Ionic strength dependence of localized contact formation between membranes: nonlinear theory and experiment.

Erythrocyte membrane surface or suspending phase properties can be experimentally modified to give either spatially periodic local contacts or continuous contact along the seams of interacting membranes. Here, for cells suspended in a solution of the uncharged polysaccharide dextran, the average lateral separation between localized contacts in spatially periodic seams at eight ionic strengths, decreasing from 0.15 to 0.065, increased from 0.65 to 3.4 micrometers. The interacting membranes and intermembrane aqueous layer were modeled as a fluid film, submitted to a disjoining pressure, responding to a displacement perturbation either through wave growth resulting in spatially periodic contacts or in perturbation decay, to give a plane continuous film. Measured changes of lateral contact separations with ionic strength change were quantitatively consistent with analytical predictions of linear theory for an instability mechanism dependent on the membrane bending modulus. Introduction of a nonlinear approach established the consequences of the changing interaction potential experienced by different parts of the membrane as the disturbance grew. Numerical solutions of the full nonlinear governing equations correctly identified the ionic strength at which the bifurcation from continuous seam to a stationary periodic contact pattern occurred and showed a decrease in lateral contact and wave crest separation with increasing ionic strength. The nonlinear approach has the potential to recognize the role of nonspecific interactions in initiating the localized approach of membranes, and then incorporate the contribution of specific molecular interactions, of too short a range to influence the beginning of perturbation growth. This new approach can be applied to other biological processes such as neural cell adhesion, phagocytosis, and the acrosome reaction.

Influence of polymer concentration and molecular weight and of enzymic glycocalyx modification on erythrocyte interaction in dextran solutions.

Erythrocytes adhere to each other when suspended in supra-threshold concentrations of dextran of molecular mass of 40 kD or greater. The plasma membranes are parallel to each other over the entire length of the contact seam at the lower effective polymer concentrations. When cells are pretreated with the proteolytic enzyme pronase or the sialidase neuraminidase the membranes are not parallel but make contact at spatially periodic locations along the membrane surface. Pronase induced reduction of cell electrophoretic mobility rapidly reaches a limiting value. Nevertheless, prolonged pre-exposure to enzyme leads to a continuing reduction in contact separations. This result taken with the observation that, for equal loss of electrophoretic mobility, a shorter contact separation results from pronase rather than neuraminidase pre-treatment implies that a non-electrostatic consequence of pronase pre-treatment dominates membrane interaction in the experimental regimes examined here. The average lateral contact separation for different enzyme regimes lay in the range 3.3 microns to a limiting lower value of about 0.7 micron. There was a good correlation between the logarithm of a contact separation index (the approach of separation distance to its limiting value) against the logarithm of a derived index related to net attractive interaction for a wide range of experimental conditions. Treatments which increased attraction or decreased repulsion (e.g. increased dextrans concentration or enzyme pre-treatment) lead to shorter lateral contact separation. This result is qualitatively consistent with the predicted behaviour for the dominant wavelength arising from interfacial instability of a thin aqueous film between adjacent membranes.

Predictability of human EEG: a dynamical approach.

The electroencephalogram recordings from human scalp are analysed in the framework of recent methods of nonlinear dynamics. Three stages of brain activity are considered: the alpha waves (eyes closed), the deep sleep (stage four) and the Creutzfeld-Jakob coma. Two dynamical parameters of the attractors are evaluated. These are the Lyapunov exponents, which measure the divergence or convergence of trajectories in phase space and the Kolmogorov or metric entropy, whose inverse gives the mean predicting time of a given EEG signal. In all the stages considered, the results reveal the presence of at least two positive Lyapunov exponents, which are the footprints of chaos. This number increases to three positive exponents in the case of alpha waves, indicating that although for very short episodes the alpha waves seem extremely coherent, the variability of the brain increases markedly over larger periods of activity. The degree of entropy/chaos increases from coma to deep sleep and then to alpha waves. The large predicting time observed for deep sleep suggests that these waves are related to a slow rate of information processing. The predicting time of the alpha waves is much smaller, indicating a rapid loss of information. Finally, with the help of the Lyapunov exponents, the attractor's dimensions are evaluated using two different conjectures and compared to values obtained previously by the Grassberger-Procaccia algorithm.

Membrane-membrane contact: involvement of interfacial instability in the generation of discrete contacts.

The classical approach to understanding the closeness of approach of two membranes has developed from consideration of the net effect of an attractive van der Waals force and a repulsive electrostatic force. The repulsive role of hydration forces and stereorepulsion glycocalyx forces have been recently recognized and an analysis of the effect of crosslinking molecules has been developed. Implicit in these approaches is the idea of an intercellular water layer of uniform thickness which narrows but retains a uniform thickness as the cells move towards an equilibrium separation distance. Most recently an attempt has been made to develop a physical chemical approach to contact which accommodates the widespread occurrence of localized spatially separated point contacts between interacting cells and membranes. It is based on ideas drawn from analysis of the conditions required to destabilize thin liquid films so that thickness fluctuations develop spontaneously and grow as interfacial instabilities to give spatially periodic contact. Examples of plasma membrane behaviour which are consistent with the interfacial instability approach are discussed and experiments involving polycation, polyethylene glycol, dextran and lectin adhesion and agglutination of erythrocytes are reviewed.

