Exploring Instructive Physiological Signaling with the Bioelectric Tissue Simulation Engine

Ions are actively transported by pumps in the cell membrane. Both passive and active transport processes generate ion fluxes (Φi). These combined fluxes can lead to changes in concentration and charge density, and can generate a system-wide ionic current density J→.

BETSE assumes passive electrodiffusive mass transport in a multicellular cluster follows three distinct pathways: (1) transmembrane, via intra- and extracellular spaces across the plasma membrane; (2) intercellular, between cellular spaces via gap junctions; and (3) extracellular, between extracellular spaces and within the global environment (Figure 3). Active transport from ion pumps is always assumed to be transmembrane. Therefore, BETSE considers the following sources of ion flux for an ion i:

Transmembrane, from passive electrodiffusive transport resulting from gradients between the local intra- and extracellular spaces using the Goldman–Hodgkin–Katz Flux equation (GHK Flux equation), which is derived from the Nernst–Planck Differential equation for the case of electrodiffusion across a cell membrane for a non-steady-state Vmem (Bowman and Baglioni, 1984): (2)Φmemi=ziVmemFDiR T dmemccelli−cenviexp−ziVmemFRT1−exp−ziVmemFRT here, F is the Faraday constant, R is the ideal gas constant, and dmem is the plasma membrane thickness (see Table 1). Positive fluxes are directing mass into cells.

Transmembrane, from active transport resulting from ion pump activity. Details of how the dynamic ion pump rates (α) are calculated are given in section 2.7: (3)Φpumpi=α(ccelli,cenvi,Vmem,t)

Intercellular, from passive electrodiffusive transport resulting from gradients between neighboring, GJ networked cells: (4)Φ→celli=−Di∇ccelli−DiziqkbTccelli∇Vcell+u→cellccelli

Extracellular, from passive electrodiffusive transport resulting from gradients between neighboring environmental spaces: (5)Φ→envi=−Di∇cenvi−DiziqkbTcenvi∇Venv+ûenvcenvi

Changes in concentration are made by assuming the concentration change in an ion i depends on the divergence of the net (Φtoti) sum of all fluxes of the ion entering or changing in a particular region of the space (i.e., cells or environment):

Net ionic charge density was calculated by summing all ion concentrations at a region of space:

The dynamics of ionic charge density were calculated from the mass flux of all ions:

The total current density of the environment or cell, J→, was calculated using the continuity equation in combination with the assumption of bulk electro-neutrality for electrolytes due to charge screening. Using the Continuity equation for current, the current density in a region follows:

As electro-neutrality (zero net charge density) must be preserved in the bulk electrolyte, the base current density calculated by BETSE (J→o) was corrected by assuming that an internal electric field develops in the bulk electrolyte as a result of charge screening, which is the negative gradient of an electric potential φint:

Substituting equation (10) into equation (9) and rearranging to solve for the internal electric potential:

After obtaining φint, it is used with equation (10) to produce the corrected current density for the system. Current density in the environment and in cell spaces was treated as separate.

Note that as movement in both concentration and electrical gradients can occur, the transport properties of bioelectrical systems cannot be strictly reduced to electrical constants, such as resistance or conductance. However, examining the Nernst–Planck equation [equation (1)] reveals that the diffusion coefficient D is able to serve as the constant of proportionality for movement in both chemical and electrical terms. In the absence of a concentration gradient, and multiplying by F z to convert mass flux to ionic current density, the Nernst–Planck Flux equation reduces to:

Noting that the definition of an electric field is E→=−∇V, equation (12) parallels the equation relating current density to electric field via media conductivity 1γ:

Therefore, BETSE makes use of diffusion constants to characterize ion transport in different regions of the multicellular cluster, but can approximate conversions between conductivity and the diffusion constant.

Note that for movement across a membrane with thickness dmem, the permeability of the membrane is simply Pmem = Dmem/dmem.

