Multicellular adaptation to electrophysiological perturbations analyzed by deterministic and stochastic bioelectrical models
Cells can compensate a disruptive change in one ion channel by compensatory changes in other channels. We have simulated the adaptation of a multicellular aggregate of non-excitable cells to the electrophysiological perturbation produced by the external blocking of a cation channel. In the biophysical model employed, we consider that this blocking provokes a cell depolarization that opens a voltage-gated calcium channel, thus allowing toxic Ca2+ levels. The cell adaptation to this externally-induced perturbation is ascribed to the multiplicity of channels available to keep the cell membrane potential within a physiological window. We propose that the cell depolarization provokes the upregulated expression of a compensatory channel protein that resets the cell potential to the correct polarized value, which prevents the calcium entry. To this end, we use two different simulation algorithms based on deterministic and stochastic methods. The simulations suggest that because of the local correlations coupling the cell potential to transcription, short-term bioelectrical perturbations can trigger long-term biochemical adaptations to novel stressors in multicellular aggregates. Previous experimental data on planarian flatworms’ adaptation to a barium-containing environment is also discussed.
Subject terms: Ion transport, Developmental biology, Computational biology and bioinformatics
Cell homeostasis results from the coupling between biochemical, biomechanical, and bioelectrical processes in a dynamic environment. Multicellular aggregates can adapt to changes in the surroundings by means of changes in regulatory networks and gene expression. This adaptation is particularly intriguing in those cases where cells and tissues encounter unfamiliar environments and novel challenges1–3, and sudden alterations provoke incremental responses to acquired tolerance4. We explore here how cells could handle the possible transcriptional responses to solve electrophysiological challenges in a window time that could not permit a purely random exploration.
One kind of biological problem-solving concerns the navigation of electrophysiological state space by neural and non-neural systems under normal conditions or when exposed to stressors5–9. The ability to respond to changes to maintain homeodynamic states is relevant not only on the time-scale of an individual embryo or adult, but also on the evolutionary time-scale of genomic responses to modifications in ion channel and pump genes. Degeneracy and complexity are ubiquitous biological properties10 and should play a central role in bioelectrical adaptation, where cells can find solutions to a disruptive change in one ion channel by compensatory changes in other channels. We study a bioelectrically-focused model based on the transcriptional changes that can occur in the ion channels expression after an external perturbation. The basic concepts provide a complementary view to biochemical models11,12 and offer new insights into adaptation mechanisms that do not rely on neural networks13. Understanding such responses is crucial for basic evolutionary developmental biology and the field of basal cognition14. It is especially relevant for biomedical efforts to correct pathological electrophysiological states, because cellular responses to interventions can limit drug efficacy (or, could be exploited to facilitate a new class of drugs that exert long-term repair)15.
The membrane potential is central to bioelectricity because it reflects both the internal cell state and the environmental conditions, including the neighboring cell states16. Our adaptation process considers that a potassium channel which regulates the physiological cell potential is blocked, e.g. by an external divalent cation or a blocking molecule. This perturbation causes a short-term cell depolarization. However, because depolarization phenomena are not rare in cell biology, mechanisms opposing this stressor perturbation should be available. Voltage-gated ion channels are characterized by their rapid responses to changes in membrane potential based on modifications of the selective ion conductances17. Thus, it is plausible that adaptive re-polarizations be evolutionary encoded by the upregulated transcription of compensatory channels. In our model, we analyze how a rescue channel that conducts outward positive currents could be upregulated to compensate for the electrophysiological stress caused by the externally-induced depolarization (Scheme 1). Experimentally, this case can be relevant to planaria adaptation to a barium environment4—a puzzling phenomenon in which planarian flatworms degenerate their heads in the presence of this potassium channel blocker, but then regenerate new functional heads despite its presence.
