Bioelectrical Signals and Ion Channels in the Modeling of Multicellular Patterns and Cancer Biophysics
The potential use of charged nanoparticles in clinical applications addressed to small tissue regions is receiving much attention44,45. Figure 8 shows that the map of electric potentials resulting from cell coupling can regulate the distribution of charged nanoparticules over the multicellular ensemble shown in Fig.7 (bottom panel). The number of nanoparticles (small circles) around each individual cell shown in Fig. 8 is scaled to the exponential function of the local electrical potential, assuming a Boltzmann equilibrium for the spatial distribution of particles9,50. As experimentally observed44,45, the positive nanoparticles tend to be concentrated around the negatively charged cells. This result indicates that the spatial map of potentials should influence the local uptake of charged nanoparticles over a tissue44. Remarkably, the binding of nanoparticles to cells can also disrupt the cell membrane potential45 and modify intercellular communication (Fig. 4). These processes suggest external methods to modify the map of potentials alternative to those shown in Figs 5 and 6.
In summary, Figs 4, 5, 6, 7, 8 suggest that the spatial map of electric potentials characteristic of multicellular ensembles can be externally modified by taking advantage of the long-range electrical coupling between cells. The results illustrate electrical normalization caused by multicellular coupling and blocking of specific channels, spatial patterning along predefined directions, and electrically-induced spatial redistribution of charged nanoparticles over small regions of a model tissue. The model considers only two generic voltage-gated channels (ion pumps and active transport, as well as additional ion channels, may be needed for a quantitative description of particular cells2,13,34,38). The slow diffusional processes34,51 which should follow the relatively fast electrical responses obtained here have also been noted (see Fig. 6 and Methods section). Even with these limitations, we have shown the importance of bioelectrical signals, providing results of qualitative value which can be connected with current experimental problems:
the spatial map of potentials in Fig. 4 and the single cell local potential domains can be imaged by membrane voltage-reporting dyes (see Figs 8 and 9 of refs 2 and 48). This information should be significant because cancer microenvironment may show long-range bioelectrical signals and gap junctional insulation is involved in tumorigenesis (refs 6, 25 and 42). Also, membrane depolarization may trigger transcriptional changes by regulating morphogens transport6;
the changes in the single cell membrane potential polarization discussed in Figs 4, 5, 6 should occur when specific channels are up-regulated, down-regulated, or physically blocked (refs 2, 6, 7, 11, 15 and 29, 30, 31). In principle, decreasing the activity of a particular channel will modify the membrane polarization state it promotes, as shown in Fig. 6. However, addressing only a specific channel may be misleading because the membrane polarization is a physiological characteristic that results from the non-linear combination of different ion transporters which operate in a changing microenvironment6,9 and suppressing a target channel may be compensated by other redundant channels48. In our simple model, this compensation could be achieved in different ways: modifying the gap junction conductance (Fig. 4), shifting the balance between the outward- and inward-rectifying channels functionalities (Fig. 5), and blocking specific channels (Fig. 6). Increasing the complexity of the model with other ion channels and pumps should lead to additional feedback mechanisms. Persistent changes in the membrane polarization may then require acting simultaneously over different transporters because bioelectrical phenomena occur at a different level than cell genetics48;
the role of ion channels in the modeling of cancer biophysics has recently been noted10,11 and ion channel blockers relevant to the effects shown in Fig. 6 are available (see e.g. refs 15 and 45). The blocking channel approach should provide key information on basic cellular processes. Because of the broad functionality of ion transporters in living systems9, however, this approach can promote also serious side effects (e. g. cardiac arrhythmias15) and is limited to the channel conducting mechanism only31,52.
the electrical characteristics associated with the spatial distribution of external morphogens over the multicellular ensemble in Fig. 7 can be of significance for patterning (see refs 2 and 32 and references therein). In particular, the results indicate that in addition to specific morphogens, interactions among cells can be crucial for positional information, as noted in experimental studies48,49;
the map of electric potentials resulting from cell coupling in Fig. 8 locally regulates the experimental uptake of charged nanoparticles over tissues (see Fig. 2 of ref. 44 and 1 of ref. 45). These nanoparticles can disrupt the cell membrane potential45 and modify intercellular communication (see Fig. 6 for the possible effects of nanoparticle blocking of specific channels).
Most theoretical descriptions of the spatial patterns characteristic of positional information and cancer processes are single-cell centered and emphasize biochemical signals and pathways. While these approaches have been useful, ionic species are crucial for cell function and then questions concerning bioelectrical signals, ion channels, cell membrane potentials, and intercellular electrical communication naturally arise2,4,8,14,18,24. Collective multicellular states may show electrical coupling mechanisms that are not readily deduced from biochemical descriptions at the individual cell level8,14,22,26.
We have explored the consequences of a simple biophysical model for electrically-connected cells based on a reduced number of concepts: the cell membrane potential (obtained from two generic voltage-gated ion channels favoring cell hyperpolarization and depolarization), the multicellular electrical coupling (supported by a network of gap junction channels), and the biochemical coupling of the cell ensemble with the external microenvironment (realized by a channel blocker). Specific biochemical signals and pathways characteristic of real biological problems are ignored but new biophysical insights amenable to experimental analysis are obtained on the basis of the building blocks characteristic of the multicellular electrical circuitry: the protein ion channels and coupling junctions of the cell membrane.
