8194
J . Phys. Chem. 1992, 96, 8194-8200
(9) Hamilton, J. A.; Small, D. M.Proc. Natl. Acad. Sei. U S A . 1981, 78, 6878. (10) Burns, R. A., Jr.; Roberts, M. F. J . Biol. Chem. 1981, 256, 2716.
(I 1) Brouillette, C.; Segrest, J. P.; Ng,T. C.; Jones, J. L. Biochemistry
1982, 21, 4569. (12) Cistola, D. P.; Walsh, M. T.; Corey, R. P.; Hamilton, J. A.; Brecher, P. Biochemistry 1988, 27, 7 1 1. (13) Stark, R. E.;Manstein, J. L.; Curatolo, W.; Sears, B. Biochemistry 1983, 22, 2486. (14) Stark, R. E.; Storrs, R. W.; Levine, S.E.; Yee, S.;Broido, M. S. Biochim. Biophys. Acta 1986,860, 399. (15) Mascioli, E. A.; Lopes, S.;Randall, S.;Porter, K. A.; Kater, G.; Hirschberg, Y.; Babayan, V. K.; Bistrian, B. R.; Blackbum, G. L. Lipids 1989, 24, 193. (16) Wang, D.; Hadipour, N. L.; Jerlin, E. A.; Stark, R. E. J . Lipid Res. 1992, 33, 431. (17) Chachaty, C. Prog. NMR Spectrosc. 1987, 19, 183. (18) Wennerstrom, H.; Lindman, B.; Soderman, 0.;Drakenberg, T.; Rosenholm, J. B. J . Am. Chem. Soc. 1979, 101,6860. (19) Walderhaug, H.; Soderman, 0.;Stilbs, P. J . Phys. Chem. 1984,88, 1655. (20) Wocssner, D. E. J. Chem. Phys. 1%2,37, 647. (21) Levy, G. C.; Craik, D. J.; Phan Viet, M. T.; Dekmenzian, A. J. Am. Chem. Soc. 1982,104, 25. (22) Doddrell, D.; Glushko, V.; Allerhand, A. J . Chem. Phys. 1972, 56, 3683. (23) Lipari, G.; Szabo,A. J . Am. Chem. SOC.1982, 104, 4546; 4559. (24) Ahlnas, T.; Soderman, 0.;Walderhaug, H.; Lindman, B. J . Phys. Chem. 1983,87, 822. (25) Soderman, 0.; Walderhaug, H.; Henriksson, U.; Stilbs, P. J . Phys. Chem. 1985,89,3693. (26) Nery, H.; Soderman, 0.; Canet, D.; Walderhaug, H.; Lindman, B. J . Phys. Chem. 1986, 90, 5802.
(27) Bovey, F. A.; Jelinski, L. W. J. Phys. Chem. 1985.89, 571. (28) Bull, L.M.; Gillies, D. G.; Matthews, S.J.; Sutcliffe, L. H.; Williams, A. J. Magn. Reson. Chem. 1991, 29, 273. (29) Levine, Y.K.; Birdsall, N. J. M.; Lee, A. G.; Metcalfe, J. C.; Partington, P.; Roberts, G. C. K. J . Chem. Phys. 1974, 60, 2890. (30) Bleich, H. E.; Glasel, J. A.; Latina, M.; Visintainer, J. Biopolymers 1979, 18, 2849. (31) Burkert, U.; Allinger, N. L. Molecular Mechanics; ACS Monograph Series 177; American Chemical Society: Washington, DC, 1982. (32) Levy, G. C.; Kumar, A.; Wang, D. J . Am. Chem. SOC.1983, 105, 7536. (33) Berger, S.;Kreissl, F. R.; Grant, D. H.; Roberts, J. D. J. Am. Chem. SOC.1975, 97, 1805. (34) Canet, D.; Levy, G. C.; Peat, I. R. J . Magn. Reson. 1975, 18, 199. (35) Freeman, R.; Hill, H. D. W.; Kaptein, R. J . Magn. Reson. 1972, 7, 327. (36) Craik, D. J.; Kumar, A.; Levy, G. C. Chem. In$ Compur.Sci. 1983, 23, 30. (37) Gajewski, J. J.; Gilbert, K. E.;McKelvey, J. In Advances in Molecular Modeling, Liotta, D., Ed.; JAI Press: Greenwich, CT, 1990; Vol. 2, pp 65-92. (38) Robeson, J.; Foster, B. W.; Rosenthal, S. N.; Adams, E. T., Jr.; Fendler, E. J. J. Phys. Chem. 1981, 85, 1254. (39) Burns, R. A., Jr.; Roberts, M. F. Biochemistry 1980, 19, 3100. (401 Doddrell. D.: Allerhand. A. J . Am. Chem. SOC.1971. 93. 1558. (41) Williams, E.; Sears, B.; Allerhand, A.; Cordes, E. H. A h . Chem. SOC.1973, 95,4871. (42) Bratt, P. J.; Gillies, D. G.; Sutcliffe, L. H.; Williams, A. J. J. Phys. Chem. 1990. 94. 2721. (43) Davjs, J: H.;Jeffrey, K. R. Chem. Phys. Lipids 1977, 20, 87. (44) Klasson, T.; Henriksson, U. In Solution Chemistry of Surfactants; Mittal, K. L., Fendler, E. J., Eds.; Plenum Press: New York, 1982; Vol. 1.
