Pattern formation of stationary transcellular ionic currents in Fucus

(2+δ)-dimensional theory of the electromechanics of lipid membranes: Electrostatics

The coupling of electric fields to the mechanics of lipid membranes gives rise to intriguing electromechanical behavior, as, for example, evidenced by the deformation of lipid vesicles in external electric fields. Electromechanical effects are relevant for many biological processes, such as the propagation of action potentials in axons and the activation of mechanically gated ion channels. Currently, a theoretical framework describing the electromechanical behavior of arbitrarily curved and deforming lipid membranes does not exist. Purely mechanical models commonly treat lipid membranes as two-dimensional surfaces, ignoring their finite thickness. While holding analytical and numerical merit, this approach cannot describe the coupling of lipid membranes to electric fields and is thus unsuitable for electromechanical models. In a sequence of articles, we derive an effective surface theory of the electromechanics of lipid membranes, called the (2+δ)-dimensional theory, which has the advantages of surface descriptions while accounting for finite thickness effects. The present article proposes a generic dimension reduction procedure relying on low-order spectral expansions. This procedure is applied to the electrostatics of lipid membranes to obtain the (2+δ)-dimensional theory that captures potential differences across and electric fields within lipid membranes. This model is tested on different geometries relevant for lipid membranes, showing good agreement with the corresponding three-dimensional electrostatics theory.

A Kirchhoff-Nernst-Planck framework for modeling large scale extracellular electrodiffusion surrounding morphologically detailed neurons

Many pathological conditions, such as seizures, stroke, and spreading depression, are associated with substantial changes in ion concentrations in the extracellular space (ECS) of the brain. An understanding of the mechanisms that govern ECS concentration dynamics may be a prerequisite for understanding such pathologies. To estimate the transport of ions due to electrodiffusive effects, one must keep track of both the ion concentrations and the electric potential simultaneously in the relevant regions of the brain. Although this is currently unfeasible experimentally, it is in principle achievable with computational models based on biophysical principles and constraints. Previous computational models of extracellular ion-concentration dynamics have required extensive computing power, and therefore have been limited to either phenomena on very small spatiotemporal scales (micrometers and milliseconds), or simplified and idealized 1-dimensional (1-D) transport processes on a larger scale. Here, we present the 3-D Kirchhoff-Nernst-Planck (KNP) framework, tailored to explore electrodiffusive effects on large spatiotemporal scales. By assuming electroneutrality, the KNP-framework circumvents charge-relaxation processes on the spatiotemporal scales of nanometers and nanoseconds, and makes it feasible to run simulations on the spatiotemporal scales of millimeters and seconds on a standard desktop computer. In the present work, we use the 3-D KNP framework to simulate the dynamics of ion concentrations and the electrical potential surrounding a morphologically detailed pyramidal cell. In addition to elucidating the single neuron contribution to electrodiffusive effects in the ECS, the simulation demonstrates the efficiency of the 3-D KNP framework. We envision that future applications of the framework to more complex and biologically realistic systems will be useful in exploring pathological conditions associated with large concentration variations in the ECS.

Effect of Ionic Diffusion on Extracellular Potentials in Neural Tissue

Recorded potentials in the extracellular space (ECS) of the brain is a standard measure of population activity in neural tissue. Computational models that simulate the relationship between the ECS potential and its underlying neurophysiological processes are commonly used in the interpretation of such measurements. Standard methods, such as volume-conductor theory and current-source density theory, assume that diffusion has a negligible effect on the ECS potential, at least in the range of frequencies picked up by most recording systems. This assumption remains to be verified. We here present a hybrid simulation framework that accounts for diffusive effects on the ECS potential. The framework uses (1) the NEURON simulator to compute the activity and ionic output currents from multicompartmental neuron models, and (2) the electrodiffusive Kirchhoff-Nernst-Planck framework to simulate the resulting dynamics of the potential and ion concentrations in the ECS, accounting for the effect of electrical migration as well as diffusion. Using this framework, we explore the effect that ECS diffusion has on the electrical potential surrounding a small population of 10 pyramidal neurons. The neural model was tuned so that simulations over ∼100 seconds of biological time led to shifts in ECS concentrations by a few millimolars, similar to what has been seen in experiments. By comparing simulations where ECS diffusion was absent with simulations where ECS diffusion was included, we made the following key findings: (i) ECS diffusion shifted the local potential by up to ∼0.2 mV. (ii) The power spectral density (PSD) of the diffusion-evoked potential shifts followed a 1/f2 power law. (iii) Diffusion effects dominated the PSD of the ECS potential for frequencies up to several hertz. In scenarios with large, but physiologically realistic ECS concentration gradients, diffusion was thus found to affect the ECS potential well within the frequency range picked up in experimental recordings.

