The Multi-Conductivity Clausius-Mossotti Factor as an Electrophysiology Rosetta Stone: Dielectrophoresis, Membrane Potential and Zeta Potential
Second, the membrane capacitance was assumed to be a measure of membrane folding, since the membrane is primarily composed of a lipid bilayer which has a defined capacitance per unit area when stretched flat [62]. Finally, the cytoplasm conductivity was assumed to represent the quantity of free ions in the bulk cytoplasm. Importantly, it was generally assumed that all three parameters should be constant, homogeneous, and unaffected by medium composition; specifically, that the membrane properties should not be affected by extracellular ion concentration (since the system is regarded as electrically passive, effectively a resistor and capacitor in parallel), though observations of red blood cells in media of different conductivity had shown changes that could be attributed to either a change on membrane conductance or capacitance as a function of σmed [49].
Similarly, the intracellular ion concentration was assumed to be retained from its state in culture, leading for example to a widely held belief that all cells exhibit only negative DEP at higher conductivities. This assumption of an unchanging cytoplasm has been challenged more frequently; it had been known for several years that the medium conductivity can play a significant role in the measurement of cytoplasm conductivity. For example, Gascoyne and colleagues observed changes in Friend murine erythroleukaemic cells after resuspension in low-conductivity medium [13] which they attributed to cell leakage. A more detailed study by Chung et al. [14,15] showed again a dependence on σmed that led to a reduction in the high-frequency crossover, and hence to the measurement of σcyto. In both cases, the effect was attributed to ion leakage from cells leading to an anomalous value of σcyto. Implicit in this analysis is the idea that cytoplasm conductivity should be independent of σmed; that the value measured by DEP should reflect the value of σcyto in normal physiological conditions; and that leak represents a deviation from that ideal state, producing results that are potentially misleading and likely meaningless. The only paper [63] to identify a pattern in such behavior was taken by a related technique (electrorotation), which suggested that not only did varying σmed lead to a change in σcyto, but that the change might be linear.
Following the work of Sukhorukov and colleagues [63], analysis of several cell types demonstrated that σcyto consistently varies as a function of σmed in a linear manner. Analysis of the RBC data showed a linear relationship (R2 > 0.99) across the five treatment conditions [16]; subsequent analysis has shown similar relationships in chondrocytes and monocytes (both shown in Figure 4), as well as platelets and cancer cells [63] in addition to the Jurkat data obtained by electrorotation [63]. This was modeled using the classical equation of a straight line, σcyto = A· σmed + B, where value B represents the value of cytoplasm conductivity independent of the medium, and which may represent the contribution of charges within the cytoplasm that are unable to equilibrate across the membrane, such as charged cytosolic macromolecules. The value of B varies between cell types, with estimates varying from 96 mSm−1 in RBCs to 300 mSm−1 in chondrocytes. HeLa and Jurkat cells were found to have B values near to zero.
The gradient value A was observed to change between cell types. By analyzing A in RBCs, Hughes et al. [64] suggested that σcyto does not reflect the average ion concentration (and hence conductivity) of the cytoplasm, but instead reflects the conductivity at the interface between the cytoplasm and the inner membrane surface, in the inner surface’s electrical double layer. They found that A could be approximated with the equation(12)A=2exp−ze(2Ψsto−Ψsti)kT+expze(2Ψsto−Ψsti)kT where ψsti and ψsto represent the surface potentials on the inner and outer surfaces of the membrane, respectively. This is because, if we consider cytoplasm content to be equilibration due to simple diffusion, we expect the ion concentrations in the cytoplasm to be the same as those in the bulk. However, the ion concentration at the outer surface—representing the ions available for diffusion across the membrane—is not the same as the bulk; it is altered by the Poisson–Boltzmann equation (Equation (7)), which (for a negatively charged surface) elevates cation concentrations and suppresses anion concentrations. This is reflected in the two exponential terms in the denominator (one for cations and one for anions), which differ from the ion concentration in the bulk (the 2 in the numerator, representing the anions plus cations). However, if the cytoplasm reflects the ion concentration in the outer double layer, then this will be further altered as the ion concentrations then interact with the inner membrane surface, which alters the ion concentration further. Since the Clausius–Mossotti factor is about interfacial relationships, it is the charge concentration at this interface which we most likely measure when we determine σcyto from Equation (2)—not the ion concentration in the bulk cytoplasm.