i.
Conformational Transitions in Aqueous Solutions of Poly(L-glutamic acid): A Radiowave Dielectric Study F. Bordi, Sezione di Fisica Medica. Dipartimento di Medicina Interna. Universita' di Roma "Tor Vergata", Rome, Italy
C. Cametti,* Dipartimento di Fisica, Universita' di Roma "La Sdpienza", Piazrale A . Mor0 5, Rome, Italy
and G. Paradossi Dipartimento di Scienze e Tecnologie Chimiche, Universita' di Roma "Tor Vergata". Rome, Italy (Received: January 15, 1992; In Final Form: April 13, 1992)
The dielectric properties of poly@-glutamic acid) aqueous solutions during the entire conformational helix to random coil transition induced by pH have been measured in the frequency range from 1 kHz to 10 MHz. The dielectric spectra have been decomposed into two separate dielectric relaxations attributed to counterion fluctuation in the neighborhood of the polyion. The characteristic parameters underlying the counterion polarization dielectric models have been estimated. In particular, at high pH values, when the polyion is no longer in a helical conformation, some evidence is shown to attribute the high-frequency dielectric dispersion to the Maxwell-Wagner effect occurring in heterogeneous systems.
Introduction Synthetic polypeptides are very interesting model systems to study the static and dynamical properties of biological charged macromolecules such as natural peptides and proteins because of the importance of electrostatic interactions between charged groups in determining the molecular codlguration. * Moreover, these polymers undergo a charge-induced helix-coil transition in solutions,2where the pH is an important parameter upon which the charge density, and hence the ion distribution, depends. Dielectric relaxation studies in appropriate frequency intervals, reflecting the local dynamic properties of the system, are expected to give detailed information on the polyion conformation and polyion-counterion interactions in solution. Charged polypeptides,
owing to their complex structure, can give rise in principle to different dielectric relaxation processes, ranging from a complete molecule orientation, involving its zwitterionic form, to a partial rearrangement of side chains beside the displacement of counterions between charged or ionized polymer groups. Moreover, owing to the fact that the distance between adjacent charged groups of the polyion depends on its conformation, these substances have attracted particular interest to elucidate the electrical properties of polyelectrolyte solutions, especially those based on the nonequilibrium small ion distribution. Linear ionic polymer solutions in dilute or semidilute regions generally exhibit a complex dielectric relaxation phenomenologyf5 which has betn analyzed on the basis of two different mechanisms
0022-3654f 9212096-8l94$03.QQf 0 0 1992 American Chemical Society
Conformational Transitions in Poly@-glutamicacid) Solutions The Journal of Physical Chemistry, Vol. 96, No. 20, 1992 8195 involving the dynamic behavior of bound counterions and Occuning on separate frequency intervals. Both the mechanisms are based on the rising of an induced dipole moment due to counterion distribution in the neighborhood of the polyion. The low-frequency dispersion (10'10" Hz) is ascribed to the movement of small ions along the whole polymer chain, whereas the high-frequency dispersion (105-10' Hz), whose origin is more questionable, is usually attributed to motion of counterions in a direction perpendicular to the polyion axis or within particular zones along the chain (polyion subunits in the Mandel model) defined by potential barriers of the order of some keT, which can exist on a short time scale6 along the chain. A somewhat different picture emerges when the polyions assume a more flexible conformation in solution. In this case, the spherical domains associated with the volume occupied by the polyion in solution, in the two-state model, have a different dielectric constant and conductivity from that of the solvent, which give rise to the well-known Maxwell-Wagner effect,' attributable to frequencydependent surface ion polarization at the interface of different media of heterogeneous systems. Among polypeptides, poly(L-glutamic acid) (PGA) has been extensively used as a model system for studying the helix-coil transition, and the dielectric properties of its aqueous solutions have been studied in the low- and high-frequency dielectric regions,8-11 under different conditions of the dissolving medium, in order to investigate the conformational changes induced by pH or the internal motions of the side groups of the polymer chain. The dielectric behavior of PGA in aqueous solution was firstly studied as a function of pH by Takashima? who observed a larger dielectric increment for the coil form than that of the helical form, clearly showing that there are two different types of relaxation, the most predominant of these involving counterion polarization in the neighborhood of the carboxylic groups. Later, Muller et al." investigated the dielectric properties of PGA of different molecular weights as a function of polymer concentration and degree of ionization, extending the frequency range up to 100 