J. Phys. Chem. 1996, 100, 3855-3860
3855
Measurement and Computation of the Dipole Moment of Globular Proteins. 4. r- and γ-Chymotrypsins Shiro Takashima Department of Bioengineering, UniVersity of PennsylVania, Philadelphia, PennsylVania 19104-6392 ReceiVed: September 18, 1995; In Final Form: NoVember 15, 1995X
The dipole moments of R- and γ-chymotrypsins are determined using the dielectric constant measurement. The results are roughly comparable to those of electric dichroism measurements despite the principle and methodology of these two techniques are entirely different. Nevertheless, the differences which exist between them seem to be beyond the experimental error. The cause of disagreement appears to be, at least partially, due to the difficulty of finding the correct internal field. A new theory which is based on an ellipsoidal particle surrounded by a hydration shell is discussed. The model was found to improve the agreement markedly. Additionally, in order to corroborate the observed dipole moments with numerical computations, the dipole moments of R- and γ-chymotrypsins were calculated using protein data bases. The dipole moment of small proteins consists of two major components, the moment due to fixed surface charges and the core moment due to polar chemical bonds. The calculation of the surface charge dipole moment consists of two parts: (1) computation of the pK shifts of polar groups in proteins and (2) computation of the dipole moments using corrected pK’s. The core moment was calculated as the vectorial summation of polar group moments in the backbone and side chains. The agreement between measured and calculated dipole moments is excellent.
Introduction In general, the dipole moment of polymer molecules such as proteins and nucleic acids in solution can be determined by two different methods. The one is the frequency domain dielectric constant technique.1-3 In this method, the measured dielectric constant of the sample is converted to the dipole moment using appropriate dielectric theories. With the time domain techniques, e.g., electric dichroism, the dipole moments of macromolecules in solution can be measured directly using the magnitude of transient dichroic ratios.4,5 However, for this technique, the correction of the measured dipole moment for the internal field is an essential step in order to obtain a true dipole moment. In view of the difficulty of finding the correct internal field, the time domain technique is not completely unequivocal. As to the frequency domain technique, the local field problem is an integral part of dielectric theories. However, there are still some uncertainties with regard to the calculation of the dipole moment using a measured dielectric constant. In this sense, neither the time domain nor frequency domain techniques are completely unequivocal. It is, therefore, very important to investigate, ideally using the same sample, the consistency or inconsistency which might exist between these two methods. However, these two techniques have been developed quite independently without mutual interactions, and no such studies have been attempted hitherto. In this paper, I will discuss the dipole moments of R-and γ-chymotrypsins which were determined by these two experimental techniques. R-Chymotrypsin was chosen for this study because this is one of the few biological macromolecules whose dipole moments were determined using the time domain technique. The dipole moment of R-chymotrypsin, obtained by Antosiewicz and Porschke6,7 using the electric dichroism technique, is 2.4 × 10-27 C‚m. This value was corrected for the internal field using the following equation:8
Ei ) [3a/(2a + i)]E0 X
(1)
Abstract published in AdVance ACS Abstracts, January 15, 1996.
