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Anionic Polyelectrolyte Adsorption on Mica Mediated by Multivalent Cations: A Solution to DNA Imaging by Atomic Force Microscopy under High Ionic Strengths David Pastre´,*,† Loı¨c Hamon,† Fabrice Landousy,‡ Isabelle Sorel,† Marie-Odile David,† Alain Zozime,† Eric Le Cam,‡ and Olivier Pie´trement‡ Laboratoire Structure et Reconnaissance des Biomole´ cules, EA 3637, UniVersite´ d’EVry, Rue du Pe` re Jarlan, 91025 EVry Cedex, France, and Laboratoire de Microscopie Mole´ culaire et Cellulaire, UMR 8126 CNRS-IGR-UPS, Institut GustaVe-Roussy, 39 rue Camille Desmoulins, 94805 Villejuif Cedex, France ReceiVed December 14, 2005. In Final Form: May 5, 2006 Adsorption of DNA molecules on mica, a highly negatively charged surface, mediated by divalent or trivalent cations is considered. By analyzing atomic force microscope (AFM) images of DNA molecules adsorbed on mica, phase diagrams of DNA molecules interacting with a mica surface are established in terms of concentrations of monovalent salt (NaCl) and divalent (MgCl2) or multivalent (spermidine, cobalt hexamine) salts. These diagrams show two transitions between nonadsorption and adsorption. The first one arises when the concentration of multivalent counterions is larger than a limit value, which is not sensitive to the monovalent salt concentration. The second transition is due to the binding competition between monovalent and multivalent counterions. In addition, we develop a model of polyelectrolyte adsorption on like-charged surfaces with multivalent counterions. This model shows that the correlations of the multivalent counterions at the interface between DNA and mica play a critical role. Furthermore, it appears that DNA adsorption takes place when the energy gain in counterion correlations overcomes an energy barrier. This barrier is induced by the entropy loss in confining DNA in a thin adsorbed layer, the entropy loss in the interpenetration of the clouds of mica and DNA counterions, and the electrostatic repulsion between DNA and mica. The analysis of the experimental results provides an estimation of this energy barrier. We then discuss some important issues, including DNA adsorption under physiological conditions.
* To whom correspondence should be addressed. Tel: (33) 1 69 47 03 03. Fax: (33) 1 69 47 01 05. E-mail:
[email protected]. † Universite ´ d’Evry. ‡ Institut Gustave-Roussy.
explain the process allowing DNA adsorption.7,13,14 Recently, we showed that the correlations between multivalent counterions are involved in DNA adsorption on mica.15 The attraction of like-charged bodies by counterion correlations has also been the subject of many studies during past decades.16-20 Counterion correlations are thought to be the mechanisms that lead to DNA condensation by multivalent cations.21-24 However, little is known about the implication of counterion correlations in DNA adsorption on negatively charged surfaces. In addition, the study of DNA/ligand complexes by AFM requires imaging DNA under different physiological conditions. A better understanding of the DNA adsorption mechanism should enable one to adapt the buffer composition to biological requirements, such as that of the ionic strength. For this purpose, we present phase diagrams of DNA molecules adsorbed on mica surface in terms of concentrations of monovalent (NaCl) and multivalent salts. Three multivalent
(1) Tripathy, S. K.; Kumar, J.; Nalwa, H. S. Handbook of Polyelectrolytes and Their Applications; Marcel Dekker: New York, 2002; Vol. 99. (2) Hansma, H. G. Annu. ReV. Phys. Chem. 2001, 52, 71-92. (3) Lindsay, S. M. Biophys. J. 1994, 67, 2134-2135. (4) Lyubchenko, Y. L. Cell Biochem. Biophys. 2004, 41, 75-98. (5) Janicijevic, A.; Ristic, D.; Wyman, C. J. Microsc. 2003, 212, 264-272. (6) Pie´trement, O.; Pastre´, D.; Fusil, S.; Jeusset, J.; David, M.-O.; Landousy, F.; Hamon, L.; Zozime, A.; Le Cam, E. Langmuir 2003, 19, 2536-2539. (7) Shao, Z.; Mou, J.; Czajkowksy, D. M.; Yang, J.; Yuan, J. Y. AdV. Phys. 1996, 45, 1-86. (8) Guthold, M.; Bezanilla, M.; Erie, D. A.; Jenkins, B.; Hansma, H. G.; Bustamante, C. Proc. Natl. Acad. Sci. U.S.A. 1994, 91, 12927-12931. (9) Samori, B.; Muzzalupo, I.; Zuccheri, G. Scanning Microsc. 1996, 10, 953960; discussion 960-952. (10) Berge, T.; Ellis, D. J.; Dryden, D. T.; Edwardson, J. M.; Henderson, R. M. Biophys. J. 2000, 79, 479-484. (11) van Noort, S. J.; van der Werf, K. O.; Eker, A. P.; Wyman, C.; de Grooth, B. G.; van Hulst, N. F.; Greve, J. Biophys. J. 1998, 74, 2840-2849. (12) Crampton, N.; Bonass, W. A.; Kirkham, J.; Thomson, N. H. Langmuir 2005, 21, 7884-7891.
