Cell Systems Bioelectricity: How Different Intercellular Gap Junctions Could Regionalize a Multicellular Aggregate

Electric potential patterns across tissues are instructive for development, regeneration, and tumorigenesis because they can influence transcription, migration, and differentiation through biochemical and biomechanical downstream processes. Determining the origins of the spatial domains of distinct potential, which in turn decide anatomical features such as limbs, eyes, brain, and heart, is critical to a mature understanding of how bioelectric signaling drives morphogenesis. We studied theoretically how connexin proteins with different voltage-gated gap junction conductances can maintain multicellular regions at distinct membrane potentials. We analyzed a minimal model that incorporates effective conductances ultimately related to specific ion channel and junction proteins that are amenable to external regulation. We also consider a bioelectrical relationship between the connexin composition of the intercellular gap junction and different stages of cancer.

Electric potential distributions can act as instructive pre-patterns for development, regeneration, and tumorigenesis in cell systems. The biophysical states influence transcription, proliferation, cell shape, migration, and differentiation through biochemical and biomechanical downstream transduction processes. A major knowledge gap is the origin of spatial patterns in vivo, and their relationship to the ion channels and the electrical synapses known as gap junctions. Understanding this is critical for basic evolutionary developmental biology as well as for regenerative medicine. We computationally show that cells may express connexin proteins with different voltage-gated gap junction conductances as a way to maintain multicellular regions at distinct membrane potentials. We show that increasing the multicellular connectivity via enhanced junction function does not always contribute to the bioelectrical normalization of abnormally depolarized multicellular patches. From a purely electrical junction view, this result suggests that the reduction rather than the increase of specific connexin levels can also be a suitable bioelectrical approach in some cases and time stages. We offer a minimum model that incorporates effective conductances ultimately related to specific ion channel and junction proteins that are amenable to external regulation. We suggest that the bioelectrical patterns and their encoded instructive information can be externally modulated by acting on the mean fields of cell systems, a complementary approach to that of acting on the molecular characteristics of individual cells. We believe that despite the limitations of a biophysically focused model, our approach can offer useful qualitative insights into the collective dynamics of cell system bioelectricity.

Patterns in biology result from the interplay between biochemical [1] and biomechanical [2] signals that establish spatio-temporal correlations in multicellular aggregates. In addition, a complementary distributed control based on the cell membrane bioelectricity is also emerging [3,4,5,6,7]. The cell potential V, defined as the electric potential difference between the cell inside and the external microenvironment, regulates the distribution of signaling cations and biochemical messengers. In turn, V is also modulated by these distributions because of their influence on the transcription of ion channel proteins, which constitutes a bioelectrical feedback control at the single-cell scale [3,4,5,8].

At the multicellular scale, two general mechanisms for cells coordination are the extracellular signaling and the intercellular gap junctions. The connexin proteins forming the gap junction channels allow the latter communication by the exchange of biochemical and electrical signals between adjacent cells [5,7,9,10,11,12,13,14]. Different junction types can be identified on the basis of their constituent proteins and distinct voltage-gated conductances. In the case of bioelectrical coupling, it has been shown that multicellular mean fields can collectively influence downstream transcriptional processes via the spatio-temporal maps of biochemical messengers which are dictated by the local electric potentials [3,4,5,7,15]. Bioelectric signaling has now been shown to be instructive for morphogenesis: specific changes in membrane potential patterns can induce new organ formation such as eye, trigger the regeneration of whole appendages, repair the brain after mutation or the exposure to teratogens, and normalize tumors [16,17,18]. While much effort has gone into identifying the transduction machinery by which voltage changes influence second messenger pathways [9], it is much less clear what factors control the shape, size, and location of specific bioelectric domains that serve as pre-patterns laying out the face, brain, and other organs [19,20].

We aim here at theoretically exploring how the spatial distribution of different gap junctions can maintain regions of cells in distinct bioelectrical states, thus regionalizing tissue into specific patterns. We consider different scenarios that concern the single-cell bioelectrical characteristics, the different gating and coupling degree of the junctions, and the relative sizes of the regions forming the multicellular aggregate. The experimental motivation for the simulations is that electric potentials can act as instructive pre-patterns for morphogenesis [21,22,23] and tumor initiation and normalization [14,24,25], as suggested by Levin and others. Our bioelectrical approach can also be applied to interconnected smooth muscle cells that propagate signals over relatively long distances [26]. In addition, it permits both stationary and oscillating patterns [5,27], thus providing useful insights into a variety of multicellular systems including syncytia of astrocytes [4,28].

