12174
J. Phys. Chem. B 2000, 104, 12174-12178
Coil to Extended Coil Transition in Polygalacturonic Acid: Conductometric Titration and Monte Carlo Simulations Marcelo Andres Fossey,† Cecilia Cristina Marques dos Santos,‡ Jorge Chahine,*,† and Joa˜ o Ruggiero Neto*,† Departamento de Fisica Instituto de Biocieˆ ncias, Letras e Cieˆ ncias Exatas UNESP, 15054-000-Sa˜ o Jose´ do Rio Preto, SP, Brazil, and Instituto Adolfo Lutz, Laborato´ rio de Bromatologia, 15060-020-Sa˜ o Jose´ do Rio Preto, SP, Brazil ReceiVed: May 31, 2000; In Final Form: September 20, 2000
The conformational transition from coil to extended coil for polygalacturonic acid has been studied by conductometric titrations and Monte Carlo simulations. The results of conductometric titrations at different polymer concentrations have been analyzed using the model proposed by Manning,1 which describes the conductivity of polyelectrolitic solutions. This experimental approach provides the transport factor and the average distance between charged groups at different degrees of ionization (R). The mean distances between charged groups have been compared with the values obtained by Monte Carlo simulations. In these simulations the polymer chain is modeled as a self-avoiding random walk in a cubic lattice. The monomers interact through the unscreened Coulombic potential. The ratio between the end-to-end distance and the number of ionized beads provides the average distance between charged monomers. The experimental and theoretical values are in good agreement for the whole range of ionization degrees accessed by conductometric titrations. These results suggest that the electrostatic interactions seem to be the major contribution for the coil to extended coil conformational change. The small deviations for R e 0.5 suggests that the stiffness of the chain, associated with local interactions, becomes increasingly significant as the fraction of charged groups is decreased.
I. Introduction The quantitative description of the charge process of weak polyacids, observed during the titration experiment, has drawn much attention.2,3 For these polymers the conformation becomes more extended as the degree of ionization increases.4,5 Moreover, for most of them the conformational change is cooperative. Upon addition of titrating base, the electrostatic repulsion takes place as a consequence of the partial neutralization of the charged groups, and the initial globular form of the polymer becomes more extended. Usually the chain extension is observed through a simple potentiometric titration experiment through the plot of apparent pK vs the ionization degree (pKapp vs R). The analysis of this process, however, is not simple. A shoulder observed in the pKapp vs R curves for low R values complicates the analysis through the conventional polyelectrolytic models. The pKapp vs R curves obtained by the Poisson-Boltzmann equation (PBE)6 show a continuous increase of the pKapp with the ionization degree (R).5 The counterion condensation model (CC) generates pKapp vs R curves with a discontinuity at R ) 1/ξ where ξ is the dimensionless charge parameter.7 In this way the anomalous (shoulder) behavior observed on the experimental pKapp vs R plots is not satisfactorily described by these two models, which provide only a partial description of the experimental pKapp vs R, even when the polymer is depicted as a semiflexible rod.8 Theoretical studies of titration of linear polyelectrolytes have also been carried out using lattice9 and off-lattice2 Monte Carlo * To whom correspondence should be addressed. Email: chahine@ df.ibilce.unesp.br;
[email protected]. † UNESP. ‡ Laborato ´ rio de Bromatologia.
simulations. Different models for titrating polyelectrolytes have also been used10 in simulations of very long chains by means of an efficient pivot algorithm.11,12 Reed et al.2 proposed an off-lattice Monte Carlo model to calculate the average ionization of the monomers as a function of pH (R vs pH) for several Debye screening lengths and Manning parameters ξ. These parameters were used by the authors in a modified DebyeHuckel potential in order to take into account the phenomenon of counterion condensation proposed by Manning.13-15 For some values of the parameters used in the simulations their results showed good correlation with the experimental titration curve of hyaluronic acid.16 In a more recent work Nishio3 used a Monte Carlo method to study a potentiometric titration of polyL-glutamic acid. In separated simulations he obtained the titration curves for the helical and coiled forms of the chain. Good agreement with the experimental data was obtained for polymer solutions at low concentrations. To study the role of electrostatic energy on the conformational properties of the weak polyacid chain we have used a simpler lattice Monte Carlo model to study the ionization process of another weak polyacid (polygalacturonic acid). In these simulations we only used the electrostatic term described by the unscreened Coulomb potential. Instead of measuring the average ionization as a function of the pH, we propose to follow the ionization process by calculating the average distance between charged monomers as a function of the ionization. This is due to the fact that it is possible to access the mean distance between charges on the polyacid by conductometric titrations. The analysis of these titrations is performed through the statistical model for polyelectrolitic conductivity proposed by Manning.1 This procedure enables us to determine the transport factor and hence the distance between charged groups in the polyelectrolyte
10.1021/jp001995a CCC: $19.00 © 2000 American Chemical Society Published on Web 11/30/2000
Coil to Extended Coil Transition in PGA
J. Phys. Chem. B, Vol. 104, No. 51, 2000 12175
during the ionization process. The aim of the comparison between simulations and experimental data is to study the role of electrostatic energy on the conformational properties of the chain. Monitoring the charge process of a weak polyacid through the transport factor, measured from conductometric titrations, presents an additional advantage: the theoretical description of the conductivity of polyacid solutions in different ionization degrees, proposed by Manning,1 does not show the discontinuity observed with the CC model for the pKapp vs R curve. This experimental approach has been used to estimate the distance between charged groups in polysaccharides17,18 and help determine the structure of ionic polysaccharides19,20 and nucleic acids.21 Recently, this method was successfully used to study the coil-to-helix transition in κ-caragenan,22 showing good agreement with circular dichroism results.
