A novel equivalent circuit model for gap-connected cells

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2 Phys. Med. Biol. 43 (1998) Printed in the UK PII: S (98) A novel equivalent circuit model for gap-connected cells E C Fear and M A Stuchly Department of Electrical and Computer Engineering, University of Victoria, Box # 3055, Stn. CSC, Victoria, BC V8W 3P6, Canada Received 29 September 1997, in final form 27 January 1998 Abstract. Gap junctions connect neighbouring cells, providing the intercellular communication that is essential for cell growth regulation, for example. There is some evidence that gap communication changes upon exposure to electromagnetic (EM) fields. In previous work, we performed detailed finite element method (FEM) modelling of gap junction connected cells exposed to EM fields. For cell configurations, the presence of gap junctions influences the transmembrane potential and its frequency behaviour. The relaxation frequency cannot be accurately predicted by previously developed simplified models. We present a novel equivalent circuit model (ECM) that incorporates more detailed models of the gaps, and compare results obtained with this ECM to finite element and leaky cable (LC) model results. Our ECM provides more accurate estimates of the frequency behaviour of cells than the leaky cable model. Also, our ECM results suggest limitations of the application of simple models to gap-connected cells: with higher gap resistivity, the current flow in the cell interiors becomes increasingly complex and is not well represented by simple models. In this case, techniques such as the finite element method are required to model accurately cell behaviour. 1. Introduction In some studies, biological effects have been observed from relatively weak low-frequency magnetic fields; however, the interaction mechanisms responsible for producing these effects have not yet been identified. It has been suggested that the observed effects are due to the induced electric fields and currents rather than directly due to the applied magnetic field (e.g. Liburdy 1995). When cells are exposed to electric fields, a potential difference is induced across the cell membrane (transmembrane potential or TMP), and this TMP may be of sufficient magnitude to be biologically significant. To estimate the induced TMP, simple cell models consisting of a thin low-conductivity membrane surrounding cell contents or cytoplasm and placed in a conductive medium are generally used (e.g. Foster and Schwan 1989). Most cells in tissues are connected by gap junctions which are basically channels through the cell membrane that connect the interiors of neighbouring cells (Yamasaki 1990). Cells connected by gaps include heart, dendritic trees, epithelia, uterine muscle and other types of smooth muscle (De Mello 1990). Cells that are not connected by gaps, such as red blood cells, tend to be highly specialized and differentiated (De Mello 1990, Kumar and Gilula 1996). Gap junctions provide a form of intercellular communication by allowing for the exchange of certain substances between cells (Kumar and Gilula 1996). This communication is important for processes such as heart muscle contraction in which gaps facilitate the rapid This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) /98/ $19.50 c 1998 IOP Publishing Ltd 1439

3 1440 E C Fear and M A Stuchly and synchronous transfer of information (De Mello 1990). In many normal cell processes in other tissues, gap junctions play an essential role. These processes include the disposal of harmful chemicals, nourishment of sick cells, and the mediation of cell growth in which cells use gap communication to share growth suppressers (Holder et al 1993). Finally, gap junctions electrically connect cells, thus increasing the effective cell size and sensitivity to externally applied electric fields (Cooper 1984). Because of its important role in normal cell function, the disruption of gap junction communication may lead to disease processes, including cancer (Holder et al 1993). Many tumour promoters have been shown to interfere with gap communication in both in vivo and in vitro studies (Holder et al 1993). It has been suggested that the absence, change or reduction of gap junction communication in tumours all prevent or limit cells from exchanging information important for the regulation of cell growth (Yamasaki 1990). Changes in gap junction communication are related to factors ranging from the number of gaps in the cell to the conductance of the