Note that some tumor and anti-tumor approaches based on connexin targeting are closely related to the bystander community effect that results from the gap junction intercellular coupling [14,29,30,31,32,33,34,35]. In particular, the differential expression of connexin subunits Cx46 and Cx43 between the cancer stem cells (CSCs) that contribute to tumor propagation and non-CSCs has been connected to distinct membrane potentials and cell differentiation processes [14]. The present simulations constitute a significant extension of those of reference [36] to the case of cell systems with different connexin proteins and gap junctions. We consider only the junctional role of connexin proteins as transducers of electrical signals, disregarding additional biochemical and biomechanical effects that can be relevant in other contexts.

The single-cell model has been experimentally motivated and described with detail previously [5,36]. It considers two generic populations of voltage-gated ion channels of maximum conductances Gpolo and Gdepo that act to establish the polarized and depolarized single-cell states. While a wide variety of channels can exist in a cell [37], voltage-gated channels are crucial to bioelectricity because they allow the counteracting dynamics typical of many physiological functions including pacemaking, neural slow-wave oscillations, circadian clocks, and bioelectrical oscillatory phenomena in artificial tissues [5,11,12,37,38]. Here, we introduce two phenomenological equations that can qualitatively describe the experimental I–V curves characteristic of voltage-gated channels [37,39]:(1)Idep=Gdep(V−Edep)=Gdepo(V−Edep)1+exp[−z(V−Vth)/VT] (2)Ipol=Gpol(V−Epol)=Gpolo(V−Epol)1+exp[z(V−Vth)/VT]

Note that, in Equations (1) and (2), we have made explicit the dependence of the voltage-gated conductances Gdep and Gpol on the cell potential V [39]. Typical order of magnitude values are Gdepo/Gpolo=1 for the ratio of conductances, z = 3 for the effective gating charge of the channel, and Vth=−VT=−(RT/F)=−27 mV for the threshold potentials, with R the gas constant, T the temperature, and F the Faraday constant [37,39]. The values introduced here for the cell polarized (pol) and depolarized (dep) equilibrium potentials are Epol=−55 mV and Edep=−5 mV [39]. Equations (1) and (2) permit to establish a bi-stable bioelectrical regime for an isolated cell, as previously described [36,39]. Note that, as shown in Figure 3a of [9], terminally differentiated somatic cells such as muscle cells tend to be polarized—they show negative membrane potentials—while embryonic, stem, and tumor cells tend to be depolarized. Thus, independently of the particular ionic species, gradients, and transporters that are involved in each case, we considered that the above potentials are representative of the pol and dep states, respectively [4,5].

Multicellular states depend not only on the single-cell channel conductances ratio Gdepo/Gpolo and concentration-dependent potentials Epol and Edep but also on the intercellular gap junctions [5,39]. The bioelectrical states of neighboring cells are coupled by these conductances whose local states define the dynamic connectivity characteristic of multicellular patterns [5,39]. We aim here at extending our previous theoretical approach [36] to the biologically relevant case [14,26,40,41] of two different junction types that can be experimentally identified on the basis of their constituent family proteins and conductances. Note that vertebrate gap junctions are usually voltage-gated and thus their conductances depend on the intercellular potential difference rather than on the absolute potentials of the adjacent cells. In particular, high conductances are obtained when the two neighboring cells have similar cell potentials while low conductances are found when these cells are at different potentials [4,5,40,41].

We will not address here the complex route between connexins, connexons, and gap junctions. Instead, we concentrate on the bioelectrical characteristics of voltage-gated gap junctions. Figure 1 shows the typical junction conductances observed for two homotypic Cx43/Cx43 and Cx45/Cx45 channels together with the case of the heterotypic Cx43/Cx45 channel [41]. Conductance changes in ion channels and junctions can be ascribed to voltage-gated structural rearrangements of the protein subunits along the pore. However, pH and ionic concentration effects are also important because they can modify the state of the pore charge moieties [37], as shown in Figure 2 for the case of a biomimetic nanopore [42,43]. Note that these pH-dependent charge groups are also present in connexin proteins.