0022-3654/96/20100-3855$12.00/0
TABLE 1: Dipole Moments (D) of r- and γ-Chymotrypsin at Various pH’s Determined by the Electric Birefringe Method and Those Obtained by Dielectric Constant Measurements present work pH 8.2-8.5 R-chymotrypsin γ-chymotrypsin
389 401
pH 8.3 499
electric dichroism pH 7 pH 5.7 pH 4.2 419.5
449.4
359.5
where Ei and E0 are internal and applied electrical fields, a and i are the dielectric constants of the external medium and of the spherical cavity. This cavity is either assumed to be vacuum or filled with dielectric material. i was assumed to be 1 by previous authors, a value for vacuum, and a dipole moment of 1.66 × 10-27 C‚m, (or 480 D) at pH 8.3 was reported (see Table 1). The dipole moment of R-chymotrypsin determined by the present author using the frequency domain technique is 390 D (at pH 8.2), as discussed below. Experiment Measuring Techniques. The electrical capacitances of chymotrypsin solutions were measured using a fully computerized impedance analyzer Hewlett-Packard 4191A between 10 kHz and 10 MHz. The dielectric constants of protein solutions were calculated substituting measured capacitances in the following equation:9
) A(w - 1) where
A ) (Cs - C0)/(Cw - C0)
(2)
and where is the dielectric constant of the protein solutions and w is the dielectric constant of water. Cs, Cw, and C0 are the capacitances of the cell with protein solutions, the capacitance with a matched electrolyte solution, and the capacitance of the dry cell, respectively. Materials. Both R- and γ-chymotrypsins were purchased from Boehringer (Manheim, Germany) and Sigma Chemical © 1996 American Chemical Society
3856 J. Phys. Chem., Vol. 100, No. 9, 1996
Takashima
Figure 1. Frequency profile of the dielectric constant of R-chymotrypsin. 0 and ∞ are the low- and high-frequency dielectric constants. ∞ is calculated using Maxwell’s mixture formula. (1 - )/(21 + ) ) p(1 - 2)/(21 + 2), where , 1 and 2 are the dielectric constant of protein solution and those of water and protein core. p is the volume fraction of solute. 1, and 2 are assumed to be 78.5 and 6, respectively.
Corp. (St. Louis, Mo). They were used without further purification. The concentration of protein solutions ranged between 60 and 5 g/L. The pH’s of the solutions were adjusted at each concentration to approximately 8.2-8.3 by the use of dilute HCl and/or NaOH solutions. The conductivity of the protein solution is equivalent approximately to that of 4 × 10-3 mol of KCl solution. The low ion concentrations were necessary in order to minimize the error due to electrode polarization. A pair of parallel platinum rods as used as electrodes with a vertical configuration. They were heavily plated with platinum black in order to further reduce electrode polarization. The temperature of the sample was controlled at 25 °C using a circulation system. The detail of this electrode configuration is found in ref 9. Results
Figure 2. Concentration dependence of the specific dielectric increment ∆/(g/L) (∆ ) 0 - ∞; g is gram concentration). The abscissa is concentration grams per liter. The intrinsic dielectric increment δ is obtained by extrapolating the curve at zero concentration.
Figure 3. Internal fields in ellipsoidal particles with a hydration shell surrounding it. Curve 1 is calculated with an applied field E0 across the major axis, and curve 2 is obtained for a longitudinal field. ka, ki, and ks (a ) external medium, i ) protein core and s ) hydration shell) are assumed to be 78.5, 6.0, and 40. The point for sphere (a ) b) was calculated using eq 7. The abscissa is the axial ratio of ellipsoids.
Figure 1 illustrates the plot of measured dielectric constants of R-chymotrysin against frequency. The intrinsic dielectric increment δ is determined by extrapolating the plot of ∆/C vs C (C is the gram concentration per liter) at zero concentration, as shown by Figure 2. The dipole moments were calculated using Kirkwood’s theory10 for the binary mixture of polar molecules.
µ2 ) (9kT/4πN)P2
(3)
P2 ) (2/9)(1000δ + V)
(4)
where
and where δ is the molar dielectric increment, V is the partial molal volume of the dipolar molecule, and is the dielectric constant of the solvent. Analysis of Data. As shown in Table 1 and Figure 4, the dipole moment of R-chymotrypsin was determined at four different pH’s by Antosiewicz and Porschke6 using the electric dichroism method, while the present measurement is limited to pH 8.2. It is noted that the dipole moments obtained by electric dichoroism seem to be larger, beyond the limit of experimental error, than the value obtained by the dielectric technique. As mentioned earlier, the dipole moments obtained by the electric dichroism technique were corrected using eq 1, which represents the “cavity field” defined by Onsager.8 However, for the present case, the cavity is filled with a protein molecule and, thus, the mean dielectric constant of protein molecules; i.e., 5-6 must be used for i (the mean dielectric constant of
Figure 4. pH dependence of the dipole moment of R-chymotrypsin: (0) dipole moment determined by electric dichroism; (9) dipole moment determined by dielectric constant. Filled circles are calculated dipole moments. The inset shows the calculated positive (curve 1) and negative (curve 2) charges at different pH.
