The interplay between genetic and bioelectrical signaling permits a spatial regionalisation of membrane potentials in model multicellular ensembles

The single cell-centred approach emphasises ion channels as specific proteins that determine individual properties, disregarding their contribution to multicellular outcomes. We simulate the interplay between genetic and bioelectrical signals in non-excitable cells from the local single-cell level to the long range multicellular ensemble. The single-cell genetic regulation is based on mean-field kinetic equations involving the mRNA and protein concentrations. The transcription rate factor is assumed to depend on the absolute value of the cell potential, which is dictated by the voltage-gated cell ion channels and the intercellular gap junctions. The interplay between genetic and electrical signals may allow translating single-cell states into multicellular states which provide spatio-temporal information. The model results have clear implications for biological processes: (i) bioelectric signals can override slightly different genetic pre-patterns; (ii) ensembles of cells initially at the same potential can undergo an electrical regionalisation because of persistent genetic differences between adjacent spatial regions; and (iii) shifts in the normal cell electrical balance could trigger significant changes in the genetic regulation.

The multicellular patterns characteristic of processes such as morphogenesis, regeneration and carcinogenesis have traditionally been described on the basis of chemical signals and concentrations gradients. However, the transport of the ions and signalling molecules which affect downstream biochemical cascades and transcriptional processes is also influenced by the electrical cell state and intercellular coupling1234567891011. The purpose of this study is to propose a highly idealised model for the integration of genetic expression and bioelectrical signals in the case of non-excitable cells, from the single-cell level to the multicellular ensemble.

Genetic networks influence and in turn may be influenced by bioelectrical signals. The proteins forming the ion channels in the cell membrane and the intercellular gap junctions regulate the bioelectric signals. These signals influence the genetic pathways via the unidirectional transport and spatial accumulation of calcium and different signaling molecules, electrically-induced conformational changes in the voltage-sensing domains of membrane proteins, and the activation of voltage-gated channels123456. In turn, these proteins are regulated by the genetic and epigenetic networks that allow transcriptional control78. This bidirectional flow is not simply described by specifying the protein cellular levels because ion channels and gap junctions are gated post-translationally by the particular dynamics of the cell environment17.

The interplay between genetic and bioelectrical signals is relevant to multicellular patterning. In Xenopus embryogenesis, the spatial regionalisation of cell electrical potentials specifies a prepattern for craniofacial patterning genes910. The disruption of the normal regionalisation of potentials can affect the expression patterns of genes critical for the patterning of the face10. Other mechanisms that transduce alterations of the cell bioelectrical states into downstream transcriptional changes include the transporters of calcium and signaling molecules (serotonin, butyrate) which are regulated by cell membrane potentials1. In particular, the voltage-gated sodium11, potassium5, and calcium2 channels are known to influence gene regulation and cancer. The potential biomedical opportunities of reprogramming cells and tissue patterning via bioelectrical pathways have recently been reviewed1.

The dominant biochemical approach emphasises reaction-diffusion processes1213 and disregards often the bioelectrical signals that influence the single-cell cycle and may provide positional information in embryogenesis, regeneration and cancer171415161718. We have developed recently a simple theoretical scheme for describing multicellular ensembles using basic concepts from membrane and ion channel biophysics19. The model ignores diffusion-reaction processes202122 by focussing on collective electrical states and multicellular patterning. We consider now the case of voltage-dependent genetic regulation, from the local single-cell level to the long range multicellular ensemble. To this end, we study the electrically-induced changes in genetic regulation using a network of intercellular gap junctions working as non-linear bioelectrical units.

The single-cell genetic regulation considered here is based on mean-field kinetic equations for the mRNA and protein relative concentrations. The transcription is assumed to depend on the cell potential, which is modulated by the voltage-gated cell ion channels and the intercellular gap junctions. We consider simple physical mechanisms to translate single-cell bioelectric states into multicellular regionalisations that can be observed using voltage-sensitive dyes232425. While the importance of chemical signals and concentrations gradients in the genetic regulation and establishment of biological patterns is well documented12132126, the regionalisation of single-cell polarisations is not well understood and contributes to the spatial distribution of signalling molecules in the ensemble142728.

The cell resting potential is usually defined as the negative potential difference that exists between the cell cytoplasm and the extracellular environment at zero total current1929. Differentiated cells tend to be more polarised, showing a high absolute value of the negative cell potential, whereas proliferating cells (embryonic, stem and tumor cells) are depolarised, showing a low absolute value of this potential171718. However, the spatio-temporal maps of electrical potential in multicellular systems should be regarded as an ensemble characteristic rather than a simple aggregate of single-cell values1930.