MHz. The dielectric properties of various polypeptides in aqueous solutions have been studied by various a u t h o r ~ , ' ~showing J~ the utility of dielectric measurements in gaining information on the polymer structure and confirming, to a more or lesser extent, the above-stated basic phenomenology. Biopolymers undergoing the helix-coil transition in aqueous solution represent a suitable model system to establish the effectivenessof various theoretical descriptions which have appeared, especially those based on counterion fluctuation, since in these systems the electrolytic character of the polyion can be varied continuously by varying the physicochemical properties of the solvent. In this work we investigate further the dielectric properties of PGA of different molecular weights in aqueous solution, over the frequency range from 1 kHz to 10 MHz, where the whole dielectric prows attributable to the whole polymer or to polymer-solvent interactions occurs. The conformation of the polymer chain in solution has been varied from the helix to the coil state by changing the pH of the aqueous phase. Our results have been analyzed on the basis of two dielectric models proposed by Mandel and co-worker~~-~ which offer the possibility of evaluating some important structural parameters from the dielectric measurements. Moreover, the ion distribution in the polymer domain, when the polyion assumes a coil conformation, has been evaluated by means of the spherical nonlinearized Poisson-Boltzmann (PB) equation which allows the electrical conductivity in the neighborhood of the polyion to be properly ~alculated.'~This provides support to the attribution of the high-frequency dielectric dispersion as due to the Maxwell-Wagner effect in light of the two-state model. Experimental Section Materials. The sodium salt of poly@-glutamic acid) (Na-PG) of different molecular weights ranging from 32 X lo3 to 51 X lo3 was purchased from Sigma Chemical Co. and was employed without further purification. In order to remove the Na+ ions,
the initial Na-PG solutions were passed throughout a column containing a severalfold excess of strong cation-exchange resin in acidic form (Amberlite IR-120, Merck). Poly(L-glutamic acid) samples (PGA) thus obtained were neutralized with a titrated 0.1 M NaOH solution, changing the pH of the solution from about 3.5 (the initial value with the polymer in full helical conformation) to about 10.5 (when the polyion is completely in random coil state). Doubly distilled, deionized water (electrical conductivity less then 1 X lod C2-I cm-' at 20 "C) was used throughout the preparation of the NaPG solutions. The polymer concentrations C, (expressed in monomol/L of the carboxylic acid groups) were determined by potentiometric titration using NaOH as titrant. The conformationaltransition from the helix to the random coil was monitored by circular dichroism measurements whose details are reported e1~ewhere.l~ Dielectric M e a " & The permittivity E' in the frequency range from 1 kHz to 10 MHz was obtained from standard impedance measurements carried out by means of a low-frequency impedance analyzer (Hewlett-Packard Model 4192A). The dielectric cell consists of two platinum electrodes 1 mm apart covered with platinum black, whose constants have been determined by calibration with standard liquids of known conductivity and dielectric constant. This procedure, based on that proposed by Bottomley,15requires measurements as a function of frequency of three different standards in order to calculate, at each frequency, the circuit elements that describe the frequency behavior of the measuring cell. To ascertain the overall accuracy of the experimental setup, we have performed measurements of simple NaCl electrolyte solutions of electrical conductivity close to that of the polymer solutions investigated, and after the data were corrected by means of the calibration procedure, no dependence on frequency was observed in the low-frequency region of the dielectric spectrum; the measured values scatter within few units around the expected ones of the electrolyte solutions. The quoted accuracy, taking into a m u n t the combined sources of errors of the frequency domain dielectric measurements, is about f3% at frequencies below 10 kHz and reduces to less than 1% at frequencies above 100 kHz. The impedance measurements allow in principle the determination of the complete behavior of the dielectric sample through the complex dielectric constant e*(w)
= e(@)
- i d'tot(w)
where C(w) is permittivity and d',(w) the total dielectric loss that will contain the contribution due to the dielectric loss, Q " ~ ~ ~ , ( W ) , and that from the dc electrical conductivity, uo/cow. At lower frequencies, this latter term dominates the first, especially in the case of high-conductivity samples as those investigated here. This means that the dielectric loss, when the frequency conductivity increment is small, cannot be accurately distinguished from the ionic loss, and consequently we have confined our analysis on the permittivity €'(a) and the dc ionic conductivity uo values only. All the dielectric measurements were performed at the temperature of 20.0 OC within 0.1 OC.