Artificial biomimetic nanopores functionalized with amino acid chains show a qualitatively similar behavior, not only because of symmetric and asymmetric cross-sectional areas [42] but also because of the different charge distributions that can be externally established at the pore mouths [43]. Figure 2 shows the case of a symmetric cigar-shaped nanopore where, due to the particular pore geometry, the carboxylic acid and amino groups of lysine are concentrated close to the pore tips. In this case, the conductance vs. voltage curves measured at 0.1 M KCl salt concentration can be made symmetrical or asymmetrical according to the pH values of the left (L) and right (R) solutions.

The conductance gating of Figure 1 can be qualitatively reproduced by the phenomenological equation:(3)Gij=G1+G2{1+exp[−(Vj−Vi−Vth,i)/V0,i]}{1+exp[(Vj−Vi−Vth,i)/V0,j]} where (Vj−Vi) is the intercellular electric potential difference between two adjacent cells j and i of potentials Vj and Vi, Vth,j and Vth,i are the respective protein subunits threshold potentials, and the reference potentials V0,j and V0,i account for the different symmetry and sharpness of Gij observed in Figure 1. The conductances G1 and G2 are related to the minimum and maximum values of Gij. The conductance of Equation (3) allows to modulate the multicellular network of coupled cells at both the single-cell transcriptional level and the intercellular post-translational level, as explained previously [5,36].

Figure 3 (top) shows that Equation (3) qualitatively reproduces the experimental symmetrical and asymmetrical shapes of the different gap junctions Gij of Figure 1 [40,41]. Note that the values of Gij are scaled to a reference junction conductance Gref to better show the model trends. The effect of decreasing the conductance parameters G1 and G2 to the values G1’=0.75G1 and G2’=0.25G2 is also shown (dashed lines).

Figure 3 (bottom) schematically shows the two cell regions with the distinct families of connexin proteins used in the simulations. We assume that these regions are initially in different single-cell polarization states and study if the inner patch of abnormally depolarized cells (low absolute value of V) can resist normalization against the surrounding bulk of normally polarized cells (high absolute value of V). Note that all cells in the two regions share the same single-cell bioelectrical parameters, the only differences being their initially distinct polarized or depolarized stable states [5,39] and the region-dependent connexins expressions (Figure 3).

The intercellular currents through the gap junction conductances of Equation (3) and Figure 3, together with the single-cell channel currents Ipol and Idep of Equations (1) and (2), establish the individual cell potentials Vi(i = 1, 2, …, N) that change with time t according to the equation [5]:(4)CidVidt=−Ipol−Idep+∑j ∈nearestneighborsGij(Vj−Vi)

We restrict the sum to the nearest neighbors, assume an average number of one effective junction between two nearest neighbor cells, and introduce the single-cell capacitance Ci=100 pF. For a reference conductance Grefo=1 nS, the single-cell electrical time is τ=Ci/Grefo=0.1 s only but the cell systems time can increase to 10–100 s for an aggregate composed of hundreds of cells [4,5].

Equation (4) gives multicellular mean fields that attempt to force the same polarization state for all cells in a highly connected region [4,5]. Experimentally, this result permits to encode instructive bioelectrical information over spatial regions of cells on the basis of their particular polarization states [3,4,5,9,15]. While the non-excitable cell dynamics of Equation (4) is slow compared with action potentials in neurons, the concepts of polarized and depolarized cell states, network connectivity, and multicellular patterns are common to both cell systems despite the differences in the information processing times.

We will address here quasi-stationary patterns only, but the case of oscillatory regions modulated by the intercellular connectivity is described elsewhere [27]. Additionally, we have not considered the transcriptional regulation of the channel conductances because we studied it with detail previously [5,44]. This downstream regulation gives comparatively slow responses of the order of hours when transcriptional [44] and diffusional [39] processes are incorporated in the simulations. Experimentally, bioelectrical pre-patterns constitute templates for the spatio-temporal distributions of signaling ions and molecules that regulate the subsequent downstream biochemical processes [3,4,7].