the protein core should be differentiated from the local permittivity near charged sites in proteins). While eq 1 gives a numerical value of Ei/E0 ) 1.50 for i ) 1, the same equation gives a smaller value of 1.44 with i ) 6. Therefore, the dipole moments reported previously should be increased from 480 to 499 D, further increasing the difference. The values shown in Figure 4 are the recalculated dipole moments. As is well-known, local fields consist of two components, i.e., internal field Ei and reaction field Er. The latter is the local field which arises from the dipole-dipole interaction between a central molecule and those surrounding it. The reaction field Er was ignored by Antosiewicz and Porschke6 in their data analyses. Takashima11 investigated the contribution of reaction
Dipole Moment of Globular Proteins. 4
J. Phys. Chem., Vol. 100, No. 9, 1996 3857
field using the bispherical interaction model developed by Stoy.12,13 He found that the contribution of Er was very small compared to that of the internal field Ei. Thus, it was concluded that the inconsistency which appears to exist between the two sets of dipole moments is almost entirely due to the inappropriate choice of the internal field Ei. While eq 1 is applicable only for spherical particles, many protein molecules are ellipsoidal. Therefore, in order for an internal field theory to be applicable to many proteins (including chymotrypsins), a model based on ellipsoidal shape must be formulated. An internal field theory which is fairly realistic for ellipsoidal rigid dipolar molecules was discussed by O’Konski and Krause some years ago.14 In the present work, in order to make the theory even more general, ellipsoidal particles surrounded by a hydration shell (or dielectric shell) was used as the model.15,16 As discussed in the Appendix, the internal field in an ellipsoid surrounded by a thin shell is given by the following equation.
KaKs Ei ) E N 0
TABLE 2: Internal Fields of Ellipsoidal Particles as a Function of the Dielectric Constant of the Hydration Shell (Axial Ratio: 2.0) ks Ei/E0(trans) Ei/E0(long) Ei/E0(random) dipole moment (D)
10
15
20
eq 1
1.905 1.370 1.723 417
1.794 1.386 1.655 433
1.618 1.356 1.530 469
1.445 499
TABLE 3: Calculated Dipole Moment (D) of r-Chymotrypsin (1) Fixed Charge Dipole Moment µx
µy
µz
µ(charge)
∆µa
246.08
281.37
60.99
378.74
88.47
(2) Core Dipole Moment µx
µy
µz
µ(core)
30.541
-4.95
-6.486
31.612
(5)
(3) Net Dipole Moment µx
µy
µz
µ(net)
µ(obs)
where E0 is the applied field and N is defined by
273.31
276.63
61.31
393.63
389.13
N ) {1/2(ka - ks)AjRo - ka}{1/2(ki - ks)BjRs - ks} -
a
1
/2(ka - ks)(ki - ks)RsAj(1/2AjRo - 1) (6)
where ka, ks, and ki are the dielectric constants of the suspending medium, the hydration shell, and the protein core. Ro and Rs are given in the Appendix. Aj and Bj are elliptic integrals which are also given in the Appendix (eq A7). For spherical particles, eq 6 reduces to eq 7, where Ri and Ra are internal and external radii of the particle.
