Proc. Nat. Acad. Sci. USA Vol. 72, No. 12, pp. 4936-4939, December 1975
Biophysics
Analogue solution for electrical capacity of membrane covered square cylinders in square array at high concentration (Laplace equation/composite systems/resistor-capacitor analogue/cylindrical tissues)
KENNETH S. COLE National Institutes of Health, Bethesda, Maryland 20014; and Marine Biological Laboratory, Woods Hole, Massachusetts 02543
Contributed by Kenneth S. Cole, September 18, 1975
Analytical solutions of Laplace equations ABSTRACT have given the electrical characteristics of membranes and interiors of spherical, ellipsoidal, and cylindrical cells in suspensions and tissues from impedance measurements, but the underlying assumptions may be invalid above 50% volume concentrations. However, resistance measurements on several nonconducting, close-packing forms in two and three dimensions closely predicted volume concentrations up to 100% by equations derived from Maxwell and Rayleigh. Calculations of membrane capacities of cells in suspensions and tissues from extensions of theory, as developed by Fricke and by Cole, have been useful but of unknown validity at high concentrations. A resistor analogue has been used to solve the finite difference approximation to the Laplace equation for the resistance and capacity of a square array of square cylindrical cells with surface ca acity. An 11 X 11 array of resistors, simulating a quarter of the unit structure, was separated into intra- and extra-cellular regions by rows of capacitors corresponding to surfaice membrane areas from 3 X 3 to 11 x 11 or 7.5% to 100%. The extended Rayleigh equation predicted the cell concentrations and membrane capacities to within a few percent from boundary resistance and capacity measure*ments at low frequencies. This single example suggests that analytical solutions for other, similar two- and three-dimensional problems may be approximated up to near 100% concentrations and that there may be analytical justifications for such analogue solutions of Laplace equations. There have been many measurements of transport in composite systems such as electrical conductance and dielectric constant, heat transfer, optical transmission and diffusion including everything from soils to emulsions to biological tissues and cell suspensions. Many of these have been analyzed, under somewhat restrictive assumptions, as solutions of a Laplace equation. Fricke (1, 2) developed theories for conductance and capacitance of suspensions of nonconducting and membrane covered spheres, which he extended for spheroids. A general impedance equation was given (3) and the special case of a square array of square cylinders was formulated (3). Fricke's conductance work was later shown (4) to be approximations of Maxwell (5), Rayleigh (6), and Wiener (7). Boiler and Cole (8) and Cole and Curtis (9) adapted Maxwell's technique to derive the transverse conductance of a suspension of circular cylinders, which was Rayleigh's first approximation. These results are summarized L1] (1 - Rl/Ro)/(x + Rl/Ro) = p/x where Ro is the resistivity for a volume concentration p of the nonconductor in a medium of resistivity R1, and the shape factor is x = 2 for spheres or x = 1 for circular cylinders. There have been extensions of theory such as for high concentrations of nonconducting and conducting spheres and cylinders (10, 11). However, the formulas for dilute suspensions of spheres and circular cylinders have been used
for erythrocytes up to 95% volume concentration, and muscle and cortical tissues at about 80% where the assumptions of uniform fields external and internal to the nonconducting obstacles are unlikely to be even approximated. With a single exception (12), there was no justification for the use of the equations or basis for alternatives, until a recent investigation of some close-packing analogues (13). Here it was found that over most of the range from zero to 100% the two-dimensional equation was valid for square and hexagonal cylinders as was the three-dimensional equation for cubes and tetrakaidecahedra. These solutions of the corresponding Laplace equations thus gave basis for some confidence in interpreting electrical and analogous data on some tissues and similar systems. Fricke's equation (14) for the capacitance of a suspension of membrane covered spheres was also an approximation from Maxwell (5, 3), and the corresponding cylindrical solution was derived similarly (8, 9) [2] C = [x2p/(x + p)2]aC or
[3] R1/Ro)ac where a is the radius and c is the membrane capacitance/ unit area in each case. Fricke's original work led him to a capacity of 0.81 MF/ cm2 and a thickness of 33 A (15) for the erythrocyte membrane. This first demonstration of the molecular thickness of a living cell membrane, and its confirmation by extraction and spreading, electron micrograph and x-ray diffraction work, led to establishment of the bimolecular lipid layer as the basic biological membrane structure. The capacity of a vast number and variety of cells and tissues has been measured and the membrane capacities have been calculated by Eqs. 2 and 3 but without any assurance, on the basis of their derivations, that they were valid above about 50% cell concentration. The need for some indication of the meaning of capacity measurements at higher concentrations has been increasingly apparent over the past thirty years (3), but it was primarily the recent analysis (16) of early published impedance dati on the squid giant axon that spurred the present preliminary attempt at an analogue solution. The high frequency dispersion was attributed to the axon sheath and the resistance was interpreted, on the basis of the earlier analogue results, as the result of a 99;75% cell volume concentration. However, it seemed rather rash to deal similarly with the capacity data without some support. C
=
(1
-
R1/Ro)(x
+
EXPERIMENTAL Analogue and Measurements. There was no obvious way to insert a capacitive boundary in a teledeltos or kymograph 4936
TTtTrT__TT T
7 o0
0
4937
Proc. Nat. Acad. Sci. USA 72 (1975)
Biophysics: Cole
0
0
0
0
0
0
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044 0
ot o oIo
oJ
__Sot
-~~ lillLeS FIG. 1. Schematic diagram of resistor-capacitor analogue Light lines outline the units approximated by a resistor star for each at its center. These resistors are shown only at the boundaries of the 7 X 7 region C, enclosed by capacitors, and at the terminals. At high concentration most of the capacitive current flows from terminal V and region A into C. The resistive current flows into region B, bounded by the adjacent figure, to the axis of symmetry at ground and some capacitive current flows into C. array.