We aim at providing qualitative insights into the fate of an abnormally depolarized inner patch initiated within a normally polarized multicellular aggregate when the intercellular coupling is changed. To this end, we describe the bioelectrical patterns that result from different decreases of the junction conductance (Figure 3, top). We might anticipate that weakening the intercellular coupling should favor the electrical regionalization initially established in Figure 3 (bottom) because of the decrease in the interfacial connectivity between the two regions. However, weakening the intercellular coupling also causes a decrease in the internal connectivity of each region and thus a decrease in the community effect that keeps the patch in the depolarized state. Thus, a delicate balance between these two opposing trends will eventually dictate the cell system fate. In addition to small cell systems, the modification of the intercellular coupling can also suggest mechanisms for establishing and modifying target patterns in tissue engineering [12,38].

Reprogramming of single-cell bioelectrical states may benefit from the mean field effects allowed by the intercellular coupling of multicellular aggregates [3,4,5,9]. Experimentally, adjacent domains of cells in different polarization states are characteristic of developmental and tumorigenic processes [3,4,9,45,46]. They can be established by local changes in gene expression [45] and spatial heterogeneities in the signaling molecules concentrations [46]. In our case, we consider an inner patch of initially depolarized cells surrounded by a bulk of polarized cells (Figure 3, bottom) at time t = 0. Subsequently, we simulate the multicellular aggregate changes that result from different junction conductance decreases affecting the system connectivity. To this end, we use Equations (1)–(4) and Figure 3.

In our case, the bioelectrical heterogeneity of Figure 3 (bottom) might be due to a distinct complement of ion channel proteins giving different conductances Gdepo and Gpolo as well as to distinct ionic concentrations giving different equilibrium potentials Edep and Epol in the two regions [5,39,47]. Note that both cases are fully accounted for in the model Equations (1) and (2). However, we will consider here that all cells have the same single-cell parameters and take profit of the polarization state bi-stability [5,39,47] to assume that the individual cells in the two regions of Figure 3 are initially at different polarization states. In this way, we can ascribe the results obtained in Figure 4 and Figure 5 to the changes in the intercellular coupling exclusively because all individual cell parameters remain identical. Figure 4 considers the effect of different G2 decreases at fixed G1 decrease while Figure 5 considers the effect of different G1 decreases at fixed G2 decrease. In each case, we show the cell system changes (Figure 4 and Figure 5, left) together with the electric potential changes for three cells located at the patch, the surrounding bulk, and the interfacial region (Figure 4 and Figure 5, right).

Figure 4 shows that a minimum G2 conductance decrease is required to weaken the community effect in the abnormally depolarized central patch just enough to be normalized by the polarized surrounding bulk. Figure 5 shows that a minimum value of G1 conductance is needed for the surrounding bulk to force the polarization of the patch. Note that decreasing the values of the conductances G1 and G2 of Equation (3) impacts on both the patch and the surrounding bulk because of their respective connectivity decreases. These decreases make more difficult not only the resistance of the patch to normalization but also the normalizing effect exerted by the surrounding bulk.

Taking together, Figure 4 and Figure 5 suggest that different changes in the junction conductances can give non-trivial results that depend not only on the intercellular connectivity but also on the different type of connexins and the particular single-cell bioelectrical states [5] assumed initially. Note also that the above results are difficult to obtain with only one type of connexin gap junction for biologically plausible single-cell and connectivity parameters (not shown here). Indeed, not only the particular values of the conductances G1 and G2 of Equation (3) but also the different sharpness of the two voltage-gated conductances Gij of Figure 3 (top) are crucial here.

The above effects are modulated further by the asymmetric junction conductance at the interfacial region (Figure 3). Indeed, Equations (3) and (4) show that the intercellular currents at the patch-bulk interfacial region of Figure 3 (bottom) that force the patch normalization should depend on the asymmetric type of heterotypic connexin formed. In particular, Figure 3 suggests that the width of the bell-shaped junction conductance Gij, which is regulated by the potentials V0,j and V0,i, should be important. For instance, no patch regionalization can exist initially if these potentials are high and give thus wide Gij bell functions (the case of Cx43 in Figure 3). These effects are further modulated by the particular values of the conductances G1 and G2 in Figure 3. Thus, complex cell system dynamics can be obtained even with a relatively simple biophysical model.