9kaksRa3 E i ) E0 3 Ra (2ka + ks)(2ks + ki) + 2Ri3(ka - ks)(ks - ki)
(7)
For the calculation of internal field using eq 6, the values of ka and ki were assumed to be 78.5 and 5, respectively. Although the dielectric constant of the hydration shell, ks, is difficult to define explicitly, there is evidence that the dielectric constant of hydration water is smaller than that of bulk water.17,18 The water molecules in the proximity of a charged surface are highly polarized, and the reduced orientational freedom decreases the dielectric constant of water considerably in this region. However, the dielectric constant increases rapidly to the normal value as the distance from the charged surface increases. As a whole, the average permittivity of water in the hydration shell is likely to be 10-15. In the present calculation, the thickness of the hydration shell is assumed to be 10 A and ks was treated as an adjustable parameter ranging between 10 and 20. Figure 3 shows the ratios Ei/E0 for particles with different axial ratios (ks ) 10). If the dielectric constant of the hydration shell, ks, is assumed to be 10, the internal field Ei for an ellipsoid with an axial ratio of 2 is found to be 1.905Eo for transverse fields and 1.37E0 for longitudinal fields. For random orientation, the internal field was calculated by Ei ) (Ep + 2Et)/3, where Ep and Et are parallel and transverse fields. Using this, we find the average internal field to be 1.72E0. Thus, the internal fields are found to be considerably larger than that mentioned above (1.55E0 or 1.44E0). The internal fields and dipole moments for other ks values are shown in Table 2. Clearly, the optimum value of ks is 10-12, where the dipole moment obtained by electric dichroism becomes very close to the one reported by the present author. The result discussed above is very encour-
Root mean square moment.
aging and indicates that both time and frequency domain techniques are credible means to investigate the dipole moments of biopolymers. Calculation of the Dipole Moment Using the Protein Data Base. As discussed, the dipole moments determined by frequency and time domain techniques agree well if the internal field is properly chosen. In order to further corroborate these results, numerical computations of the dipole moments were performed. The calculation is based on accurate knowlege of the structure of proteins, and recent calculations have been shown to produce excellent agreement with measured values. With the advent of X-ray crystallography and the availability of protein data bases, it became possible to perform the numerical computation of the dipole moments of globular proteins with considerable accuracies.6,19-27 The dipole moments of proteins consist of two major components: the one is the core moment due to the vector sum of chemical bond moments, and the other is the moment due to fixed surface charges. These two components are not necessarily additive, and often their vectors are directed in such a way that they partially cancel each other. Dipole Moment Due to Fixed Surface Charges. The three dimensional coordinates of amino acid residues of R- and γchymotrypsins were obtained from the Brookhaven Protein Data Bank. Positively charged amino acid residues used for the computation are lysine, arginine, and histidine in side chains and N-terminals. On the other hand, negatively charged residues are aspartic acid, glutamic acid, tyrosine, and C-terminals. Calculation of pK Shifts. The magnitude of the surface charge dipole moment depends on the pK’s of polar amino acids. The pK’s of polar residues are shifted from the intrinsic value when they are placed in protein molecules due to electrostatic interactions between neighboring charged sites. The pK shifts of amino acid residues were not incorporated in most of the previous computations. Antosiewicz and Porschke6 may be the first one to incorporate pK shifts explicitly in the calculation. Although the calculated dipole moments do not critically depend on small pK shifts, nevertheless, use of the corrected pK’s yields better agreements. Although this calculation is cumbersome and time consuming, pK shifts of all of the ionizable residues
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Takashima
was carefully calculated in this work, in order to improve the agreement as much as possible. The pK shifts were calculated using the theory by Tanford and Kirkwood28 modified by Warshel and Russell29 and also by Antosiewicz and Porschke.6 In addition, there are different methods by Matthew et al.30 and Bashford and Karplus.31 The details of these calculations were already discussed in previous publications6,11,24,2 and will not be repeated. The calculated pK’s are in reasonable agreement with the acid-base titration data by Marini and Martin.32 Calculation of Dipole Moment. Dipole Moment Due to Fixed Surface Charges. The three dimensional coordinates of amino acid residues of both R- and γ-chymotrypsins were obtained from the Brookhaven Protein Data Bank (Brookhaven, NY). The x-component of the dipole moment produced by a group of positive and negative charges is defined by the following equation:
µx ) ∑nje(X+ - X-)