(13) Hansma, H. G.; Laney, D. E. Biophys. J. 1996, 70, 1933-1939. (14) Rivetti, C.; Guthold, M.; Bustamante, C. J. Mol. Biol. 1996, 264, 919932. (15) Pastre´, D.; Pie´trement, O.; Fusil, S.; Landousy, F.; Jeusset, J.; David, M. O.; Hamon, L.; Le Cam, E.; Zozime, A. Biophys. J. 2003, 85, 2507-2518. (16) Lau, A. W.; Pincus, P. Phys. ReV. E: Stat. Nonlinear, Soft Matter Phys. 2002, 66, 041501. (17) Kjellander, R.; Marcelja, S. Chem. Phys. Lett. 1985, 114, 124. (18) Arenzon, J. J.; Stilck, J. F.; Levin, Y. Eur. Phys. J. B 1999, 12, 79-82. (19) Angelini, T. E.; Liang, H.; Wriggers, W.; Wong, G. C. L. Proc. Natl. Acad. Sci. U.S.A. 2003, 8634-8637. (20) Rouzina, I.; Bloomfield, V. A. J. Phys. Chem. 1996, 100, 9977-9989. (21) Matulis, D.; Rouzina, I.; Bloomfield, V. A. J. Mol. Biol. 2000, 296, 10531063. (22) Bloomfield, V. A. Biopolymers 1991, 31, 1471-1481. (23) Pelta, J.; Livolant, F.; Sikorav, J.-L. J. Biol. Chem. 1996, 271, 56565662. (24) Pastre, D.; Pietrement, O.; Landousy, F.; Hamon, L.; Sorel, I.; David, M. O.; Delain, E.; Zozime, A.; Le Cam, E. Eur. Biophys. J. 2006, 35, 214-223.
1. Introduction Adsorption of polyelectrolytes onto charged surfaces is an important aspect of numerous technologies.1 In the present paper, the adsorption of highly charged polyelectrolytes to like-charged surfaces is considered. In particular, this study deals with DNA adsorption on mica, which is the most popular substrate for imaging biomolecules by atomic force microscopy (AFM).2-12 Since DNA and mica are both highly negatively charged, a weak electrostatic attachment of DNA on mica is obtained by adding multivalent counterions such as Mg2+ to the buffer. Generally, authors have referred to an unclear array of binding sites to
10.1021/la053387y CCC: $33.50 © 2006 American Chemical Society Published on Web 06/14/2006
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Figure 1. AFM images of pUC19 DNA molecules deposited on mica for three different concentrations of spermidine and NaCl: (a) Tris 10 mM, pH 7.5, NaCl 100 mM, spermidine 20 µM; (b) Tris 10 mM, pH 7.5, NaCl 200 mM, spermidine 20 µM; (c) Tris 10 mM, pH 7.5, NaCl 400 mM, spermidine 20 µM. Scan area: 3 × 3 µm2; z range: 5 nm; scan frequency: 1.5 Hz. It can be noted that DNA molecules are gradually released from the surface by increasing the monovalent salt concentration. Loosely bound molecules appear as bright dots in panels b and c. Progressive DNA release from the surface by increasing NaCl concentration is also observed for DNA molecules adsorbed on mica by divalent counterions (Mg2+).