Bioelectrical potentials are involved in the intercellular communication and collective information processing required for multicellular processes such as developmental patterning and carcinogenesis131. Experimentally, these potentials can be resolved in vivo by using appropriate voltage-reporters (see Fig. 4 of ref. 1 for voltage-sensitive fluorescent dyes revealing spatio-temporal patterns of bioelectrical gradients in vivo., Fig. 1 of ref. 31 for membrane potential gradients across the anterior-posterior axis of planarian flatworms, Fig. 6 of ref. 23 for a voltage reporter assay demonstrating long-term changes of the bioelectrical connectivity in octanol-exposed planaria, Figs 2 and 8 of ref. 24 for the experimental distribution of membrane potentials in Drosophila ovarian follicles, and ref. 25 for methods of visualising gap junction networks). The spatio-temporal regionalisation of cell polarisation is a crucial step to patterning and tumorigenesis because it regulates the transport of signalling molecules and ions such as serotonin, butyrate, and calcium which influence different biochemical pathways. Changes in the cell potential are usually transduced into downstream cascades and transcriptional processes (see Fig. 3b of ref. 1 for different mechanisms transducing modifications of the cell potential into downstream cascades and transcriptional changes).

The electrically-driven transport of signalling molecules through the intercellular gap junctions regulates the differences between the left and right sides in the Xenopus embryo (see Fig. 2 of ref. 27 for the case of morphogen electrophoresis through these junctions and Fig. 3 of ref. 32 for serotonin signaling in the patterning of the left-right axis in chick and frog embryos). Also, the transport to and accumulation in polarised cells of serotonin, together with the membrane potential regionalisation, is involved in ectopic eye innervation (see Fig. 6 of ref. 28 for the induction of nerve growth via modulation of host resting potential). In planaria, the action of depolarising ion channels and gap junctions results in the spatial map of electric potentials that regulate regeneration (see Fig. 4 of ref. 1 for the influence of bioelectrical signals on head regeneration). The membrane potential regulates also the calcium flux, intracellular concentration, and intercellular transport33. In turn, the calcium signaling influences gene expression1. Recently available experimental tools for monitoring and modifying membrane potentials have shown the complex feedback between bioelectrical and genetic signals191023.

Bioelectric patterning has also been studied during oogenesis in Drosophila ovarian follicles using fluorescent indicators and inhibitors. The spatial distributions of membrane potentials, intracellular pH and specific membrane channel proteins modulate the observed developmental processes. Significant similarities between the membrane potential patterns and the spatial activity of voltage-gated calcium channels have been found, suggesting a mechanism for transducing bioelectric signals into cellular responses24.

The electric potential-dependent rates of serotonin influence also the transcriptional downstream responses observed in the metastatic behaviour of melanocytes (see Fig. 6 of ref. 14 for a mechanism transducing voltage changes into transcriptional cascades with the serotonin transporter SERT) and the electrically-driven transport of butyrate appears to be involved in tumour suppression in Xenopus embryos (see Fig. 8 of ref. 14 for the particular bioelectric mechanism). Intercellular communication33 is also crucial: a decreased gap junction activity is associated with the initial stages of some cancers in animal models and humans (see Fig. 1 of ref. 34 for the formation of induced tumour like structures that are gap junctionally connected to the host and refs 35 and 36 for the characterisation of defective gap junctional intercellular communication in carcinogenic processes). However, the role of gap junctions depends on the states of the neighbouring cells and the particular signals to be transferred and this context-dependent behaviour constitutes a problem for practical applications3537.

The above experimental facts suggest that multicellular electrical regionalisation should be coupled with local genetic regulation. Figure 1 shows a simple scheme illustrating the two feedback scenarios considered for the genetic and bioelectrical descriptions together with the relevant biophysical equations. The simplest modeling of genetic regulation is based on the mean-field equations3839404142434445. As to the bioelectrical regulation, we assume that a particular ion channel in the cell membrane, the voltage-gated outward-rectifying channel, is formed by a specific protein whose concentration p depends on the intracellular concentration m of the corresponding mRNA. The conductance Gout regulated by this channel protein contributes to the total membrane conductance, thus allowing the modulation of the cell potential V. In turn, the channel protein concentration p in the cell is regulated by V because of the potential-dependent transcription assumed in Fig. 1. Note that we consider relative values for p and m, calculated with respect to a particular reference value, because the absolute concentrations should depend on the particular biochemical system considered.