The multicellular electrical patterns are spatio-temporal maps of single cell potentials. These maps are of significance for positional information processes based on the coupling of the individual potentials with the spatial distribution of a blocker1,53. Also, the electrical patterns show the importance of ion channels in the modeling of defective intercellular communication and cancer biophysics10,25,27. Knowledge of the local map of potentials should facilitate procedures to collectively change the bioelectrical characteristics of small tissue regions, e.g. by the external modulation of the single cell states by blockers of specific ion channels9,15,28,54. The spatial distribution of charged nanoparticles over tissues, a subject of current clinical interest, is also regulated by the electric potential map: the membrane potential modulates the binding of the nanoparticles to the cell surface44. All in all, the present simulations are based on a reduced number of concepts and show significant connections with current biophysical problems.
The inward-rectifying channels allow large inward currents at potentials more negative than the equilibrium potential (low outward currents are obtained at potentials less negative than 9,14. The phenomenological I–V curve is
where R, F, and T are the gas constant, the Faraday constant, and the temperature, respectively. Vth,in is the threshold potential (the potential at which the average relative conductance of the channels is 0.5), Gin is the maximum channel conductance, and z is the effective number of charges for gating9,14. The dimensionless current allows to write the above equation as
where the dimensionless potentials are defined as , , and . The thermal potential VT = RT/F = 27 mV (T = 310 K). Typical dimensionless values are = 2 for V = 54 mV and = 2 for I = 54 pA, obtained with a channel conductance Gin = 1 nS14. The I–V curve of the outward-rectifying channel is similar to that of Eq. 2, except for the changes for , for , Vth,out for Vth,in, and −z for z. Finally, the cell membrane potential Vm results from the zero current equation14:
where is given by Eq. 1. The cell capacitance is (constant; see Fig. 4). The conductance and capacitance of the gap junction which couples cells i and j in Fig. 3 are and , respectively (see Fig. 4). The summations are restricted to the cell nearest neighbors (nn) in Fig. 3. The reference potential in the extracellular microenvironment is assumed to be zero. Note that the local potentials are regulated by the single cell inward- and outward-rectifying ion channels (the first two terms of Eq. 4) and the cell coupling with the other cell potentials in the multicellular ensemble (the last two terms of Eq. 4). Using dimensionless variables, the above equation is
The dot indicates derivation with respect to the dimensionless time , where the characteristic time τ = C0/Gin (see Fig. 4). In most cases, the coupling conductance term dominates over the capacitance term for typical values of G and C14 and is then omitted in the simulations.
The blocking of the outward-rectifying channel is simulated by changing locally the conductance ratio function [Gout/Gin](x) with the axial position x according with the blocking concentration functions c(x) shown in Fig. 6. In each case, the functional parameters are chosen to give [Gout/Gin](x = 0) = 0.05 (left position in the multicellular ensemble) and [Gout/Gin](x = L) = 2.50 (right position in the multicellular ensemble of length L). In the case of Fig. 7, Gout/Gin is changed according to exponential functions of the spatial coordinates x and y whose parameters are chosen to give the minimum and maximum values shown in this figure.
The potentials in Eq. 5 are to be solved for the cells in order to obtain the spatial map of cell potentials as a function of time. The numerical solution of Eq. 5 for potential can be obtained using matrix algebra in the form:
Calculate the cell potential of the isolated cells assuming a constant equilibrium potential and the parameters introduced in Fig. 2.
The initial values are set to either the hyperpolarized or the depolarized potential, which correspond to the stable potentials of the cells.
The numerical calculations proceed until a prescribed final time is reached, with the change for each time step, following the scheme:
Equations (1)–(8) do not include diffusional terms. The diffusion of the blocker and the local equilibrium reaction causing the blocking of the outward-rectifying channel can be included in the model using the equations:
where D is the blocker diffusion coefficient and K is the blocking constant. An additional equation for the mass conservation should be added if the blocker availability is limited. Note that the blocker concentration c(x,t) and the conductance ratio in Eq. (10) are now functions of the position x and time t. Equation (10) assumes that the blocking process forces the conductance Gout to take values between zero (maximum blocking) and Gin (minimum blocking). Introducing dimensionless variables in Eq. (9) leads to the characteristic diffusional time τd = L2/D. For an ensemble length of 100 cell diameters, L is of the order of 10−3 m. Assuming a typical diffusion coefficient of the order of D = 10−10 m2/s, we obtain τd = 104 s which is about 3 h. As expected, diffusion times should be much larger than electrical times for the multicellular ensemble.
How to cite this article: Cervera, J. et al. Bioelectrical Signals and Ion Channels in the Modeling of Multicellular Patterns and Cancer Biophysics. Sci. Rep. 6, 20403; doi: 10.1038/srep20403 (2016).
Financial supports by the Generalitat Valenciana (Program of Excellence Prometeo/GV/0069), the Spanish Ministry of Economic Affairs and Competitiveness (MAT2015-65011-P and FIS2013-40473-P), and FEDER are gratefully acknowledged. We thank the anonymous reviewers for their valuable comments.