cations (Mg2+, spermidine (Spd3+), and cobalt hexamine (CoHex3+)), which interact differently with DNA and mica, are considered. Mg2+ ions tend to bind via nonspecific electrostatic interaction with the phosphate groups.25 Spermidine is also thought to interact with the DNA chain mainly via nonspecific electrostatic interactions.13,26,27 On the other hand, cobalt hexamine produces a direct interaction with DNA bases and binds preferentially to specific sequences.28,29 The phase diagrams of DNA molecules adsorbed on mica should indicate whether the cations interact specifically with DNA and (or) mica, since their diagrams should then deviate from the diagram of DNA adsorption via purely electrostatic interactions. From AFM images of adsorbed DNA molecules, phase diagrams of DNA molecules interacting with a mica surface are plotted in terms of concentrations of monovalent and multivalent cations. We then discuss the general features of these phase diagrams, which share some similarities with the diagrams of DNA condensation in the presence of polyamines.23 A simple analytical model is also developed to describe the DNA adsorption mechanism. This model incorporates the influence of the counterion correlations30,31 and the influence of the binding competition between monovalent and multivalent counterions.32,33 In addition, we introduce an energy barrier that needs to be overcome to adsorb DNA on a negatively charged surface. Its value is estimated, and the phenomena contributing to the energy barrier are described. The main features of the DNA adsorption process, observed by AFM, are then discussed, and the critical concentrations of multivalent counterions necessary to activate DNA adsorption are predicted. The last part of the paper takes the form of answers to practical questions that need to be addressed when considering AFM experiments. In particular, we discuss the possibility of adsorbing DNA on mica under physiological conditions (i.e., high ionic strength). Additionally, we wonder if it is possible to observe stable DNA toroids or rods on a mica surface. This section shows how the model, combined with the AFM experiments, provides (25) Duguid, J.; Bloomfield, V. A.; Benevides, J.; Thomas, G. J., Jr. Biophys. J. 1993, 65, 1916-1928. (26) Gier, S.; Johns, W. D. Appl. Clay Sci. 2000, 16, 289-299. (27) Braunlin, W. H.; Bloomfield, V. A. Biochemistry 1988, 27, 1184-1191. (28) Ouameur, A. A.; Tajmir-Riahi, H. A. J. Biol. Chem. 2004, 279, 4204142054. (29) Rouzina, I.; Bloomfield, V. A. Biophys. J. 1998, 74, 3152-3164. (30) Shklovskii, B. I. Phys. ReV. E 1999, 60, 5802-5811. (31) Grosberg, A. Y.; Nguyen, T. T.; Shklovskii, B. I. ReV. Mod. Phys. 2002, 74, 329-317. (32) Rouzina, I.; Bloomfield, V. A. J. Phys. Chem. 1996, 100, 4292-4304. (33) Rouzina, I.; Bloomfield, V. A. J. Phys. Chem. 1996, 100, 4305-4313.
further insight into the adsorption of a highly charged polyelectrolyte on a like-charged surface. 2. Materials and Methods 2.1. Sample Preparation. The chemical compounds (MgCl2, spermidine, and cobalt hexamine) and pUC19 plasmid DNA were purchased from Sigma-Aldrich. The DNA molecules were diluted to a concentration of 2 µg/mL in a buffer solution containing 10 mM Tris, pH 7.5, and different concentrations of NaCl, divalent (MgCl2), or trivalent (spermidine or cobalt hexamine) salts. Solutions were incubated at room temperature for 5 min. A 5 µL droplet of DNA solution was deposited onto the surface of freshly cleaved mica (muscovite) for 1 min. Then, a 100 µL droplet of 0.02% diluted uranyl acetate solution was added to fix the previously adsorbed DNA molecules34 or to induce the monomolecular compaction of loosely adsorbed DNA (see section 2.3). The sample was then rinsed with pure water (Millipore) to obtain a clean surface after drying with filter paper. It is worth noting that the results presented here are not dependent on the uranyl acetate concentration for a large concentration range (0.2% and 0.02% uranyl acetate solutions have been tested). 2.2. Atomic Force Microscopy. Imaging was performed in tapping mode with a multimode AFM (Veeco) operating with a Nanoscope IIIa controller. We used silicon cantilevers (AC160TS, Olympus) with resonant frequencies of about 300 kHz. The scan frequency was 1.5 Hz, and the applied force was minimized as much as possible. 2.3. Image Analysis. The phase diagrams of DNA molecules interacting with a mica surface versus multivalent and monovalent salt concentrations (NaCl) were obtained by analyzing AFM images of DNA molecules on mica. The measurements were concentrated on the transitions between adsorption and nonadsorption. For each point of the diagrams, we analyze the DNA adsorption of at least three different mica surfaces. DNA molecules loosely adsorbed on mica are considered “nonadsorbed” in this study. These molecules appear as bright dots in AFM images because they are transformed into globules by the addition of uranyl acetate solution. Indeed, the addition of diluted uranyl acetate solution triggers DNA aggregation in bulk solution.35 Since loosely adsorbed DNA molecules adopt a nearly threedimensional conformation on the surface, uranyl acetate leads to monomolecular DNA compaction on the surface. On the other hand, DNA molecules well spread on mica are considered “adsorbed”. Indeed, the addition of uranyl acetate solution leads to the strong adsorption of previously adsorbed DNA molecules, as observed by AFM in liquid.36 This distinction between adsorbed DNA and loosely bound DNA allows for a semiquantitative study of the influence of (34) Revet, B.; Fourcade, A. Nucleic Acids Res. 1998, 26, 2092-2097. (35) Zobel, C. R.; Beer, M. J. Biophys. Biochem. Cytol. 1961, 10, 335-346. (36) Pastre´, D.; Pie´trement, O.; Zozime, A.; Le Cam, E. Biopolymers 2005, 77, 53-62.