The highly non-linear characteristics of the equations describing genetic383940414245 and bioelectrical networks192229 suggest significant feedback scenarios. The cell potential may influence the rate of gene transcription into mRNA, eventually resulting in changes of the rate of mRNA translation into proteins that should impact in the function of the protein ion channels. The effects of the membrane potential on gene expression are indirect and mediated by the following processes which are regulated by the potential difference V between the intracellular and extracellular media: (i) the calcium cell entry and subsequent regulation of genetic pathways256, (ii) the transport of different signaling molecules to the cell123741, (iii) the electrically-induced conformational changes in the voltage-sensing domains of membrane proteins14, and (iv) the activation of voltage-gated potassium54647 and calcium26244849 channels. These experimental facts constitute functional links between the genetic and bioelectric descriptions, establishing clear qualitative connections between the model and experimental results.

The existing experimental data for processes (i–iv) suggest an exponential dependence for V in most cases. Under quasi-equilibrium conditions, the intracellular concentrations of calcium and other positively charged molecules should increase exponentially with the absolute value of the negative cell potential |V|. Indeed, because the gradient of electric potential acts as a driving force for cell entry in processes (i) and (ii), the integration of the Nernst-Planck flux equation under quasi-equilibrium conditions gives an exponential dependence29 of the species concentration on the potential V. If we assume Hill kinetics for the dependence of the transcription process with the calcium concentration, the effective value of rm in Fig. 1 should incorporate then the appropriate exponential functions for the cases of positive and negative regulation. Note also that the electrically-induced changes in the protein voltage-sensing domains (process (iii)) and the activation of voltage-gated channels (process (iv)) should also depend exponentially on the cell potential because of the respective Boltzmann factors that govern these processes192938. Typically, the Boltzmann factors are exponentials of the electrical energy term, which includes the cell potential, divided by the thermal energy29.

Because the membrane ion channels controlling the potential V are proteins, they can be regulated by the protein concentration p of Fig. 1. This regulation is complex and may be due to structural changes such as phosphorylation, involving different spatial and time scales. In our phenomenological model, we incorporate the effect of p on the ion channel conductance by considering that the protein inserts into the cell membrane to form an outward-rectifying ion channel of conductance Gout. This fact implicitly assumes that the ion channel response occurs on a time scale much faster than the gene expression, which is a realistic condition383940. The above assumptions are simple and permit to introduce the genetic control in the bioelectric regulation of Fig. 1 using basic biophysical concepts.

Considering again a Hill kinetics scheme, the dimensionless conductance should follow the dependence on the protein concentration p shown in Fig. 1, where is the maximum value of this conductance and the reference concentration p0 corresponds to the conductance . Note that Gout is an output parameter of the genetic regulation but an input parameter for the bioelectric regulation of Fig. 1 because the cell potential V is regulated by the outward- and inward-rectifying channels of conductances Gout and Gin in the bioelectrical model19 of Fig. 1.

There are many ion channels and pumps relevant for cell biology222950. We consider only two generic voltage-gated channels because these channels are central to cell signalling2951 and provide useful qualitative results with moderate mathematical complexity1952535455. More complete computational models for the electrical conductance regulation of the resting membrane potential have also emphasised the dominant contributions of inward and outward-rectifying channels5153. Increasing the complexity of the theoretical model by adding more channels and pumps should lead to additional feedback mechanisms that will invariably obscure the primary goal of this study: to show that bioelectrical phenomena occur at a different level than cell genetics and that the view of channel function as a direct gene product should be revised (see ref. 7 and Fig. 1 here).

Although multistability is inherent to many biological networks, membrane potential bistability52 is not a necessary condition for an efficient coupling between the genetic and bioelectrical descriptions but it allows a clear distinction between single-cell states in the spatial pattern of potentials19. It should be noted, however, that electrical bistability is crucial for the dynamics of neural cells and that long-term changes in the neuron function are influenced by the interplay of membrane potential pulses and gene expression39. Therefore, it is of biological interest to explore the consequences of bistability for the case of non-neural cells. Experimentally, bistability has been observed in lysenin channels inserted into lipid bilayer membranes54, the membranes of saccular hair cells in the green frog55, and skeletal and mouse lumbrical muscle cells53. Recently, resting potential bistability has been systematically studied in the modelling of a wide variety of amphibian embryos and mammalian cells using the Channelpedia ion channel database56. These biologically-oriented simulations emphasised the importance of voltage-gated channels for mammalian channels which can stably maintain one of several resting potentials56.