Among the electrophysiological parameters that can be obtained by DEP, the most conspicuous absence is Vm, the primary electrophysiological parameter derived from whole-cell patch clamp and many other methods. Previous correlations had been made between σcyto and Vm [65]; this is logical, since both relate to the ion concentration in the cytoplasm. However, there was no direct analytical link between them.
This was first addressed using the large RBC electrome study, in which measurements of Vm and σcyto were made over a range of conductivities and chemical treatments. This yielded the surprising result that the two scaled linearly with medium concentration—and that the difference between the different conditions could be accounted for by parameter A, the slope of σcyto as a function of σmed from Equation (10). This data set demonstrated that Vm in RBCs in media of low conductivity varies substantially—something which had been demonstrated previously using the carbonylcyanide-m-chlorophenylhydrazone (CCCP) method of measuring pH following treatment with a proton ionophore [66]; this cannot be obtained directly by patch clamp, which requires cells to be in high-conductivity physiological medium to function. If we examine the simplest conventional model of membrane potential, the Nernst equation (Equation (5)), we see that there is an inbuilt ratio of external and internal ions. If we consider that a change in extracellular ion concentration δcout causes a commensurate change in intracellular ion concentration δcin, then substituting these into Equation (5) suggests that this ratio may produce a value of Vm, assuming that Vm remains constant across that range of conductivities (as has been observed in neurons, for example [43]). Indeed, the equation would also suggest that if Vm remains constant over a range of extracellular ion concentrations, then cytoplasm conductivity would be expected to change.
This model was tested in a second study [67] which included many other cell types including HeLa cancer cells, chondrocytes, and THP-1 monocytes (examples of which can be seen in Figure 4), as well as the Jurkat data published by Sukhorukov et al. [63]. This data set used DEP to measure σcyto at multiple conductivities, and compared the derived slope of change in cytoplasm conductivity σcyto for a given change in medium conductivity σmed. When this was compared to the literature values of Vm acquired in physiological media (about 1.8 Sm−1) determined with patch clamp, a relationship was determined between that slope and Vm. The two data sets led to the development of two models, with different purposes. This larger data set led to an equation, adapted from Equation (5), which delivers a highly accurate (typically to within 1–2 mV) estimation of the values of published Vm measured in physiological media, as taken by patch clamp. The DEP data are of course not taken at the same conductivity that the patch clamp data were measured at (DEP data were typically taken at conductivities below 500 mSm−1) yet yielded accurate predictions of patch clamp-measured values. The expression was based on the Nernst equation (Equation (5)), but substituting ion concentrations with derived conductivity values [67]:(13)Vm=−RTFlnδσmed δσcyto+Ψsto = −RTFlnA+Ψoffset where Ψoffset was found to be a constant −12 mV for all the cell types examined (RBCs, monocytes, platelets, cervical cancer, leukemia, stem cells, primary chondrocytes). As shown in Figure 5, this yielded estimates of Vm very similar to measurements provided by patch clamp. The equation was found to work for all cell types examined, and represents the key Rosetta stone translation between DEP and conventional electrophysiology.
The −12 mV offset was found to be universal across all cell types, and may represent the surface potentials of the two cell surfaces (which are predominately lipid in origin, and do not vary much between cell types). The model predicts that Vm is largely uniform across different medium conductivities, in line with observations of squid axons [43].