gap channel which are influenced by the cell and its environment. For example, the conductance of the channel may be voltage dependent or depend on the proteins constructing the channel (Bennett and Verselis 1992). The presence of certain substances such as calcium may decrease or increase gap conductance (Kumar and Gilula 1996). Additionally, the appearance or turnover of gaps may be influenced by certain chemicals or physical states (Holder et al 1993). Gap communication has also been shown to change with the application of electromagnetic fields (Schimmelpfeng et al 1995). We investigate the behaviour of gap junction connected cells in weak low-frequency fields. This may provide information useful for experimental dosimetry (e.g. for electric or magnetic field exposure studies) and perhaps help the search for interaction mechanisms related to weak low-frequency field exposure. Previously, we modelled configurations of cells connected by gap junctions with the finite element method (FEM) (Fear 1997, Fear and Stuchly 1997). Specifically, we investigated TMP variations in cell chains and clusters of cells in which all neighbouring cells were connected by gaps. Chains and clusters comprised of cells of various sizes and with different gap junction properties were examined. The d.c. TMP and relaxation frequency, or the frequency at which the TMP was 3 db less than the d.c. value, were computed. Our findings indicated that, for very low frequencies, many configurations of cells could be modelled as single cells having the same overall shape as the original configuration (equivalent cells). The presence of the gaps did, however, influence the frequency response of gap-connected cells. A single or equivalent cell behaved as a low-pass filter, while gap-connected cell configurations added a bandstop filter in series with the low-pass filter that described the equivalent cell. The bandstop filter characteristics were related to the characteristics of the gap junctions, such as gap size and conductivity. Several approaches to simplified modelling of gap-connected cells have previously been reported. Gap-connected cells can be represented by an equivalent ellipsoidal cell with increased cytoplasm resistivity to account for the gaps (Gailey 1996). The TMP can be estimated by analytically solving Laplace s equation (e.g. Bernhardt and Pauly 1973). The leaky cable model has been used to represent gap-connected chains in several ways: (i) with equivalent cells (Cooper 1984); (ii) with the gaps included by decreasing cytoplasm conductivity (Gailey 1996); and (iii) with gaps represented as secondary sources (Plonsey and Barr 1986). These approaches can be used effectively to predict d.c. TMPs of cell configurations, but do not accurately predict the frequency behaviour shown by our FEM computations for models of gap-connected cells (Fear 1997). The finite element method, while accurate, is computationally intensive, so it is of interest to develop simpler models

4 Novel ECM for gap-connected cells 1441 of the behaviour of gap-connected cells. In this paper, we present a novel equivalent circuit model (ECM) that adequately predicts the behaviour of various gap-connected cells. 2. Models and methods The FEM and ECM cell models are based on a simplified biological cell consisting of a thin, low-conductivity membrane that surrounds the cytoplasm and is placed in a conductive medium (Foster and Schwan 1989). The FEM cell configurations consist of several simple cells connected by gap junctions (e.g. figure 1). An ECM of a chain of two cells is shown in figure 2. Our ECM is similar to the circuit used to describe the leaky cable model. That is, resistances representing the cytoplasm and medium are connected in parallel by membrane impedance. The membrane is represented by a resistance and capacitance connected in parallel. The number of resistors used to represent the cytoplasm (n + 1) defines the discretization of the model, and the accuracy increases as the number of discretizations increases. Some leaky cable models terminate the cell with endcaps of membrane (e.g. Gaylor et al 1988). Our results did not indicate that these endcaps improved the model, so endcaps are not included in the ECM. The ECM is solved using graph theory techniques, Figure 1. Chain of three cells connected by gap junctions. Electrical properties of the cell materials are indicated (Foster and Schwan 1989). Figure 2. Equivalent circuit model: (a) two cells connected by a gap junction; (b) gap junction model.