conducting paper, such as was used before for the two-dimensional resistance analogue solutions. Nor were there ideas or data at hand to design capacitive surfaced conducting figures to replace the plastic forms used in the electrolytic tank for three-dimensional solutions. The simplest approach was to measure the impedance of a square resistor array as a finite difference approximation for the solution of the Laplace equation in the case of square cylinders in a square lattice (17, 3). The quarter of the square containing a single cell was represented between the lattice and symmetry boundaries by 'k W 10 kQ I 5% resistors soldered between brass posts in an 11 X 11 array with row of similar 5.1 kg resistors at the electrode and equipotential midline as shown in Fig. 1. The 242 resistors were not measured individually, but several tests showed no marked lack of uniformity. The resistances of the uniform array, and at high frequencies with capacitors, were 10.17 I 0.03 kQ; deviation was attributed to small variations of temperature from the mean 22.5°. The veteran Wheatstone bridge (18), except with input and output interchanged because the shielded and balanced input transformers were not available, was used to measure the impedances in parallel with a standard 10 kQ as parallel R and C over the range from 0.002 to 200 kHz. Procedure. After measurement of the complete array, resistor connections were opened so that central quarter squares of from 3 X 3 to 10 X 10 units could be isolated and the resistance of the analogous nonconducting square cylinders and the electrical measure of their volume concentration obtained. Similar capacitors were then wired into the breaks, as shown in Fig. 1 for the 7 X 7 array, so that a uniform membrane surface was simulated and a frequency run was made.
FIG. 2. Low frequency capacity Co from Eq. 4 as a fraction of capacity Cloo,.for concentration pa = 1.0, versus po (solid curve). Experimental points adjusted to fit curve and give C10o in each case. Experimental circuit (CKT) calculations interpolate between n = 10 and n = 11 by Eqs. 6 and 7. Note singular point at po = 1.0 as discussed in text.
The limiting low frequency parallel resistance, Ro, and
cawere estimated from the data. The data were (Fig. 3), also plotted on the complex plane, as Z = + and the capacity was calculated from the frequency of maxA). imum reactance, but this was The first experiment with nominal 0.01 ,uF ceramic capacitors was not entirely satisfactory because of difficulty in
pacity, Co,
R,
iXs
questionable (Appendix
estimating the low frequency capacity asymptote. In the second experiment, with nominal 0.0024 tLF silver mica capacitors, the low frequency capacity was measured with good precision partly because of the shorter time constant and also probably because of better electrical characteristics.
RESULTS Volume concentration and resistances The first test of the data is the comparison with the earlier confirmation of the Rayleigh equation (16) by conducting paper analogue. The agreement is close except for a systematic difference: the concentrations calculated from the resistances are consistently higher than the geometric values by a maximum of 2% units. A satisfactory explanation for these differences has not been found (Appendix B), so in the calculations to follow it seems necessary to use the resistances as measured and ignore the values calculated from the geometric concentrations. However, because of this discrepancy, more weight has been given to results at higher volume concentrations where its effect is less. Capacities The low frequency parallel capacity Co, as derived for low concentrations of circular cylinders (Eq. 2), was rewritten Co
=
[4po /(1
+
po)2117c
[4]
where po is the geometric concentration and c is the capaciCo/C1oo versus po is shown as the solid
tance per mesh unit.