The interplay between genetic and bioelectrical signaling permits a spatial regionalisation of membrane potentials in model multicellular ensembles

The single cell-centred approach emphasises ion channels as specific proteins that determine individual properties, disregarding their contribution to multicellular outcomes. We simulate the interplay between genetic and bioelectrical signals in non-excitable cells from the local single-cell level to the long range multicellular ensemble. The single-cell genetic regulation is based on mean-field kinetic equations involving the mRNA and protein concentrations. The transcription rate factor is assumed to depend on the absolute value of the cell potential, which is dictated by the voltage-gated cell ion channels and the intercellular gap junctions. The interplay between genetic and electrical signals may allow translating single-cell states into multicellular states which provide spatio-temporal information. The model results have clear implications for biological processes: (i) bioelectric signals can override slightly different genetic pre-patterns; (ii) ensembles of cells initially at the same potential can undergo an electrical regionalisation because of persistent genetic differences between adjacent spatial regions; and (iii) shifts in the normal cell electrical balance could trigger significant changes in the genetic regulation.

Bioelectrical Signals and Ion Channels in the Modeling of Multicellular Patterns and Cancer Biophysics

Bioelectrical signals and ion channels are central to spatial patterns in cell ensembles, a problem of fundamental interest in positional information and cancer processes. We propose a model for electrically connected cells based on simple biological concepts: i) the membrane potential of a single cell characterizes its electrical state; ii) the long-range electrical coupling of the multicellular ensemble is realized by a network of gap junction channels between neighboring cells; and iii) the spatial distribution of an external biochemical agent can modify the conductances of the ion channels in a cell membrane and the multicellular electrical state. We focus on electrical effects in small multicellular ensembles, ignoring slow diffusional processes. The spatio-temporal patterns obtained for the local map of cell electric potentials illustrate the normalization of regions with abnormal cell electrical states. The effects of intercellular coupling and blocking of specific channels on the electrical patterns are described. These patterns can regulate the electrically-induced redistribution of charged nanoparticles over small regions of a model tissue. The inclusion of bioelectrical signals provides new insights for the modeling of cancer biophysics because collective multicellular states show electrical coupling mechanisms that are not readily deduced from biochemical descriptions at the individual cell level.

Electrical coupling in ensembles of nonexcitable cells: modeling the spatial map of single cell potentials

We analyze the coupling of model nonexcitable (non-neural) cells assuming that the cell membrane potential is the basic individual property. We obtain this potential on the basis of the inward and outward rectifying voltage-gated channels characteristic of cell membranes. We concentrate on the electrical coupling of a cell ensemble rather than on the biochemical and mechanical characteristics of the individual cells, obtain the map of single cell potentials using simple assumptions, and suggest procedures to collectively modify this spatial map. The response of the cell ensemble to an external perturbation and the consequences of cell isolation, heterogeneity, and ensemble size are also analyzed. The results suggest that simple coupling mechanisms can be significant for the biophysical chemistry of model biomolecular ensembles. In particular, the spatiotemporal map of single cell potentials should be relevant for the uptake and distribution of charged nanoparticles over model cell ensembles and the collective properties of droplet networks incorporating protein ion channels inserted in lipid bilayers.

Membrane potential bistability in nonexcitable cells as described by inward and outward voltage-gated ion channels

The membrane potential of nonexcitable cells, defined as the electrical potential difference between the cell cytoplasm and the extracellular environment when the current is zero, is controlled by the individual electrical conductance of different ion channels. In particular, inward- and outward-rectifying voltage-gated channels are crucial for cell hyperpolarization/depolarization processes, being amenable to direct physical study. High (in absolute value) negative membrane potentials are characteristic of terminally differentiated cells, while low membrane potentials are found in relatively depolarized, more plastic cells (e.g., stem, embryonic, and cancer cells). We study theoretically the hyperpolarized and depolarized values of the membrane potential, as well as the possibility to obtain a bistability behavior, using simplified models for the ion channels that regulate this potential. The bistability regions, which are defined in the multidimensional state space determining the cell state, can be relevant for the understanding of the different model cell states and the transitions between them, which are triggered by changes in the external environment.