In non-excitable cells, cell potential windows can be established by multiple ion channels with opposing polarization trends18. Thus, the restoration of the physiological potential after blocking must be feasible by changes in the expression of other compensatory channels. Experimentally, the diversity of these channels allows a multitude of self-regulation phenomena based on electro-genetic feedback processes9,19–22. This fact makes the inverse problem difficult, because different channel conductance combinations should be possible to counter-act cell depolarization. However, the cell would not need to fine-tuning the transcription of every channel protein to establish precise individual conductances, but only to reset forward the physiological cell potential (Scheme 1). To this end, we assume here that the membrane potential acts as a master regulator16,21, both in metabolic and osmotic processes occurring at the single-cell level and in developmental and regenerative processes occurring at the multicellular level4,19–21,23.
Experimentally, bioelectrical cell states can be instructive for biochemical downstream processes5,24–28. Multicellular potentials convey short-term bioelectrical information to long-term transcriptional processes due to the coupling between the electric potentials and the spatio-temporal distributions of signaling ions (e.g., calcium) and molecules (e.g., serotonin)5. In model animals, voltage-sensitive dyes evidence that distinct anterior-posterior morphologies can be obtained by changing the axial electric potential24,25, which is locally regulated by the ion channel conductances. The intercellular gap junctions, which couple cells into electrical networks29,30, can also be important for adaptive phenomena: different planarian head morphologies are observed after junction blocking by octanol26. Thus, our model must couple the single-cell potentials at the multicellular aggregate level.
For the sake of generality, we employ two different simulation algorithms that make use of either a deterministic or a stochastic method in the adaptation process. They are based on a well-established electrophysiological fact: the cell self-regulation provided by the available multiplicity of ion channels31. In particular, the model explicitly accounts for the coupling between the bioelectricity of three channel families and their respective genetic regulator networks, thus offering new insights into adaptation mechanisms that do not rely on neural networks13. Also, we pay attention to the intercellular connectivity because the adaptation involves a community effect at the multicellular scale4,26.
To focus on the adaptive process (see Scheme 1), we have grouped together the model equations used in a final, mathematically-oriented, section. This procedure permits to frame the simulation methods in terms of the biological phenomena studied.
The diversity of channel families observed in the transcriptome of model animals32 suggests that the compensation of a particular blocked channel should be feasible but anticipates also a complex multidimensional space for modeling. While multiple channels may influence the membrane potential, the fact is that only a relatively small number of them are central to each experimental case19,20,33–35. Thus, only a minimal set of generic conductances for the potassium, calcium, and rescue channels (Scheme 1) is considered here. In this way, multiple transmembrane currents are lumped into an operationally tractable but plausible bioelectrical model (Fig. 1).
For the sake of concreteness, we consider that the externally-induced depolarization caused by the potassium (pol) channel blocking provokes the opening of the calcium (dep) channel, increasing the Ca2+ level inside the cell and triggering downstream degenerative processes4. While this depolarization can proceed through the multicellular aggregate later, we will concentrate first on the single-cell polarized (pol) and depolarized (dep) states, suggesting how transcriptional changes can eventually occur along the adaptive process.
Figure 1a shows the I−V curves of two counteracting channels that reproduce the different rectifications observed in potassium and calcium channels. These hyperpolarizing and depolarizing generic channels are characterized by the different rectification parameters explained in the section Methods: biophysical model and simulations. The two generic channels have dep and pol conductances that act to establish the equilibrium potentials and , which correspond to the zero current conditions and , and characterize the dep and pol cell states, respectively16,31. Note that, in this model, the ionic currents and the voltage-gated conductances and of the dep and pol channels depend on the cell potential V; see the section Methods: biophysical model and simulations for details.