5 1442 E C Fear and M A Stuchly specifically the mixed nodal tableau (Chandrashekar and Savage 1992). Maple (Waterloo Maple Software, Waterloo, Ontario) is used to implement the model. With this approach, cell chain dimensions and discretization can be easily adjusted and additional components can be added to the model without changing the method of solution. The dimensions of the cells and cell configurations examined in this work are summarized in table 1. Unless otherwise indicated, the gap junction conductivity is 0.5 Sm 1 and material properties are as indicated in figure 1. Table 1. Cell models. Number of Cell Total connected diameter length Gap radius Model cells (µm) (µm) (nm) Elongated cells Short chains , 15, Eight chain Seven chain A B Five chain Three chain Ten chain A B 200 C 300 Nine chain A B 100 Models representing the passive properties of gap junctions usually consist of a resistivity corresponding to the dimensions and material of the gap channel. The conductivity of the gap junctions in our models is varied from 10 7 to1sm 1 which corresponds to a gap conductance range of 2 fs to 7 µs. Biological gap junctions have conductivities of up to 10 ms (Peracchia 1994); however, increasing the gap conductivity above that of the cytoplasm (0.5 Sm 1 ) does not have a significant effect on the induced TMP, as shown by the FEM models (Fear and Stuchly 1997). Gap junction channels have a diameter of about 1.5 nm, and gaps typically comprise up to 3% of the cell surface area (Bennett and Verselis 1992). The gap junctions in our models have a minimum diameter of 10 nm, so our gap models represent a plaque or cluster of gap junctions. In the ECM, the gap junctions are modelled as a resistance representing the gap channel; however, several other components are added in order to better represent the FEM model and cell behaviour in EM fields (figure 2(b)). First of all, a capacitance is added to the gap channel. The membrane

6 Novel ECM for gap-connected cells 1443 in which the gap is embedded (interior membrane) is also included, and modelled as a resistance and capacitance connected in parallel. Finally, access resistance is added in series with the gap model. Access resistance is related to the fact the current squeezing through the gap junction must also squeeze through a small volume of cytoplasm near the gap entrance (Hall and Gourdie 1995). For a circular gap junction with the channel end represented as a thin disc, access resistance can be calculated as (Hall and Gourdie 1995) R access = 1 2πσ g R g. (1) FEM and ECM results are compared with results obtained with a leaky cable model that considers a chain of cells as an equivalent cell (primary problem) and gap junctions which act as secondary sources (secondary problem) (Plonsey and Barr 1986, Cartee and Plonsey 1992). In order to adapt this leaky cable model to better represent our FEM cell models, we add access resistance and a resistance corresponding to the interior membrane to the gap junction model. Three variations of this leaky cable model are examined: one with only gap resistance (LC 1), one with gap and access resistances (LC 2) and one with gap, access and interior membrane resistances (LC 3). TMPs are calculated for applied field strengths of 1Vm 1 (the results scale linearly), and are the maximum TMPs induced in the cell configuration. For the FEM and the leaky cable model, relaxation frequencies are estimated by evaluating TMPs at frequencies near the relaxation and using linear interpolation to approximate the relaxation frequencies. For the ECM results, the expression for TMP is in terms of frequency and can be solved for the relaxation frequency. 3. Verification Our ECM without gap junctions is the discrete version of a leaky cable model, specifically Cooper s (1984) model at d.c. and Pilla et al (1992) model for time harmonic excitations. Results for elongated cells modelled with the FEM, the leaky cable models and our ECM are compared in table 2. For shorter cells, ECM and leaky cable results for d.c. TMPs match. Errors in the ECM and LC results when compared with the FEM results are due to geometrical differences between the models and the assumption of an unperturbed applied field inherent in the ECM and LC. The effect of this neglected field perturbation in the ECM and LC models is clearly visible in the results for shorter cells (e.g. 2 and 3 µm in length). With sufficiently long cells, the leaky cable models and the ECM predict both the d.c. TMPs and relaxation frequencies of the FEM cell models to within 10%, as the field perturbation decreases. The difference between the ECM and LC results decreases with an increased number of discretizations in the ECM model. Error sources inherent in the FEM models include the discretization and truncation of the problem space, and the linear interpolation used to estimate the relaxation frequency. For the ECM, error is introduced by the finite number of discretizations used in the model, the limitation to cylindrical geometries and the assumption that the external electric field is unperturbed by the presence of the cells. For the leaky cable model, error results from interpolation of relaxation frequencies, the restriction to cylindrical geometries and the assumption of an unperturbed applied field.