Changes in gravity rapidly alter the magnitude and direction of a cellular calcium current

In single-celled spores of the fern Ceratopteris richardii, gravity directs polarity of development and induces a directional, trans-cellular calcium (Ca(2+)) current. To clarify how gravity polarizes this electrophysiological process, we measured the kinetics of the cellular response to changes in the gravity vector, which we initially estimated using the self-referencing calcium microsensor. In order to generate more precise and detailed data, we developed a silicon microfabricated sensor array which facilitated a lab-on-a-chip approach to simultaneously measure calcium currents from multiple cells in real time. These experiments revealed that the direction of the gravity-dependent polar calcium current is reversed in less than 25 s when the cells are inverted, and that changes in the magnitude of the calcium current parallel rapidly changing g-forces during parabolic flight on the NASA C-9 aircraft. The data also revealed a hysteresis in the response of cells in the transition from 2g to micro-g in comparison to cells in the micro-g to 2-g transition, a result consistent with a role for mechanosensitive ion channels in the gravity response. The calcium current is suppressed by either nifedipine (calcium-channel blocker) or eosin yellow (plasma membrane calcium pump inhibitor). Nifedipine disrupts gravity-directed cell polarity, but not spore germination. These results indicate that gravity perception in single plant cells may be mediated by mechanosensitive calcium channels, an idea consistent with some previously proposed models of plant gravity perception.

Changes in gravity rapidly alter the magnitude and direction of a cellular calcium current

In single-celled spores of the fern Ceratopteris richardii, gravity directs polarity of development and induces a directional, trans-cellular calcium (Ca(2+)) current. To clarify how gravity polarizes this electrophysiological process, we measured the kinetics of the cellular response to changes in the gravity vector, which we initially estimated using the self-referencing calcium microsensor. In order to generate more precise and detailed data, we developed a silicon microfabricated sensor array which facilitated a lab-on-a-chip approach to simultaneously measure calcium currents from multiple cells in real time. These experiments revealed that the direction of the gravity-dependent polar calcium current is reversed in less than 25 s when the cells are inverted, and that changes in the magnitude of the calcium current parallel rapidly changing g-forces during parabolic flight on the NASA C-9 aircraft. The data also revealed a hysteresis in the response of cells in the transition from 2g to micro-g in comparison to cells in the micro-g to 2-g transition, a result consistent with a role for mechanosensitive ion channels in the gravity response. The calcium current is suppressed by either nifedipine (calcium-channel blocker) or eosin yellow (plasma membrane calcium pump inhibitor). Nifedipine disrupts gravity-directed cell polarity, but not spore germination. These results indicate that gravity perception in single plant cells may be mediated by mechanosensitive calcium channels, an idea consistent with some previously proposed models of plant gravity perception.

Effective zero-thickness model for a conductive membrane driven by an electric field

The behavior of a conductive membrane in a static (dc) electric field is investigated theoretically. An effective zero-thickness model is constructed based on a Robin-type boundary condition for the electric potential at the membrane, originally developed for electrochemical systems. Within such a framework, corrections to the elastic moduli of the membrane are obtained, which arise from charge accumulation in the Debye layers due to capacitive effects and electric currents through the membrane and can lead to an undulation instability of the membrane. The fluid flow surrounding the membrane is also calculated, which clarifies issues regarding these flows sharing many similarities with flows produced by induced charge electro-osmosis (ICEO). Nonequilibrium steady states of the membrane and of the fluid can be effectively described by this method. It is both simpler, due to the zero thickness approximation which is widely used in the literature on fluid membranes, and more general than previous approaches. The predictions of this model are compared to recent experiments on supported membranes in an electric field.

Ephaptic conduction in a cardiac strand model with 3D electrodiffusion

We study cardiac action potential propagation under severe reduction in gap junction conductance. We use a mathematical model of cellular electrical activity that takes into account both three-dimensional geometry and ionic concentration effects. Certain anatomical and biophysical parameters are varied to see their impact on cardiac action potential conduction velocity. This study uncovers quantitative features of ephaptic propagation that differ from previous studies based on one-dimensional models. We also identify a mode of cardiac action potential propagation in which the ephaptic and gap-junction-mediated mechanisms alternate. Our study demonstrates the usefulness of this modeling approach for electrophysiological systems especially when detailed membrane geometry plays an important role.