However, RBCs are unusual, in that studies have shown that Vm is highly affected by medium conductivity, becoming positive at low ion concentrations when measured using the CCCP method [66]. When investigated using CCCP in parallel with DEP, Vm of RBCs was found to typically be around +26 mV in 16 mSm−1 solution, and around +20 mV in ca. 160 mSm−1 solution [16]. A model was developed based on the capacitance between cytoplasm and medium, which suggested that an excess of cations on the inner leaflet of the membrane, with the charge neutrality in the bulk acting as a 0 V reference, acts as a charged capacitor:(14)Vm= 1κσmedΛeA2C+Vx where C is the capacitance of the membrane and associated electrical double layers, e is the electronic charge, and Λ is the number of ions per cubic meter for a conductivity of 1 Sm−1. Vx was a constant which held values between +21 mV and +27.5 mV. This model is different to the general model of Equation (13), though both yield similar results at physiological conditions. The difference is that whilst Equation (14) is proportional to medium conductivity, Equation (13) is not. One way that they can be reconciled is by altering the value of Ψoffset in Equation (13) to account for different media; for RBCs, Equation (13) yields similar results to Equation (14) if this is changed to +28 mV for media with conductivity 160 mSm−1 and +35 mV for conductivity 16 mSm−1. The physics underpinning this offset is as yet unresolved; for those wishing to explore this, these three offsets are given by either of these empirical expressions:(15)Ψoffset=−σmed42.2+0.02 or RTF1−σmed
Whether this effect is observed solely in RBCs or is present in other cells, we are unlikely to determine due to the difficulties of direct measurement on Vm in low-salt media. However, −12 mV remains consistent for all cell types in Equation (13), which in light of the finding about Vm in neurons [43] remaining stable across a range of values, suggests this may be an RBC-specific phenomenon.
As described earlier, one of the reasons for the selection of RBCs as a model for electrome study was prior observation of unusual behavior in Gspec; Gascoyne and coworkers [49] noted that it increased significantly with σmed, an observation confirmed by Hughes et al. in 2021 [16]. The other unusual measurement was the observation of circadian rhythms in RBCs, where rhythmic behavior in Gspec and σcyto was in antiphase. This was attributed to the membrane conductance rising due to active channel pumping to reduce the cytoplasm content, though comparison with patch clamp data suggests that the amplitude of Gspec was much higher than might be expected from channel activity.
The conductance of the membrane as determined by DEP has long since been known to contain both transmembrane (along the bulk-to-cytoplasm axis) and tangential (conduction around the cell, through the electrical double layer) components [60]. This has been explored most deeply in the behavior of latex nanoparticles, which (due to their size, and being nonconductive) are far more dependent on tangential conduction as the primary source of conduction.
Based on existing models of nanoparticles, the value of Gspec was anticipated to be(16)Gspec=1tσmem+2Ksrcell where the σmem component represents the transmembrane contribution and the Ks component represents the tangential contribution. The latter is well-characterized from colloid science [44] via the Bikerman equation:(17)Ks=4F2cz2D(1+3m/z2)RTκcoshzqζ2kT−1 where R, T, and F are as before, z the valence of the counterion, Dd is the ion diffusion coefficient in the diffuse layer, ζ is the ζ-potential, and m is given by:(18)m=RTF22εm3ηDd and where η is the viscosity. However, this did not fit the observed behavior in RBCs or other cells (such as shown in Figure 6)—in particular, the dependence of observed values of Gspec at higher conductivities, which have a strong σmed correlation. Empirical fitting to data [68] revealed that an excellent fit was achieved when:(19)Gspec=σmed2rcell−Ks2trcell where the expression for Ks was adapted from Equation (8), replacing ζ with the Stern layer potential Ψsto and Ξ Vm, where Vm is the membrane potential and Ξ represents the proportion of Vm observed at the shear plane, as described previously [16]:(20)Ks=4F2cz2Dd(1+3m/z2)RTκcoshzqψst+ΞVm2kT−1
This model is intriguing for many reasons, the most obvious of which is the absence of a σmem term; it is simply too small to play a detectable role in most cases. The second point of note is that the value of Ks in Equation (20) depends not on ζ but on ζ + ΞVm, showing how the capacitive coupling outlined in previous sections results in another measurable change in a surface parameter. However, the most intriguing part is in Equation (19), where the conductance equals the medium conductivity divided by the radius (suggesting conduction around the particle through the bulk medium), less that of the surface conduction value. This suggests that the capacitive coupling actively reduces surface conduction, for example by reducing the number, availability, or polarity of charge carriers in the double layer.
