Bioelectric memory: modeling resting potential bistability in amphibian embryos and mammalian cells

Importantly, the initial conditions that lead to particular equilibria do not always form connected regions in voltage space - in other words, after release, a clamped system need not evolve to the nearest voltage memory but instead may deterministically overshoot a nearby attractor due to the presence of slow variables (Figs. 3b, 4a, 6c). If these models were reproduced to reasonable accuracy, driving a system toward a low voltage (for instance by opening a strongly expressed light-sensitive potassium channel population; see, e.g. [76]) should cause the system to jump to a high-voltage memory, rather than the nearby low-voltage memory, after cessation of the driving current.

For completeness, we remark on some channel combinations where multistability did not occur, or where convergence was not observed or inferable within a 10 s timeframe. A majority of these were monostable, e.g. the system in Fig. 5a, where slow channel activation and inactivation led to a transient depolarization but eventual convergence on equilibrium. Other models, however, did not appear to converge on any particular asymptotically stable equilibrium, a phenomenon that often occurred in the presence of Kv channels in amphibian models (see e.g. Fig. 5b). Indeed, with the exception of Kv1.4, Kv channels always disrupted bistability in amphibian models and often disrupted bistability in mammalian models as well. This points to a heuristic where suppression of Kv channels should in general aid in the experimental realization of membrane potential multistability.Fig. 5Examples 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

Finally, we illustrate controlled switching between stable memories, as might be experimentally realized by sequences of applied transient currents. Here, we simulated one continuous 3.5 s period divided into 700 ms epochs. In each epoch, the voltage was clamped for 200 milliseconds and released for 500 milliseconds, with the following sequence of clamped voltages chosen, in mV: −140, −70, 0, −100, −30. As above, the membrane potential rapidly evolves to stable fixed points after the clamp is released (Fig. 6), demonstrating that it is possible to reversibly drive the voltage from one attractor to the other, implementing a true memory element.Fig. 6Switching 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

Bistability has been described previously [50, 77–82], although data have been focused on neurons. Our findings extend prior knowledge not only by examining a much larger array of ion channels, but also by comparing the contributions of these channels to bistability in both amphibian oocytes and mammalian systems beyond neurons. In so doing, we highlight a major difference in voltage dynamics between these classes arising due to differences in reversal potential loci. The work we have presented here may also be seen as a rough guide for the synthesis of cells capable of bistable voltage patterning through appropriate choices of ion channel expression.

Although our models assert a specific 2 mS/cm2 maximal conductance for its nonleak ion channels and a 10:1 nonleak to leak conductance ratio, our results are likely to hold approximately when either Nav1.6 or Kir2.1 channels are overexpressed relative to a cell’s other ion channels, although the values of Vmem may differ. We furthermore expect that small differences in reversal potentials should not make a major difference in stability properties (e.g. in Fig. 2, changing a reversal potential by a few millivolts will not affect the number of roots); the profound difference seen between amphibian oocyte and mammalian models is not due to minute differences in reversal potentials; the reversal potentials for mammals and frog oocytes are not even in the same order.

Interestingly, the specific channels we identified are already implicated in examples of bioelectric control of patterning. For example, Kir2.1 is known to regulate muscle cell differentiation [83–86] and craniofacial patterning (being responsible for Andersen Tawil Syndrome and its craniofacial deformities [87–89]). Likewise, the NaV1.6 channel is implicated in neural development [90–92].

The ion channels we have simulated, of course, do not represent channels in totality, and our work highlights the need for further contributions to databases such as Channelpedia; Nav1.2 and Nav1.5, for instance, are also strongly implicated in patterning [33] and cellular controls [93–95] but not found in this database. More complete repositories of channel models should facilitate further discoveries based on combinatorial approaches such as this one. However, given the set we have simulated, the relative rarity of channels capable of generating bistable voltage patterns raises the possibility that three- and higher-fold stability is not possible for uncoupled cells in uniform bath conditions without the aid of regulatory processes or modulators that alter the level of ion channel expression or efficacy. This hypothesis would be falsified if a combination of fast ion channels could be found such that the Equation (1) has 5 or more zero crossings with derivatives of alternating sign. If membrane voltage multistability indeed represents a switch affecting cell metabolism, then a deeper search for higher-fold stability would certainly be worthwhile. This, however, may depend on the relative expression levels of different ion channel populations.

