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“Like-Charge Attraction” between Anionic Polyelectrolytes: Molecular Dynamics Simulations Ferenc Molnar* and Jens Rieger BASF Aktiengesellschaft, Polymer Physics, Carl-Bosch Str. 38, 67056 Ludwigshafen, Germany Received August 2, 2004. In Final Form: October 14, 2004 “Like-charge attraction” is a phenomenon found in many biological systems containing DNA or proteins, as well as in polyelectrolyte systems of industrial importance. “Like-charge attraction” between polyanions is observed in the presence of mobile multivalent cations. At a certain limiting concentration of cations, the negatively charged macroions cease to repel each other and even an attractive force between the anions is found. With classical molecular dynamics simulations it is possible to elucidate the processes that govern the attractive behavior with atomistic resolution. As an industrially relevant example we study the interaction of negatively charged carboxylate groups of sodium polyacrylate molecules with divalent cationic Ca2+ counterions. Here we show that Ca2+ ions initially associate with single chains of polyacrylates and strongly influence sodium ion distribution; shielded polyanions approach each other and eventually “stick” together (precipitate), contrary to the assumption that precipitation is initially induced by intermolecular Ca2+ bridging.
Introduction Charged polymers (polyelectrolytes) dissolved in water play an essential role in biology and in industrial and household applications.1,2 Polyelectrolytes strongly interact with water and other charged species present in solution. The sodium salt of poly(acrylic acid) (PAA-Na) is a polyelectrolyte polymer, which dissociates in water into negatively charged macroions and mobile Na+ counterions. This polymer is applied, for example, as a dispersing agent for pigments, as an encrustation inhibitor in laundry detergents3 and seawater desalination,4 and in a cross-linked form as a superabsorber in diapers. Water treatment relies on the interaction of PAA with calcium ions (Ca2+). A detailed look at the PAA-Ca2+ system reveals that solutions of polyelectrolytes, such as PAA, exhibit complex phase behavior, depending on the nature and the concentration of the counterions.5-7 Addition of CaCl2 to a solution of PAA-Na results in precipitation of the polyelectrolyte after reaching some limiting concentration of calcium cations.8-13 A great interest in the detailed understanding of charged molecules in water also stems from the fields of biochemistry and molecular biology. Proteins and DNA are polyelectrolytes that interact with * To whom correspondence may be addressed. Phone: +49 621 6052687. Fax: +49 621 6092281. E-mail: ferenc.molnar@ basf-ag.de. (1) Hara, M., Ed. Polyelectrolytes: Science and Technology; Marcel Dekker: New York, 1993. (2) Dautzenberg, H.; et al. Polyelectolytes; Hanser Verlag: Mu¨nchen, 1994. (3) Rieger, J. Tenside, Surfactants, Deterg. 2002, 39, 221. (4) Rieger, J.; Ha¨dicke, E.; Bu¨chner, K.-H. Proceedings of the IDA World Congress on Desalination and Reuse, Manama, Bahrain, 2002; Int. Desalination Association: Topsfield, MA, 2002 (CD-ROM). (5) Solis, F. J.; Olvera de la Cruz, M. Eur. Phys. J. E 2001, 4, 143. (6) Schweins, R.; Hollmann, J.; Huber, K. Polymer 2003, 44, 7131. (7) Boisvert, J.-P.; Malgat, A.; Pochard, I.; Daneault, C. Polymer 2002, 43, 141. (8) Fantinel, F.; Rieger, J.; Molnar, F.; Hu¨bler, P. Langmuir 2004, 20, 2539. (9) Michaeli, I. J. Polym. Sci. 1960, 48, 291. (10) Sabbagh, I. Ph.D. Thesis, Universite´ Denis Diderot Paris VII, Paris, 1997. (11) Pochard, I.; Foissy, A.; Couchot, P. Colloid Polym. Sci. 1999, 277, 818. (12) Schweins, R.; Huber, K. Eur. Phys. J. E 2001, 5, 117. (13) Schweins, R.; Lindner, P.; Huber, K. Macromolecules 2003, 36, 9564.