Another intriguing aspect of this is that whilst the DEP-derived values of Gspec are large for cells that are typically in suspension in the body (blood cells, macrophages, etc.), they are generally small-to-zero for cells found in tissues. Furthermore, the suspension cells often have relatively depolarized values of Vm (typically at −20 mV or less), whereas adherent cells often have more polarized values (nearer the oft-quoted −70 mV). It is possible that the reason for this might be the active suppression of surface conduction, or rather of the conditions which are generated by the mechanisms responsible for surface conduction, such as manipulation of the anion/cation balance immediately outside the cell.
The final component of the Clausius–Mossotti spectrum is arguably no such thing; it is an anomalous effect that appears in DEP spectra, particularly those extending to frequencies below 10 kHz. In this frequency range, DEP spectra often exhibit an upswing in polarizability, creating an additional very-low-frequency dispersion (typically below 10 kHz) shown schematically in Figure 7. This is not strictly a Clausius–Mossotti phenomenon—it is not based on interfacial polarization—but does appear as a component of DEP spectra taken and is commonly referred to as part of the “Clausius–Mossotti spectrum”.
Analysis of the RBC data set, as well as HeLa, neurons, chondrocytes, THP-1 cells, and platelets suggested that the effect is due to the polarization of the electrical double layer itself. This effect has been previously observed in low-frequency polarization in the DEP spectra of nanoparticles at high conductivity [70]. This additional polarization term was found to resemble a classical resistor–capacitor single-order filter characterized by a time constant, in line with observations of double layer polarization made by other methods [44].
The low-frequency dispersion (often referred to as the low frequency “tick”) is an additional term which acts like a low-pass filter added to the standard C-M model (Equation (4)), thus:(21)FDEP∝Reεcell*−εmed*εcell*+2εmed*+a1+jωτ where τ is the time constant and a is the amplitude. When τ was examined, it was found to exhibit a highly linear relationship to the double layer length 1/κ (examples of which are shown in Figure 8) which varies with medium conductivity when observed in RBCs under the range of conditions described previously, plus THP-1 monocytes before and after treatment with potassium channel blocker TEA, as well as primary neurons and HeLa cervical cancer cells.
Earlier work using homogeneous nanoparticles [70] suggested that this low-frequency effect arises from the polarization of the double layer; this work suggested that it in fact represented the polarization of a combination of the electrical double layer and the cell membrane. If one considers the single-order filter as arising from the capacitance and conductance of these layers, forming a filter with the classical time constant as follows:(22)τ=C/G then we can fit this to the linear data produced for the different cell types. As straight lines, these are characterized by their slopes and y-intercepts.
Since τ is proportional to double layer length 1/κ, we can examine these two components separately. The capacitance of the electrical double layer is usually calculated via the linearized Poisson–Boltzmann expression [44]:(23)C=ε0εdlκ where εdl is the permittivity of the double layer. It can be seen from Equation (8) that κ is related to the square root of medium ion concentration c, and hence in turn to the medium conductivity. The G term was found to fit best when Equation (19) was used. Since (from Equation (23)) capacitance C is proportional to κ and hence to c0.5 and thus to σmed0.5, and as Gspec is proportional to σmed (in Equation (19)), then from Equation (22) we observe that τ would indeed be proportional to σmed−0.5 and hence 1/κ.
