraction, transmitter release and neurosecretion, and the control of membrane permeability. Recent advances have now allowed [Ca2+]i to be measured in non-excitable cells.     

Photogeneration of membrane potential hyperpolarization and depolarization in non-excitable cells

We monitored femtosecond laser induced membrane potential changes in non-excitable cells using patchclamp analysis. Membrane potential hyperpolarization of HeLa cells was evoked by 780 nm, 80 fs laser pulses focused in the cellular cytoplasm at average powers of 30–60 mW. Simultaneous detection of intracellular Ca²⁺ concentration and membrane potential revealed coincident photogeneration of Ca²⁺ waves and membrane potential hyperpolarization. By using non-excitable cells, the cell dynamics are slow enough that we can calculate the membrane potential using the steady-state approximation for ion gradients and permeabilities, as formulated in the GHK equations. The calculations predict hyperpolarization that matches the experimental measurements and indicates that the cellular response to laser irradiation is biological, and occurs via laser triggered Ca²⁺ which acts on Ca²⁺ activated K⁺ channels, causing hyperpolarization. Furthermore, by irradiating the cellular plasma membrane, we observed membrane potential depolarization in combination with a drop in membrane resistance that was consistent with a transient laser-induced membrane perforation. These results entail the first quantitative analysis of location-dependent laser-induced membrane potential modification and will help to clarify cellular biological responses under exposure to high intensity ultrashort laser pulses.     

Simulation of chemical oscillations in membranes

A computer simulation model has been developed to follow chemical oscillations in a membrane for immobilized enzyme systems. It is a discrete particle type model which follows the spatial and temporal fluctuations of the concentrations in a reaction involving two substrates. The parameters can be readily varied to allow dissipative structures to result from the sustained nonlinear reaction kinetics and to determine which parameters cause damping of the oscillations. The nature of the diffusion mechanism allows extension to more than one dimension.     

Light-induced generation of electric potential difference in membranes of purple and green sulfur bacteria

When associated with a planar phospholipid membrane, chromatophores isolated from photosynthetic sulfur bacteriaChromatium minutissimum, Ectothiorhodospira shaposhnikovii, andChlorobium limicola f. thiosulfatophilum were shown to generate a light-induced transmembrane electric potential difference measured by a direct method using macroelectrodes and a voltmeter. The maximal photoelectric responses were observed upon the addition of 1,4-naphthoquinone in combination with phenazine methosulfate (or TMPD) and ascorbate. The photoeffects were inhibited by CCCP and gramicidin. The data demonstrate that similar mechanisms of photoelectric generation function in membranes of the different bacteria studied.     

Minimal models of multidimensional computations

The multidimensional computations performed by many biological systems are often characterized with limited information about the correlations between inputs and outputs. Given this limitation, our approach is to construct the maximum noise entropy response function of the system, leading to a closed-form and minimally biased model consistent with a given set of constraints on the input/output moments; the result is equivalent to conditional random field models from machine learning. For systems with binary outputs, such as neurons encoding sensory stimuli, the maximum noise entropy models are logistic functions whose arguments depend on the constraints. A constraint on the average output turns the binary maximum noise entropy models into minimum mutual information models, allowing for the calculation of the information content of the constraints and an information theoretic characterization of the system's computations. We use this approach to analyze the nonlinear input/output functions in macaque retina and thalamus; although these systems have been previously shown to be responsive to two input dimensions, the functional form of the response function in this reduced space had not been unambiguously identified. A second order model based on the logistic function is found to be both necessary and sufficient to accurately describe the neural responses to naturalistic stimuli, accounting for an average of 93% of the mutual information with a small number of parameters. Thus, despite the fact that the stimulus is highly non-Gaussian, the vast majority of the information in the neural responses is related to first and second order correlations. Our results suggest a principled and unbiased way to model multidimensional computations and determine the statistics of the inputs that are being encoded in the outputs.     