Experimentally, pairs of opposing voltage-gated channels can be found in neural cells, human cardiomyocytes, pancreatic islets, and biosynthetic tissues, as discussed elsewhere16. Figure 1a−c show how a third res channel eventually takes the role of the blocked pol channel (Scheme 1). The I−V curve of the res channel shows the general qualitative characteristics of an outward-rectifying ion channel that favor cell repolarization, e.g. by a potassium efflux, over a broad voltage window. We make no reference to the channel selectivity here because we associate the adaptive response to the cell membrane repolarization rather than a particular ionic flux. It is conceivable that the compensating res channel of voltage-gated conductance played a similar role to that of the compensated potassium channel that is blocked. Certainly, additional ion pumps and transporters can also be essential for maintaining ionic homeostasis but relatively small changes in the concentrations of the relevant ionic species can still give significant changes in the cell potential when they occur together with lasting changes in the ion channel conductances16,31. Thus, it seems plausible to associate the main adaptive counteracting actions to ion channels rather than to pumps. To test further this assumption, it was experimentally checked that some particular channels were upregulated in BaCl2-treated animals and that the blocking of these channels by specific agents prevented adaptive transcriptional changes4. We must admit, however, that other changes could occur at the level of physiology, not transcription, and it is likely that additional mechanisms of plasticity are also present4.
Figure 1a corresponds to the initially unblocked potassium channel. At physiological cell polarizations, the res channel current (yellow curve) is lower than the absolute values of the individual pol and dep currents, whose algebraic sum is close the total current (black curve). Figure 1b shows the I−V curves after the blocking of the potassium channel. The resulting decrease in the pol channel conductance causes the cell potential depolarization, thus opening the calcium channel and increasing the calcium concentration in the cell. This deviation from homeostasis provokes the bioelectrical stress that elicits a cell response in the form of the res channel protein upregulation (Fig. 1c), which provides the adaptive response here. The sequence of Fig. 1a−c corresponds to different experimental times, as shown in Fig. 1d. Typical electrical times can be in the range 0.1–1 s for a reference channel conductance and a cell capacitance C = 10 - 100 pF16. On the contrary, transcriptional and translational processes are relatively slow: rate constants of the order of and degradation rate constants of the order of suggest time responses of hours36. Thus, model times of the order of 105 in Fig. 1d, expressed in units, correspond to times in the range 3–30 h.
Because a broad diversity of channels and ion transporters can be found in real cells31,32, Fig. 1a−1c can illustrate a rather general mechanism: the res channel compensates for the blocking of the pol channels and re-establishes the cell polarization state, restoring the cell bioelectrical homeostasis. Note also that a hypothetical blocking of the res channel would cause a new cell depolarization from the re-established polarized state because in this case. Thus, the model suggests that a specific blocking of the res channel could help identifying the particular compensatory channel involved in each adaptation4.
We next describe how the upregulated expression of the res channel (Fig. 1c) could be triggered by the blocking-induced cell depolarization (Fig. 1b). Depolarization processes occur e.g. during certain periods of the cell cycle, which is characterized by oscillatory membrane potentials37. Thus, cells should have feedback mechanisms to compensate for a depolarizing perturbation and maintain a physiological polarization. The ion channel activity can collectively be influenced at the level of mRNAs by means of the membrane potential36, as observed also in neurons9,38. If the cell potential can be coupled to multiple channel regulatory networks16, it may be plausible that the downstream response to an externally-forced depolarization is an increase in the effective transcription rate of the res channel mRNA.
Experimentally, the transcription rate of a channel protein can be regulated by a signaling ion or molecule whose concentration S depends on the cell potential16,36. Here, different events could be involved in the cell depolarization process. The initial blocking of the potassium channel by the Ba2+ ions can initiate this process because of the decreased K+ efflux and then the resulting depolarization can open the voltage-gated Ca2+ ions causing a calcium influx that may contribute further to depolarization. Thus, it seems plausible that the transcription rates of the res channel opposing this depolarization should depend on the cell potential V. In the deterministic model, we introduce an effective negative regulation for the cell potential-dependent transcription rate of the res channel. This approximation lumps a series of intermediate complex contributions from different signaling actors into a phenomenological voltage-dependent rate, which is assumed to be the final outcome of all these contributions.