Maximizing the number of memories for an arbitrary set of fast ion channel models is equivalent to the following optimization problem: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\underset{\overline{\mathbf{g}}}{ \arg \max}\left|r\left(f\left(V,\overline{\mathbf{g}}\right)\right)\right|$$\end{document}argmaxg¯|rfVg¯|, where f is the right hand side of equation (1), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\mathbf{g}}$$\end{document}g¯ is a vector of maximal conductances, and |r(⋅)| counts the real roots of a function.

However, it is not necessary that individual cells shoulder all the burden for multistability. Given that evidence indicates many locally stable voltages existing concurrently in frog embryos, it is likely that gap-junction coupling plays a much larger role in vivo than in the relatively simple ion channel dynamics we have depicted here. Moreover, the memories we have approached here are fixed points: zero-dimensional subspaces of the phase space. There is certainly no reason that higher-dimensional attractors - stable voltage oscillations, for instance - should not also be considered as candidates for functional memories, although the means of transduction between membrane potential oscillations and cell metabolic processes is less obvious. Finally, it should be noted that in vivo, some cells are able to regulate both internal and external ion levels as well as the complement of ion channels, allowing them to fine-tune the stability points of any voltage memories. Future studies must examine the constraints and capabilities of such self-tuning electric networks in the context of multicellular tissues.

Our analysis has specified some conditions under which stable voltage memories can exist. We have also identified specific ion channels that can form the elements of a synthetic biology toolbox for implementing rewritable bioelectric memories. Moreover, our analyses have yielded a number of surprising findings that should be kept in mind when formulating models of developmental bioelectric patterns or exploiting voltage states in bioelectricity. Foremost is the difference found between ion channel roles in amphibian oocytes and mammalian cells, an interesting standalone result from the perspective of comparative physiology; it may also be a useful guide for future bioengineering and regenerative medicine studies that develop bioelectric strategies in Xenopus embryos and work to move them towards mammalian applications. Following this is the intriguing finding of hyperpolarization-evoked evolution to depolarized memories, which underscores the need for quantitative modeling: it cannot be assumed that induced expression of a normally hyperpolarizing channel will necessarily lead to stable hyperpolarization. A third interesting finding is the ability of Kv channels to disturb bistability, although this was not entirely consistent across the models assessed here.

Our work here was restricted to ion channels with first-order kinetic models for their activation and inactivation variables; further research might include channels like Kv1.5, which cannot be modeled effectively in this way [96]. Additional factors that influence Vmem, to be incorporated into future models, could include additional conductances (leak channels, pumps, gap junctions, calcium stores), modulators of lipid bilayer capacitance, and physical regulators of channel activity such as pressure/tension, temperature, and chemical ligands. It is also crucial to extend this analysis to the study of multicellular bioelectric networks – cells coupled with gap junctions (electrical synapses [97–99]), which will doubtlessly have even more interesting and complex properties than single cell models. Likewise, it will be important to understand the dynamics of interacting membrane domains within individual cells [100, 101], which arise from diverse ion channel protein cargo on lipid rafts as well as from ion sequestration caused by complex cell geometries. Although reagents for functionally addressing such subcellular domains are currently limiting, the development of optogenetics and other high-resolution techniques will soon make it essential to test hypotheses of information coding by distinct voltage states on the surface of single cells (as occurs in the more familiar and linearly-extended neurons).

In general, bioelectric state is an important instructive, tractable regulator in cancer, embryonic patterning, regeneration, and stem cell regulation [9, 12, 36, 42, 45, 102]. Our main points are the following. 1) Bistable switches may be constructed by overexpressing Kir2.1 in amphibian oocytes and Nav1.6 in mammalian cells, and by limiting expression of many of the non-inward-rectifying forms of voltage-gated potassium channels. 2) Cells with strongly expressed fast channels will tend to have memory loci near the reversal potentials for those channels. 3) Memory elements may still be constructed using slow channels, but these should not be expected to have contiguous basins of attraction. These basic principles suggest a number of predictions that will be tested in vivo in developmental bioelectricity experiments to understand endogenous patterning and also serve as guidelines for bioengineering applications. Under the right conditions, many types of somatic cells should be able to operate as physiological memory elements. Thus, the quantitative, predictive understanding of bioelectric dynamics in single cells and multicellular circuits outside of the central nervous system will be an enabling step for new pattern control applications in regenerative biomedicine and synthetic bioengineering.