Minimal size of cell assemblies coordinated by gamma oscillations

In networks of excitatory and inhibitory neurons with mutual synaptic coupling, specific drive to sub-ensembles of cells often leads to gamma-frequency (25-100 Hz) oscillations. When the number of driven cells is too small, however, the synaptic interactions may not be strong or homogeneous enough to support the mechanism underlying the rhythm. Using a combination of computational simulation and mathematical analysis, we study the breakdown of gamma rhythms as the driven ensembles become too small, or the synaptic interactions become too weak and heterogeneous. Heterogeneities in drives or synaptic strengths play an important role in the breakdown of the rhythms; nonetheless, we find that the analysis of homogeneous networks yields insight into the breakdown of rhythms in heterogeneous networks. In particular, if parameter values are such that in a homogeneous network, it takes several gamma cycles to converge to synchrony, then in a similar, but realistically heterogeneous network, synchrony breaks down altogether. This leads to the surprising conclusion that in a network with realistic heterogeneity, gamma rhythms based on the interaction of excitatory and inhibitory cell populations must arise either rapidly, or not at all. For given synaptic strengths and heterogeneities, there is a (soft) lower bound on the possible number of cells in an ensemble oscillating at gamma frequency, based simply on the requirement that synaptic interactions between the two cell populations be strong enough. This observation suggests explanations for recent experimental results concerning the modulation of gamma oscillations in macaque primary visual cortex by varying spatial stimulus size or attention level, and for our own experimental results, reported here, concerning the optogenetic modulation of gamma oscillations in kainate-activated hippocampal slices. We make specific predictions about the behavior of pyramidal cells and fast-spiking interneurons in these experiments.     

Minimal model for membrane oscillations in the pancreatic beta-cell.

Following the experimental findings of Atwater et al. (In Biochemistry Biophysics of the Pancreatic-beta-Cell, George Thieme Verlag, New York, 100-107), we have formulated a mathematical model for ionic and electrical events that take place in pancreatic-beta-cells. Our formulation incorporates a Hodgkin-Huxley type gating mechanism for Ca2+ and K+ channels, in addition to Ca2+ gated K+-channels. Consistent with the experimental observations, our model generates spikes and bursts in beta-cell membrane potentials and gives the correct responses to additions of glucose, quinine, and tetraethylammonium ions. The response of the oscillations to ouabain and changing concentrations of external K+ can be incorporated into the present model, although a more complete treatment would require inclusion of the Na+/K+ pump.     

The kinetics of aggregation phenomena. I. Minimal models for patch formation of lymphocyte membranes.

Numerous biological processes involve the assembly of one or more monomers into aggregates or networks of interconnected units. In this paper we present the initial aspects of a mathematical theory for network formation on lymphocyte membranes. We assume the fluid mosaic membrane model is valid, that a lymphocyte possesses a homogeneous set of mobile but membrane bound receptors and that these receptors can form bimolecular complexes with antigen. We show that these complexes tend to aggregate and derive expressions for their size distribution as a function of time, antigen valence and concentration, and antigenreceptor affinity.At early times, the mass of the system (receptors plus antigen) is in very small aggregates. However under appropriate conditions, a critical time is reached at which they coalesce in such a way that the mass shifts, becoming concentrated predominantly in large aggregates. We assume that this coalescence (“patch” formation) is a necessary condition for lymphocyte triggering and briefly pursue the consequences.It is shown that the time required for patch formation is a sensitive function of affinity (K), antigen valence and antigen concentration (C), and that if KC is either too high or too low patch formation will not be possible. Moreover within the range of binding constants which can lead to patching, there will be an optimum value which leads to the fastest rate of triggering, and this optimum shifts to higher affinity as the concentration of free antigen surrounding the cell decreases. For optimum KC values we estimate times typically of the order of (10–100) seconds for patch formation. The theory also suggests that if antigen valence is too low, triggering will not be possible within times of interest, without introducing other factors. It thus leads naturally to a requirement for auxiliary cells which would tend to present low valence antigens in such a way that the B lymphocytes see an effective, increased valence. The theory, although primitive, thus meets some minimal requirements in that it distinguishes binding reactions from triggering reactions, makes predictions consistent with observations on affinity maturation and the nonresponsiveness to high doses and low doses of antigen, and suggests the need for helper cells (or their products) in order for low valence antigens to be effective in lymphocyte triggering.     

Minimal models of multi-site ligand-binding kinetics

Mathematical models based on the current understanding of co-operativity in ligand binding to the (macro) molecule and relating the dose-response (saturation) curve of the (macro) molecule ligation to intrinsic dissociation constants characterizing the affinities of ligand for binding sites of both unliganded and partly liganded (macro) molecule have been developed. The simplified models disregarding the structural properties and considerations concerning conformational changes of the (macro) molecule retain the ability to yield sigmoid curves of ligand binding and reflect the co-operativity. Model 1 contains only three parameters, parameter κ (a multiplier characterising the change in the affinity) reflects also the existence and type of co-operativity of ligand binding: κ<1 corresponds to positive co-operativity, κ>1 to the negative and κ=1 to the absence of any co-operativity. Model 2 contains an extra parameter, ω, equilibrium constant for the T₀↔R₀ transition but fails to produce dose-response, which would suggest negative co-operativity. For any fixed n>1, the deviation of the dose-response (saturation) curve from the Henri hyperbola depends either solely on parameter κ (Model 1) or also on parameter ω (Model 2). The (macro) molecule being a receptor, both models yield a diversity of dose-response curves due to possible variety of efficacies of the (macro) molecule. The models may be considered as extensions of the Henri model: in case the dissociation constants remain unchanged, the proposed models are reduced to the latter.     