(8)
where e is elementary charge and X+ and X- are the X coordinates of positive and negative charge centers and are defined by
X+ ) ∑{Lj+Xj}
X- ) ∑{Lj-Xj}
(9)
Likewise, Y+, Y-, Z+, Z- are given by
Y- ) ∑{Lj+Yj}
Y- ) ∑{Lj-Yj}
(10)
Z+ ) ∑{Lj+Zj}
Z- ) ∑{Lj-Zj}
(11)
The coordinates Xj, Yj and Zj are found in the protein data base. In these equations, Lj is given by the Henderson-Hasselbach factor (see Bray and White33); i.e.,
Lj+ ) 1/(1 + B) Lj- ) B/(1 + B)
for Lys, Arg, His, and NH2 terminals for Asp, Glu, Tyr, and COOH terminals
where B ) 10pH-pK. Likewise, other components, µy and µz, are computed. The pK values are calculated by the method discussed earlier. The computations were first carried out at the isoelectric point where the effective positive and negative charges are equal; i.e.,
∑nj+Lj+ ) ∑nj-Lj-
(12)
where nj is the number of the jth amino acid residues. The dipole moment at other pH’s will be discussed later. The Henderson factors Lj+ and Lj- represent the effective charge of amino acid residues. If the pH of the solution is far removed from the pK of certain polar groups, the Henderson factors are either 0 or 1, i.e., they are either fully protonated or unprotonated. Only when the pH is close to the pK of the group, the Henderson factor will have a fractional value between 0 and 1. This means (a) a uniform increase or decrease of the effective charge of a given group with the change in pH or (b) the Henderson factor representing the fraction of protonated or unprotonated residues. The latter concept was found to give a better agreement with the experimental value despite that, at times, a very large number of iterations are required. This is because the fractional ionization generates a number of different charge configurations. Chymotrypsin has an isoelectric point around 9.2, and only amino acids which may be partially ionized at this pH are lysine (pK ) 9.746) and tyrosine (pK ) 9.91).
The Henderson factor indicates that the fraction of ionized lysine residues at the isoelectric point is 12 out of 14 and the fraction of ionized tyrosine is about 1 out of 4. A simple calculation shows that the number of charge configurations would be 364. Core Dipole Moment. In addition to the fixed charge dipole moment, the core dipole moment due to polar groups must be computed. The major contribution to the core moment is the bond moment of the carbonyl groups (2.7 D).34 The 3D coordinates of carbonyl groups are well-documented in the database, and the core dipole moment can be readily calculated as the vector sum of individual CO bond moments. In addition, the moment of the NsH bond (1.31 D) cannot be ignored. If CdO bonds are colinear with the NsH bond such as in the R-helix, the group moment of CdO‚‚‚HsN would be 3.4 D,35 and the contribution of the NsH bond would be significant if the orientation of CdO bonds is uniaxial. However, in protein molecules, the predominant structure is the random coil configuration and the aforesaid colinearity between CdO and NsH would not be a dominant conformation. If CdO and NsH bonds are not conjugated by hydrogen bonds, the coordinates of H atoms in NsH bonds cannot be determined because of the inability of the X-ray crystallography to find the coordinates of H atoms. Thus, the contribution of the NsH moment was neglected assuming that the NsH bonds contribute only a negligibly small net dipole moment. Thus, only the vectorial sum of the CdO bond moments was computed in this work. It was found that the core moment is much smaller than the fixed surface charge dipole moment, indicating that the orientation of the CdO bonds is also nearly random. The Results of the Computation. R-Chymotrypsin. The dipole moments of R-chymotrypsin calculated using the method discussed earlier are summarized in Table 2. R-Chymotrypsin consisting of more than one subunits has two distinctly identifiable C- and N-terminals that must be included in the computation. Although a segment of data base 9-13 (val-leuser-gly-leu) is missing, none of these residues is charged and does not affect the value of the surface charge dipole moment, although it may cause small errors in the core moment. The calculated dipole moments are in good agreement with the measured value, i.e., 394 vs 389 D. This is a surprisingly good agreement. Figure 5a shows the 2D plot of R-chymotrypsin in the X-Y plane. The arrow in this figure shows the vector of the fixed surface charge dipole moment. The vector of the core dipole moment is not shown in this figure. However, the angle between the two dipole components can be easily calculated to be 61.8°. Likewise, the calculated dipole moment of γ-chymotrypsin was in good agreement with the measured one, i.e., 414.8 (calcd) vs 401.2 D (measured). However, the angle between the surface charge and the core dipole moment is found to be 135°, an angle entirely different than that found in R-chymotrypsin. Calculation of the Dipole Moment at Other pH’s. At pH’s other than the isoelectric point, the center of gravity is shifted according to the relative magnitudes of negative and positive charges. Dipole moments at these pH’s are computed using the following equation.
