Bioelectrical Signals and Ion Channels in the Modeling of Multicellular Patterns and Cancer Biophysics

Bioelectrical signals and ion channels are central to spatial patterns in cell ensembles, a problem of fundamental interest in positional information and cancer processes. We propose a model for electrically connected cells based on simple biological concepts: i) the membrane potential of a single cell characterizes its electrical state; ii) the long-range electrical coupling of the multicellular ensemble is realized by a network of gap junction channels between neighboring cells; and iii) the spatial distribution of an external biochemical agent can modify the conductances of the ion channels in a cell membrane and the multicellular electrical state. We focus on electrical effects in small multicellular ensembles, ignoring slow diffusional processes. The spatio-temporal patterns obtained for the local map of cell electric potentials illustrate the normalization of regions with abnormal cell electrical states. The effects of intercellular coupling and blocking of specific channels on the electrical patterns are described. These patterns can regulate the electrically-induced redistribution of charged nanoparticles over small regions of a model tissue. The inclusion of bioelectrical signals provides new insights for the modeling of cancer biophysics because collective multicellular states show electrical coupling mechanisms that are not readily deduced from biochemical descriptions at the individual cell level.

The modeling of spatial patterns in multicellular ensembles is relevant to positional information processes (e.g. embryogenesis) and cancer initiation and progress1,2,3. Much progress has been achieved on the basis of genetic concepts and specific biochemical signals. Recently, different studies have noted a number of difficulties with existing models, encouraging non-specific, biophysically-oriented approaches which may add new ideas to the field4,5,6,7,8. Bioelectrical signals should be included in these approaches because ionic species and electrical phenomena are crucial for cell function. Indeed, cells have a significant membrane potential of the order of tens of millivolts, the cell inside being at a negative electrical potential with respect to the extracellular medium2,6,9. This potential can be involved in the cell cycle and is regulated by the protein ion channels in the cell membrane (in particular, by the voltage-gated inward and outward-rectifying channels9,10,11).

We propose a highly-idealized approach for an electrical network of non-neural model cells where the biochemical coupling of the multicellular ensemble with the external microenvironment is realized by an ion channel blocker. The approach is based on a reduced number of biological assumptions: i) the membrane potential value of a single cell, assumed to be regulated by the concerted action of hyperpolarizing (inward rectifying) and depolarizing (outward rectifying) voltage-gated ion channels9,12, characterizes the cell electrical state (polarization); ii) the long-range electrical coupling of the cells is realized by a network of gap junction channels between neighboring cells13,14; and iii) the spatial distribution of a biochemical agent in the external microenvironment can modify locally the conductance of the ion channels (and then the cell membrane potential)9,15. The spatial patterns emerging from this model are characterized by the map of cell potentials, which are ultimately regulated by the voltage-gated channels because of their externally tunable electrical conductance9,10,11,12,16,17. We explore the consequences of this bioelectrical map on spatial patterning, the role of ion channels in the modeling of cancer biophysics, and the distribution of charged nanoparticles over multicellular ensembles. We concentrate on the electrical effects which may occur in small multicellular regions, ignoring the slow diffusional processes (e.g. changes in the ionic concentrations) which should follow the relatively fast electric responses.

This biophysical approach constitutes a significant extension of a previous model14 which is applied here to new problems of current interest. The model is an oversimplification of real biological systems but it incorporates an experimental fact which has traditionally been disregarded: the coupling between cells involves not only biochemical but also bioelectrical signals2,11,18,19,20. Indeed, morphogenetic fields acting on the tissue morphology (embryo), morphostatic fields keeping the tissue microarchitecture (adult), and cancer resulting from the disruption of this architecture have usually been studied on the basis of biochemical concepts at the single cell level1,21. In particular, cell genetics and diffusion-reaction processes are the basic approaches for the modeling of spatial patterns in multicellular ensembles. While these dominant approaches have shown the crucial role of genetics and biochemical signals, recent studies have also noted the importance of bioelectrical signals and communication at the multicellular level2,4,8,14,22,23,24, especially for the modeling of cancer initiation and progress6,11,22,25,26.