An unusual complementation in non-excitable mutants in Paramecium

A non-excitable behavioural mutant, d4-662, was previously characterized as the fourth pawn locus mutant pwD in Paramecium tetraurelia. We now provide data demonstrating that d4-662 is in fact controlled by a pwB allele that has the unusual feature of complementing other pwB alleles in heterozygous F1 progeny. Neither the cytoplasm nor the nucleoplasm of d4-662 cured the mutational defects of pwB and in the reverse combination of d4-662 and pwB, the result was the same. On the other hand, pwA, another non-excitable mutant, was cured upon cross-injection with d4-662 and mutants carrying trichocyst non-discharge marker genes were also cured. This evidence suggests that d4-662 is a new mutant belonging to pwB, and would be better designated as pwB662. Extensive crossbreeding analyses, however, showed an unusual genetic relationship between d4-662 and pwB (pwB95 or pwB96). When d4-662 was crossed with pwB mutants, many progeny expressing wild-type phenotype or mixed clones of wild-type and pawn cells were obtained in the F1. Less than 12.5% expressed the pawn phenotype. The appearance of wild-type progeny in this F1 strongly suggests that an inter-allelic interaction between pwB662 and other pwB alleles may occur during development of the macronucleus.     

Minimal models of top-down control of phytoplankton

1. A set of models describing the dynamics of top-down control of phytoplankton by Daphnia in lakes is reviewed. The basis of these models is a simple and well-known model that has been used, among other things, to demonstrate the paradox of enrichment.
2. We discuss minimal extensions that allow this model to mimic the effects of spatial heterogeneity, planktivory, seasonality and inedible algae.
3. These models generate hypotheses about mechanisms that may cause patterns observed in the field such as:

4. We discuss the way in which such very simple models may contribute to the building of theories about plankton dynamics in the field, and the caveats of interpreting wrongly the message from models.     

Current-voltage characteristics and self-sustained oscillations in dioleyl phosphate-millipore membranes

For an artificial membrane prepared by infiltrating dioleyl phosphate (DOPH) into pores of a Millipore filter, we propose a theoretical model for explaining observed data on electric behavior, such as d.c. current-voltage characteristics and self-sustained oscillations of the electric potential. The model consists of a simple electric circuit composed of electric resistances and capacitances as given functions of internal variables which represent conformational states of DOPH molecules and salt concentration inside the pore concerned. The kinetic equations for these variables are the same as those presented previously for describing a phase transition of DOPH. except for a slight modification taking account of effects of salt accumulation inside the pore. The present theory can describe well the I-V hysteresis and various features of spike-like oscillations with long periods of up to a few hours in the absence of external force and also short-period oscillations with periods of the order of 1 s under pressure difference.     

Studies of electric capacitance of membranes

Summary The electric capacitance and conductance of a model membrane composed of a hydrophobic filter paper and a synthetic lipid analogue, i.e., dioleylphosphate, immersed in an electrolyte solution were observed with various frequencies ranging from 20 to 3×10⁶ Hz. With successive increase of salt concentration in the external solution, the capacitance and conductance of the membrane increased discontinuously at a certain critical value of the external salt concentration. This variation of the capacitance and conductance of the membrane with the salt concentration was found to be reversible, and the critical value of salt concentration was independent of the adsorbed quantity of the lipid, and of the pore size of the filter paper as far as the adsorbed quantity of the dioleylphosphate was large.A theoretical analysis based on the membrane model for the filter paper-phospholipid system proposed in Part I of this series revealed that the dioleylphosphate impregnated in the filter paper changed its conformation from oil droplets or globular micelles to a number of bilayer membranes when the salt concentration reached the critical value for a given pair of electrolyte species and the membrane. The conformational change of the lipid analogue in the filter paper is discussed in connection with the ability of formation and stability of a black bilayer membrane of the dioleylphosphate.