For the sake of simplicity, we have introduced the effective negative regulation for the cell potential-dependent transcription rate of the res channel as . In this equation, and are reference values for the transcription rate and the concentration, respectively, VT = 27 mV is the thermal potential36, and we have introduced α = 3 in the calculations of Fig. 1. Note that cell depolarization is characterized by less negative values of V (Fig. 1b), thus decreasing the absolute value and increasing the transcription rate . The resulting upregulated expression of the res channel eventually increases its effective conductance, thus repolarizing the cell potential to a physiological value (Fig. 1c). While bioelectrical changes can occur in the range of seconds to minutes, the transcriptional changes associated with the transcription rate increase could be in the range of hours to days, as shown in Fig. 1d39,40.
The above phenomenological equation for describes the coupling between the bioelectrical and transcriptional processes in the deterministic model16. It converts the exceedingly difficult inverse problem of the relationship between genes and cell bioelectricity into the forward problem of keeping a master regulator, the cell potential V, within physiological levels. Admittedly, multiple complexes and intermediate steps may be involved in the genetic regulatory networks of adaptation. In this context, the effective rate constant , which allows increasing the conductance in Fig. 1c, can be seen as a coarse-graining approximation for the complex connection between the bioelectrical and transcriptional spaces. Note that this adaptation is based on the evolutionary available bioelectrical space of the cell and can occur within its living time.
We next extend the single-cell polarization states of Fig. 1 to the multicellular system of Fig. 2, which is interconnected by voltage-gated gap junctions of maximum conductance and average number n ≈ 4 of nearest-neighbor cells36,41. These intercellular conductances allow the current between two neighboring cells i and j, as explained in the section Methods: biophysical model and simulations. We consider that the depolarization process is initiated in the left (anterior) region of the multicellular aggregate. This would be the case if the ion or molecule that blocks the pol channel were not distributed homogeneously in the multicellular system environment. Alternatively, the system could show a left/right morphological asymmetry reflected by an axial distribution of one of the two dominant channels. The latter experimental possibility is simulated here by introducing an inhomogeneous distribution for the dep channel conductance, from to , according to the position of each cell along the left-right axis (Fig. 2).
The axial profiles of V and , averaged over the system cross-section, are also given below each time snapshot. For the sake of concreteness, we consider that cell depolarization is more intense in the left than in the right region of the multicellular aggregate. To simulate this effect, we introduce a linear profile for the dep channel conductance, from to , for the central position of each cell along the left-right axis. The resulting cell potential profile (bottom inset), averaged over the system cross-section, is shown for different simulation times (left scale). As in Fig. 1, the pol channel blocking occurs at time t = 0 while the repolarization due to the res channel upregulation occurs at times of the order of t = 105.
The multicellular system of Fig. 2 consists of a planar monolayer composed of N = 1165 cells initially at steady-state cell potentials , i = 1,…, N. In general, the cell potentials evolve according to the balance between: (1) the relative contributions of the dep, pol, and res conductances and (2) the intercellular junction conductances16. At time t > 0, we set the pol conductance to zero and let the aggregate to evolve. Figure 2 shows the multicellular states obtained for the axial profile of cell potentials (Fig. 2a) and res channel transcription rates (Fig. 2b) as functions of time. Figure 2c gives the vs. V plots obtained with the above phenomenological equation for the res channel as a function of the cell potentials in each snaphot.
Experimentally, the multicellular patterns can be visualized by voltage-sensitive dyes4,5,25,42. Because of the axial profile assumed for the dep channel conductance in Fig. 2, the cell depolarization caused by the pol channel blocking is stronger in the left region than in the right region of the aggregate. The extension of the depolarization towards the right region, which occurs at intermediate times, is opposed by the repolarizing action of the res channel at long times, so that the multicellular adaptation to the perturbation occurs at times t > 105.