The inward-rectifying channel1929 gives low outward currents for potentials V > Ein, where Ein is the channel equilibrium potential, and large inward currents for V < Ein. On the contrary, the outward-rectifying channel1929 gives low inward currents for V < Eout, where Eout is the channel equilibrium potential, and large outward currents and for V > Eout. The potentials Eout and Ein are constant provided that the ionic concentrations are kept approximately constant in the extracellular environment and the intracellular solution1957. Note in this context that the number of ions to be transferred in order to set up typical potential differences is very small compared with the total number of ions in the cell58.

Consider now the genetic regulation. The dynamics of mRNA transcription is complex and can be affected by heterogeneity effects445960. The values of the rate constants in Fig. 1 are difficult to establish because the transcription, translation and degradation processes depend on multiple factors383942434461. The transcription rate constants38424361 are in the range 0.1–5 min−1 while the translation rate constants384243 are in the range 0.1–2 min−1. The time scales may vary between minutes and hours for mRNA and protein degradations, which gives rate constants3842434461 in the range 0.003–0.1 min−1. We have considered the intermediate, biologically reasonable values and dm = 0.05 min−1 = dp to obtain the steady-state values of the mRNA and protein relative concentrations of Fig. 2a. We introduce the reference value p0 = 15 in the equations of Fig. 1 to obtain the significant changes of the channel conductance Gout on the cell potential V of Fig. 2b.

The numerical solutions for V obtained from the cell potential equation of Fig. 1 are shown in Fig. 2c as a function of the equilibrium potential Ein for different conductance ratios . The solutions correspond to those points where the horizontal line Ein = constant intersects the different curves in the Ein − V plane. For some values of the system parameters, there are three numerical solutions corresponding to the polarised (high value of |V|) normal state, the depolarised (low value of |V|) abnormal state, and the unstable (intermediate value of |V|) state. The model cell acts as a dynamical system undergoing transitions between the above bioelectric states. These transitions are regulated by the relative contributions of the inward (cell polarising) and outward (cell depolarising) channels to the total membrane conductance1952.

Figure 2c shows that an increase in the conductance ratio decouples V from the normal polarised value Ein allowing the cell potential to enter the bistability regime where abnormal depolarised values can be attained. Since the electrical conductance can be gated post-translationally730 e.g. by channel blocking19, the ratio in Fig. 2c could be decreased externally, reverting the cell potential to the normal value. Alternatively, this effect could also be achieved by changing the external concentrations to shift the equilibrium potential29 Ein, as shown in Fig. 2c. We note that up-regulation of sodium channels has been related to some types of cancer626364. The up-regulation of the outward channels and down-regulation of the inward channels can be simulated by increasing the ratio in the model19.

In summary, Fig. 2a–c show the feedback scenario between the cell potential and the protein forming the outward-rectifying ion channel. The value of |V| may increase (positive regulation) or decrease (negative regulation) the transcription of the mRNA that encodes this channel protein (Fig. 2a,b). In turn, the protein channel conductance can modulate the cell potential (Fig. 2c).

The cell is not isolated but connected to other cells in a multicellular system65. Bioelectrical signals extend beyond the single-cell states because of the intercellular channels (gap junctions) that allow the transfer of electric currents and signaling molecules between neighbouring cells. Decreased intercellular communication enhances autonomous cell behaviour and has been related to tumorigenesis363765. We assume that because of the low frequency signals characteristic of non-excitable cells, the conductive coupling dominates over the capacitance coupling19. Figure 3a,b show the intercellular coupling model together with the relevant biophysical equations that permit multicellular electrical states. The coupling is accounted for by effective gap junctions of conductance Gij which can be due to the connexin protein family35366667. Despite the high the number of gap junctions per cell, only a very small fraction of open channels participate effectively in intercellular coupling275668. We assume that the effective gap junction conductance per cell is of the same order of magnitude as the total cell conductance due to the inward and outward-rectifying channels. This condition assures that the two conductances are relevant in the model equation of Fig. 3b. Note that too low gap junction conductances would give essentially isolated cells while too high values of these intercellular conductances would produce an isopotential ensemble with no spatial regionalisation. The introduction of the dimensionless conductances (Fig. 2c) and Go/Gin(caption of Fig. 3b) allows showing the qualitative trends of the model by comparing the relative values of the gap junction and ion channel conductances.