4938
Biophysics:
Cole
Proc. Nat. Acad. Sci. USA 72 (1975)
Fig. 2, where Cloo is the capacity for po = 1.0. The experimental values of Co were calculated, with n being the number of mesh units on a side of a cylinder and iz the maximum number,
curve,
[5] Co = [(1 + R1/RoX1 - RI/Ro)](n/b)Cloo These values were compared, on logarithmic scales, with the theory to determine the values of Cloo for best fit. The results as shown in Fig. 2 are C100 = 0.126 ,gF for experiment I and C0oo = 0.0345 jOF for experiment II. The gap between the data for the 10 X 10 and 11 X 11 capacitor enclosed arrays (po = 0.83 and 1.0) requires further investigation because the measurements gave capacity ratios of 0.78 and 0.76, whereas Eq. 5 gives a ratio of unity at po = 1.0. The direct procedure, to use finer meshed grids, seemed unduly laborious, particularly since the result in the final interval would probably still be in question. There are continuous solutions for potential and current distributions at low frequencies in regions A and B, but it was necessary to show that they approximated the analogue results for the 10 X 10 array. The easier ladder solution and measurement for region A showed an appreciable current flow into B but the reduction of the effective resistance of region B was about 1%. This gave a volume concentration by Eq. 1, of p = 0.818 to agree well with the analogue p = 0.831 and the geometrical po = 0.818. The reduction of potential across the capacitors near the corner was negligible so the A contribution to the measured Co was CA
=
nc
[6]
The effective capacity of the uniform line, n = 10 in B, was calculated from the Guillemin (19, 3) equivalent T, and the Zobel (20) conversion gave the B contribution to the measured capacity [7] nc/[l + 2(n - n)/n] These components, with the expected Cmoo, gave Co/Cloo = 0.888 as compared with 0.901 obtained by Eq. 4, and the analogue values of 0.881 and 0.907. Thus, extrapolations of Ro and Co above the analogue values for n = 10 seem justified and may be considered as extensions of them. The intermediate calculations for n = 10.5, 10.75, and 10.99 gave analogue concentrations from Ro close to the geometric as well as the excellent values for Co/Cioo shown in Fig. 2. It is now apparent that Co/Cloo has a singular point at n = 11 or po = 1.0. This is to be expected since region B has vanished leaving no conductance path between the membranes of adjacent cells. The only current flow is then through the n = 11 capacitors of region C facing the electrode (without region A which has also disappeared) to give a value of Co/Cloo = 0.8 which is in good agreement with the analogue data as shown in Fig. 2. Thus, Eq. 5 has been demonstrated to be valid for all values of PO from less than 0.08 up to 0.998 and presumably any value of po < 1.0. Membrane capacities Having established an approximate validity for Eq. 5 over the entire range from PO = 0, by extrapolation from po = 0.073 to po = 1.00, we are now in a position to make the final and crucial calculations of the values of the mesh unit capacitances, c = Cloo/1.25 fi from the experiments. Thus, with ri = 11 for experiment I, where C100 = 0.126 OF, c = 92 nF as compared with c = 89.5 I 0.58 nF measured for the 20 capacitors used. For experiment II with C100 =
CB
=
0.0345 ,F, the mesh unit calculated c = 2.52 nF whereas direct measurement gave c = 2.413 i 0.034 nF. These agreements are about as good as had been expected (Appendix C). The assumption that the squid axon sheath capacity of 0.084 MiF/cm2 at 1 kHz (16) is attributed to a Schwann layer 0.8 Am thick leads to a capacitance 6.72 10'12F/cm = 1.25 nc. The layer structure is complicated and it may not be possible to arrive at a representative cell dimension. But, although it may be -only a formality, we may take c = 1.0 ,uF/cm2 as found for the plasma membrane, to arrive at a characteristic cell dimension 2n = 0.11 um or a seven-cell thickness for the sheath. -
SUMMARY AND CONCLUSION The surprising accuracy of analogue extensions of the Rayleigh and Maxwell equations for the resistances of conductors around two- and three-dimensional close-packing insulators from the low concentration into the medium and high concentration ranges led to the hope that similar extensions might be found for the capacity with conducting forms having capacitive boundaries. The results of the present experiments with a resistor-capacitor analogue for solutions of the finite difference Laplace equation for square cylinders are in good agreement with theory for volume concentrations from 8% to 83% and extrapolations to zero and 100%. With all of the two- and three-dimensional solutions of the Laplace equation for the resistance which follow the low concentration analytical solutions up to close packing and now the single example for the capacity of membrane covered forms, it is very tempting to assume the same result for other such problems. But, since the only common factor of these problems is that they are solutions of Laplace equations, which we have not obtained directly by analysis at high concentrations, we have no valid reason to expect that they will always be extrapolated from the low concentration results. It may be interesting to make similar analogue measurements for high concentrations of hexagonal, cylindrical cells in a triangular array as a more probable tissue structure, although no marked difference from the present results is expected (21). However, it should be more useful to investigate a three-dimensional analogue. On the other hand, the present extension into a more complicated field adds to the importance of finding a broad theoretical approach to such problems, perhaps along the lines of Keller's theorem (12). APPENDIX A. Impedance. Many of the impedance versus frequency data, as taken with the analogues in parallel with the 10 k12 standard resistor, were calculated as Z = R, + jX8 and plotted on the complex plane, R8 versus X. As for the theory of small circular cylinders, which gives a bilinear equation, the loci for the smaller values of n closely approximated semicircles and the circuit capacity was unambiguous. At the middle and higher values of n the loci reflected the complex nature of the analogues seen in Fig. 3 for n = 8. This was to be expected as an approach to the case of n = 10 where, as has been shown, there are two distinct and virtually independent capacitive current paths, from regions A to C and from regions B to C. Such a two reactive element circuit is represented by a biquadratic equation and on the complex
Biophysics:
Proc. Nat. Acad. Sci. USA 72 (1975)
Cole
4939
are to be the same. Since C1oo was found entirely in terms of square parameters, a and n are equivalent at low concentrations. The explanation of this discrepancy probably must await either an analytical solution or a more accurate analogue solution for the square cylinder problem.
FIG. 3. Complex plane locus for R, versus Xs, series resistance series reactance, where Z = R8 + jX,, in kg at labeled frequencies in kHz for 8 X 8 square mesh. Note asymmetric devia-
versus
tions from approximating circular arc.
plane by the pedal curve of an ellipse (3). Such analyses have been attempted, and without them any interpretations of the intermediate and high frequency capacities are unnot
certain.