Results
Three different polymer solutions of different molecular weights, 32 X lo3,36 X lo3, and 51 X lo3, hereafter referred to as PGA-1, PGA-2, and PGA-3, respectively, have been investigated in the frequency range from 1 kHz to 10 MHz, where well-pronounced dielectric dispersions occur. The entire conformational change of the polymer from the helix to the random coil state was studied, varying the pH of the solution from 3.5 to 10.5. Typical dielectric spectra of PGA-1 at four selected values of pH are shown in Figure 1. Similar spectra have been obtained for PGA-2 and PGA-3 solutions. For all the samples investigated, the polymer concentration C was kept moderately small, in the range C = 0.5-1 mg/mL, to ensure negligible polymerpolymer interactions. To verify the dilute regime, where only intramolecular interactions and polymersolvent interactions should occur, additional
8196 The Journal of Physical Chemistry, Vol. 96, No. 20, 1992 130 L
E'
,
I
Bordi et al.
1
f
120
71) .-
-5
t
1
,
1
1o5
io4
io3
:
0
1
1o6
I
I
I
10'
Frequency [Hz] Figure 1. Permittivity of PGA (molecular weight 32 X 10)) aqumus solutions as a function of frequency at the temperature of 20 O C , for different selected values of pH, before and after the helix-coil transition. ( 0 )pH = 4.82; (A) pH = 5.15; (e) pH = 5.90; (B) pH = 8.72. The palymer concentration is 0.95 mg/mL. The solid curves represent the calculated values according to eq 1 with the dielectric parameters shown in Figures 3-5.
measurements at different polyion contents from C, = 7.4 X to 6.4 X lo4 monomol/L (corresponding to concentrations from 1.1 to 0.097 mg/mL) and at two different values of pH (pH = 8.70 and 4.10), for all the molecular weights investigated, were performed. In this polymer concentration range, the low- and high-frequency dielectric increments Aei into which the dielectric spectra have been decomposed, as will be discussed in detail as follows, show a concentration dependence Ai ( i = 1, 2 ) 1 + BjC,
AEi
C,
where Ai and B, are two constants, similar to those observed by Vreugedenhil et al.3 and van der Touw et ala4for synthetic polyelectrolytes and DNA of similar characteristics. In the present case, however, the interaction parameters Bi ( i = 1, 2) are very small, of the order of 2 X lo3 and 0.5 X IO3 L/monomol, respectively, indicating that intermolecular interactions should be negligible. The experimental points t' = f ( w ) were fitted with the help of a nonlinear least-squares fitting procedure to an expression corresponding to a superposition of two independent Cole-Cole equations e'
= e,
(e,
+ 1+
- q)(1 + ( u T ~ ) ~ sin - ~ I(7ra1/2))
+
(el
1
+
- c,)(1 + ( ~ 7
(WT2)2(1-Qz)
+
sin (?ral/2) ~ ) l - sin ~ 2 (7ra2/2))
2(WTl)1-QI
+ 2(u2)I-a2 sin ( 7 4 2 )
(1)
as usually done in the case of most polyelectrolyte solutions. Here e$ and e, are the limiting values of the permittivity e' at low and high frequency, respectively, el is the intermediate value which defines the dielectric increments of two dispersions, T , and T~ are the relaxation times, and a,and aZare the C o l d o l e parameters describing the spread of the relaxation times. For systems with two single relaxation times, ai = 0 ( i = 1, 2), and eq 1 reduces to the sum of two Debye-type dispersions. At polymer concentrations as low as those employed, the high-frequency permittivity does not deviate appreciably from that of the solvent which, in turn, can be considered equal to that of pure water. Therefore, the fitting procedure was performed assuming c, = c, = 80.01. The full lines in Figure 1 represent the calculated values with the dielectric parameters obtained from the fitting procedure. The agreement between the observed and calculated values is quite good over the entire frequency range investigated. A quantitative picture of this agreement is given in Figure 2, where the residuals for PGA-1 solutions at different
3
4
5
6
1
8
9
1
0
PH Figure 3. Dielectric increments of the low-frequency ( 0 )and high-frequency (A)dielectricdispersions of S A - 1 aqueous solution as a function of pH at the temperature of 20 O C . The polyion concentration is 0.95 mg/mL. The full lines are the spline curves through experimental data drawn for visual purposes only.