7 1444 E C Fear and M A Stuchly Table 2. TMPs and relaxation frequencies for 1 µm diameter elongated cells. n refers to the number of discretizations used in the ECM model. The leaky cable models are Cooper (1984) for d.c. TMPs and Pilla (1992) for relaxation frequencies. TMP (µv) f relax (MHz) Length Error Error Error Error (µm) FEM LC (%) ECM (%) FEM LC (%) ECM (%) (n = 10) (n = 40) (n = 15) (n = 30) (n = 15) (n = 35) (n = 20) (n = 25) (n = 40) (n = 50) (n = 70) Results and discussion Cell chains much shorter than the characteristic length and exposed to d.c. fields can be represented using equivalent cells. Simple models, such as the leaky cable model, can then be used to estimate the TMP (Fear and Stuchly 1997). Because of the agreement between ECM and leaky cable models, the ECM can also be used to estimate d.c. TMPs. However, in this application the ECM does not offer any significant advantages over the LC method. The relaxation frequencies of gap-connected chains are lower than those of equivalent cells due to the presence of the gaps. The gaps act as a bandstop filter in series with the equivalent cell and thus lower the relaxation frequency (Fear 1997). The relaxation frequencies computed by FEM and ECM are summarized in table 3. Results obtained with Cartee and Plonsey s (1992) leaky cable model (LC 1) and our modified versions of this model (i.e. LC 2 with access resistance and LC 3 with access and interior membrane resistances) are also provided. Without the addition of access resistance, LC 1 has 90 to 140% error in relaxation frequency prediction for the cells in table 3. For these relatively short, smalldiameter chains, our ECM predicts relaxation frequencies generally within 10% of the FEM results. The modified leaky cable models are not as effective at estimating the relaxation frequencies of these chains. This is due to the use of linear interpolation and the absence of capacitance in the gap junction model. For these chains, the interior membrane has an insignificant influence on the results (i.e. LC 2 and LC 3 results are not significantly different). Table 3. Relaxation frequencies in MHz for short cell chains computed with FEM, ECM and leaky cable models based on Cartee and Plonsey (1992) (LC 1 = gap resistance only; LC 2 = gap + access; LC 3 = LC 2 + membrane). Length (µm) FEM ECM Error LC 1 Error LC 2 Error LC 3 Error (n = 19) (n = 11) (n = 15) (n = 13) (n = 17) (n = 15)

8 Novel ECM for gap-connected cells 1445 ECM and LC results for larger diameter chains are presented in tables 4 and 5. For cell dimensions and designation, refer to table 1. The influence of the gaps on d.c. TMP is more significant for these larger diameter cells, so the gaps must be included in the d.c. models (Fear 1997). With the addition of access resistance to the leaky cable model, this model (LC 2) predicts the d.c. TMPs of the larger diameter chains to within 15%. The ECM provides similar results, but at greater computational expense. Table 4. Larger-diameter cell chain d.c. TMPs (µv) and error (%) computed with FEM, ECM and the leaky cable model LC 1 (Plonsey and Barr 1986) and our modified versions LC 2 (with access resistance) and LC 3 (with access and interior membrane resistances). Model FEM LC 1 Error LC 2 Error LC 3 Error ECM Error Eight chain (n = 13) 3 Seven chain A (n = 20) 20 B Five chain (n = 19) 15 Three chain (n = 17) 12 Ten chain A (n = 17) 9 B C Nine chain A B Table 5. Relaxation frequencies (Hz) and error (%) for larger-diameter cells computed with FEM, ECM and the leaky cable model LC 1 (Cartee and Plonsey 1992) and our modified versions LC 2 and LC 3. Model FEM LC 1 Error LC 2 Error LC 3 Error ECM Error Eight chain (n = 23) 33 Seven chain A (n = 13) 16 B Five chain (n = 24) 22 Three chain (n = 13) 25 Ten chain A (n = 19) 26 B C Nine chain A (n = 31) 5 B For relaxation frequency estimates, the inclusion of access resistance in the gap model is necessary in order to obtain reasonable results. Similarly to the smaller-diameter chains, the ECM estimates of relaxation frequencies are slightly more accurate, and the inclusion of the interior membrane resistance does not greatly influence the results Gap size and conductivity The FEM models show variations in frequency behaviour with changes to gap junction properties. Specifically, relaxation frequencies decrease with increased gap resistivity. Figure 3 compares the changes in relaxation frequency with gap conductivity computed with FEM and ECM models for a chain of two cells. The ECM results show a linear

