Stable spatially periodic patterns of ion channels in biomembranes

Hexagonal, square, and stripe patterns of the ion channel density in biomembranes

Transmembrane ion flow through channel proteins undergoing density fluctuations may cause lateral gradients of the electrical potential across the membrane giving rise to electrophoresis of charged channels. A model for the dynamics of the channel density and the voltage drop across the membrane (cable equation) coupled to a binding-release reaction with the cell skeleton [P. Fromherz and W. Zimmerman, Phys. Rev. E 51, R1659 (1995)] is analyzed in one and two spatial dimensions. Due to the binding release reaction spatially periodic modulations of the channel density with a finite wave number are favored at the onset of pattern formation, whereby the wave number decreases with the kinetic rate of the binding-release reaction. In a two-dimensional extended membrane hexagonal modulations of the ion channel density are preferred in a large range of parameters. The stability diagrams of the periodic patterns near threshold are calculated and in addition the equations of motion in the limit of a slow binding-release kinetics are derived.

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.

Pattern formation driven by nematic ordering of assembling biopolymers

The biopolymers actin and microtubules are often in an ongoing assembling-disassembling state far from thermal equilibrium. Above a critical density this leads to spatially periodic patterns, as shown by a scaling argument and in terms of a phenomenological continuum model, which meets also Onsager's statistical theory of the nematic-to-isotropic transition in the absence of reaction kinetics. This pattern forming process depends much on nonlinear effects and a common linear stability analysis of the isotropic distribution of the filaments is often misleading. The wave number of the pattern decreases with the assembling-disassembling rate and there is an uncommon discontinuous transition between the nematic and periodic states.

Energy-landscape-model analysis for irreversibility and its pulse-width dependence in cells subjected to a high-intensity ultrashort electric pulse

We provide a simple, but physical analysis for cell irreversibility and apoptosis in response to an ultrashort (nanosecond), high-intensity electric pulse. Our approach is based on an energy landscape model for determining the temporal evolution of the configurational probability function p(q). The primary focus is on obtaining qualitative predictions of a pulse width dependence to apoptotic cell irreversibility that has been observed experimentally. The analysis couples a distributed electrical model for current flow with the Smoluchowski equation to provide self-consistent, time-dependent transmembrane voltages. The model captures the essence of the experimentally observed pulse-width dependence, and provides a possible physical picture that depends only on the electrical trigger. A number of interesting features are predicted.

Pattern formation of ion channels with state-dependent charges and diffusion constants in fluid membranes

A model of mobile, charged ion channels in a fluid membrane is studied. The channels may switch between an open and a closed state according to a simple two-state kinetics with constant rates. The effective electrophoretic charge and the diffusion constant of the channels may be different in the closed and in the open state. The system is modeled by densities of channel species, obeying simple equations of electrodiffusion. The lateral transmembrane voltage profile is determined from a cable-type equation. Bifurcations from the homogeneous, stationary state appear as hard-mode, soft-mode, or hard-mode oscillatory transitions within physiologically reasonable ranges of model parameters. We study the dynamics beyond linear stability analysis and derive nonlinear evolution equations near the transitions to stationary patterns.

Glutamatergically induced pattern of Ca2+ driving potential as a mechanism of postsynaptic plasticity

Simulation studies were performed in a model of neuronal dendrite with Na+ and K+ channels and with ionotropic and metabotropic glutamate receptors. The ionotropic receptors were either N-methyl-D-aspartate (NMDA)-sensitive, voltage-dependent, and permeable to Ca2+, Na+, and K+, or non-NMDA-sensitive, voltage-independent, and permeable to Na+ and K+. The metabotropic receptors provided a catalytic effect on Ca2+-induced Ca2+ release from intracellular stores. Local intracellular concentration [Ca2+]i in the cytoplasm was changed because of exchange with the stores, axial diffusion, and transmembrane inward passive and outward pump fluxes. Tonic activation of ionotropic and metabotropic receptors in a particular range of intensities triggered the formation of spatially periodic [Ca2+]i hot and cold bands arising from an initial uniform state. The period and width of the bands were smaller at higher levels of tonic NMDA activation and higher metabotropically controlled rates of Ca2+-induced Ca2+ release. The bandwidths also depended on the dendrite diameter, the specific membrane, and cytoplasm resistivity. This activity-induced pattern led to long-term, spatially inhomogeneous change in local excitatory postsynaptic potentials (EPSPs) of NMDA synapses phasically activated with the same presynaptic intensity. The phasic EPSPs were potentiated if the synapse occurred in the hot band.