other charged species. In many cases the interaction with counterions has a strong effect on the structure of the biomolecules.14-18 Theoretical investigations and simulations of polyelectrolyte systems so far have mainly focused on simplified models. Theoretical models are based on Poisson-Boltzmann19,20 and counterion condensation theory21 with wellknown limitations.22 Simulations use Brownian dynamics,23-25 Monte Carlo,20,26 or (Langevin) molecular dynamics (MD)27-29 approaches. Usually coarse-grained model systems with explicit counterions and continuum electrostatics (for the solvent) are applied in the simulation studies. Going beyond the above approaches we carried out fully atomistic classical MD simulations taking into account explicitly the solvent structure, that is, water molecules, to study the behavior of PAA-Na oligomers solvated in water. The goal of these simulations was to obtain a detailed structural understanding of the interac(14) Lyubartsev, A. P.; Tang, J. X.; Janmey, P. A.; Nordenskio¨ld, L. Phys. Rev. Lett. 1998, 81, 5465. (15) Tang, J. X.; Janmey, P. A.; Lyubartsev, A.; Nordenskio¨ld, L. Biophys. J. 2002, 83, 566. (16) Das, R.; Mills, T. T.; Kwok, L. W.; Maskel, G. S.; Millett, I. S.; Doniach, S.; Finkelstein, K. D.; Herschlag, D.; Pollack, L. Phys. Rev. Lett. 2003, 90, 188103-1. (17) McFail-Isom, L.; Sines, C. C.; Dean Williams, L. Curr. Opin. Struct. Biol. 1999, 9, 298. (18) Li, A. Z.; Huang, H.; Re, X.; Qi, L. J.; Marx, K. A. Biophys. J. 1998, 74, 964. (19) Deserno, M.; Holm, C.; May, S. Macromolecules 2000, 33, 199206. (20) Pack, G. R.; Wong, L.; Lamm, G. Biopolymers 1999, 49, 575. (21) Porasso, R. D.; Benegas, J. C.; van den Hoop, M. A. G. T.; Paoletti, S. Phys. Chem. Chem. Phys. 2001, 3, 1057. (22) Lamm, G. In Reviews in Computational Chemistry; Lipkowitz, K. B., Larter, R., Cundari, T. R., Eds.; Wiley-VCH: Hoboken, 2003; Vol. 19, p 147. (23) Chang, R.; Yethiraj, A. J. Chem. Phys. 2003, 118, 11315. (24) Chang, R.; Yethiraj, A. J. Chem. Phys. 2002, 116, 5284. (25) Sottas, P.-E.; Larquet, E.; Stasiak, A.; Dubochet, J. Biophys. J. 1999, 77, 1858. (26) Lyubartsev, A. P.; Nordenskio¨ld, L. J. Phys. Chem. B 1997, 101, 4335. (27) Messina, R.; Holm, C.; Kremer, K. J. Chem. Phys. 2002, 117, 2947. (28) Liu, S.; Gosh, K.; Muthukumar, M. J. Chem. Phys. 2003, 119, 1813. (29) Jusufi, A.; Likos, C. N.; Lo¨wen, H. J. Chem. Phys. 2002, 116, 11011.
10.1021/la048057c CCC: $30.25 © 2005 American Chemical Society Published on Web 12/14/2004
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tion between the polyanion, water, and different types of counterions. Specifically, the mode of action of Ca2+ in this system as a function of concentration was investigated. This investigation sheds light on the mechanism of precipitation of PAA with Ca2+, which is closely related to the phenomenon of “like-charge attraction”. Using a realistic atomistic description is indispensable to understand the details of the interaction, such as correlation of counterion movement or exchange of explicit water molecules with carboxylate groups in the hydration spheres of the ions. With regard to more complex polymers such as proteins it is evident that detailed understanding at the molecular level is even more necessary. Methods Our model PAA is an oligomer consisting of 20 monomers. The polyanion is completely deprotonated, which corresponds to a pH value larger than 11. The negative charges of the 20 carboxylate groups are compensated by sodium cations. The neutral molecule is then placed in a cubic water box of about 60 × 60 × 60 Å3 (containing about 7900 water molecules). The system is equilibrated for 300 ps at a temperature of 300 K and a pressure of 1 atm. The typical number of atoms in the system is around 24 000. The constant pressure-constant temperature (NPT) MD equilibration run is carried out with a time step of 2 fs. O-H and C-H bond lengths are fixed using the SHAKE algorithm. All MD runs were carried out using the program NAMD.30 NAMD was developed by the Theoretical and Computational Biophysics Group in the Beckman Institute for Advanced Science and Technology at the University of Illinois at Urbana-Champaign. After equilibration, the box volume is fixed and the simulation is continued at a constant density in the microcanonical (NVE) ensemble for at least 1 ns. The trajectories of these production runs on the nanosecond time scale are used in the analysis of the simulation results. The CHARMM force field, with default TIP3P type water, is applied.31 CaCl2 molecules are added consecutively to the system to simulate the concentration dependence of the PAA behavior in the presence of divalent cations. The addition of CaCl2 is always followed by an equilibration phase of 300 ps and a production run of at least 1 ns, as described above. Figures displaying atomistic pictures of molecules were generated using VMD.32