9 1446 E C Fear and M A Stuchly Figure 3. Variation in relaxation frequency with gap conductivity in a chain of two cells (R = 0.5 µm, R g = 20 nm, n = 13 for ECM) computed with FEM and ECM. change in relaxation frequency with changes in gap conductivity, while the FEM results have a more complex variation. For gap conductivity of 0.5 Sm 1, the ECM results are in error by 5%. As the conductivity is decreased, the error increases and is 58% for a gap conductivity of 0.01Sm 1. The relaxation frequencies computed for various gap sizes are compared with gap area ratios in table 6. The FEM results show that the relaxation frequency changes with, but not in direct proportion to, changes in gap area. The changes in the LC and ECM results are close to those of the FEM results due to the inclusion of the access resistance. The error in ECM relaxation frequency prediction increases with decreasing gap size (from 7% for the 40 nm diameter gap to 15% for the smallest gap). Table 6. Relaxation frequencies computed with FEM, leaky cable and ECM for various gap areas connecting chains of two cells (R = 0.5 µm). For the ECM, n = 13. f relax (MHz) Frequency ratio Gap radius (nm) FEM LC 3 ECM Area ratio FEM LC 3 ECM G1: G1 : G2 = G1:G3= G2: G2 : G3 = G3: Gap shape The addition of access resistance allows for various gap shapes to be considered in the ECM. For example, the access resistance of the disc-shaped gap shown in table 7 can be

10 Novel ECM for gap-connected cells 1447 Table 7. Relaxation frequencies (MHz) and error (%) computed with FEM, LC and ECM for two cells connected by a single, circular gap and disc-shaped gaps of various radii (R = 5 µm, n = 9 for ECM models). Model FEM LC 3 Error ECM Error Central gap Disc gap 1 (R 1 = 2 µm) Disc gap 2 (R 1 = 1 µm) computed from the capacitance of the disc as 1 R access = (2) 2π(R g,1 + R g,2 )σ g where R g,1 is the inner disc radius and R g,2 is the outer disc radius. Computations for chains of two cells connected by various gaps are summarized in table 7. Although access resistance makes the leaky cable and ECM models sensitive to gap shape, the FEM results are predicted less accurately for the disc-shaped gaps. The treatment of the gap in the leaky cable model, even with the addition of access resistance, results in greater error than with the ECM predictions. With more complex gap configurations, the current flow in the chain interior is not correctly modelled, as the ECM does not fully reflect the electric field perturbation in the cytoplasm near the gap. The access resistance compensates somewhat for the perturbation of the current flow, but the ECM is less accurate with more complex gap configurations which correspond to a more complex current flow pattern. This is also true for smaller gaps and gaps of lower conductivity. In principle, it may be possible to develop more complex expressions for the access resistance to better predict the relaxation frequencies. 5. Conclusions The ECM is the most effective model for predicting relaxation frequencies of the FEM cell of those methods examined by us. Our ECM predicts relaxation frequencies of short (less than the characteristic length), small-diameter cell chains to within 10%. For largerdiameter chains, the error is less predictable and can reach 80%. These errors in prediction are slightly less than those obtained with a leaky cable model (Cartee and Plonsey 1992) modified by the addition of access resistance. This modified leaky cable model does, however, provide reasonable results at less computational cost. Our results indicate that it is necessary to include the access resistance in models of the gap junctions in order to accurately predict relaxation frequencies and behaviour of chains of gap-connected cells. The access resistance compensates for the electric field perturbations that result from the gaps. This has implications for models of gap-connected cells used to study phenomena such as defibrillation. Although the addition of access resistance makes the LC and ECM more sensitive to changes in gap shape and conductivity, these models are less effective at predicting behaviour for more complex gap shapes or low gap conductivities. This is because the interior current flow becomes spatially complicated, and is not well predicted