and σLi, measured in these titrations. In our experiments the conductometric and potentiometric titrations were carried out under identical conditions so that the a transport factor could be calculated at each value of R as given by
f(R) )
103RCp(λ0K - λ0Li)
f)1-
[
]
0.55(Zξ)2 (π + Zξ)
[XOH] + 10-pH Cp
(1)
Measurements. Conductometric titrations of PGA solutions, at different concentrations, were carried out at the same conditions of the potentiometric titrations, using Merck standard solutions of LiOH and KOH whose concentrations were 0.1 N. The solution conductivity was measured with a MetrohmMicronal Wien bridge operating at 1 kHz equipped with a platinized electrode characterized by a cell constant of 1.00 cm-1. The cell constant was calibrated with KCl solutions after each measurement. The titrations were carried out in a jacketed glass cell and a Neslab RTE 221 bath; the temperature was maintained at 25.0 ( 0.1 °C. During the titrations, dried N2 was bubbled in the solution in order to avoid changes in conductivity due to dissolved CO2. The equivalent conductivity of a polyelectrolitic solution in the absence of support salt can be written as1,24
σ ) fCp(λ0c + λp)
(2)
where σ is the equivalent conductivity of the solution, λ0c is the limit equivalent conductivity of the counterion free in pure solvent, λp is the polymer equivalent conductivity, Cp is the equivalent polymer concentration, determined by potentiometric titrations, and f is the transport factor. The transport factor can be determined experimentally through conductometric titrations of the polyacid solution with different bases19,24 such as KOH and LiOH. f is then determined from conductivity values σK
ξ e |1/Z|
(4)
and
f ) 0.87(Zξ)-1
Samples. Polygalacturonic acid (PGA) purchased from Sigma was dissolved in water, neutralized by addition of NaOH, and centrifuged at 15 000g. The supernatant was precipitated with ethanol and resuspended in water thrice and dialyzed exhaustively against water. The acid form was obtained by dialysis against acetic acid (0.1 N) until the pH of the polymer solution was constant and then dialyzed against water to remove the excess acid. All reagents used (salt, bases, acids, etc.) were analytical grade and purchased from Merck. Water used was double distilled in quartz (∼0.6 µΩ-1 cm-1). The equivalent polymer solution concentrations Cp were determined by potentiometric titrations of the polysaccharides initially in the acid form as described elsewhere.19,23 The ionization degrees R at each base concentration were calculated from the pH values and the polymer concentrations as the sum of the neutralization (Rn) and the self-dissociation (RH) degrees:
(3)
Through the relations proposed by Manning1 it was possible to calculate the polymer charge density from the values of the transport factor:
II. Materials and Methods
Ri ) Rn + RH )
σK(R) - σLi(R)
ξ g |1/Z|
(5)
Here Z is the counterion valence and ξ is the linear charge density calculated as
ξ)
e2 kTb
(6)
e is the electric charge of the charged groups in the polyelectrolyte, is the solvent dielectric constant, k is the Boltzmann constant, T is the temperature, and b is the distance between the charged groups in the polyelectrolyte. Monte Carlo Simulation. The polymer is modeled as a chain of N beads represented by a self-avoiding walk in a periodic cubic lattice. The beads interact through the unscreened (no salt added) Coulombic potential. The lattice spacing a is chosen in the following way: with the chain fully ionized we perform simulations where the lattice spacing is varied until the mean distance b between the charged beads matches the experimental value obtained from the conductometric titrations as described in section II. The mean distance between charged groups is calculated by dividing the end-to-end distance by N - 1. Just a few simulations are performed to obtain the appropriate value of a. The potential used in the simulations is given by
U)
e2
N i-1
∑∑ i)2 j)1
δiδj rij
(7)
The quantity δi is 1 or 0, depending if the monomer i is charged or not, with the electronic charge e, and rij is the distance between monomers i and j. The dielectric constant is 78.5. The chain with N ) 40 beads is placed in a periodic cubic lattice, and after choice of an initial configuration, the system undergoes several types of internal movements followed by reptation,9 reaching equilibrium after 4 × 104 Monte Carlo steps. To calculate the mean values of the end-to-end distance, 2 × 106 additional steps were used. The degree of ionization R of the chain is varied from 1.0 to 0.3 to obtain b as a function of R. These results are compared with experimental data. III. Results Conductometric Titrations. Conductometric titrations of polygalacturonic acid with different bases KOH and LiOH, displayed in a plot of electrical conductivity (σ) vs neutralization degree (Rn), present a characteristic profile as shown in the Figure 1. In this figure a slow decrease in conductivity is observed in the beginning of the neutralization process. The
12176 J. Phys. Chem. B, Vol. 104, No. 51, 2000
Fossey et al.