pH values are plotted as a function of frequency. As can be seen, no systematic deviations can be evidenced in any part of the frequency range. Nevertheless, some different dielectric functions have been used in the attempt to reduce the number of the free parameters in the fit. In particular, attempts to fit our data with a single Cole-Cole and Debye or conversely a Debye and Cole-Cole functions did not yield sisnifcantly better results, or no significant improvement of the fit was obtained. The details of this analysis of the experimental data have been omitted to make the discussion of the results more concise. The dielectric parameters (the dielectric increments Ael = (es - el), Ae2 = (el - e,)), the relaxation frequencies vI and uZ, and the spread parameters a,and u2characterizing the two ColtCole dielectric dispersions of the PGA-1 solutions as a function of pH are shown in Figures 3-5. The dielectric relaxation of PGA in fully helical conformation, in the pH range below 5 , is characterized by a small dielectric increment, well below that o k e d in the coil conformation. This finding was firstly observed by Takashima,* Nakamura and Wada? and Muller et al." As the conformational transition proceeds, the low-frequency dielectric increment markedly increases, up to values of Acl/C = 40 mL/mg showing a marked change in correspondence of pH = 5-6, whereas the high-frequency dielectric increment regularly increases with pH values. A similar behavior can be observed in the dependence of the parameters al and a2on pH even if, in this case, a shift of about 1 pH unit appears. The relaxation times of both the dielectric
Conformational Transitions in Poly(L-glutamic acid) Solutions The Journal of Physical Chemistry, Yo/. 96, No. 20, 1992 8197 interactions, the low- and high-frequency dielectric dispersions are given by
I
3
4
1
I
5
6
7
8
9
1
0
PH Figure 4. Relaxation frequencies of the low-frequency ( 0 ) and highfrequency (A) dielectric dispersion of PGA-1 aqueous solution as a
function of pH at the temperature of 20 OC. The polymer concentration ia 0.95 mg/mL. The full lines are the spline curyes through experimental data drawn for visual purposes only.