Results and Discussion Polyelectrolyte Solvation. The simulations reveal that Na+ ions dissociate rapidly (on the picosecond time scale) from the negatively charged PAA chain after immersion in water. From the integrated radial distribution function of the Na+-carboxylate oxygen atom pair it is derived that no contact ion pairs remain (data not shown). Sodium ions are efficiently solvated by water. No clear-cut coordination number for sodium and carboxylate groups is found. On the basis of the radial distribution function of water hydrogen and carboxylate oxygen, PAA is well-solvated. To assess the orientation of carboxylate groups with respect to each other the radial distribution function for the carboxylate carbon atoms is evaluated. Figure 1 displays the results for the radial distribution function of carboxylate-carbon atom pairs. Two main peaks are discernible, which correspond to conformations with and without a water bridge (Figure 1). Interaction of Polyacrylate and Ca2+. The lifetime of Ca2+-water contacts in the first solvation shell of the (30) Kale´, L.; Skeel, R.; Bhandarkar, M.; Brunner, R.; Gursoy, A.; Krawetz, N.; Phillips, J.; Shinozaki, A.; Varadarajan, K.; Schulten, K. J. Comput. Phys. 1999, 151, 283. (31) MacKerell, A. D., Jr.; Brooks, B.; Brooks, C. L., III; Nilsson, L.; Roux, B.; Won, Y.; Karplus, M. In The Encyclopedia of Computational Chemistry; Schleyer, P. v. R., et al., Eds.; John Wiley & Sons: Chichester, 1998; Vol. 1, p 271. (32) Humphrey, W.; Dalke, A.; Schulten, K. J. Mol. Graphics 1996, 14, 33.
Figure 1. Radial pair-distribution function g(C-C) for carboxylate carbon interactions. The two main peaks correspond to the most favorable, water-mediated, interactions. The first peak at about 3.5 Å corresponds to the minimum distance between carboxylate groups with a direct water bridge between them. The second peak at 4.5 Å is characteristic for atom pairs without a direct water bridge. The other peaks correspond to “next nearest neighbors”. The occurrence of the peak at 3.5 Å seems at first sight counterintuitive because neighboring carboxylate groups are negatively charged and should repel each other. The reason for the tendency to overcome the repulsion is the possibility to form intermittent bridging hydrogen bonds to water molecules.
divalent cation is estimated to be on the order of hundreds of picoseconds (ps) on the basis of independent simulations for CaCl2 solvated in water. Therefore, only a limited number of exchange events between bulk and solvation shell water molecules can be observed during the course of the nanosecond simulation. Placing a Ca2+ ion along with two Cl- anions randomly into the simulation box results in a migration of the hydrated cation toward the PAA chain (Figure 2). Initially, on the time scale of the simulation (1 ns) no exchange between water molecules in the first solvation shell and carboxylate oxygen atoms was observed. However, after extending the simulations to 7 ns for the case of five added CaCl2 units, the exchange of water in the first solvation shell with carboxylate groups from PAA is observed after about 2 ns, and eventually all Ca2+ ions end up in direct contact with the PAA chain. At low concentrations Ca2+ ions are able to migrate along the polymer backbone. With increasing calcium concentration the movement of divalent cations becomes more and more localized and confined to a small number of carboxylate units. To speed up the simulations, additional Ca2+ ions were inserted into the simulated system close to the PAA oligomer, and production runs of several nanoseconds were carried out, resulting in a variety of interactions with the carboxylate groups. Examples of Ca2+ ions forming intrachain bridges are displayed in Figure 2. Direct intrachain bridges between carboxylate groups are relatively stable (lifetime >1 ns) and remain intact during a large part of the simulation run. Bridging to “remote” (nonneighboring) carboxylate occurs initially only through solvent-separated configurations. These “indirect” bridges exhibit reduced stability. After simulation times of 4 ns,
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Figure 2. Ca2+-PAA interaction. (a) Hydrated Ca2+ [light blue, van der Waals (vdW) sphere; water oxygen, red vdW sphere) diffusion toward PAA chain. Randomly placed Ca2+ migrates toward the polyanion. It initially remains solvent separated on the 1-ns time scale of the simulation. Complete water exchange in the first solvation sphere is observed after 400 ps. The first water solvation sphere around Ca2+ was recalculated at 600 and 1000 ps. Exchange of water against carboxylate groups in the first solvation sphere occurs on a longer time scale (2 ns). (b) Stable Ca2+ intrachain bridges: direct bridging between neighboring carboxylates. Color code: Ca2+, light blue vdW; water oxygen, red vdW; polyacrylate chain, ball-and-stick; sodium, dark blue. Only water molecules associated with Ca2+ are shown.