(
µ ) er
2σ+σσ+ + σ-
)
(13)
where r is the distance between positive and negative charge centers. σ+ and σ- are effective positive and negative charges; i.e., σ+ ) ∑nj+Lj+ and σ- ) ∑nj-Lj-. The results of these calculations are shown in Figure 4. This figure demonstrates that the dipole moment is nearly constant between pH 5 and 9.
Dipole Moment of Globular Proteins. 4
J. Phys. Chem., Vol. 100, No. 9, 1996 3859
Only below pH 5 and above pH 9 do the dipole moments change abruptly. The pH profile of calculated negative and positive charges is shown in the inset of Figure 4. As shown, the numbers of both negative and positive charges are nearly constant in this pH range, corroborating the constancy of observed and calculated dipole moments. Figure 5 illustrates the 2D projection of chymotrypsins on the X-Y plane. Crosses and circles represent positive and negative charges on side chains and N- and C-terminals. Arrows show the dipole vectors of surface charge moments. The vectors of core moments are not shown. Discussion The aim of the experimental part of this work is to investigate the consistency or inconsistency which may exist between the time domain and frequency domain dipole moment determinations. Using R-chymotrypsin, it was found that the agreement between the dipole moments obtained by these two methods was less than satisfactory; i.e., the difference could not be explained solely by experimental errors. First of all, in view of the observation that the reaction field was too small to be the major cause of disagreement, it was concluded that the choice of the internal field was the cause of the difference. The internal field expressed by eq 1 was originally derived for a simple model, i.e., a spherical cavity without a dipolar molecule in it. Thus, this model is far from the reality of the protein molecules which are surrounded by a layer of hydration water and external bulk medium. A more realistic internal field theory was formulated for ellipsoidal particles having a hydration shell surrounding it. As discussed, this model improves the agreement markedly with a proper choice of the dielectric constant of the hydration shell, ks. In any event, the important finding is that the dipole moments of proteins determined by two entirely different techniques agree reasonably well with the recalculation of internal fields. The model is applicable for spherical particles as well as infinitely long rods. However, the model is, at present, not applicable for oblate ellipsoids, although the model can be easily converted for oblate shapes if necessary. Although the consistency between frequency and time domain methods seems to indicate that the observed dipole moments are, by and large, acceptable, we still need to corroborate these resuslts with numerical computation of the dipole moment using the accurately known internal structure of this protein. The result of the calculation indicates an excellent agreement between observed and calculated dipole moments. The numerical calculation of the dipole moment based on the X-ray data is, at present, highly reliable and provides us with an excellent means for supporting or reject the experimental results. Appendix. Calculation of the Internal Fields for Ellipsoids The calculation of internal fields is based on an ellipsoidal particle which is surrounded by a hydration shell. The dielectric constants of three regions are designated by ka for the external medium, ks for the hydration shell, and ki for the protein core. The electrical potentials in each region are given by the following equations.15,16
φ1 ) F1(χ) F2(η) F3(ζ)[C1 + C2Aj]
(A1)
φ2 ) F1(ξ) F2(η) F3(ζ)[C4 + C5Aj]
(A2)
φ3 ) F1(ξ) F2(η) F3(ζ)C3
(A3)
where ξ, η, and ζ are variables of the ellipsoidal coordinates. ξ
Figure 5. (A) 2D plot of the chain configuration of R-chymotrypsin in the X-Y plane. Circles are negative charges and crosses are positive charges on side chains (side chains are not plotted). The distance between the positive and negative charge centers is 4.19A. (B) 2D plot of γ-chymotrypsin.