Computational modelling of epithelial patterning

The generation of polar cell polarity (PCP) can be regarded as a pattern-forming process. Pattern formation requires local self-enhancement and long-range inhibition that can take place either within a cell or between adjacent cells. A comparison of this general condition with implementations in molecular terms in recent PCP models facilitates an understanding of inherent similarities and differences between them. In addition, it is important to integrate the most interesting and still valid results of classical transplantation experiments that were made some 40 years ago. They remind us that the global polarizing signal is based on graded positional identities carried by the individual cells whose molecular nature is still unknown.

Microscopic calculations of local lipid membrane permittivities and diffusion coefficients for application to electroporation analyses

Interaction of electric fields with biological systems has begun to receive considerable attention for applications that include field-assisted drug delivery, medical interventions, and genetic engineering. External fields induce the strongest effects at membranes with electroporation being a common feature. Membrane transport in this context of poration is often based on continuum approaches utilizing macroscopic parameters such as the permittivity, diffusion coefficients, and mobilities. In such modeling, field dependences, local inhomogeneities, and microscopic details are usually ignored. Here, a molecular dynamics (MD) scheme is used for a more rigorous and physically realistic evaluation of such parameters for potential application to electroporative transport model development. A suitable membrane structure containing a nanopore derived from MD analysis is used as the initial geometric configuration. Both static and frequency dependent diffusion coefficients have been evaluated. Permittivities are also calculated and shown to be dramatically non-uniform in the vicinity of membranes under high external fields. A positive feedback mechanism leading to enhanced membrane fields is discussed.

Traveling ion channel density waves affected by a conservation law

A model of mobile, charged ion channels embedded in a biomembrane is investigated. The ion channels fluctuate between an opened and a closed state according to a simple two-state reaction scheme whereas the total number of ion channels is a conserved quantity. Local transport mechanisms suggest that the ion channel densities are governed by electrodiffusionlike equations that have to be supplemented by a cable-type equation describing the dynamics of the transmembrane voltage. It is shown that the homogeneous distribution of ion channels may become unstable to either a stationary or an oscillatory instability. The nonlinear behavior immediately above threshold of an oscillatory bifurcation occurring at finite wave number is analyzed in terms of amplitude equations. Due to the conservation law imposed on ion channels, large-scale modes couple to the finite-wave-number instability and have thus to be included in the asymptotic analysis near the onset of pattern formation. A modified Ginzburg-Landau equation extended by long-wavelength stationary excitations is established, and it is highlighted how the global conservation law affects the stability of traveling ion channel density waves.

Parity-breaking bifurcation and global oscillation in patterns of ion channels

Stationary spatiotemporal pattern formation emerging from the electric activity of biological membranes is widespread in cells and tissues. A known key instability comes from the self-aggregation of membrane channels. In a two-dimensional geometry, we show that the primary pattern undergoes four secondary instabilities: Eckhaus-like, period-halving, drift instabilities, and a global oscillation. The stability diagram is determined. The parity-breaking (drift) bifurcation of channel density is characterized analytically and numerically.

Cotransport-induced instability of membrane voltage in tip-growing cells

A salient feature of stationary patterns in tip-growing cells is the key role played by the symports and antiports, membrane proteins that translocate two ionic species at the same time. It is shown that these cotransporters destabilize generically the membrane voltage if the two translocated ions diffuse differently and carry a charge of opposite (same) sign for symports (antiports). The orders of magnitude obtained for the time and length scale are in agreement with experiments. A weakly nonlinear analysis characterizes the bifurcation.

Marine plants may polarize remoteFucus eggs via luminescence

Fucus zygotes can be polarized by many environmental vectors. These include those created by pieces of all the intertidal marine plants tested. At distances of up to 5-10 mm away from such pieces, Fucus zygotes form their intitial outgrowths or germinate towards them. Earlier papers had inferred that this so-called 'thallus effect' is mediated by diffusing molecules. The present reinvestigation indicates that the thallus effect is exerted by influences that can go right through glass barriers. This indicates action via luminescence. This luminescence may come from bacteria growing in biofilms on the similar surfaces of these otherwise unrelated source plants. Moreover, this directive luminescence is inferred to lie at wavelengths in the red or longer and may take the form of more or less coherent biophotons.