Figure 1. Conductivity vs neutralization degree for PGA titrated with KOH (9) and LiOH (b) both at 0.1 N and 25 °C. The dashed line indicates the neutralization of the carboxyl groups. Cp ) 0.87 mequiv/ L.
relatively low initial electrical conductivity of acid polymer is due to the lability of the protons bound to the carboxyl groups. When the titrating base is increased, the conductivity decreases due to the substitution of protons by its cation which have lower ionic intrinsic conductivity than protons. This decrease is observed until the number of neutralized groups becomes higher than the nonneutralized ones. From this point a slow increase in the conductivity is observed up to the point where complete neutralization of the polymer occurs at Rn ) 1.0 as illustrated in Figure 1. Beyond this point the linear increase in the conductivity is due to the base excess. Dilution Experiments. The electrical conductivity of polygalacturonic acid neutralized by KOH and LiOH, where the polymer ionization degree was maintained equal to one (R ) 1), was measured in different polymer concentrations. These dilution experiments were analyzed through a Vink plot where the electrical conductivity (σ) of PGA in both salt forms PGA-K and PGA-Li is plotted as a function of polymer concentration (Cp) as shown in Figure 2. The experimental data were adjusted numerically through a curve fitting using the polynomial function proposed by Vink25 and Kusnetsov:21
σ ) σ0 + Λ0Cp - ACp3/2
(8)
In this equation σ0 is the electrolytic conductivity of the solvent obtained at zero polymer concentration, Λ0 is the molar equivalent conductivity at infinite dilution, and A is a constant. The parameters σ0, Λ0, and A obtained from a best fit of the experimental data in Figure 2 were the following: σ0 ) 5.0 ( 0.2µΩ-1 cm-1, Λ0 ) 93.2 ( 0.5 Ω-1 cm2 equiv-1, A ) 9.7 ( 0.3 µΩ-1 cm-1 mequiv2/3 for potassium; σ0 ) 4.3 ( 0.2 µΩ-1 cm-1, Λ0 ) 72.6 ( 0.3 Ω-1 cm2 equiv-1, A ) 8.3 ( 0.2 µΩ-1 cm-1 mequiv2/3 for lithium. The transport factor estimated using the values of the equivalent conductivity at infinite dilution is f ) 0.57 ( 0.03, and the distance between charged groups b ) 4.7 ( 0.2 Å in reasonable agreement with b ) 4.45 Å from crystallographic data.26 The conductivity values measured in the titrations with KOH and LiOH at each R value displayed in Figure 1 were used to calculate the transport factor using eq 3. The dimensionless charge parameter, ξ, and the distance between charged groups, b, were calculated from the values of the transport factor with
Figure 2. Vink plot: conductivity (σ) vs Cp obtained from dilution experiments of PGA-K (9) and PGA-Li (b) at 25 °C, where the initial polymer concentration is Cp ) 1.91 mequiv/L. The lines represent the best fitting obtained from nonlinear least-squares fitting using eq 8.