We check now the above deterministic model by introducing a stochastic algorithm where individual cells have a more general capacity for acting on the res channel transcription needed to counteract depolarization. In this stochastic model, no phenomenological equation for the relationship between the cell potential V and the rate constant of the res channel is introduced a priori. In addition, we consider the possibility that those stochastic changes that are not able to avoid a significant long-term depolarization lead to cell death. This additional feature allows a better visualization of the multicellular adaptive process.
As in the deterministic model, the depolarization causes a significant deviation from the cell bioelectrical homeostasis. However, we assume now that this external stressor makes the cell to relax the initially precise control of the channel expression, thus allowing changes in the regulation of the res channel following the pol channel blocking. This view implicitly assumes a decreased control of the representational map between cell electrophysiology and transcription5,36, which is a plausible assumption under stress conditions.
The cell exploratory dynamics is regulated by the stress-driven feedback between the depolarized and the polarized cell potentials. Thus, we assume that the stochastic change allowed in the res channel transcriptional rate is proportional to the difference between the current cell potential and the physiological target potential to be restored. The res channel expression is thus iteratively changed to minimize the difference (error signal) between these potentials. This adaptive exploration is implemented in the single-cell stochastic algorithm described in the section Methods: biophysical model and simulations.
Figure 3 considers the case of no intercellular coupling (isolated cells). The simulations show the time-dependent multicellular potential (Fig. 3a) and res channel rate constant (Fig. 3b). The axial profiles of V and , averaged over the system cross-section, are also given. Note the significant number of non-viable cells (white squares) whose membrane potentials went over the maximum depolarization allowed by the stochastic adaptive process and suffered thus non-physiological depolarizations for a long time. The vs. V stochastic plot obtained for the living cells (Fig. 3c) suggests a significant decrease of the res channel transcription rate with the absolute value of the cell potential V, in qualitative agreement with the phenomenological equation assumed in the deterministic model.
The axial profiles of V and , averaged over the system cross-section, are also given below each time snapshot. The non-viable cells (white squares) have local potentials that went over the maximum depolarization allowed during the stochastic adaptive process. As in Fig. 2, we introduce a linear profile for the dep channel conductance, from to , for the central position of each cell along the antero-posterior axis.
Figure 4 reconsiders the simulations of Fig. 3 for non-zero intercellular coupling. In this case, the adaptation is not a single-cell but a multicellular process in the sense that the community effect of the polarized cells in the right region counteracts the depolarization process occurring in the left region of the aggregate. This effect limits both the spatial extension of the depolarization (Fig. 4a) and the increase in the res channel transcription rate (Fig. 4b and c) with respect to those obtained in absence of intercellular coupling (Fig. 3). Note also the relatively low number of non-viable cells obtained in Fig. 4 compared with that of Fig. 3. This result arises from the bioelectrical buffering16,43 caused by the polarized potentials of the cells in the right region, as they can resist better the depolarization wave caused by the left region cells. This bioelectrical effect add to the biochemical buffer effect caused by the now interconnected cell cytoplasm solutions, so that the cells of Fig. 4 behave as a multicellular aggregate to establish and maintain the new adaptive single-cell state of Fig. 1.
To better understand the effects of the intercellular gap junctions in Fig. 4, note that electrically connecting the cells increases the total system capacitance, thus making the multicellular ensemble more resistant to an electrical change (the externally-induced depolarization here). Thus, by avoiding the cell depolarization, the Ca2+ influx through voltage-gated calcium channels should be decreased. However, in the hypothetical case (not considered here) of chemical rather than electrical signals, it might be possible to limit the spread of depolarization during excitotoxic cascades to a small local region by inhibiting junctional conductance. In this case, junction inhibition could make it more difficult the intercellular diffusion of a toxic agent. Thus, the effects due to the intercellular junctions are context-dependent and it should be understood that the simulation results presented here concern only the bioelectrical buffering effect caused by the normally polarized tissue over the depolarized head region.