is the surfaces of confocal ellipsoids, η is a hyperboloid of one sheet and ζ is a hyperboloid of two sheets. C1 ) -E0/[(b2 a2)(c2 - a2)]1/2 and a, b, and c are the axes of the ellipsoid. Aj is the elliptic integral (eq A7). Other unknown constants can be computed with the following boundary conditions,
φ1 ) φ2 ka
(A4)
[ ] [ ] 1 ∂φ1 1 ∂φ2 ) ks h1 φξ h1 ∂ξ
(A5)
at the outer surface of the ellipsoid, ξ1, where h1 dξ (h1 ) 1/ [(ξ - η)(ξ - ζ)1/2/R (R is given by eq A8)) represents the 2 ξ ξ distance in curvilinear coordinates and ξ1 is the area of the outer surface. Likewise similar boundary conditions hold at ξ ) 0, i.e., the inner surface of the ellipsoid. The resultant simultaneous equations can be expressed in the matrix form given in eq A6, where Aj and Bj are elliptic integrals
[
-Bj 0
(
)
0 -1
1 1 -ka Bj 2 Rs 0 0
1 1
Bj Aj
( (
) )
1 1 /2ks ks 2Bj - R s 1 1 -1/2ki 1/2ks ks 2Aj - R o 1
][ ] [ ]
C1 C2 0 C3 1 ) /2kaC1 C4 0 C5 (A6)
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Takashima
as follows:
Aj ) ∫0
∞
ds (s + aj2)Rs
Bj ) ∫ξ
∞ 1
ds (s + aj2)Rs
(A7)
and Ro and Rs are given by
Ro ) abc
Rξ ) [(ξ + a2)(ξ + b2)(ξ + c2)]1/2 (A8)
where a, b, and c are the axes of ellipsoids and ξ is the surfaces of confocal ellipsoids. ξ is zero on the surface of the ellipsoid. The elliptic integrals cannot be solved analytically for general ellipsoids. However, they can be solved for the ellipsoid of revolution. The Bj for the prolate elliposoid of revolution for transverse and longitudinal electrical fields are as follows:
Bj )
{
[ξ1 + n2b2]1/2 1 1 + × 2 2 2 2 b (n - 1) ξ1 + b 2b[n - 1]1/2 ln
[
]}
[ξ1 + n2b2]1/2 - b[n2 - 1]1/2
[ξ1 + n2b2]1/2 + b[n2 - 1]1/2
(A9)
for transverse fields and
Bj )
{
2 -1 1 + × 2 2 2 1/2 2 b (n - 1) [ξ1 + n b ] b[n - 1]1/2 2
ln
[
]}
[ξ1 + n2b2]1/2 - b[n2 - 1]1/2
[ξ1 + n2b2]1/2 + b[n2 - 1]1/2
(A10)
for longitudinal fields. b is minor axis, n is the axial ratio, and ξ1 is the surface of the ellipsoid at the outer boundary. The elliptic integral Aj for the surface of the ellipsoid can be readily derived replacing ξ1 with 0 in eq A6. The manual computation of the matrix equation is very tedious and time consuming. However, the availability of a software package makes the solution much faster and less prone to computational errors. The internal field Ei is shown earlier by eqs 6 and 7. References and Notes (1) Takashima, S. Electrical Properties of Biopolymers and Membranes; Adam Hilger: Bristol, United Kingdom, 1989.
