Nernst-Planck-Gaussian modelling of electrodiffusional recovery from ephaptic excitation between mammalian cardiomyocytes.

Introduction: In addition to gap junction conduction, recent reports implicate possible ephaptic coupling contributions to action potential (AP) propagation between successive adjacent cardiomyocytes. Here, AP generation in an active cell, withdraws Na+ from, creating a negative potential within, ephaptic spaces between the participating membranes, activating the initially quiescent neighbouring cardiomyocyte. However, sustainable ephaptic transmission requires subsequent complete recovery of the ephaptic charge difference. We explore physical contributions of passive electrodiffusive ion exchange with the remaining extracellular space to this recovery for the first time. Materials and Methods: Computational, finite element, analysis examined limiting, temporal and spatial, ephaptic [Na+], [Cl-], and the consequent Gaussian charge differences and membrane potential recovery patterns following a ΔV∼130 mV AP upstroke at physiological (37°C) temperatures. This incorporated Nernst-Planck formalisms into equations for the time-dependent spatial concentration gradient profiles. Results: Mammalian atrial, ventricular and purkinje cardiomyocyte ephaptic junctions were modelled by closely apposed circularly symmetric membranes, specific capacitance 1 µF cm-2, experimentally reported radii a = 8,000, 12,000 and 40,000 nm respectively and ephaptic axial distance w = 20 nm. This enclosed an ephaptic space containing principal ions initially at normal extracellular [Na+] = 153.1 mM and [Cl-] = 145.8 mM, respective diffusion coefficients D Na = 1.3 × 109 and D Cl = 2 × 109 nm2s-1. Stable, concordant computational solutions were confirmed exploring ≤1,600 nm mesh sizes and Δt≤0.08 ms stepsize intervals. The corresponding membrane voltage profile changes across the initially quiescent membrane were obtainable from computed, graphically represented a and w-dependent ionic concentration differences adapting Gauss's flux theorem. Further simulations explored biological variations in ephaptic dimensions, membrane anatomy, and diffusion restrictions within the ephaptic space. Atrial, ventricular and Purkinje cardiomyocytes gave 40, 180 and 2000 ms 99.9% recovery times, with 720 or 360 ms high limits from doubling ventricular radius or halving diffusion coefficient. Varying a, and D Na and D Cl markedly affected recovery time-courses with logarithmic and double-logarithmic relationships, Varying w exerted minimal effects. Conclusion: We thereby characterise the properties of, and through comparing atrial, ventricular and purkinje recovery times with interspecies in vivo background cardiac cycle duration data, (blue whale ∼2000, human∼90, Etruscan shrew, ∼40 ms) can determine physical limits to, electrodiffusive contributions to ephaptic recovery.

Finite element analysis predicts Ca2+ microdomains within tubular-sarcoplasmic reticular junctions of amphibian skeletal muscle.

A finite element analysis modelled diffusional generation of steady-state Ca2+ microdomains within skeletal muscle transverse (T)-tubular-sarcoplasmic reticular (SR) junctions, sites of ryanodine receptor (RyR)-mediated SR Ca2+ release. It used established quantifications of sarcomere and T-SR anatomy (radial diameter [Formula: see text]; axial distance [Formula: see text]). Its boundary SR Ca2+ influx densities,[Formula: see text], reflected step impositions of influxes, [Formula: see text] deduced from previously measured Ca2+ signals following muscle fibre depolarization. Predicted steady-state T-SR junctional edge [Ca2+], [Ca2+]edge, matched reported corresponding experimental cytosolic [Ca2+] elevations given diffusional boundary efflux [Formula: see text] established cytosolic Ca2+ diffusion coefficients [Formula: see text] and exit length [Formula: see text]. Dependences of predicted [Ca2+]edge upon [Formula: see text] then matched those of experimental [Ca2+] upon Ca2+ release through their entire test voltage range. The resulting model consistently predicted elevated steady-state T-SR junctional ~ µM-[Ca2+] elevations radially declining from maxima at the T-SR junction centre along the entire axial T-SR distance. These [Ca2+] heterogeneities persisted through 104- and fivefold, variations in D and w around, and fivefold reductions in d below, control values, and through reported resting muscle cytosolic [Ca2+] values, whilst preserving the flux conservation ([Formula: see text] condition, [Formula: see text]. Skeletal muscle thus potentially forms physiologically significant ~ µM-[Ca2+] T-SR microdomains that could regulate cytosolic and membrane signalling molecules including calmodulin and RyR, These findings directly fulfil recent experimental predictions invoking such Ca2+ microdomains in observed regulatory effects upon Na+ channel function, in a mechanism potentially occurring in similar restricted intracellular spaces in other cell types.