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Plant Physiol
2015 Oct 15;12:22. doi: 10.1104/pp.15.00499.
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Bioelectric memory: modeling resting potential bistability in amphibian embryos and mammalian cells.
Law R
,
Levin M
.
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Bioelectric gradients among all cells, not just within excitable nerve and muscle, play instructive roles in developmental and regenerative pattern formation. Plasma membrane resting potential gradients regulate cell behaviors by regulating downstream transcriptional and epigenetic events. Unlike neurons, which fire rapidly and typically return to the same polarized state, developmental bioelectric signaling involves many cell types stably maintaining various levels of resting potential during morphogenetic events. It is important to begin to quantitatively model the stability of bioelectric states in cells, to understand computation and pattern maintenance during regeneration and remodeling. To facilitate the analysis of endogenous bioelectric signaling and the exploitation of voltage-based cellular controls in synthetic bioengineering applications, we sought to understand the conditions under which somatic cells can stably maintain distinct resting potential values (a type of state memory). Using the Channelpedia ion channel database, we generated an array of amphibian oocyte and mammalian membrane models for voltage evolution. These models were analyzed and searched, by simulation, for a simple dynamical property, multistability, which forms a type of voltage memory. We find that typical mammalian models and amphibian oocyte models exhibit bistability when expressing different ion channel subsets, with either persistent sodium or inward-rectifying potassium, respectively, playing a facilitative role in bistable memory formation. We illustrate this difference using fast sodium channel dynamics for which a comprehensive theory exists, where the same model exhibits bistability under mammalian conditions but not amphibian conditions. In amphibians, potassium channels from the Kv1.x and Kv2.x families tend to disrupt this bistable memory formation. We also identify some common principles under which physiological memory emerges, which suggest specific strategies for implementing memories in bioengineering contexts. Our results reveal conditions under which cells can stably maintain one of several resting voltage potential values. These models suggest testable predictions for experiments in developmental bioelectricity, and illustrate how cells can be used as versatile physiological memory elements in synthetic biology, and unconventional computation contexts.
Fig. 1. Modeling of voltage stability for ion channel combinations. Our approach for retrieving, calibrating, and simulating is diagrammed in (a). Currents (suppresing a factor of 2) for an arbitrary ion channel are schematized geometrically in (b); inward or outward currents are represented by wedge areas inside or outside the circle, respectively (c) Currents diagrammed using the method illustrated in b at three voltages above (top), below (bottom) and at (middle) a stable equilibrium for the Nav1.6 + leak model system, in mammals, with a 10:1 ratio of maximal conductances between Nav1.6 and leak channels. Arrows indicate the direction of voltage evolution. Note that in these diagrams, V and E have been shifted by +150 mV to be strictly positive. d Simulations of voltage evolution in the mammalian Nav1.6 + leak model after a 50 ms initial voltage-clamping period, with corresponding currents as indicated in c
Fig. 2. Phase-space diagrams and simulations demonstrate Nav1.6 + leak is bistable for mammalian cells but not for amphibian oocytes. We compared the phase spaces and membrane voltage dynamics for the Nav1.6 + leak model (with 10:1 maximal conductance ratio) under mammalian and amphibian oocyte conditions. a Phase plane for the timescale-reduced Nav1.6 + leak model for mammals and b corresponding simulations of voltage evolution. c Phase plane of the same ion channels expressed in amphibian oocyte models with corresponding simulation in (d). Black arrows in (a) and (c) indicate the direction of voltage evolution. Red curves indicate dV/dt as a function of V for the combined system, while blue (resp. green) indicates dV/dt supposing the Nav1.6 channel (resp. leak channel) were expressed alone. In the simulations, 30 initial voltages were chosen ranging from −140 to 150 mV and clamped for 50 ms before release. The model is bistable in the mammalian case, but monostable in the amphibian case
Fig. 3. Simulations demonstrating bistability in two mammalian models. We searched for voltage bistability in X + Y + leak mammalian models (10:10:1 maximal conductance ratio) by simulation. Two such channel combinations uncovered by this search were the a Kv1.1 + Nav1.6 + leak and b HCN1 + Nav1.6 + leak models. As before, simulations were initialized at 30 voltages ranging from −140 to 150 mV and clamped for 50 ms before release. In both of these models, voltages initialized in the physiological range converge on one of two locally stable voltages
Fig. 4. Simulations demonstrating bistability in three amphibian models. We searched for voltage bistability in X + Y + leak amphibian oocyte models (10:10:1 maximal conductance ratio) by simulation. Simulations were initialized at 30 voltages ranging from −140 to 150 mV and clamped for 50 ms before release. Three such channel combinations were the a Kir2.1 + leak, b Cav2.1 + Kir2.1 + leak, and c Cav2.2 + Kir2.1 + leak models. Note that Cav2.1 is a fast channel, and that A and C have effectively identical memory loci. In all of these models, voltages initialized in the physiological range converge on one of two locally stable voltages. The basins of attraction for these stable voltages are however, disconnected (in V), as initial hyperpolarization to –140 mV resulted in evolution to the higher-voltage memory
Fig. 5. Examples of simulations not exhibiting bistability. We examined instances of models generating non-bistable behavior in the X + Y + leak amphibian case (10:10:1 maximal conductance ratio). Simulations were initialized at 30 voltages ranging from −140 to 150 mV and clamped for 50 ms before release. a The Kv1.2 + Kir2.1 + leak model exhibits a hyperpolarization-evoked transient caused by clamping at –140 mV. b The Cav2.2 + Kv2.2 + leak model does not appear to be asymptotically stable
Fig. 6. Switching between stable voltages. We simulated a sequence of voltage clamps at the following values: −140 mV, −70 mV, 0 mV, −100 mV, −30 mV, for several models: a mammalian Kir2.1 + Nav1.3 + leak, b mammalian Nav1.3 + Nav1.6 + leak, c amphibian oocyte Kir2.1 + Nav1.3 + leak, and d amphibian oocyte Nav1.3 + Nav1.6 + leak. Clamping periods are in yellow and phase-space diagrams are overlaid, with the grey line indicating dV/dt = 0 (see also Fig. 2). In the bistable models (b and c) the current introduced by the voltage clamp may be seen to act as a switch between stable equilibria, while in the monostable models (a and d), releasing the clamp always leads to the same equilibrium point
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