Cell membranes after malignant transformation. Part II: Dynamic stability with surface chemical reactions.

The dynamic stability of cell membranes in presence of chemical reactions is analysed using the same hydrodynamic cell model as in Part I, with a spherical geometry. Chemical reactions give an additional contribution leading to instability even for positive total surface tension. The mechanical properties of the surface change drastically via the gradient of the surface tension (mechano-chemical coupling). An enzymatic regulation of cell division is proposed, via cAMP. Loss of contact-inhibition of division in cancer cells is interpreted as a lowering of the threshold for cell division, which is not modified at confluence. In that sense, failure of control mechanism in cancer cells is of more significance than rapid growth.

Cell membranes after malignant transformation. Part I: Dynamic stability at low surface tension.

A hydrodynamic cell model is introduced to analyze the dynamic stability of the cell membrane after malignant transformation. The cell membrane is considered as a two-dimensional charged interface between intra- and extra-cellular fluids. Employing a first order stability analysis, conditions are established under which growth of surface fluctuations can occur (leading to microvilli formation or cell division). The system is unstable if the total surface tension, i.e. the pure surface tension plus the free energy of formation of the double layers, is negative. Following that criterion, cell division is promoted in cancer cells; moreover, as cancer cells are more fluid than normal cells, they will divide more rapidly. The model also predicts that microvilli (protrusions of the cell membrane) will have a diameter of the order of the dominant wavelengths of perturbation (0.1 - 1 mu) which supports the view that such protrusions are consequences of amplified cell surface fluctuations.

Repulsive stabilization in black lipid membranes. A hydrodynamic model.

A linear stability analysis is performed for a black lipid membrane. The hydrodynamic model consists of a viscous hydrocarbon film sandwiched between two aqueous phases. Attractive forces (van der Waals and electrical) and repulsive forces (steric) are expressed as body forces in the equations of fluid motion in the three phases. The steric repulsion due to overlap of the hydrocarbon chains of the lipids at small film thicknesses is described via an exponentially decaying interaction potential. The dispersion equation displays two modes of vibrations: the bending mode with the two Film surfaces transversely in phase, and the squeezing mode with the two surfaces 180 degrees out of phase. For symmetrical films, these two modes are uncoupled, and the squeezing mode (with thickness variations) is stabilized by the repulsive interactions. For nonsymmetrical films (different surface tensions, surface charges, etc.). these two modes are coupled and the asymmetry induces a shift of the marginal stability curve to shorter wavelengths.

Kinetic properties of electrostatic pores with orientable dipoles, for Na+ and K+ transport through biological membranes.

The model, used previously to account for the transport of K+ ions through squid axon membranes under steady-state conditions, is extended to the description of the kinetic behavior of Na+ and K+ currents, for sudden variations of the applied potential. Theoretical curves are obtained by numerical integration of the electrodiffusion equation for ions within pores, with variable boundary conditions resulting from a progressive reorientation of dipoles at the pore surface. The pores are supposed to be selective and the dipole parameters are allowed to be different for Na+ and K+ pores. The K+ current varies with time, in agreement with the K+ dipole parameters deduced from the steady-state results of Gilbert and Ehrenstein (1969). The dipole parameters for Na+ current are deduced from the steady-state results of Armstrong, Bezanilla & Rojas (1973), where the inactivation phase of the Na+ current is suppressed by introducing pronase in the inside solution. The dipole reorientation is relavent to explain the sigmoid shape of the activation phase of the Na+ current, while the inactivation phase seems to resort to another physical mechanism. The predictions based on this model agree with the experimental results for the steady-state negative resistance and the gating current, associated both with a reorientation of surface dipoles, as well as the activation phase of the Na+ current using a consistent set of parameters for all these comparisons.

The role of proteins in a dipole model for steady-state ionic transport through biological membranes.

The steady-state current-voltage characteristics of biological membranes are analyzed for means of an application of the electrodiffusion theory to the passage of ions through "dielectric pores", with orientable dipoles at the pore-water interfaces. A detailed evaluation of the electrostatic potential barrier shows, indeed, that the ions have practically no chance to penetrate into the phospholipid bilayer, but that they can cross the membrane through local protein inclusions, of high dielectric constant. A "gating mechanism" can be provided, moreover, by a change of the potential barrier, resulting from a dipole reorientation at the pore-water interface. Dipole-dipole interactions are opposed to the orienting effect of an applied field, but they can be neglected when the separation between the dipoles exceeds a certain critical value. The high polarizability of the pore material leads to an amplification of the effect of an applied field on the orientable dipoles. It is therefore possible to achieve a satisfactory agreement with the experimental results of Gilbert and Ehrenstein (Biophys. J., 9: 447, 1969) for the squid axon, and, in particular, to account for the width of the negative resistance regions with a relatively small value for the length of the orientable dipoles.