B. Resistance Shape Factors. The average difference, p 0.01, between the nonconducting volume concentracalculated by Eq. 1 for x = 1 and the geometric fraction po = (n/-i)2, is systematic and probably significant. The shape factor x has been calculated and tested extensively for spheroids and ellipsoids (22) but no analysis is known for elliptic, hexagonal, or the present square two-dimensional cylinder problems. However, for x = 1.2 the average p po was reduced to approximately zero, but systematic divergences up to 0.01 remained after the omission of two irregular values. Thus, it appears that the effect of the shape of the cross section may need a different formulation. C. Capacitance Shape Factor. Before interpreting the values of Cloo obtained in the two experiments it is necessary to consider the relation between the radius, a, of the small cylinder for which the theory was derived and the number, n, of mesh units representing a half side of the square cylinder that has been used as a basis for structure measurements and analysis of the analogue. Fricke (2) calculated shape constants for spheroids that appeared as factors in his approximation to Eq. 3 for this capacity. A similar analysis for the square cylinder does not seem possible without a solution for the potential theory problem. The analogue capacities were more closely approximated to Eq. 4 for po < 0.3 with values of C1oo = 0.128 and 0.0355 MF for experiments I and II, respectively, but these are not importantly different from the values used for the entire range of po so they give no indication of a significant effect of shape between the circular and square cross section in the range for which the circle derivation should be valid, or beyond it. The dependent variable Co/Cloo = 4po Vo/(1 + PO)2 of Fig. 2 was chosen to present, as clearly as possible, comparisons of the theoretical and analogue dependence on volume concentration without the intervention of shape factors and the substitution of n in Eq. 5 for a from Eq. 3 was entirely arbitrary. However, for any given value of P0 it is necessary that -ra2/4 = n2 or n = xfir/4 a = 0.886 a, if the capacities - po = tion p
-
1. Fricke, H. (1924) "A mathematical treatment of the electric conductivity and capacity of disperse systems. I. The electric conductivity of a suspension of homogeneous spheroids," Physic. Rev. 24,575-585. 2. Fricke, H. (1925) "A mathematical theory of the electric conductivity and capacity of disperse systems. II. The capacity of a suspension of conducting spheroids surrounded by a nonconducting membrane for a current of low frequency," Physic. Rev. 26, 678-681. 3. Cole, K. S. (1968, 1972) Membranes, Ions and Impulses (Univ. Calif. Press, Berkeley, Calif.). 4. Cole, K. S. (1928) "Electric impedance of a suspension of spheres," J. Gen. Physiol. 12, 29-36. 5. Maxwell, J, C. (1873) Treatise on Electricity and Magnetism (Clarendon Press, Oxford). 6. Rayleigh, Lord (J. W. Strutt) (1892) "On the influence of obstacles arranged in a rectangular order upon the properties of the medium," Philos. Mag. 34,481-502. 7. Wiener, 0. (1912) "Die Theorie des Mischkorpers fur das Feld der Stationkren Strbmung," Abh. K. Saechs Ges. Wiss., Math. Phys. Ki. 32,509-604. 8. Bozler, E. & Cole, K. S. (1935) "Electric impedance and phase angle of muscle in rigor," J. Cell. Comp. Physiol. 6,229-241. 9. Cole, K. S. & Curtis, H. J. (1936) "Electric impedance of nerve and muscle," Cold Spring Harbor Symp. Quant. Biol. 4, 73-89. 10. Meredith, R. E. & Tobias, C. W. (1962) "Conduction in heterogeneous systems," Adv. Electrochem. Electrochem. Eng. 2, 15-43. 11. Keller, J. B. (1963) "Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders," J. Appl. Phys. 34, 991-993. 12. Keller, J. B. (1964) "A theorem on the conductivity of a composite medium," J. Math. Phys. (N.Y.) 5,548-549. 13. Cole, K. S., Li, C-l. & Bak, A. F. (1969) "Electrical analogues for tissues," Exp. Neurol. 24, 459-473. 14. Fricke, H. (1925) "Electrical capacity of suspensions with special reference to blood," J. Gen. Physiol. 9, 137-152. 15. Fricke, H. (1923) "The electric capacity of cell suspensions," Physic. Rev. 21, 708-709. 16. Cole, K. S. (1975) "Electrical properties of the squid axon sheath," Biophys. J., in press. 17. de Packh, D. C. (1947) "A resistor network for the approximate solution of the Laplace equation," Rev. Sci. Instrum. 18, 798-799. 18. Cole, K. S. & Curtis, H. J. (1937) "Wheatstone bridge and electrolytic resistor for impedance measurements over a wide frequency range," Rev. Sci. Instrum. 8, 333-339. 19. Guillemin, E. A. (1935) Communication Networks (Wiley, New York), Vol. II. 20. Zobel, 0. J. (1923) "Theory and design of uniform and composite electric wave-filters," Bell Syst. Tech. J. 2, 1-46. 21. Nicholson, P. W. (1965) "Specific impedance of cerebral white matter," Exp. Neurol. 13,386-401. 22. Velick, S. & Gorin, M. (1940) "The electrical conductance of suspensions of ellipsoids and its relation to the study of avian erythrocytes," J. Gen. Physiol. 23, 753-771.