Cancer cells tend to be depolarized (low membrane potential in absolute value), which has been connected with overexpression of specific ion channels27. In particular, upregulation of sodium channels and downregulation of potassium channels has been ascribed to carcinoma cell lines and tissues7. The blocking of specific channels by an external agent could reverse the membrane depolarization, acting to reduce the oncogenic process. In addition, depolarization of the cell membrane caused by channel overexpression produces significant changes in the spatial distribution of negatively charged lipids and their interactions with positively charged proteins27. These changes may activate biochemical pathways which promote uncontrolled cell proliferation (see Fig. 1 of ref. 27). The above experimental facts suggest that changes in specific ion channels regulating the single cell membrane potential can promote long-range processes because of the biochemical and bioelectrical couplings underlying multicellular organization2,13,20. In an effort to illustrate the consequences of the bioelectrical coupling, our model simulations explore the cases of ion channel upregulation and blocking, defective multicellular communication, and spatial patterning. The conclusions obtained are related with current experimental approaches and suggest that bioelectrical signals and ion channels should be of particular significance for the modeling of cancer biophysics.6,7,10,11

The cell electrical state is described by the membrane potential Vm < 0, which is defined as the potential difference between the cell cytoplasm and the extracellular microenvironment under zero current conditions. This potential difference regulates the entry of sodium, potassium, calcium, and biologically-relevant molecules to the cell7,9,10. Figure 1 schematically shows the model cell as a dynamical system undergoing transitions between the low (depolarized, abnormal) and high (hyperpolarized, normal) values of Vm12. These transitions may be associated with changes in biological parameters such as the pH and the ionic concentrations of the salt solutions9,25 but we concentrate here on the electrical characteristics of two model voltage-gated channels. These channels are involved in membrane hyperpolarization/depolarization processes and cancer, as shown in experimental studies6,10,11,28,29,30,31. Indeed, the negative membrane potential value is a physiologically-relevant cell characteristic: anomalous inward-rectifying potassium channels found in tumor cell lines give values of Vm different than those found in normal cells6,25 and changes in the regulation of voltage-gated sodium channels are associated with depolarized values of Vm and cancer7,28,29. Note however that different ion channels and pumps, in addition to potassium and sodium ions, are implicated in cancer (see e.g. Table 1 of ref. 6 and references therein). Abnormally-low absolute values of Vm correspond to plastic cells while high absolute values of Vm are found in terminally differentiated cells25,32.

While there is a multiplicity of ion channels which make contributions to Vm9,11, the basic characteristics of the membrane potential bi-stability shown in Fig. 1 are illustrated using the physiologically significant outward and inward-rectifying channels in Fig. 2a12,14. These channels can mimic the bioelectrical properties typical of sodium10,11 and potassium10,11,30,31 channels and are introduced here as a simple model for membrane potential regulation. Note that we ignore the effects of membrane ion pumps in Fig. 2a, assuming that passive ion transport determines the electrical potential across the plasma membrane of animal cells to a large extent13.

The current (I)-voltage (V) curve in Fig. 2a is determined by adding the contributions of the inward and outward-rectifying channels (see the Methods section)14. The membrane potential is then defined as Vm = V(I = 0). The channel electrical conductances Gout (outward) and Gin (inward) are of the order of 1 nS9. For the sake of simplicity, the equilibrium potentials are fixed to Ein = −60 mV (hyperpolarized value) and Eout = 0 (depolarized value)32 in Fig. 2a. These equilibrium potentials depend on the ionic concentrations, which are approximately constant for the short times characteristic of electrophysiological experiments33. The number of effective charges involved in the channel gating is z = 3 and the threshold potential (midway voltage of activation, i.e. the potential at which the open probability is 0.5) is Vth,in = Vth,out = −25 mV in Fig. 2a. These channel characteristics are typical of voltage-gated ion channels9,16,34,35.