(2) Cole, R. H. Annu. ReV. Phys. Chem. 1989, 40, 1-18. (3) Schwan, H. P.; Takashima, S. Encyclopedia of Applied Physics; 1992; Vol. 5, pp 177-200. (4) Yoshioka, K.; Watanabe, H. Dielectric Properties of Proteins II. In Physical Principles and Techniques of Protein Chemistry, Part A; Leach, S. J., Ed.; Academic Press: New York, 1969; Chapter 7, pp 335-368. (5) Jennings, B. R.; Stoylov, S. P. Proceedings of the Conference on Colloid and Molecular Electro-Optics; IOP Publication: Varna, Bulgaria, 1991. (6) Antosiewicz, J.; Porschke, D. Biochemistry 1989, 28, 10072-10078. (7) Porschke, D. In International Symposium on Colloid and Molecular Electro-Optics; Jennings, B. R., Stoylov, S. P., Eds.; IOP Publication: Varna, Bulgaria, 1991. (8) Onsager, L. J. Am. Chem. Soc. 1936, 58, 1486-1493. (9) Schwan, H. P. Determination of biological impedances. In Physical Techniques in Biological Research; Nastuk, W. L., Ed.; Academic Press: New York, 1962; Vol. 6. (10) Kirkwood, J. G. J. Chem. Phys. 1939, 7, 911-919. (11) Takashima, S. In International Symposium on Colloid and Molecular Electrooptics, Bielefeld, Germany; Schwarz, G., Neumann, E., Eds., in press. (12) Stoy, R. J. Appl. Phys. 1989, 65, 2611-2615. (13) Stoy, R. J. Electrost. 1994, 33 (3), 385-392. (14) O’Konski, C. T.; Krause, S. J. Phys. Chem. 1970, 74, 3243-3250. (15) Bernhardt, J.; Pauly, H. Biophysik 1973, 10, 89-98. (16) Ashe, J. W. Ph.D. Dissertation, University of Pennsylvania, 1990. (17) Hasted, J. B.; Ritson, D. M.; Collie, C. H. J. Chem. Phys. 1948, 16, 1-21 (Parts I and II). (18) Takashima, S.; Casaleggio, A.; Giuliano, F.; Morando, F.; Ridella, S. Biophys. J. 1986, 49, 1003-1008. (19) Schlecht, P. Biopolymers 1969, 8, 757-765. (20) South, G. P.; Grant, E. H. Proc. R. Soc. London, A 1972, 328, 371-387. (21) Orttung, W. H. J. Phys. Chem. 1969, 73, 418-423. (22) Barlow, D. J.; Thornton, J. M. Biopolymers 1986, 25, 1717-1733. (23) Takashima, S.; Asami, K. International Symposium on Colloid Molecular Electro-Optics; IOP Publication: Varna, Bulgaria, 1991, pp 187195. (24) Takashima, S.; Asami, K. Biopolymers 1993, 33, 59-68. (25) Takashima, S. Biophys. J. 1993, 64, 1550-1558. (26) Antosiewicz, J.; Porschke, D. Biophys. J. 1995, 68, 655-664. (27) Antosiewicz, J. Biophys. J. 1995, 69, 1344-1354. (28) Tanford, C.; Kirkwood, J. G. J. Am. Chem. Soc. 1957, 79, 53335339. (29) Warshel, A.; Russell, S. T. Q. ReV. Biophys. 1984, 17, 283-422. (30) Matthew, J. B.; Hanania, G. I. H.; Gurd, F. R. N. Biochemistry 1979, 18, 1919-1928. (31) Bashford, D.; Karplus, M. Biochemistry 1993, 29, 10219-10225. (32) Marini, J. B.; Martin, C. J. Euro. J. Biochem. 1971, 19, 162-168. (33) Bray, H. G.; White, K. Kinetics and Thermodynamics in Biochemistry; Academic Press: New York, 1966. (34) Wada, A. In Polyamino Acids, Polypeptides and Proteins; Stahman, M. A., Ed.; University of Wisconsin Press: Madison, WI, 1962; pp 131146. (35) Birkroft, J. J.; Blow, D. M. J. Molec. Biol. 1972, 68, 187-240.
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