Experimentally, effective decreases in the gap junction conductance can be obtained by downregulating the connexin protein expression and by post-translational blocking of the junctions by external agents. In general, low conductances cause restricted intercellular connectivity and can facilitate the spatial regionalization of signaling molecules characteristic of embryogenesis [3,4]. On the contrary, high conductances enhance intercellular interconnectivity and might contribute to the normalization of abnormal cell states in tumorigenesis caused by local changes in the single-cell characteristics [31,39]. However, the fact is that tumorigenic processes are complex and may show apparently contradictory facts regarding the connexin–carcinogenesis association [29,30,32,33,34,35]. In particular, the transference of signaling agents through intercellular junctions may result in different outcomes depending on the target cell state and the molecule to be transferred. We emphasized here a different collective effect: Figure 4 and Figure 5 show that the single-cell responses can also depend on multicellular mean fields regulated by the intercellular coupling conductances [4,36].

Additionally, the results suggest the existence of optimal connectivity values and connexin types for bioelectrical normalization, although the complex dynamics makes practical application difficult. For example, enhancing the junctional role, e.g., via increased protein expression in the patch, may result in an enhanced intra-connectivity and community effect, thus increasing the patch resistance to bioelectrical normalization. Additionally, inhibiting the junctional role at the initial stages of tumorigenesis might decrease the patch-surrounding bulk inter-connectivity, making the bioelectrical normalization more difficult and thus giving more opportunities for further growth and dissemination. Therefore, a complex scenario emerges even in the highly simplified case of considering bioelectrical signals only.

Figure 6 considers the effect of the patch size on the bioelectrical normalization at constant junction conductance decrease. While the size effect may seem trivial, it can be complicated further by spatial heterogeneities. For instance, no initial electric potential regionalization should be possible for too small patch sizes. In addition, if spatial arrangements of small dispersed patches locally distributed exist in the multicellular aggregate [36], these spots will show low community resistances to polarization, thus enhancing the normalization effect (not shown here). On the contrary, if the depolarized patch of Figure 3 is located in the outer rather than in the inner region of the multicellular aggregate, the normalization effect is significantly diminished, takes more time to be completed and, eventually, cannot occur for big enough patches (not shown here). As to the two-dimensional nature of the cell system used in the simulations, we would expect the mean-field effects reported here to be even stronger for three-dimensional multicellular aggregates because of the increased intercellular connections.

To further understand the relation between the different contributions G1 and G2 to the total junction conductance (Equation (3) and Figure 3, Figure 4 and Figure 5) and the patch size effect (Figure 6), Figure 7 shows the bioelectrical phase space obtained for the normalization (polarization here) of the depolarized patch. In order to force the bioelectrical normalization of the patch, Figure 7 suggests increasing the minimum conductance term G1 and decreasing the maximum conductance term G2. In the first case, the increased connectivity within the polarized bulk and at the interfacial region between the bulk and the patch will give intercellular currents high enough to force the normalization (Equation (4)). In the second case, the decreased community effect within the depolarized patch will not allow it to resist the normalization by the polarized bulk. As expected, the above normalizing effects are less effective as the patch size increases.

It is important to note that increasing both the minimum conductance term G1 and V0,j=V0,j=V0 in Equation (3) and Figure 3 makes the voltage-gated Cx45 junction conductance to become close to that of the Cx43 conductance. This result emphasizes further the enhanced connectivity of the Cx43 junction compared with that of the Cx45 junction (see Figure 8 and the respective experimental curves of Figure 1). This fact suggests that the Cx43 junction could facilitate the bioelectrical normalization of the small patch by the surrounding bulk at the initial stages of the abnormal local event. On the contrary, the Cx43 junction might enhance the influence of a growing abnormal patch on the normal surrounding bulk at late stages of this event. Experimentally, the context-dependent role of gap junctions in tumorigenic processes has been noted previously [29,32,33,34,35,48,49].

Note that we concentrated here on the different spatial distribution of electrical conductances only, ignoring the fact that distinct connexins might show different transcriptional and half-life times. In other words, not only the spatial but also the dynamical changes in the connexin subunit composition of the respective hemichannels can influence the enhancement or shutdown of the interfacial communication. The problem of a time and voltage-dependent protein transcription has also been analyzed by us recently [5,36]. In fact, additional simulations for different single-cell parameters, connexin types (Figure 3, top), junction conductance changes, and region characteristics can be conducted and the results of Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 constitute only representative examples of the possible model outcomes [5,36].