Biophysics
Analogue solution for electrical capacity of membrane covered square cylinders in square array at high concentration (Laplace equation/composite systems/resistor-capacitor analogue/cylindrical tissues)
KENNETH S. COLE National Institutes of Health, Bethesda, Maryland 20014; and Marine Biological Laboratory, Woods Hole, Massachusetts 02543
Contributed by Kenneth S. Cole, September 18, 1975
Analytical solutions of Laplace equations ABSTRACT have given the electrical characteristics of membranes and interiors of spherical, ellipsoidal, and cylindrical cells in suspensions and tissues from impedance measurements, but the underlying assumptions may be invalid above 50% volume concentrations. However, resistance measurements on several nonconducting, close-packing forms in two and three dimensions closely predicted volume concentrations up to 100% by equations derived from Maxwell and Rayleigh. Calculations of membrane capacities of cells in suspensions and tissues from extensions of theory, as developed by Fricke and by Cole, have been useful but of unknown validity at high concentrations. A resistor analogue has been used to solve the finite difference approximation to the Laplace equation for the resistance and capacity of a square array of square cylindrical cells with surface ca acity. An 11 X 11 array of resistors, simulating a quarter of the unit structure, was separated into intra- and extra-cellular regions by rows of capacitors corresponding to surfaice membrane areas from 3 X 3 to 11 x 11 or 7.5% to 100%. The extended Rayleigh equation predicted the cell concentrations and membrane capacities to within a few percent from boundary resistance and capacity measure*ments at low frequencies. This single example suggests that analytical solutions for other, similar two- and three-dimensional problems may be approximated up to near 100% concentrations and that there may be analytical justifications for such analogue solutions of Laplace equations. There have been many measurements of transport in composite systems such as electrical conductance and dielectric constant, heat transfer, optical transmission and diffusion including everything from soils to emulsions to biological tissues and cell suspensions. Many of these have been analyzed, under somewhat restrictive assumptions, as solutions of a Laplace equation. Fricke (1, 2) developed theories for conductance and capacitance of suspensions of nonconducting and membrane covered spheres, which he extended for spheroids. A general impedance equation was given (3) and the special case of a square array of square cylinders was formulated (3). Fricke's conductance work was later shown (4) to be approximations of Maxwell (5), Rayleigh (6), and Wiener (7). Boiler and Cole (8) and Cole and Curtis (9) adapted Maxwell's technique to derive the transverse conductance of a suspension of circular cylinders, which was Rayleigh's first approximation. These results are summarized L1] (1 - Rl/Ro)/(x + Rl/Ro) = p/x where Ro is the resistivity for a volume concentration p of the nonconductor in a medium of resistivity R1, and the shape factor is x = 2 for spheres or x = 1 for circular cylinders. There have been extensions of theory such as for high concentrations of nonconducting and conducting spheres and cylinders (10, 11). However, the formulas for dilute suspensions of spheres and circular cylinders have been used
for erythrocytes up to 95% volume concentration, and muscle and cortical tissues at about 80% where the assumptions of uniform fields external and internal to the nonconducting obstacles are unlikely to be even approximated. With a single exception (12), there was no justification for the use of the equations or basis for alternatives, until a recent investigation of some close-packing analogues (13). Here it was found that over most of the range from zero to 100% the two-dimensional equation was valid for square and hexagonal cylinders as was the three-dimensional equation for cubes and tetrakaidecahedra. These solutions of the corresponding Laplace equations thus gave basis for some confidence in interpreting electrical and analogous data on some tissues and similar systems. Fricke's equation (14) for the capacitance of a suspension of membrane covered spheres was also an approximation from Maxwell (5, 3), and the corresponding cylindrical solution was derived similarly (8, 9) [2] C = [x2p/(x + p)2]aC or
[3] R1/Ro)ac where a is the radius and c is the membrane capacitance/ unit area in each case. Fricke's original work led him to a capacity of 0.81 MF/ cm2 and a thickness of 33 A (15) for the erythrocyte membrane. This first demonstration of the molecular thickness of a living cell membrane, and its confirmation by extraction and spreading, electron micrograph and x-ray diffraction work, led to establishment of the bimolecular lipid layer as the basic biological membrane structure. The capacity of a vast number and variety of cells and tissues has been measured and the membrane capacities have been calculated by Eqs. 2 and 3 but without any assurance, on the basis of their derivations, that they were valid above about 50% cell concentration. The need for some indication of the meaning of capacity measurements at higher concentrations has been increasingly apparent over the past thirty years (3), but it was primarily the recent analysis (16) of early published impedance dati on the squid giant axon that spurred the present preliminary attempt at an analogue solution. The high frequency dispersion was attributed to the axon sheath and the resistance was interpreted, on the basis of the earlier analogue results, as the result of a 99;75% cell volume concentration. However, it seemed rather rash to deal similarly with the capacity data without some support. C
=
(1
-
R1/Ro)(x
+
EXPERIMENTAL Analogue and Measurements. There was no obvious way to insert a capacitive boundary in a teledeltos or kymograph 4936
TTtTrT__TT T
7 o0
0
4937
Proc. Nat. Acad. Sci. USA 72 (1975)
Biophysics: Cole
0
0
0
0
0
0
0
0
0
044 0
ot o oIo
oJ
__Sot
-~~ lillLeS FIG. 1. Schematic diagram of resistor-capacitor analogue Light lines outline the units approximated by a resistor star for each at its center. These resistors are shown only at the boundaries of the 7 X 7 region C, enclosed by capacitors, and at the terminals. At high concentration most of the capacitive current flows from terminal V and region A into C. The resistive current flows into region B, bounded by the adjacent figure, to the axis of symmetry at ground and some capacitive current flows into C. array.
conducting paper, such as was used before for the two-dimensional resistance analogue solutions. Nor were there ideas or data at hand to design capacitive surfaced conducting figures to replace the plastic forms used in the electrolytic tank for three-dimensional solutions. The simplest approach was to measure the impedance of a square resistor array as a finite difference approximation for the solution of the Laplace equation in the case of square cylinders in a square lattice (17, 3). The quarter of the square containing a single cell was represented between the lattice and symmetry boundaries by 'k W 10 kQ I 5% resistors soldered between brass posts in an 11 X 11 array with row of similar 5.1 kg resistors at the electrode and equipotential midline as shown in Fig. 1. The 242 resistors were not measured individually, but several tests showed no marked lack of uniformity. The resistances of the uniform array, and at high frequencies with capacitors, were 10.17 I 0.03 kQ; deviation was attributed to small variations of temperature from the mean 22.5°. The veteran Wheatstone bridge (18), except with input and output interchanged because the shielded and balanced input transformers were not available, was used to measure the impedances in parallel with a standard 10 kQ as parallel R and C over the range from 0.002 to 200 kHz. Procedure. After measurement of the complete array, resistor connections were opened so that central quarter squares of from 3 X 3 to 10 X 10 units could be isolated and the resistance of the analogous nonconducting square cylinders and the electrical measure of their volume concentration obtained. Similar capacitors were then wired into the breaks, as shown in Fig. 1 for the 7 X 7 array, so that a uniform membrane surface was simulated and a frequency run was made.
FIG. 2. Low frequency capacity Co from Eq. 4 as a fraction of capacity Cloo,.for concentration pa = 1.0, versus po (solid curve). Experimental points adjusted to fit curve and give C10o in each case. Experimental circuit (CKT) calculations interpolate between n = 10 and n = 11 by Eqs. 6 and 7. Note singular point at po = 1.0 as discussed in text.
The limiting low frequency parallel resistance, Ro, and
cawere estimated from the data. The data were (Fig. 3), also plotted on the complex plane, as Z = + and the capacity was calculated from the frequency of maxA). imum reactance, but this was The first experiment with nominal 0.01 ,uF ceramic capacitors was not entirely satisfactory because of difficulty in
pacity, Co,
R,
iXs
questionable (Appendix
estimating the low frequency capacity asymptote. In the second experiment, with nominal 0.0024 tLF silver mica capacitors, the low frequency capacity was measured with good precision partly because of the shorter time constant and also probably because of better electrical characteristics.
RESULTS Volume concentration and resistances The first test of the data is the comparison with the earlier confirmation of the Rayleigh equation (16) by conducting paper analogue. The agreement is close except for a systematic difference: the concentrations calculated from the resistances are consistently higher than the geometric values by a maximum of 2% units. A satisfactory explanation for these differences has not been found (Appendix B), so in the calculations to follow it seems necessary to use the resistances as measured and ignore the values calculated from the geometric concentrations. However, because of this discrepancy, more weight has been given to results at higher volume concentrations where its effect is less. Capacities The low frequency parallel capacity Co, as derived for low concentrations of circular cylinders (Eq. 2), was rewritten Co
=
[4po /(1
+
po)2117c
[4]
where po is the geometric concentration and c is the capaciCo/C1oo versus po is shown as the solid
tance per mesh unit.
4938
Biophysics:
Cole
Proc. Nat. Acad. Sci. USA 72 (1975)
Fig. 2, where Cloo is the capacity for po = 1.0. The experimental values of Co were calculated, with n being the number of mesh units on a side of a cylinder and iz the maximum number,
curve,
[5] Co = [(1 + R1/RoX1 - RI/Ro)](n/b)Cloo These values were compared, on logarithmic scales, with the theory to determine the values of Cloo for best fit. The results as shown in Fig. 2 are C100 = 0.126 ,gF for experiment I and C0oo = 0.0345 jOF for experiment II. The gap between the data for the 10 X 10 and 11 X 11 capacitor enclosed arrays (po = 0.83 and 1.0) requires further investigation because the measurements gave capacity ratios of 0.78 and 0.76, whereas Eq. 5 gives a ratio of unity at po = 1.0. The direct procedure, to use finer meshed grids, seemed unduly laborious, particularly since the result in the final interval would probably still be in question. There are continuous solutions for potential and current distributions at low frequencies in regions A and B, but it was necessary to show that they approximated the analogue results for the 10 X 10 array. The easier ladder solution and measurement for region A showed an appreciable current flow into B but the reduction of the effective resistance of region B was about 1%. This gave a volume concentration by Eq. 1, of p = 0.818 to agree well with the analogue p = 0.831 and the geometrical po = 0.818. The reduction of potential across the capacitors near the corner was negligible so the A contribution to the measured Co was CA
=
nc
[6]
The effective capacity of the uniform line, n = 10 in B, was calculated from the Guillemin (19, 3) equivalent T, and the Zobel (20) conversion gave the B contribution to the measured capacity [7] nc/[l + 2(n - n)/n] These components, with the expected Cmoo, gave Co/Cloo = 0.888 as compared with 0.901 obtained by Eq. 4, and the analogue values of 0.881 and 0.907. Thus, extrapolations of Ro and Co above the analogue values for n = 10 seem justified and may be considered as extensions of them. The intermediate calculations for n = 10.5, 10.75, and 10.99 gave analogue concentrations from Ro close to the geometric as well as the excellent values for Co/Cioo shown in Fig. 2. It is now apparent that Co/Cloo has a singular point at n = 11 or po = 1.0. This is to be expected since region B has vanished leaving no conductance path between the membranes of adjacent cells. The only current flow is then through the n = 11 capacitors of region C facing the electrode (without region A which has also disappeared) to give a value of Co/Cloo = 0.8 which is in good agreement with the analogue data as shown in Fig. 2. Thus, Eq. 5 has been demonstrated to be valid for all values of PO from less than 0.08 up to 0.998 and presumably any value of po < 1.0. Membrane capacities Having established an approximate validity for Eq. 5 over the entire range from PO = 0, by extrapolation from po = 0.073 to po = 1.00, we are now in a position to make the final and crucial calculations of the values of the mesh unit capacitances, c = Cloo/1.25 fi from the experiments. Thus, with ri = 11 for experiment I, where C100 = 0.126 OF, c = 92 nF as compared with c = 89.5 I 0.58 nF measured for the 20 capacitors used. For experiment II with C100 =
CB
=
0.0345 ,F, the mesh unit calculated c = 2.52 nF whereas direct measurement gave c = 2.413 i 0.034 nF. These agreements are about as good as had been expected (Appendix C). The assumption that the squid axon sheath capacity of 0.084 MiF/cm2 at 1 kHz (16) is attributed to a Schwann layer 0.8 Am thick leads to a capacitance 6.72 10'12F/cm = 1.25 nc. The layer structure is complicated and it may not be possible to arrive at a representative cell dimension. But, although it may be -only a formality, we may take c = 1.0 ,uF/cm2 as found for the plasma membrane, to arrive at a characteristic cell dimension 2n = 0.11 um or a seven-cell thickness for the sheath. -
SUMMARY AND CONCLUSION The surprising accuracy of analogue extensions of the Rayleigh and Maxwell equations for the resistances of conductors around two- and three-dimensional close-packing insulators from the low concentration into the medium and high concentration ranges led to the hope that similar extensions might be found for the capacity with conducting forms having capacitive boundaries. The results of the present experiments with a resistor-capacitor analogue for solutions of the finite difference Laplace equation for square cylinders are in good agreement with theory for volume concentrations from 8% to 83% and extrapolations to zero and 100%. With all of the two- and three-dimensional solutions of the Laplace equation for the resistance which follow the low concentration analytical solutions up to close packing and now the single example for the capacity of membrane covered forms, it is very tempting to assume the same result for other such problems. But, since the only common factor of these problems is that they are solutions of Laplace equations, which we have not obtained directly by analysis at high concentrations, we have no valid reason to expect that they will always be extrapolated from the low concentration results. It may be interesting to make similar analogue measurements for high concentrations of hexagonal, cylindrical cells in a triangular array as a more probable tissue structure, although no marked difference from the present results is expected (21). However, it should be more useful to investigate a three-dimensional analogue. On the other hand, the present extension into a more complicated field adds to the importance of finding a broad theoretical approach to such problems, perhaps along the lines of Keller's theorem (12). APPENDIX A. Impedance. Many of the impedance versus frequency data, as taken with the analogues in parallel with the 10 k12 standard resistor, were calculated as Z = R, + jX8 and plotted on the complex plane, R8 versus X. As for the theory of small circular cylinders, which gives a bilinear equation, the loci for the smaller values of n closely approximated semicircles and the circuit capacity was unambiguous. At the middle and higher values of n the loci reflected the complex nature of the analogues seen in Fig. 3 for n = 8. This was to be expected as an approach to the case of n = 10 where, as has been shown, there are two distinct and virtually independent capacitive current paths, from regions A to C and from regions B to C. Such a two reactive element circuit is represented by a biquadratic equation and on the complex
Biophysics:
Proc. Nat. Acad. Sci. USA 72 (1975)
Cole
4939
are to be the same. Since C1oo was found entirely in terms of square parameters, a and n are equivalent at low concentrations. The explanation of this discrepancy probably must await either an analytical solution or a more accurate analogue solution for the square cylinder problem.
FIG. 3. Complex plane locus for R, versus Xs, series resistance series reactance, where Z = R8 + jX,, in kg at labeled frequencies in kHz for 8 X 8 square mesh. Note asymmetric devia-
versus
tions from approximating circular arc.
plane by the pedal curve of an ellipse (3). Such analyses have been attempted, and without them any interpretations of the intermediate and high frequency capacities are unnot
certain.