Figure 2a shows three values of Vm = V(I = 0) corresponding to the hyperpolarized and depolarized stable values (which are reminiscent of a two-state biological memory) and the central unstable value. Electrical bi-stability is a key characteristic of neural cells36 and it is meaningful to consider it in the case of non-neural cells operating over times longer than those characteristic of excitable cells. Moreover, bi-stability has been obtained previously in theoretical37,38 and experimental studies (lysenin channels inserted into lipid bilayer membranes39, hair cell membranes40, and skeletal and mouse lumbrical muscle cells34).

Figure 2b shows transitions between the hyperpolarized and depolarized membrane potentials induced by modifying the values of the equilibrium potential Ein (this potential depends on the ionic concentrations9). Remarkably, the transition characteristics can be regulated by the conductance ratio Gout/Gin14, which suggests that cells expressing differently the outward and inward rectifying channels characteristics (and then the ratio Gout/Gin) may show different membrane potentials Vm.

Recent reviews on clinical and experimental oncology have considered the concurrent upregulation of sodium channels and downregulation of potassium channels as essential steps to cancer progress via increased sodium inward currents and decreased potassium outward currents (see e.g. Fig. 2 of ref. 7 and references therein). As an illustrative example showing the consequences of specific channel upregulation at the single cell level, consider the physiologically-significant electric potentials V < 0 in Fig. 2a. In the present model, upregulation of outward channels and downregulation of inward channels would correspond to the case Gout/Gin > 1 in Fig. 2b. In this case, the membrane potential Vm decouples from the normal hyperpolarized value Ein, entering the bi-stability and cell depolarization regime in Fig. 2b. Depolarization is a characteristic of abnormal cells that promotes the initiation of biochemical signal cascades7,27 which will be ignored in this biophysically-centered approach.

The cell microenvironment characteristics should also be incorporated in the model. We consider here the case of an ion channel blocker because it has been suggested that metastatic activity could be decreased by externally blocking the activity of specific ion selective channels15,28. This suggestion can be simulated in the model by introducing the blocker concentration c(x,y) in the external cell microenvironment, where x and y are the spatial coordinates in the cell ensemble (see the Methods section for details). As an illustrative example, the blocking of the outward-rectifying channel can be described by decreasing the channel conductance ratio Gout/Gin with the blocker concentration. In this way, channel blocking would correspond to the case Gout/Gin < 1 in Fig. 2b.

Intercellular communication has been found defective in abnormal tissues22,23,41,42. Hence, in addition to the single cell and microenvironment characteristics, the model should incorporate the effects of intercellular coupling. The long-range electrical coupling in the multicellular ensemble can be simulated by effective conductances (G) and capacitances (C) arranged in parallel (Fig. 3)14. These bioelectrical elements can be realized by the protein channels which act as gap junctions between neighboring cells13,42,43.

The cell ensemble consists of N identical model cells simulating a small patch of tissue (we do not consider the problem of tissue growth here). The central cell (i) in Fig. 3 has a time (t)-dependent electric potential Vi(t) which may have an intermediate value between the low (depolarized) and high (hyperpolarized) membrane potentials in Fig. 2b. The spatial map of electric potentials changes with time because of the local ionic currents established through the junctions of neighboring cells in the multicellular network14.

In the multicellular ensemble simulations, we assume that the single cell potential changes only because of the electrical coupling between neighboring cells and ignore local changes in the ionic concentrations (the equilibrium potentials Ein and Eout are then constant). This assumption should be valid for the extracellular environment acting as a buffer which fixes the external ionic concentrations34. The intracellular ionic concentrations should also be approximately constant in the case of electrical relaxation processes because the number of ions transferred across the membrane to set up typical potential differences is very small compared with the total number of ions in the cell13. The above assumptions are equivalent to assuming that the multicellular ensemble dynamics are dictated by the individual cell potentials, which should be valid for short enough experimental times13,14,33,38.