The Poisson equation (V=−ρeε, where ρe is electronic charge density and ε is medium electrical permittivity) is typically used to determine voltage from charge density. In air, a charged object will emanate a voltage gradient (electric field) into the space around it according to the Poisson equation. However, electrolytes are more complex. Due to the presence of mobile, oppositely charged ions in electrolytes, objects with steady-state voltages or bound charges collect an opposite surface charge from the electrolyte to form an electrical double layer approximately 1-nm thick in biological systems, which screens the voltage/charge of the object and a prevents long-range electrical field from developing at macroscopic distances into the electrolyte (Bazant and Squires, 2004). Moreover, biological systems are highly heterogeneous, with opposite-sign charge distributed at intra- and extracellular interfaces of the plasma membrane. This means opposite sign charges are separated by the small membrane thickness (3.5–9 nm), and that in a collective of many cells with closely interfacing membranes, charges are present in the low-volume extracellular space that is approximately 5–50-nm wide between cells. Therefore, a new technique was adopted to model voltage in the biological tissue. Voltages in the intra- and extracelluar spaces (Vintra, Vextra), and the related Vmem = Vintra − Vextra, were calculated from net ionic surface charge distributions using a formulation called the Maxwell Capacitance Matrix (Clements et al., 1975; Heinzel, 2008).

Capacitance is typically known in terms of an electrical device characterized by two metal plates (electrical conductors) with equal and opposite charge (±Q) on either plate, which are separated by a layer of insulating material (Figure 4A). The capacitance (C) is defined by the ratio of the voltage (V) between the plates in relation to the charge Q on each plate (Figure 4A):

However, arrangements of conductors can involve capacitance via multiple insulator-separated conductors with variable amounts of charge and voltage on each conductor (Figure 4B). For the case of multiple conductors, the basic capacitance relation shown in equation (4) must be extended to parameterize more complex arrangements. For instance, for the three conductors shown in Figure 4C, the charge Q on each conductor can be related to voltages and capacitances of the system as:

here, Cm is a capacitance connecting the conductors, and Csa, Csb, and Cso are the self-capacitances of the conducting objects.

The self-capacitance of a conductor describes how much charge is acquired on the conductor/unit voltage applied to its surface. For electrolytes, self-capacitance is related to the ability of the electrolyte to screen voltage on a submerged object (see Figure 4B). Using Boltzmann relations for low voltages, in an electrolyte the ionic charge density ρe forming near the surface of an object with a surface voltage φo can be expressed as a function of distance, x, away from the surface as:

where κ is the inverse Debye length for the electrolyte, which assuming the approximation for a symmetric monovalent electrolyte with total molar concentration Ctot for a typical BETSE system is expressed:

Integrating equation (16) with respect to x from the surface at 0 to infinity, and dividing by the surface voltage to approximate a self-capacitance/unit surface area for surfaces in the electrolyte:

The system of linear equations derived when considering more complex arrangements of insulator-separated conductors can be expressed in a matrix form (Heinzel, 2008). For the highly simplified system shown in Figure 4B, and described by equation (15), the Maxwell Capacitance matrix (M) interrelating charge Qk and voltage Vk on each conductor is:

For the case where the set of charges Q¯ are known, the corresponding set of voltages V¯ can be found by calculating the pseudo-inverse of the Maxwell Capacitance matrix (Minv) using a singular-value decomposition method.

In the biological system, we propose every point of contact between two cells represents a situation similar to the one shown in Figure 4C. In order to calculate voltage within the closely-spaced intra- and extracellular regions, and to thereby derive Vmem for a cell cluster, each cell–cell interface is reduced to a capacitive electrical system consisting of three conductors: two intracellular spaces and one extracellular space, which are each separated by two cell membranes with capacitance Cmem. Each space has net ionic charge Qk and voltage Vk. The self-capacitance of each space is related by equation (18). This allows a Maxwell Capacitance Matrix identical to the one defined in equation (19) to be constructed for a single cell–cell junction (Figure 4D).

In BETSE, a typical cell cluster consists of many hundred to thousand cell–cell interfaces and, therefore, has a very large M, the pseudo-inverse of which was used to calculate voltage in each intra- and extracellular space from net charge Qk in each region. To complete the calculation, Vmem are calculated by taking the difference between the intra- and extracellular voltages at a respective membrane point.

Note that the use of the Maxwell Capacitance Matrix to derive Vmem is only one component of the computation of bioelectrical variables – a simplified bioelectrical “circuit” is shown in Figure 1, and must also include electrodiffusive transport of ions via transmembrane, intercellular, and extracellular networks, in addition to active transport of ions by pumps, as described in section 2.4.

It is important to clarify that while it is well known that the ions of electrolytes screen voltages arising from static charge distributions, thereby preventing electric field (voltage gradient) from static charge distributions from being seen past the electrical double layer, any net ionic current density (J→) arising from ion fluxes in the biological system is known to generate a small magnitude observable macroscopic electric field (E→global) according to: