The Multi-Conductivity Clausius-Mossotti Factor as an Electrophysiology Rosetta Stone: Dielectrophoresis, Membrane Potential and Zeta Potential

Dielectrophoresis (DEP) has been used for decades to estimate the passive electrical properties of cells. However, the body of work on cell electrophysiology derived from Clausius–Mossotti analysis of DEP-derived data pales to insignificance against the wider backdrop of cell electrophysiology based on the Goldman–Hodgkin–Katz equation measured by patch clamp, which focuses on membrane potential Vm—a parameter which does not appear in the Clausius–Mossotti model—and values of patch clamp-derived membrane conductance which, shorn of double-layer conductivity, are often orders of magnitude lower than those derived from DEP. Conversely, the body of work on DEP analysis is more substantial than that reporting the electrical properties of the extracellular (ζ) potential. To address this, several studies have recently been published into the interconnections between the electrical properties determined by the Clausius–Mossotti model, Vm, and ζ-potential, which analyzed the effect of varying the suspending medium conductivity over a wide range, from below 50 mSm−1 to above 1.5 Sm−1. The results of these studies identified relationships between the cytoplasm conductivity, Vm, membrane conductance and capacitance, surface conductance, whole-cell resistance, and ζ-potential. Significantly, many of these relationships only become apparent when analyzed as a function of the conductivity of the suspending medium. This paper assembles these interconnections, using several separate studies approaching different parameter connections, to draw together a set of equations which collectively form a “cellular electrome”.

This demonstrates that analysis of the Clausius–Mossotti factor across multiple conductivities allows determination of not only passive electrical properties, but also the membrane and ζ-potential, and accurately predicts DEP behavior at higher conductivity for the first time.

For over three decades, researchers in cell biophysics have used dielectrophoresis (DEP) and related phenomena such as electrorotation to determine the electrical properties of cells [1]. This is usually performed by analyzing the motion of either single cells or cell ensembles as they interact with an electric field across a range of frequencies. This is then interpreted via the Clausius–Mossotti factor, a mathematical operation which relates DEP behavior to the electrical properties of a particle and medium as a function of frequency [1,2]. From this, researchers then produce values of membrane conductance and capacitance, surface conductance, and cytoplasm conductivity, which are then used to explain biophysical phenomena, such as differences between cell types [3,4,5,6,7,8] or the response of cells to stimuli [9,10,11,12,13]. Those unfamiliar with the principles of DEP can find an introduction to the subject in Appendix A.

However, there have been two issues with the use of DEP in this manner, which have generally received little attention. The first is that it has been known for many years that the ionic strength of the medium in which DEP takes place has a bearing on the measured value of cytoplasm conductivity. This was first reported by Gascoyne and co-workers in 1993 [13] and received more rigorous treatment by Pethig and coworkers in subsequent studies [14,15], but this relationship was not analyzed in detail for many years. The second issue is that whilst DEP produces passive values of resistance and capacitance, it does not produce data one would normally obtain via classical electrophysiology methods such as patch clamp. Whilst the measurement of single ion channel activity would always be beyond the reach of any DEP-based analysis, there are three whole-cell measures that can be determined using patch clamp; whole-cell capacitance and conductance (related to the DEP-derived membrane permittivity and conductivity, as well as the membrane thickness and cell surface area) and cell membrane potential. The last of these is the gold standard of cell electrophysiology yet has no direct measurement in DEP.

A study in 2021 [16] showed significant interactions between multiple bioelectric phenomena including membrane potential, zeta (ζ) potential (the voltage as the hydrodynamic plane of shear, approximately 1 nm outside the cell and related to cell surface charge) [17], surface conductance, membrane conductance, membrane capacitance, and cytoplasm conductivity, as well as the ionic strength of the medium. This led to an exploration of the connection of each of these with the other and has built into a substantial model which is presented here in toto for the first time.

The combination of these parameters, and their interconnections, has been referred to as a cellular electrome. The word “electrome” began to gain traction in the early 2010s and first appeared in the published literature in 2017 [18]. That paper did not, however, define what the electrome might be. Subsequent uses have been applied to ionic behavior in plants [19,20,21], and the term recently appeared in the title of a book describing bioelectric phenomena more generally [22]. An electromic approach might be best regarded as taking a holistic approach to the interaction of various cellular phenomena, all of which are rooted in the effects of charge (either fixed on a surface, or mobile charges such as ions) and the potentials and currents these charges cause. The effects of these charges appear in different scientific traditions—electrophysiology, dielectrophoresis, and surface science as well as classical cell biology—but these different manifestations are often caused by the same underlying phenomena, and by understanding these connections we can better understand both the way in which the cell functions electrostatically, and the implications this has for wider cellular function. For this paper (and posterity), the electrome is defined as “the totality of the electrical charges (fixed, mobile, and induced) across a biological system, and the associated electrical potentials, conductances, and capacitances through which they interact. It has implications from the nanoscale (ion channels and double layers) to macro scale (EEG and ECG) and impacts both on how cells function in isolation, how they interact with their environment and other cells, and how ensembles of cells perform multiple biological functions”.

This paper collects recent work uniting a range of electrophysiological parameters to the DEP response, in order to describe the cellular electrome. In particular, it addresses the different ways in which the Clausius–Mossotti factor response changes as a function of medium conductivity, and how these changes can be used to determine other electrical parameters beyond those present in the standard model, creating a “Rosetta stone” connecting DEP, patch clamp, and zeta potential, and allowing determination of membrane and zeta potentials, as well as whole-cell resistance comparable to that obtained by conventional methods. Given the simplicity and rapidity of DEP measurement, this offers a potential new tool for direct measurement of cell electrophysiology.

The aim of this work has been to unify different schools of thought in cellular bioelectric phenomena—DEP, electrophysiology, and surface science—into a single cellular electrome model, comprising the measures shown schematically in Figure 1. This benefits our understanding of cell function, and also allows one analytical method to be used to measure other electrical phenomena—for example, allowing DEP to estimate the zeta and membrane potentials, by using the Clausius–Mossotti factor as a “Rosetta stone” connecting multiple phenomena. To understand this, we must begin by considering what the different languages of our Rosetta stone might be.

Ottaviano-Fabrizio Mossotti (1791–1863) was an Italian physicist who made a name for himself in Buenos Aires and researched in dielectrics and neuroscience; Rudolf Clausius (1822–1888) was a Prussian physicist whose primary contributions formed the cornerstone of modern thermodynamics. Their roles in the derivation of the expression that bears their name remains something of a matter of contention, as described in Pethig’s excellent historical review [23]. Mossotti’s contribution came in the paper “Analytical discussion on the influence that the action of a dielectric medium has on the surface distribution of electricity of several electric bodies scattered in it” in 1850 [24], refined to its canonical form in Clausius’ book “The Mechanics of Electricity” in 1879 [25]. However, whilst these two names are the ones that we associate with the model, its origins also lie with contemporary British scientists such as Green, Faraday and Maxwell, and French scientists Laplace and Poisson [23]. Nevertheless, the term Clausius–Mossotti factor emerged in the late 1980s and has become the standard terms for the relative polarizability of a suspensoids, at least within the DEP community [23,26].

Mathematical treatments exist for the analysis of ellipsoids (which can be found in Appendix B) [26,27], and indeed, any arbitrary shape [28]. However, most researchers in cell DEP classically approximate the cell to a sphere. For a homogeneous sphere, the DEP force acting upon it is given by:(1)F=2πεmedrcell3ReKω∇E2 where εmed is the absolute permittivity of the suspending medium, rcell is the radius of the cell, ∇ is the gradient operator, E is the magnitude of the electric field, Re denotes the real part, and K(ω) is the relative polarizability of the particle with respect to the suspending medium. This is the Clausius–Mossotti factor, given by the expression:(2)K(ω)=εcell*−εmed*εcell*+2εmed* where the subscript cell refers to the whole cell, med to the suspending medium, and ε* denotes complex permittivity, given by(3)ε*=ε−jσω where ε is the permittivity, σ the conductivity, ω the angular frequency of the applied field, and j the complex operator √−1. Assuming a suspensoid does not change its size or medium composition, the force in Equation (1) is dependent on two things; the local electric field geometry and the value of K(ω). If the distribution of ∇E2 remains constant—that is, the RMS electric field geometry remains fixed within the analysis volume [29]—then the DEP force is dependent only on constants (2, π, εmed, rcell, ∇E2) and on K(ω). Consequently, analysis of variation in DEP force as a function of frequency yields a spectrum that is dependent on constants and the value of Re[K(ω)] at each tested frequency. This means that the particle properties can be determined where the properties of the medium are known, by fitting the DEP spectrum from Equation (2), multiplied by a scaling factor.

Equation (2) describes a single transition (known as a dielectric dispersion) from the steady-state value at low frequencies, to the steady-state value at high frequencies. The frequency where K(ω) = 0 as it crosses over from positive to negative (termed the crossover frequency fx) has been used as a means of determining the dielectric parameters [1] with the use of suitable approximations (since only one parameter is measured, it is impossible to determine both permittivity and conductivity and usually only the former is measured). A typical dispersion following Equation (2) can be seen in Figure 2a.

For more complex structures such as cells, an extension to the Clausius–Mossotti factor allows a more complete understanding of the makeup of the cell. This works by considering a sequence of nested shells surrounding a central core. The process begins with the innermost core and the first shell, where the core is envisaged suspended in a medium comprising the shell; the net result is then imagined nestling within a medium comprising the next shell, and onwards until all shells and the medium itself are included. Whilst some practitioners have used multi-shelled models to predict the dielectric properties of up to four cell components (nucleus, nuclear membrane, cytoplasm, and cell membrane), there are too few unique features of the Clausius–Mossotti spectrum to determine so many unique parameters. In general, the spectrum is reduced to four key points; the starting value of the spectrum at low frequency, the end value at high frequency, and the frequencies of the two dielectric dispersions; the first of which causes the spectrum to rise with increasing frequency, and the second causing it to fall at high frequencies. To achieve this, we need to substitute for εcell in Equation (2), using the following expression [1,2,30,31]:(4)εcell*=εm*rcellrcell −t3+2εcyto*−εmem*εcyto*+2εmem*rcellrcell−t3−εcyto*−εmem*εcyto*+2εmem* where t is the membrane thickness, and subscripts cyto and mem refer to the properties of the cytoplasm and membrane, respectively. The combined values of Equations (2) and (4) yield a spectrum of Re[K(ω)] such as the one in Figure 2b.

Whilst Equation (4) yields the membrane conductivity and permittivity, accurate determination of both values is dependent on accurate measurement of the membrane thickness t, which is very difficult to measure (requiring electron microscopy or better). Consequently, it is usual to assume a value, with 7 nm being common.

The values of membrane permittivity and conductivity as derived from Equation (4) both scale with t; using a value of t of 14 nm yields values of membrane permittivity and conductivity that are twice as large as those determined where t = 7 nm, for the same spectrum. We can eliminate this dependence by dividing the derived parameters by t, which yields the specific membrane capacitance (that is, capacitance per unit area) Cspec (=εmem/t) and specific membrane conductance (conductance per unit area) Gspec (=σmem/t). These two values are sometimes referred to as the effective specific membrane capacitance Ceff and conductance Geff.

These represent values per unit area of membrane as determined from Equation (4), which assumes a spherical cell with a smooth membrane. We can then use the area of that sphere (that is, 4πrcell2 where rcell is the radius used in Equation (4)) to produce values of whole cell capacitance Cwc and conductance Gwc. Measured values of Ccw have been shown to be similar to those measured using patch-clamp [32,33]; the value of Gspec is however different to that derived from whole cell resistance Rwc. The reasons for this are discussed in Section 4.4 and Section 4.5.

To compare DEP-derived electrophysiology data to other, more common electrophysiological methods, it is important to consider the phenomena they study. The most commonly used whole-cell electrophysiological measure is the cell membrane potential Vm, which typically takes a range of values from around 0 mV to −100 mV in mammalian cells, with many (particularly the most-studied, “excitable” cells of muscle and nerve tissue that employ Vm to function) in the −70 mV to −85 mV range. Determination of the membrane potential was a focus of electrophysiology throughout the nineteenth century, with the role of electricity in biological function having been identified by Galvani in the late 1700s [34]. It was in the 1800s that Emile duBois Reymond [35] identified the origin of Vm as arising from the partition of ion species across the cell membrane; the cytoplasm has much higher levels of potassium (K+) and lower levels of sodium (Na+) compared to the extracellular fluid, and these electrochemical gradients drive the membrane potential [36]. Since the process is ultimately diffusive, the expression for Vm derived by Julius Bernstein [37] drew heavily on the general work of Nernst on diffusion [38], which led to the equation bearing Nernst’s name rather than Bernstein’s. The expression for a monovalent cation is as follows:(5)Vm=RTFlncoutcin where cout and cin describe the external and internal ion concentrations, and R, T, and F have their usual thermodynamic meanings. The denominator and numerator in the logarithm term are exchanged in the case of an anion being dominant. In the case of most cells, the dominant ion was believed to be K+ and the expression was generally written with K+in and K+out to reflect this.

However, by the 1940s, with better instrumentation and models built around the physically large giant squid axon, it became apparent that other ions also played a role [39,40,41]. This led to the extended version of Equation (5) for the three primary monovalent ions, K+, Na+, and Cl− (chloride) posited by David Goldman in his famous “constant field” model [42], later adapted by Hodgkin and Katz to describe Vm in muscle tissue [43] in the now-canonical form of the equation (for this reason, it is referred to either as the Goldman equation, or the Goldman–Hodgkin–Katz (GHK) equation). The model considers membrane potential Vm creating a uniform electric field within the cell membrane, with zero electric field outside it. The GHK equation is typically written as follows:(6)Vm=RTFlnPNa+Na+out+PK+K+out+PCl−Cl−inPNa+Na+in+PK+K+in+PCl−Cl−out where Px is the ion permeability for ion x (either Na+, Cl−, or K+). If conduction is dominated by one ion, then Equations (5) and (6) are similar.

Another electrical measurement which has been applied to cells—albeit far less frequently than measurement of Vm or DEP—is the zeta potential (ζ). ζ is the electrical potential arising due to the charges on the surface of any object in an aqueous solvent [44]. To understand this, we must consider the complex situation arising at the interface between a charges surface and a suspending medium. This complexity arises because there are two processes which occur in parallel at this interface: an electrical process and a hydrodynamic process. Let us consider these separately.

In electrical terms, the charge on the surface attracts countercharge (either counterions or the charged ends of polar molecules) and repels coions with like charge. This results in a layer of charge around the cell in which the ion concentrations deviate from the bulk; this is known as the electrical double layer. Immediately outside the cell surface, some counterions will be electrostatically bound to the surface, and immobilized; as will the positive ends of some water dipoles. These form a thin aqueous shell around the surface called the Stern layer, which terminates at the Helmholtz plane. Beyond this is a second zone (the Debye layer) where the medium ions are free to move but vary in concentration from the bulk with elevated counterions and reduced coions. The concentration of any ion ci(0) with valence z deviates from the bulk ion concentration cb according to the Poisson–Boltzmann equation:(7)ci0=cbexp−zieΨs,okT where Ψs,o is the potential at the surface of the outer membrane surface, and e, k, and T have their usual thermodynamic meanings. The boundary for the electrical double layer is generally defined as the Debye length κ−1, given by the expression:(8)1/κ=εRT2czF2 where c is the electrolyte concentration (mol m−3). The changes in ion concentration described by Equation (8) gives rise to an electrical potential ψ that begins at the Helmholtz plane with value ψst and reduces with distance r, thus:(9)ψ(r)=ψst e−(kr)

In parallel to the electrical structure of the extracellular space, there is also the second, hydrodynamic structure. It is a principle of fluid dynamics that fluids are immobile when in contact with solids and move at the same speed as the surface; that is, at the point of interface the fluid is stagnant, and only moves freely a few molecules beyond the interface [45]. The plane delineating the stagnant layer from free-moving solution is referred to as the “hydrodynamic plane of shear”, or the “slip plane”. This is independent of the electrical structure, and the slip plane is usually found beyond the Helmholtz plane, within the Debye layer. The ζ potential is the value of ψ at the location of the hydrodynamic plane of shear, typically 0.5–2 nm from the surface of the cell. It is worth noting that whilst the Debye length of Equation (8) depends on medium ion composition, the location of the slip plane does not. Consequently, increasing the ionic content of the medium reduces the Debye length, and compresses the exponential describing the potential from Equation (9) into a shorter distance. Since the slip plane is a fixed distance from the surface, this means a raised medium conductivity will lead to a reduced value of ζ.

ζ is a common measurement in materials science, colloid chemistry, and chemical engineering. It is used to measure the stability of solutions; that is, in determining whether suspended particles have enough charge to repel each other against the attractive van der Waals force. Typically, if a solution contains particles with |ζ| > |±10| mV the solution will be stable; if |ζ| < |±5| mV, it will tend to flocculate, and the suspensoids will clump together. The classical example of this is milk, which contains billions of lipid droplets (micelles), whose ζ = −17.5 mV; adding lemon juice to milk moves the micelles towards their isoelectric point, reducing ζ to −3.5 mV and causing the milk to curdle [46].

ζ is infrequently studied in cells, since the membrane composition (and hence surface charge) is unlikely to change significantly, leading to the misapprehension that it is unlikely to be significantly different from one cell type to the next. However, a comprehensive study of published values of ζ in cells [17] suggested that cells manifest a range of values, from −10 mV for platelets and red blood cells to −31 mV for metastatic breast cancer. It has been shown that cancerous cells are more depolarized than their healthy counterparts, and that macrophages and platelets depolarize on activation. Bacteria occupy an even wider range from +14 mV to −49 mV depending on the strain and conditions, with it being noted that different values of ζ corresponded to different bacterial behavior—whether monodisperse, clustered, or filamentous [17].

The Hungarian physicist, Nobel prizewinner, and proponent of bioelectric research Albert Szent-Gyorgyi kept a shark hook in his office as a reminder that, to catch a big fish, one needs a big hook; and to catch a big answer, one needs a big question [47]. The big question posited here is “can DEP be used to determine other electrophysiological parameters such as Vm, ζ, Cwc, and Rwc?” To address this, a substantial data set was assembled, initially around a single cell type; the erythrocyte or red blood cell (RBC). RBCs have many advantages: they are highly regular in size and shape; they are well-characterized by a variety of electrical methods; lacking gene expression and mitochondria, they are simple and are amenable to unusual methods of direct measurement of membrane potential; and they are plentiful. RBCs had also previously demonstrated interesting behavior, such as circadian rhythms exhibiting contrapuntal circadian rhythms in Gspec and σcyto [33,48]; indeed, it was the synchronicity between these two parameters, and the extent of Gspec movement that would be beyond the explanation of simple channel activity, that in part inspired the initial development of this model. It was also known that Gspec or Cspec (or, possibly, both) exhibited an unexpectedly strong dependence on σmed [49]. The data set comprised DEP data (Cspec, Gspec, σcyto), ζ-potential, and Vm in three conductivities ranging from 17 mSm−1 to 1.7 Sm−1, and with four chemical treatments (DMSO, DMSO plus valinomycin; neuraminidase; and DMSO plus valinomycin and neuraminidase). Subsequently, other cell types were added to this, providing additional data for platelets, neurons, monocytes, cancer cells, stem cells, and so on.

It is also to be noted that an advantage of these studies was that they were performed using measurement apparatus that were able to take measurements to such high conductivities [32]. Most DEP experiments are performed using planar thin-film gold electrodes, which are unable to sustain substantial currents and which often electrolyze at medium conductivities much above 0.1 Sm−1. Conversely, the data set here was taken using a platform using much thicker copper electrodes [32], which allowed measurements at conductivities up to 1.7 Sm−1, the first time cell DEP spectra had been measured at such conductivities. The ability to measure across a much wider range of conductivities than before was instrumental in the development of the electrome model.

The key findings of this work are expanded upon in the following sections: the most important discovery was that DEP parameters, far from remaining static across media of different conductivities, change in ways that yield valuable electrophysiological data. It is these gradients that form the interconnection between the different electrophysiological forms.

As a starting point, we can consider a connection between two potentials associated at the cellular (rather than organelle) level: the membrane potential Vm and the zeta potential ζ. Notionally, these two are functionally separate; one arises purely because of ion compartmentalization and diffusion, whilst the other arises purely from static charges on the cell surface. As such, they should be unconnected and independent; however, it has been known since the 1970s that this is not the case, where cancer cells [50], mitochondria [51], and slime molds [52] were shown to exhibit a relationship between Vm and ζ. This was later elaborated by Voight and Donath [53], who hypothesized the relationship arose from capacitive division of Vm across membrane and double layers, and described an equation for the contribution of Vm to ζ, referred to here as Δζ, thus:(10)ζ=ζ′+Δζ;Δζ=VmCmemCmem+Cdl where ζ′ is the zeta potential due solely to the surface chemistry, and Cdl is the double layer capacitance. This was later refined analytically by Dukhin in 1990, who considered the underlying mechanisms and extended the equation to account for contributions from charged surfaces [54,55]. However, this work was circulated primarily among the colloid science community, and despite the potentially significant impact this may have on cell biology, it never reached that audience.

The work was largely forgotten, until the phenomenon was rediscovered in 2021 [16]. In that work, Hughes and colleagues described the additional component of ζ as ΞVm, where Ξ was best fit by the empirical expression(11)ζ=ζ′+ΞVm; Ξ=6.71+Cshear2 where Cshear is the capacitance of the part of the double-layer between the hydrodynamic plane of shear and the cell surface. Significantly, the values of Ξ observed in cells are substantially larger than might be expected where only the capacitances of the membrane (assumed around 10 mFm−2) and double layer (much closer to 1 Fm−2) are considered, which would suggest that Δζ would be less than 0.01. Instead, values have been measured between around 0.2 in cardiomyocytes [56] and platelets [57] to around 0.37 in red blood cells [16] and algae [58].

The implications for this interdependence are wide-ranging. In both models, the capacitance of the double layer is key, suggesting that the membrane potential Vm is dropped across a capacitive potential divider that encompasses both the membrane and the electrical double layer, and suggests that the model of ζ, Vm, and passive properties of conductance and capacitance form an inter-related whole with significant implications for cell behavior. The finding also demonstrated a new use of measurement of ζ as a marker for Vm. This was demonstrated by Chacar et al. [56] who used an off-the-shelf ζ-potential instrument to measure Vm of cardiomyocytes in polarized (Vm = −85 mV), depolarized (Vm= +20 mV), and permeabilized (Vm = 0 mV) states and demonstrated that ζ offers a new method of performing cell electrophysiology. Work on platelets where ζ-potential and Vm were measured, as well as DEP parameters and antibody binding [57] showed that modulation of Vm altered both ζ-potential and antibody binding, suggesting that cells can mechanistically alter extracellular interactions by altering Vm. A similar effect was observed in red blood cells [16] where alteration of ζ by altering Vm on the extracellular ion concentration has potential to confer Vm-gating on ion channels without the requirement for a molecular mechanism. In the platelet study, notable connections were observed between electrical properties related to the cell interior (σc, Vm) and exterior (Gspec, ζ-potential, antibody binding); in particular, connections were observed between internal and external properties under all circumstances, whereas relationships between internal and external properties were only observed under conditions where Ξ was stable [57].

This leads to the crux of this paper; the intersection of Clausius–Mossotti-derived parameters and their interactions with each other, the medium, and the associated cellular potentials.

As discussed in Section 1, the Clausius–Mossotti model has been used for determining the passive electrical properties of cells for three decades. This is performed by measuring the DEP response over a frequency range (typically 10 kHz to 10 MHz, with some researchers extending this up or down by a decade or more), and then fitting Equation (4) to the DEP spectrum by varying σcyto, σmem, εcyto, and εmem to fit for a given value of medium conductivity σmed (the permittivity εmed of the medium is generally fixed at 78·ε0 or 80·ε0, though in reality the sugar content may make this slightly lower) [59]. Analysis of Equation (4), as summarized in Figure 3, showed that the low-frequency, starting plateau value is mostly governed by the membrane conductivity σmem; the frequency of the low-frequency dispersion is governed by the membrane permittivity εmem; the frequency of the high-frequency dispersion is governed by cytoplasm conductivity σcyto; and the high-frequency plateau (often reached at frequencies beyond the limit of most signal generators used in DEP analysis, making accurate evaluation difficult) is governed by εcyto [29]. For the three parameters that were determined most easily, narratives were provided that described what the values signified. First, the membrane conductance primarily describes the ion transport through the membrane (the transmembrane conductance), though an additional term was identified suggesting an effect through the electrical double layer (the tangential membrane conductance) was present in some cells [60]. However, comparison to patch clamp-acquired data did show some close comparisons for some cells between DEP-derived membrane conductance and those derived from canonical methods [61].