Figure 3 shows that the individual mechanisms acting at the cell level (Fig. 2) are coupled together by the electrical network at the multicellular level, which produces the spatial patterns over the cell ensemble. Note that the cells in the ensemble can be at different potentials because of the finite conductance values typical of biological junctions (see Fig. 2)14. More details on the dynamic model equations can be found in the Methods section and ref. 14. We will consider now three problems of current interest: the spatial patterns of single cell electric potentials obtained for different degrees of intercellular bioelectric communication, the role of ion channels in the modeling of cancer biophysics, and the spatial distribution of charged nanoparticles over a model tissue2,6,7,11,44,45.

The gap junctions in Fig. 3 allow the communication between cells by converting the electrical signals at the single cell level into multicellular states. Figure 4 (bottom panel) shows that the normalization of small regions with abnormal depolarized cells is possible for high enough coupling conductances G. However, this normalization process is not achieved for defective intercellular communication (top panel) because the low values of G produce cell isolation in this case. The results in Fig. 4 suggest also that a non-uniform distribution of gap junctions may allow the coexistence of spatial regions having increased intercellular communication (high G values) with other regions having decreased communication (low G values). This question should have clear implications on the long-range gap junctional signaling characteristic of patterning2,6 and tumorigenesis46.

For strong coupling (high G values in Fig. 4), the hyperpolarized potential state is forced by the normal cells acting as an electrical buffer for the relatively small number of abnormal cells. As it could be expected, this self-correction of the physiologically corrupted left pattern in Fig. 4 should be difficult when a high number of abnormal cells are present: the local majority rule favors the depolarized, abnormal state in this case14.

The low response times obtained suggest that slow changes in the ionic concentrations34 should follow the rapid electrical relaxation in Fig. 4. We have not described these diffusional processes which should occur over longer times and larger spatial regions than those characteristic of the rapid electrical relaxation occurring in the small multicellular region in Fig. 4. While the above processes should be incorporated in more complete tissue models, changes in the individual cell potentials can initiate and influence the comparatively slow biochemical pathways which regulate the cell state and proliferation2,6,27.

Experimentally, non-functional junctions and defective intercellular communication are typical of uncontrolled growth regulation in tissues6,23,41. In our model, abnormally low gap junction conductances can isolate depolarized potential cells from the inhibitory electrical signals characteristic of the hyperpolarized potential neighboring cells (Fig. 4, top). Remarkably, the conductance of the gap junctions can be modulated by external agents13,23,42,43, which suggests methods to reverse this process. Note finally that isolated cells could eventually proliferate and expand8,23 but this process would take times much larger than those considered here because of the relatively slow biochemical13 and diffusional34 processes occurring at the single cell and multicellular levels.

Overexpression and upregulation of specific ion channels in the membrane can alter the normal cell electrical balance, stimulating uncontrolled proliferation7,25,28,29. In particular, ion channel upregulation can lead to persistent depolarization of the cell electric potential, modifying the spatial distribution of negatively charged lipids and provoking the clustering of signaling proteins with positive residues around them.27 This clustering causes the initiation of biochemical pathways which promote cell proliferation (see Fig. 1 of ref. 27 and references therein).

The upregulation of the outward rectifying channel, simulated here by increasing the conductance Gout with respect to Gin, promotes depolarization (as shown in Figs 2b and 5). The initially depolarized central region cannot be normalized for high enough values of Gout/Gin. Instead, this abnormally electrical region expands and invades the normal cell region when the outward rectifying channel is up-regulated (bottom row).

Taken together, Figs 2b and 5 indicate that models incorporating specific ion channels with different conductances will show different mono and bi-stability regions12,34,38,39,40. Figure 5 suggests also that the membrane potential depolarization characteristic of abnormal cells should be possible when certain key channels are up-regulated, in agreement with experiments27,28,29. While these specific channels will depend on the particular biological cell studied, a simple model with only two generic voltage-gated channels qualitatively captures this fact in Fig. 5.

Functional tissues are open systems subject to biochemical, mechanical, and electrical changes in the microenvironment. In Fig. 6, we show that normalization and patterning can be promoted by an external blocking agent whose different concentration profiles c(x) are superimposed to the map of local potentials. In this case, the blocking of the outward rectifying channel locally decreases the conductance Gout with respect to Gin, an effect opposite to that shown in Fig. 5 (see the Biophysical Model section for the blocking model and Fig. 2b for the effects of changing the conductance ratio Gout/Gin).

Note that the establishment of the electrical pattern is fast because we do not describe the diffusional relaxation of the blocker, assuming a fully developed steady-state concentration profile c(x) and a fast channel blocking reaction (the channel responds immediately to the blocker presence). The coupling between cells allows the propagation of the local perturbation caused by the channel blocker at the single cell level to the multicellular ensemble. In this case, the electrical signal propagation through the axial position promotes the gradual transition from the abnormal depolarized state to the normal hyperpolarized potential cell state.

The results suggest that different blockers acting on the conductance of specific ion channels should be able to restore the normal cell polarization by suppressing the upregulation of these channels (compare Fig. 6 with Fig. 5, top row). The extension of the hyperpolarized potential region through the multicellular ensemble should be facilitated by normal, non-defective gap junctions (see Fig. 4). Taken together, Figs 4 and 6 show that the concerted action of an external agent acting at the single cell level and the long-range communication acting at the multicellular level can reduce the extension of the depolarized, electrically abnormal region. Conversely, the blocking of specific channels promoting membrane hyperpolarization rather than depolarization should increase the extension of the depolarized region. It has been observed that cationic nanoparticle-induced blocking of the potassium channels that are responsible for maintaining the membrane potential leads to significant depolarization of CHO and HeLa cells45.

We have ignored diffusional effects in Fig. 6. In order to describe the diffusional relaxation of the blocker, we should not use the assumption of a fully developed steady-state concentration profile in Fig. 6. The Methods section presents a simple approach for the coupling of the blocker diffusion equation to the channel blocking reaction. In this approach, the fast electric time in Fig. 4 is replaced by the slow diffusional time τd = L2/D where L is the multicellular ensemble length and D is the blocker diffusion coefficient. As it could be expected, time τd is of the order of hours for typical values of L and D (see the Methods section) suggesting that the results in Fig. 6 should involve times much longer than those in Fig. 4 when the diffusion processes are included in the model.

The results in Fig. 6 also indicate that the coordinated action of external perturbations acting along predefined spatial directions could produce patterning14. Figure 7 addresses this question: the multicellular patterns (right) result now from a two-dimensional spatial profile c(x,y) for the blocking agent (left) in Fig. 6. Note that we assume a fast channel blocking reaction which is locally dictated by the steady-state concentration profile c(x,y): the steepness in the spatial map of electric potential follows closely that of the blocker concentration. Remarkably, the symmetry breaking characteristic of the spatio-temporal patterns shown in Fig. 7 requires no anisotropic electrical couplings14 between the neighboring cells in Fig. 3.

Incidentally, it is tempting to speculate that the external blocker can provide a mechanism for establishing a positional information scheme based on bioelectrical signals. All individual cells in the multicellular ensemble shown in Fig. 7 are equivalent but a given cell would be able to read its relative spatial position with respect to the neighboring cells provided that it can sense the locally different electric potentials and fields imposed by the spatial distribution of the blocking agent. In addition to transport processes through the neighboring gap junctions47, this electrical sensing could be based on the charged lipids and proteins in the cell membrane which show significant spatial redistributions in response to local electrical fields9,13,27. The multicellular ensemble can then be regarded as a bioelectrically coupled network supported by the intercellular gap junctions allowing the transmission of bioelectrical signals14,46,48. It should be cited here that, in addition to specific morphogens, interactions among cells can be crucial for positional information48,49.

The present model of slow electrically excitable network incorporates external biochemical signals (blocking agents), individual cell properties (voltage-gated ion channels) and collective coupling mechanisms (gap junctions), giving electrical patterns reminiscent of morphogenetic fields. Remarkably, these mechanisms incorporate not only the spatial distribution of absolute membrane potentials but also the relative differences of these potentials across the gap junctions and their time-dependent changes, as suggested in experimental studies46,48. We must admit, however, that diffusion-reaction processes with long characteristic times should also be included in the model for quantitative descriptions of patterning1,2,21,32.