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

Endogenous Bioelectric Signaling Networks: Exploiting Voltage Gradients for Control of Growth and Form

Living systems exhibit remarkable abilities to self-assemble, regenerate, and remodel complex shapes. How cellular networks construct and repair specific anatomical outcomes is an open question at the heart of the next-generation science of bioengineering. Developmental bioelectricity is an exciting emerging discipline that exploits endogenous bioelectric signaling among many cell types to regulate pattern formation. We provide a brief overview of this field, review recent data in which bioelectricity is used to control patterning in a range of model systems, and describe the molecular tools being used to probe the role of bioelectrics in the dynamic control of complex anatomy. We suggest that quantitative strategies recently developed to infer semantic content and information processing from ionic activity in the brain might provide important clues to cracking the bioelectric code. Gaining control of the mechanisms by which large-scale shape is regulated in vivo will drive transformative advances in bioengineering, regenerative medicine, and synthetic morphology, and could be used to therapeutically address birth defects, traumatic injury, and cancer.

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.