also direct (not solvent-separated) bridging to “remote” carboxylate groups is observed. Additionally, solventseparated bridging occurs to more than two carboxylates after 4 ns. We also carried out further simulations on a system containing two PAA oligomers, but no spontaneous intermolecular Ca2+ bridging could be observed. This is not surprising because, for entropic reasons, intramolecular bridging should be much more probable. Effect of Ca2+ Concentration. Adding an increasing number of CaCl2 units (1-5 CaCl2) has a marked effect on the distribution of monovalent sodium cations around the polyanion. This effect is demonstrated by the Na+carboxylate carbon radial pair-distribution function as a function of Ca2+ concentration (Figure 3): a discontinuity is observed when changing from three added Ca2+ to four added Ca2+. The drop in the first peak of Figure 3 can be rationalized by assuming a shielding effect: the divalent
Figure 3. Radial pair-distribution function g(Na-C) of the Na+-carboxylate carbon atom pair. Sodium ions “evaporate” from PAA after the addition of the fourth calcium ion.
calcium cations “stick” closely to PAA and at a certain concentration effectively shield the negative charge of the polyanion.
Anionic Polyelectrolytes Like-Charge Attraction
To rule out the possibility that the effects of sodium ion “evaporation” and Ca2+ shielding of the polyanion are an artifact of this particular parametrization of the force field, additional simulations were carried out with partial charges based on charges derived from density functional theory (DFT) calculations. DFT charges were obtained from the Mulliken population analysis at the B-P86/TZVP// B-P86/SVP level of theory. It turned out that results for Na+ “evaporation” are not very sensitive to changes in the partial charges of the carboxylate group. The Ca2+ ions are strongly associated with carboxylate groups; they display reduced mobility along the PAA chain on the nanosecond time scale and do not dissociate from the macroanion. As a result of this association, PAA becomes less negatively charged, that is, more “hydrophobic”, and the monovalent sodium cations are “evaporating” from the polyanion. The “evaporation” of Na+, therefore, marks the onset of precipitation. The more “hydrophobic” PAA oligomers will be less efficiently solvated by water and will show an increased tendency to “stick” together and finally precipitate. This result is in accordance with recent experimental data.33 Because the simulated system is relatively small, the observed discontinuity in the Na+carboxylate carbon distribution function can only be interpreted as the “signature of a phase transition” and not as the observation of an actual phase transition. The underlying molecular mechanism is, however, probably identical to the one causing the real phase transition in the experiment (namely, precipitation). To examine the validity of these conclusions, additional extended simulations (7 ns) were carried out for a system with two PAA oligomers (Figure 4). It is deduced from Figure 4 that after the addition of 14 CaCl2 units (the two oligomers carry 20 anionic charges each), the oligomeric chains are efficiently shielded and can approach each other more closely on average. This effect corresponds to the so-called “like-charge attraction” phenomenon. The observation of restricted movements of the Ca2+ ions along the PAA chain indicates that the “like-charge attraction” between the two oligomers on the nanosecond time scale is due to local charge neutralization resulting in increasing hydrophobic attractive interaction. The attractive interaction, thus, is not due to “correlated Ca2+ motion” or (33) Sinn, C. G.; Dimova, R.; Antonietti, M. Macromolecules 2004, 37, 3444.
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Figure 4. Radial pair-distribution function g(C-C) of carboxylate carbon atoms on different oligomers in a system containing two PAA oligomers. Ca2+ ion addition shields the polyanions and allows the oligomer chains to approach each other more closely on the average. The approximate distance for an intermolecular Ca2+ bridge is indicated. Production runs were carried out for 10 ns in the case of 5 CaCl2 and for 7 ns in the cases of 10 and 14 CaCl2.
“Wigner lattice formation”.34,35 To our knowledge this is the first time that the mechanism of “like-charge attraction” can be interpreted reliably by an atomistic model. Summary We have shown, with the help of fully atomistic MD simulations, that at a limiting concentration of mobile divalent cations, negatively charged macroions cease to repel each other and even an attractive force between the anions occurs. Ca2+ ions associate with single chains of PAA and strongly influence the original sodium ion distribution around the polymer. With increasing Ca2+ loading the polyanions become shielded and are able to approach (“attract”) each other. This effect does not require exact compensation of charges. As a function of Ca2+ concentration, shielded polyanions approach each other and eventually “stick” together (precipitate), contrary to the assumption that precipitation is initially induced by intermolecular Ca2+ bridging. LA048057C (34) Angelini, T. E.; Liang, H.; Wriggers, W.; Wong, G. C. L. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 8634. (35) Butler, J. C.; Angelini, T.; Tang, J. X.; Wong, G. C. L. Phys. Rev. Lett. 2003, 91, 028301.