Figure 3. b vs R plot. The mean distances between charged groups, represented by open circles, were calculated from the titration of Figure 1 with eqs 3-6. Solid triangle are the results from the simulations. The dashed lines is a guide to the eyes.
eqs 4-6. The results are presented in Figure 3 along with the simulation results. Monte Carlo Simulations. With the chain fully ionized a number of simulations were performed for some values of the lattice parameter in order to determine which ones could provide the experimentally obtained mean distance of 4.7 Å between charged groups. After performing a few simulations, we have found that this mean charge distance can be reproduced with a lattice parameter a ) 5.8 Å. [Recently a method of converting crystallographic data of B-DNA into a bead model found values ranging from 5.0 to 5.7 Å for the bead radius.31 Nevertheless it should be noted that these values are far from being universal and are dependent on the particular system under investigation. For instance, the mentioned method found a bead radius equal to 4.5 Å for proteins.] Once the lattice parameter is determined, the value of R is varied in order to calculate b as function of R. The results obtained from simulations are compared with experimental data in Figure 3. The mean distance between charged groups is plotted as a function of the degree of ionization R. The solid triangles represent results obtained from simulations, and open circles represent results obtained from
Coil to Extended Coil Transition in PGA conductometric titrations. The average distances between charged groups are related to the chain conformation whose changes are rather significant. The value of the end-to-end distance for the fully ionized chain is 182.5 Å, twice the value of 85 Å, obtained at a low degree of ionization (R ) 0.35). The number of beads (N ) 40) used in these simulations can be justified as follows: for R ) 0 the end-to-end distance is 62 Å, which is higher than the Gaussian value27 (N3/5b ) 52 Å). It would be preferable to use longer chains in order to reach the Gaussian limit for the end-to-end distance. However, for the simple lattice model used here, the differences do not seem to be significant, mainly due to the fact that the lowest value of the ionization degree is greater than zero; i.e., R ) 0.4. In the Manning theory for conductivity, the distance between charged groups is defined for a rigid and stretched chain through the ratio of the contour length and the number of charged groups. However, the contour length is a fixed value and therefore is not sensitive to changes in the degree of ionization as is the end-to-end distance. The use of the contour length for calculating b gives the trivial dependence b ∼ 1/R, which does not describe the conformational change studied in present work. It is worth noting that if we had used, for the fully ionized chain, the mean distance between charged monomers b ) 4.45 Å obtained from crystallographic data26,28 in order to determine the lattice parameter a, this value would be a ) 5.4 Å instead of 5.8 Å. The smaller lattice parameter would shift the distance between charged groups to slightly lower values (data not shown). The simulation data in the Figure 3 were obtained with unscreened potential eq 7. For the whole range of degree of ionization values the charge density is higher than unity and should be expected from the effect of condensed counterions. However from simulations, using a screened potential, over the range of polymer concentration used in the present work (less than 6-7 mM) the screening effects due to condensation are small data not shown. Similar results were reported by simulations of polymaleic acid in the presence of added salt.29 IV. Conclusion The conformational change following the charge process of a weak polyacid is understood as a transition from a coil to an extended coil conformation. The changes in the conductometric profile as a function of the degree of ionization were observed to be a characteristic of a weak polyacid. The plot of the transport factor or the distance between charged groups suggests that the conductometric profile reflects this conformational change. The results in Figure 3 show values of charge density ξ greater than 1 (i.e., ξ varies from 1.21 to 1.53 in the range of R between 0.4 and 1.0). Monte Carlo simulation studies using the Debye-Hu¨ckel potential with added salt have shown30 that the chain remains highly flexible for charge density up to 1.78. The results obtained in the present work suggest that this flexibility is related to nearly coiledlike conformations, at low degree of ionization, and to a nearly rodlike conformation for the fully ionized chain. Evidence for this fact is provided by the distribution functions of the distance between monomers and the center of mass of the chain, i.e., the frequencies of the distances between monomers and the centers of mass. This distribution is nearly Gaussian for coillike conformations and is independent of the bead location for a rodlike chain.29 Figure 4 shows the behavior of the distributions for three values of R (0, 0.35, 1). For R ) 0 the distribution is Gaussian-like. For R ) 1 it is nearly rodlike, and it shows a decreased, but still significant, population of the coil conformations for R ) 0.35.
J. Phys. Chem. B, Vol. 104, No. 51, 2000 12177
Figure 4. Distributions (or frequencies) of the distances between monomers and the center of mass of polymer for three values of the degree of ionization: R ) 0 (- ‚ -); R ) 0.35 (- - -); R ) 1 (s).