11 1448 E C Fear and M A Stuchly by models in which current flow is limited to the axial direction. The ECM is, however, still useful as the first approximation. For more complex gap configurations, a method such as FEM, which accounts for the electric field and current perturbation in the vicinity of the gap junctions, is expected to provide more accurate results. Therefore, it is important to assess the properties of the gaps to be modelled and utilize an appropriate model. For example, more complex models may be required to assess phenomena such as the influence of opening and closing of gap junctions. Our results indicate that the ECM is an adequate model for the frequency behaviour of gap-connected cell chains computed with FEM. The modified leaky cable model (i.e. with access resistance included) can be used to estimate cell behaviour but with greater error. Both the ECM and FEM models should be extended to examine larger clusters of gapconnected cells in order to determine if simpler ECM models of these cell configurations can be developed. Experimental verification through, for example, the use of potentiometric dyes (Loew 1992) would indicate if the modelled phenomena occur in real biological cells. This is important, for example, for assessing the effects of EM field exposure on cells, as our results indicate that the induced TMPs in large cell configurations decrease dramatically with increases in frequency after a few hundred Hz. References Bennett MVLandVerselis V K 1992 Biophysics of gap junction channels Semin. Cell Biol Bernhardt J and Pauly H 1973 On the generation of potential differences across the membranes of ellipsoidal cells in an alternating electrical field Biophysik Cartee L A and Plonsey R 1992 The effect of cellular discontinuities on the transient subthreshold response of a one-dimensional cardiac model IEEE Trans. Biomed Chandrashekar M and Savage G J 1992 Engineering Systems: Analysis, Design and Control (Ontario: University of Waterloo) Cooper M 1984 Gap junctions increase the sensitivity of tissue cells to exogenous electric fields J. Theor. Biol De Mello W C 1990 The way cells communicate Cell Communication (Boca Raton, FL: CRC) pp 1 20 Fear E C 1997 Modelling biological cells exposed to electric fields MASc Thesis University of Victoria Fear E C and Stuchly M A 1997 Biological cells with gap junctions in low frequency electric fields IEEE Trans. Biomed. accepted Foster K R and Schwan H P 1989 Dielectric properties of tissues and biological materials: a critical review Crit. Rev. Biomed. Eng Gailey P C 1996 Comparison of voltage signals induced by power frequency fields to thermal electrical noise at the cell membrane PhD Thesis University of Utah Gaylor D C, Prakah-Asante K and Lee R C 1988 Significance of cell size and tissue structure in electrical trauma J. Theor. Biol Hall J E and Gourdie R G 1995 Spatial organization of cardiac gap junctions can affect access resistance Microsc. Res. Tech Holder J W, Elmore E and Barrett J C 1993 Gap junction function and cancer Cancer Res Kumar N M and Gilula N B 1996 The gap junction communication channel Cell Liburdy R P 1995 Cellular studies and interaction mechanisms of extremely low frequency fields Radio Sci Loew L M 1992 Voltage-sensitive dyes: measurement of membrane potentials induced by DC and AC electric fields Bioelectromagnetics (Suppl. 1) Peracchia C 1994 Handbook of Membrane Channels (New York: Academic) Pilla A A, Nasser P R and Kaufman J J 1992 The sensitivity of cells and tissues to weak electromagnetic fields Charge and Field Effects in Biosystems 3 ed J Allen, S F Cleary, A E Sowers, D D Shillady (New York: Plenum) pp Plonsey R and Barr R C 1986 Inclusion of junction elements in a linear cardiac model through secondary sources: application to defibrillation Med. Biol. Eng. Comput Schimmelpfeng J, Stien J-C and Dertinger H 1995 Action of 50 Hz magnetic fields on cyclic AMP and intercellular communication in monolayers and spheroids of mammalian cells Bioelectromagnetics Yamasaki H 1990 Gap junctional intercellular communication and carcinogenesis Carcinogenesis