If we examine Equation (19), we find that Gspec is described by the sum of two components; the first (σmed/2r) contains no variables relating to the cell other than the cell radius rcell. We can examine the contribution of this component to the Tick effect by calculating Equation (22) using Equation (19) to calculate Gspec and setting ζ to 0 mV. A graph of τ vs. κ−1 shows a linear relationship, with a slope that is very similar to that observed for cells with a low value of Vm, such as the RBCs in Figure 8. Where cells had more polarized Vm, the second component deriving from Equation (20) acts to decrease the gradient, with THP-1 cells (Vm ≈ −25 mV), having a lower gradient, and neurons (Vm ≈ −70 mV) being even lower [68]. Modeling of the effect using this set of equations was found to accurately predict the time constant when Equation (13) was used to first predict Vm from the standard Clausius–Mossotti model, and then using this to calculate τ.
Another key feature of the relationship between τ and κ−1 is the y-intercept; that is, the predicted value of τ when κ−1 is 0 nm. Examination of this suggests that, if a standard value of membrane capacitance of 9 mFm−2 (the capacitance of a flat membrane) [62] is used, then the whole-cell resistance value Rwc can be obtained from the estimated value of τ for κ−1 = 0. This, combined with the Cwc values which have already been demonstrated using conventional DEP modeling [32,33], means that all of the resting electrical parameters associated with patch clamp and other canonical electrophysiology measures have been identified.
The amplitude of the low-frequency dispersion (parameter a in Equation (21)) was observed to vary between cell types, and sometimes between conductivities, but was not numerically tied to any specific parameters; suggestions were made that it might relate to cell surface area, or eccentricity (the degree to which it is prolate or oblate rather than spherical), depending on whether the amplitude was compared to that of the CM factor itself, or taken as an absolute. Certainly, a measure of the latter might be a useful tool and additional parameter to extract, and as such may be worth future pursuit.
Capacitance has not been studied in the same depth as the conductivity- and potential-related parameters, but it has been observed to change with double layer length in RBCs [16,49,69]; the model for low-frequency dispersion also used the value based on the double-layer capacitance from Equation (23). However, it is also known that capacitance can vary between cell types at the same conductivity and indicates membrane folding (e.g., [13]). We also know that, at low conductivities, conventional DEP modeling yields values similar to those observed by other methods [61]. A useful model may be a series combination of the two, which would be similar to the (much lower) membrane value at low medium conductivity; this fits the RBC models presented here well but needs to be explored across other models before a definitive report can be made.
Electrophysiology as a science has fascinated scientists for decades, and yielded Nobel prizes for several of them (including Erwin Neher, Bert Sakmann, Alan Hodgkin, John Eccles, and Andrew Huxley). However, despite DEP being touted as an electrophysiology tool for several decades, it has never gained traction with the wider electrophysiology community despite offering relatively easy access to several passive electrophysiological parameters. This cannot simply be attributed to the electrophysiology community being too conservative to adopt new methods; beyond patch clamp, there are many other methods in regular use, including voltage-sensitive fluorescence markers, pH changes, or rubidium radioactivity [71]. Given this diversity of methods, and the absence of DEP, we must instead look to the shortcomings of our method; if we are not heard, it is because we do not speak the language. Whilst DEP has been used to study the differences in biology between cells, or study the effects of interventions on cells, it has done so in a way that is not meaningful to most electrophysiologists. This is even the case where the distance is small; parameters such as Gspec and Cspec have been referred to as “those DEP parameters” by scientists who routinely use Cwc and Rwc, despite the two being readily convertible by multiplying by the presumed cell surface area. If DEP is to be taken seriously beyond its practitioner base, it is necessary to learn the language and present results in a more widely interpretable way. This is beginning to happen (e.g., [33]) but is still not widespread.
The principal issue with the broader adoption of DEP has been its inability to reproduce some of the electrophysiological parameters that standard methods have used for decades—in particular Vm, and to a lesser degree values of Rwc in line with those observed by patch clamp; these simply do not appear on the Clausius–Mossotti spectrum in any meaningful way, though links have previously been drawn between cytoplasm conductivity and membrane potential [66].