Qualitatively it is reasonable to suggest that as titration increases the degree of ionization R from 0.4 to 1, the chain switches from a flexible conformation to a nearly rodlike conformations. Our results indicate that the Coulombic interactions seem to be responsible for the changes observed in the charge density of PGA. In the lower charge density region (R < 0.5) the results in Figure 3 show that the simulations produced slightly higher values for the mean distance between charged monomers when compared with the experimental results. As the degree of ionization decreases, the stiffness of the chain associated with local interactions is likely to become increasingly significant. Even after taking into consideration the simplicity of the lattice model used to represent the chain, good agreement between the simulations and experimental results for the mean distances b between charged monomers was obtained. This simple model lacks short-range potentials such as angle and torsional terms and also does not take into account the geometry of the monomer, which probably would describe the behavior of an uncharged PGA chain. In the opposite limit, for R values close to one, it is known that pectin chains can be stiffened by intramolecular hydrogen bonds26 which could be responsible for a rather different behavior of the dependence of b against R. However, simulations from Hirose et al.,29 who used simple models to study the behavior of maleic acid, showed that, as the concentration of the screening charges (salts) decreased, the conformational parameters were not affected by the hydrogen bonds. This is the case of the present study where salts were absent. The present results show that, for a large range of ionization degree, from 0.4 to 1.0 the conformational behavior of the PGA chain is mainly dictated by the long-range Coulombic potential. Acknowledgment. We are grateful to Brazilian agencies FAPESP and CNPq for financial support. References and Notes (1) Manning, G. S. J. Phys. Chem. 1975, 79, 262. (2) Reed, C. E.; Reed, W. F. J. Chem. Phys. 1992, 96, 1609. (3) Nishio, T. Biophys. Chem. 1998, 71, 173. (4) Rice, S. A.; Nagasawa, M. Polyelectrolytes in Solution; Academic Press: New York, 1961. (5) Oosawa, F. Polyelectrolytes; Marcel Dekker Inc.: New York, 1971. (6) Rinaudo, M. Polyelectrolytes; Se´le`gny, E., Ed.; D. Reidel Publishing Co.: Dordrecht, Holland, 1974. (7) Manning, G. S. J. Phys. Chem. 1981, 85, 870.
12178 J. Phys. Chem. B, Vol. 104, No. 51, 2000 (8) Cesaro A.; Paoletti, S.; Benegas, J. C. Biophys. Chem. 1991, 39, 1. (9) Sassi, A. P.; Beltran, S.; H. H. Hooper, H. H.; Blanch, H. W.; Prausnitz, J. M.; Siegel, R. A. J. Chem. Phys. 1992, 97, 8767. (10) Ullner, M.; Jonsson, B.; Soderberg, B.; Peterson, C. J. Chem. Phys. 1996, 104, 3048 (11) Lal, M. Mol. Phys. 1969, 17, 57. (12) Madras, N.; Sokal, A. D. J. Stat. Phys. 1988, 50, 109. (13) Manning, G. S. J. Chem. Phys. 1969, 51, 924. (14) Manning, G. S. J. Chem. Phys. 1969, 51, 3249. (15) Manning, G. S. Biophys. Chem. 1977, 7, 95. (16) Cleland, R. L.; Wang, J. L.; Detweiler, D. M. Macromolecules 1982, 15, 386. (17) Rochas, C. Food Hydrocolloids 1987, 1, 215. (18) Rochas, C.; Rinaudo, M. Biopolymers 1980, 19, 1675. (19) Ruggiero, J.; Vieira, R. P.; Moura˜o, P. A. S. Carbohydr. Res. 1994, 256, 275. (20) Ruggiero, J.; Fossey, M.; Santos, J. A.; Moura˜o, P. A. S. Carbohydr. Res. 1998, 306, 545.
Fossey et al. (21) Kuznetsov, I. A.; Vorontsova, O. V.; Kozlov, A. G. Biopolymers 1991, 31, 65. (22) Ciszkowska, M.; Osteryoung, J. G. J. Am. Chem. Soc. 1999, 121, 1617. (23) Agostinho Neto, A.; Drigo F°, E.; Fossey, M. A.; Ruggiero Neto, J. J. Phys. Chem B 1997, 101, 9833. (24) Eisenberg, H. J. Polym. Sci. 1958, 30, 47. (25) Vink, H. J. Chem. Soc., Faraday Trans. 1 1981, 77, 2439. (26) Walkinshaw, M. D.; Arnot, S. J. Mol. Biol. 1981, 153, 1055. (27) Doi, M. Introduction to Polymer Physics; Clarendon Press: Oxford, U.K., 1996. (28) Atkins, E. D. T.; Nieduszynski, I. A.; Mackie, W.; Parker, K. D.; Smolko, E. E. Biopolymers 1973, 12, 1865. (29) Hirose, Y.; Onodera, M.; Kawaguchi, S.; Ito, K. Polym. J. 1995, 27, 519. (30) Seidel, C. Macromol. Symp. 1995, 100, 175. (31) Banachowicz, E.; Gapinski, J.; Patkowiski, A. Biophys. J. 2000, 78, 70.