‘ &
8
0.5
I
(4)
and, if the polyion is in a largely extended configuration, this quantity is related to the mean-square end-to-end distance ( L2) through
0.3
In this way, the low-frequency dielectric dispersion is proportional to the distance (L2)of the whole polyion and hence to its molecular weight, whereas the magnitude of the high-frequency dielectric dispersion, depending on the distance b, is largely independent of the molecular weight. The total number of counterions per unit volume, nCo,may be expressed through the effective degree of ionization a as nCo = 103aNCp,where N is the Avogadro number. Both the dielectric increments depend linearly on the fraction of condensed counterions. More recently, the above has been modified by the same authors in order to take into account the contribution of the proton fluctuation. In this case, the average fraction of counterions effectively bond to the polyion, fa,should be replaced with fa + a ( 1 - a)
0.2
2
0.1
2
+ b2/12
(S2) = (1/12)(L2)
8
. I
( S 2 )= R,2
0.4
.C( c)
.CI Y
where f is the average fraction of associated counterions, n the total number of ionizable groups per polyion, ( l e ) the charge of counterions, Cothe numeric polyion concentrationper unit volume, E, the dielectric constant of free space,and kBTthe thermal energy. The polymer configuration and its geometry are defined by the parameters R and b that represent the square root of the mean-square c h a n c e of the subunits with respect to the center of mass of the polyion and the length of the rigid rodlike subunits, respectively. The conventionally defined mean-square radius of gyration (S2)is related to R: by the relation
0.0
3
4
5
6
7
8
9
10
PH Figure 5. ColtCole parameters indicating the spread of the relaxation time of the low-frequency ( 0 ) and the high-frequency (A) dielectric dispersions of PGA-1 aqueous solutions as a function of pH, at the temperature of 20 OC. The polymer concentration is 0.95 mg/mL. The full lines are the spline curves through experimental data drawn for visual purposes only. dispersions show a moderate decrease as the pH is increased. It must be noted that, while the helix-coil transition occurs at pH = 5.8 (the value at which the fractionfh of the helix conformation reaches one-half of the total helix content, as measured for the samples investigated by circular dichroism experiments), the increase of the dielectric increment is shifted by about 1 pH unit toward lower pH values. This effect may be explained by the conformational change which passes from the helix state to a more compact structure at pH = 5.0 and then expands in the random coil state at higher values of pH, owing to Coulombic interactions between charged carboxyl groups. A similar circumstance has bean evidenced by Tsutsumi et al.,16 who show from magnetic r m a n c e studies that the average distance of proton to nitroxide group passes through a minimum at pD = 5.5 (pH 5.1). As pointed out by these authors, this observation is also consistent with the pH dependence of the viscosity of PGA in dioxantwater solution” which shows a minimum at the helix-coil transition.
-
Tbcoretical Background Within the counterion fluctuation based on the polarization of the ionic atmosphere of ionized polymers,18 the lowand high-frequency dielectric dispersions are attributed to local ion distribution along the subunit which tends to be equilibrated by a counterion redistribution, along the whole polyion chain. The two mechanisms cause two different procese*1characterized by two different relaxation times, the smaller of which is essentially independent of the polymer molecular weight. In this theory, for a dilute solution of monodisperse polyions, neglecting counterion
where the correction term originates by multiplying the average fraction of ionized groups a by the probability (1 - a) that a monomeric group carries, on average, a proton. However, in the present case, as pointed out by Nakamura and Wada? the contribution of the proton fluctuation, whose relevance decreases with increasing degree of ionization a,could be negligible, owing to the large binding constant of protons in comparison with that of Na+ ions.20 As far as the two relaxation times are concerned, the model predicts, to a first approximation, a dependence given byz1
and
where 7 is the viscosity of the solvent, L the length of the polymer, d its diameter, y a constant of slightly different values in the various treatments underlying eq 5 (y = 0.822in the present case), and u the mobility of associated counterions moving along the subunit of length b. Equation 5 gives the rotational relaxation time of a rigid prolate ellipsoidal particle rotating about the short axis in a continuous medium, whereas eq 6 derives from the counterion diffusion within a confined distance of the order of the subunit b. Within the “two-phase model”,’*the influence of the dynamics of individual ions through the association-dissociation kinetics of counterions on the dielectric relaxation of rodlike polyions has
.Bora1.. et. ai..
8198 The Journal of Physical Chemistry, Vol. 96, No. 20, 1992
been taken into account by van Dijk, van der Touw, and Mande1.23 The unperturbed counterions distribution along the polymer chain represented as a thin rigid rod is characterized by a linear density Po
= (1
-&);
,
50 I
1 1
(7)
where z, and zi are the valences of the polyion charged groups and counterions, respectively, and 6 is the charge density parameter defined in the Manning theory24as
The perturbation of the counterion concentration due to the extemal electric field, described by a rate constant K for the exchange between the free and bound phases, gives rise to an induced dipole moment whose magnitude depends on a characteristic length Z = (ukET/K)'/'
(9)
The dielectric increment Ae can be written as
01 50
-
.-.._ .- .-..-. .-.
,
I
1
85
120
155
-...
I--I
4
190
\ 225
polyion domain [A] Figure 6. Conductivity upof the polymer domain normalized to the bulk conductivity of the external medium u, as a function of the radius R, of the polymer domain, derived from the Poisson-Boltzmann equation, in the spherical cell model. Curves: (a) pH = 5 ; (b) pH = 7; (c) pH = 8. The electrical potential inside the cell has been obtained from the PB equation
that, when the condition Z / L