However, it has emerged through the works covered here that the Clausius–Mossotti factor is not the two-dimensional graph we have assumed it to be, with DEP response mapped against frequency alone, as shown in Figure 2b or extended to Figure 7. Instead, we should consider it as a three-dimensional plot such as in Figure 9, where DEP response is plotted against both frequency and medium conductivity. This provides a plot that, as well as showing the landmarks of initial and final value and the two dispersion frequencies yielding the usual parameters of Gspec, Cspec, and σcyto (and occasionally εcyto), also identifies gradients in the low-frequency dispersion frequency, Gspec and σcyto, that give the parameters of Rwc, ζ-potential, Ξ, and most importantly Vm. Since these relationships are generally linear, two conductivities may be sufficient, though there is benefit in studying more in order to assess reproducibility. Furthermore, since the parameters vary linearly across a wide range of σmed values (we have investigated this over three orders of magnitude), it is only necessary to study a relatively small conductivity window to predict the response elsewhere. Empirically, we have found that the 150–250 mSm−1 range works well; below 50 mSm−1 it becomes harder to accurately maintain a measurable solution conductivity against factors such as resuspension carryover, whilst above 250 mSm−1 some highly conductive cells have high-frequency dispersions beyond the range of many signal generators (which commonly go to 20 MHz). This is however a rule of thumb and is not intended to discourage exploration beyond these limits. For example, the RBC spectra published by Hughes et al.
in 2021 [16] can be accurately modeled across two orders of magnitude in σmed, as shown in Figure 10, by using Equations (2)–(4) adapted with Equation (21) to include the low-frequency dispersion; Equations (18)–(20) to model Gspec, using values of Vm derived from σcyto using Equation (13), to calculate σmem; using this value of Gspec together with Equations (22) and (23) to model the low-frequency dispersion; and using a series model of membrane capacitance and double-layer capacitance determined from Equation (23) to determine Cspec and hence εmem. As can be seen, this yields highly representative curves of DEP behavior across a wide range of both frequencies and conductivities.
It is also worth noting that the change in our assumptions about the DEP response may have implications for the design of DEP apparatus, in particular for DEP separators. The revised model predicts much lower negative DEP forces than had previously been predicted, as evidenced by the lack of significant negative DEP response in Figure 9. Whilst this does not deny that negative DEP response exists and has been successfully used for DEP separation for decades, it does potentially recalibrate how we design DEP separators. In particular, it means that separators based on a strong negative DEP response might require stronger field gradients than those principally exploiting positive DEP. It is also worth noting that negative DEP exhibits a minimum just below the frequency of the first dispersion, rather than being similar from that point down to DC, suggesting that precise tuning of separation frequency may be required.
The model presented here is incomplete; there are some parameters which have not been fully explored. For example the amplitude of the low-frequency dispersion has been described qualitatively in terms of its likely origin, but there is no expression yet that might yield useful physiological data; the relationship between Cspec and the actual membrane capacitance has yet to be studied in detail, though the way in which it varies does align with predictions from Equation (23). The parameter εcyto remains beyond the reach of many generators and it is not known if this varies with medium composition. Other questions arise from the way in which we construct our models and what it tells us about the electrical structure of the cell and associated double layers. Of particular note is that Gspec appears to be the result of summing the membrane and double layer conductances, a model that dates back to Arnold and Zimmermann [60]. There is also evidence that the capacitances of membrane and double layer are summed [16,64,69]. However, any high school physics student knows that capacitances and conductances only sum when in parallel; when in series, the reciprocal is given by the sum of the reciprocals of the components. This suggests that these components act tangentially—through the double layers parallel to the field lines—more than they do from cell exterior to interior. Finally, the offset values Ψoffset used for determination of Vm in different conductivities are as yet determined only empirically. Since we have values that appear to work across all cell types, this does not need to be understood in order to make the model work; instead, it warrants investigation to understand what this tells us about the nature of Vm itself.