We consider a system of N = 2524 cells initially at potentials Vi(t = 0), i = 1, …, N. The initial concentrations of protein pi(0) and mRNA mi(0) are those obtained from the steady-state solution of the genetic regulation equations (see caption of Fig. 1) for Vi(0). Then, we let the system to evolve with time, solving the N equations of Fig. 3a for Vi(t) under the different conditions considered in each case study. The time evolution of these single-cell potentials induces changes in the transcription rate constant rm of cell i (see Fig. 1). The feedback mechanism of Fig. 1 couples the genetic and bioelectric changes over the multicellular system because the outward-rectifying channel conductance depends on the cell protein concentration (Fig. 2a,b).

To better understand the ensemble dynamics, we note that there are two characteristic times in the single-cell and multicellular schemes of Fig. 1 and 3. The electrical relaxation is fast because of its relatively small characteristic time13 Ci/Gin = 0.1 s for Gin = 1 nS and Ci = 100 pF. On the contrary, transcription and translation rate constants in the range 0.1–1 min−1 give characteristic times between 1 and 10 min while degradation rates constants in the range 0.003–0.1 min−1 give times between 0.1 and 5 h38424357. Diffusion is not incorporated in the model, but it should give a characteristic time L2/D greater than 1 h for an ionic diffusion coefficient D = 10−10 m2/s and a distance L = 10−3 m, which corresponds to about 100 cell diameters19. Thus, the longer characteristic time is that of the genetic regulation if diffusion processes are ignored. These processes may significantly increase the time response if they are incorporated in the model192022. Recently, a finite volume method simulator accounting for diffusion and ionic concentration changes has been proposed by Pietak and Levin to study the spatio-temporal dynamics of bioelectric patterns22.

Genetic processes occur on a time scale much longer than electrical relaxations for the biologically plausible parameters introduced here. The significantly different time scales obtained suggest the following procedure for the system simulation. A single-cell in the ensemble should be initially in one of the bioelectric states of Fig. 2c determined by the value of V and the protein concentration p in Fig. 2a,b. Because of the coupling provided by the gap junction conductances of Fig. 3a, the other cells can rapidly change the cell potential V to a new value and then the protein concentration p will relax slowly to a new value compatible with this potential. Since the electrical relaxation is much faster than the genetic relaxation, we can obtain first the steady-state values of V and take them as constants in the calculation of the p values. This procedure allows formally decoupling the time-dependent equations for the bioelectrical and genetic regulations of Fig. 1.

Consider now the ensemble dynamics for different case studies. At time t = 0, the cell potentials in the two small patches of Fig. 4a are in a region corresponding to the depolarised potential in Fig. 2c while the rest of the cells in the system have a polarised potential. For the case of positive regulation in Fig. 1, the outward-rectifying channel conductance is high (Fig. 2b) and then the depolarised cell state is dominant in the bistability region of Fig. 2c. Thus, the cell electrical coupling forces the depolarised state over the whole ensemble of Fig. 4a after a short time. This effect shows clearly the ensemble nature of the electric potential in this model because the final isopotential state is a consequence of the concerted collective behaviour30.

Due to the positive regulation and the low single-cell potentials attained in Fig. 4a, the genetic relaxation gives low values of the protein concentration at sufficiently long times (Fig. 4b; see Fig. 2a for the correspondence between p and V). As expected, the electrical relaxation is fast compared with the protein concentration changes, as shown in Fig. 4c for the time dependence of V and p.

Figure 5a considers the case of negative regulation and different genetic characteristics between the two small regions and the rest of the system. As in Fig. 4a, the cell potentials are initially depolarised in these regions while they are polarised in the rest of the ensemble. For the case of the negative regulation, however, it is the polarised cell state that is dominant (Fig. 5a) rather than the depolarised one (Fig. 4a). Thus, the electrical relaxation gives now a polarised system after a short time. The subsequent genetic relaxation gives again very low values of the protein concentration (Fig. 5b,c) because of the negative regulation and the high values of |V| attained by the single-cell potentials (see Fig. 2a). Figure 5a,b show that a slightly different genetic pre-pattern established over adjacent spatial regions can be smoothed out by bioelectrical signals, which suggest that endogenous bioelectrical networks may influence patterning during development and regeneration7. Note also that gene expression is intrinsically probabilistic4345 and Fig. 5a,b suggest that the intercellular coupling may favour the stabilization of this expression over adjacent spatial regions.