B. Resistance Shape Factors. The average difference, p 0.01, between the nonconducting volume concentracalculated by Eq. 1 for x = 1 and the geometric fraction po = (n/-i)2, is systematic and probably significant. The shape factor x has been calculated and tested extensively for spheroids and ellipsoids (22) but no analysis is known for elliptic, hexagonal, or the present square two-dimensional cylinder problems. However, for x = 1.2 the average p po was reduced to approximately zero, but systematic divergences up to 0.01 remained after the omission of two irregular values. Thus, it appears that the effect of the shape of the cross section may need a different formulation. C. Capacitance Shape Factor. Before interpreting the values of Cloo obtained in the two experiments it is necessary to consider the relation between the radius, a, of the small cylinder for which the theory was derived and the number, n, of mesh units representing a half side of the square cylinder that has been used as a basis for structure measurements and analysis of the analogue. Fricke (2) calculated shape constants for spheroids that appeared as factors in his approximation to Eq. 3 for this capacity. A similar analysis for the square cylinder does not seem possible without a solution for the potential theory problem. The analogue capacities were more closely approximated to Eq. 4 for po < 0.3 with values of C1oo = 0.128 and 0.0355 MF for experiments I and II, respectively, but these are not importantly different from the values used for the entire range of po so they give no indication of a significant effect of shape between the circular and square cross section in the range for which the circle derivation should be valid, or beyond it. The dependent variable Co/Cloo = 4po Vo/(1 + PO)2 of Fig. 2 was chosen to present, as clearly as possible, comparisons of the theoretical and analogue dependence on volume concentration without the intervention of shape factors and the substitution of n in Eq. 5 for a from Eq. 3 was entirely arbitrary. However, for any given value of P0 it is necessary that -ra2/4 = n2 or n = xfir/4 a = 0.886 a, if the capacities - po = tion p
-
1. Fricke, H. (1924) "A mathematical treatment of the electric conductivity and capacity of disperse systems. I. The electric conductivity of a suspension of homogeneous spheroids," Physic. Rev. 24,575-585. 2. Fricke, H. (1925) "A mathematical theory of the electric conductivity and capacity of disperse systems. II. The capacity of a suspension of conducting spheroids surrounded by a nonconducting membrane for a current of low frequency," Physic. Rev. 26, 678-681. 3. Cole, K. S. (1968, 1972) Membranes, Ions and Impulses (Univ. Calif. Press, Berkeley, Calif.). 4. Cole, K. S. (1928) "Electric impedance of a suspension of spheres," J. Gen. Physiol. 12, 29-36. 5. Maxwell, J, C. (1873) Treatise on Electricity and Magnetism (Clarendon Press, Oxford). 6. Rayleigh, Lord (J. W. Strutt) (1892) "On the influence of obstacles arranged in a rectangular order upon the properties of the medium," Philos. Mag. 34,481-502. 7. Wiener, 0. (1912) "Die Theorie des Mischkorpers fur das Feld der Stationkren Strbmung," Abh. K. Saechs Ges. Wiss., Math. Phys. Ki. 32,509-604. 8. Bozler, E. & Cole, K. S. (1935) "Electric impedance and phase angle of muscle in rigor," J. Cell. Comp. Physiol. 6,229-241. 9. Cole, K. S. & Curtis, H. J. (1936) "Electric impedance of nerve and muscle," Cold Spring Harbor Symp. Quant. Biol. 4, 73-89. 10. Meredith, R. E. & Tobias, C. W. (1962) "Conduction in heterogeneous systems," Adv. Electrochem. Electrochem. Eng. 2, 15-43. 11. Keller, J. B. (1963) "Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders," J. Appl. Phys. 34, 991-993. 12. Keller, J. B. (1964) "A theorem on the conductivity of a composite medium," J. Math. Phys. (N.Y.) 5,548-549. 13. Cole, K. S., Li, C-l. & Bak, A. F. (1969) "Electrical analogues for tissues," Exp. Neurol. 24, 459-473. 14. Fricke, H. (1925) "Electrical capacity of suspensions with special reference to blood," J. Gen. Physiol. 9, 137-152. 15. Fricke, H. (1923) "The electric capacity of cell suspensions," Physic. Rev. 21, 708-709. 16. Cole, K. S. (1975) "Electrical properties of the squid axon sheath," Biophys. J., in press. 17. de Packh, D. C. (1947) "A resistor network for the approximate solution of the Laplace equation," Rev. Sci. Instrum. 18, 798-799. 18. Cole, K. S. & Curtis, H. J. (1937) "Wheatstone bridge and electrolytic resistor for impedance measurements over a wide frequency range," Rev. Sci. Instrum. 8, 333-339. 19. Guillemin, E. A. (1935) Communication Networks (Wiley, New York), Vol. II. 20. Zobel, 0. J. (1923) "Theory and design of uniform and composite electric wave-filters," Bell Syst. Tech. J. 2, 1-46. 21. Nicholson, P. W. (1965) "Specific impedance of cerebral white matter," Exp. Neurol. 13,386-401. 22. Velick, S. & Gorin, M. (1940) "The electrical conductance of suspensions of ellipsoids and its relation to the study of avian erythrocytes," J. Gen. Physiol. 23, 753-771.