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Figure 2. AFM images of pUC19 DNA molecules deposited on mica for two different concentrations of spermidine: (a) Tris 10 mM, pH 7.5, NaCl 300 mM, spermidine 300 µM; (b) Tris 10 mM, pH 7.5, NaCl 300 mM, spermidine 800 µM; (c) Tris 10 mM, pH 7.5, NaCl 300 mM, spermidine 1.5 mM. Scan area: 2.5 × 2.5 µm2; z range: 5 nm; scan frequency: 1.5 Hz. When the spermidine concentration exceeds 0.8 mM, DNA molecules are released from the surface. This transition between nonadsorption and adsorption is weakly sensitive to NaCl concentration.
Figure 4. Phase diagrams of DNA molecules interacting with a mica surface versus NaCl and cobalt hexamine concentrations. The symbols are the same as those in Figure 3. Black squares show the ionic conditions where numerous condensed structures were observed. The gray dashed line indicates the transition between coil and globule obtained in solution in a previous study.21
Figure 3. Phase diagrams of DNA molecules interacting with a mica surface. DNA concentration: 2 µg/mL pUC19. (a) Phase diagrams in terms of NaCl and MgCl2 concentrations. Symbols: open circle, DNA nonadsorption from mica due to the binding competition between Na+ cations and the multivalent cations on the mica surface; close circle, DNA adsorbed on mica, which means that at least 50% of the DNA molecules are spread on the mica surface; open triangle, DNA nonadsorption on mica, which means that less than 50% of the DNA molecules are spread on the surface. Hatched surfaces represent the continuous transition between nonadsorption and adsorption. (b) Phase diagrams in terms of NaCl and spermidine concentrations. The symbols are the same as those in panel a. the multivalent and monovalent salt concentration on DNA adsorption. To construct the phase diagrams of DNA adsorption, we measured the percentage of adsorbed molecules on the surface (see Figures 3 and 4). The distinction between adsorption and nonadsorption is that adsorption is considered to have occurred when the percentage of adsorbed DNA is larger than 50%, and nonadsorption is considered in the opposite case. The transitions between nonadsorption and adsorption are not abrupt, but they spread over a concentration interval in which DNA molecules adsorbed on the surface and DNA molecules released from the surface both coexist (see Figures 1 and 2). The hatched
areas in the diagrams of Figures 3 and 4 represent the transition regions between adsorption and nonadsorption. A continuity of the transition has also been observed for DNA condensation by multivalent cations. Indeed, at the single-molecule level, there is a discrete transition between coil and globule, but because measurements generally involved many molecules, the transition spreads over a concentration interval.37 AFM images of DNA molecules deposited on mica are presented in Figure 2 for [NaCl] ) 300 mM and for various spermidine concentrations. In Figure 2a, the contours of the DNA molecules adsorbed on mica are well-defined ([Spd3+] ) 300 µM). This indicates that the molecules are adsorbed on mica. We observe an adsorption/nonadsorption transition when the concentration of multivalent counterions exceeds a critical value C*, ([NaCl] ) 300 mM, [Spd3+] ) 0.8 mM, and [Spd3+] ) 1.5 mM; see Figure 2b,c). This transition is also continuous, but the point is that this transition is independent of the NaCl concentration over a large range of NaCl concentrations (from 10 to about 300 mM). At concentrations of multivalent counterions larger than C*, a surface depletion of DNA is observed, which means that the DNA molecules no longer interact with the mica surface (see Figure 2c). This indicates that the DNA/surface interaction is repulsive, which may be explained by an overcharging effect. In addition, the depletion is observed even though the polyamine concentration is lower than the critical concentration required for DNA condensation in solution. Thus, the depletion could not be attributed to multimolecular DNA aggregation in solution. (37) Yoshikawa, K.; Takahashi, M.; Vasilevskaya, V. V.; Khokhlov, A. R. Phys. ReV. Lett. 1996, 76, 3029-3031.
6654 Langmuir, Vol. 22, No. 15, 2006 For concentrations of multivalent counterions lower than C*, and for low NaCl concentrations ([Spd3+] ) 20 µM, [NaCl] < 200 mM; see Figure 1a.), DNA molecules are well-adsorbed on the surface for AFM observations. However, the DNA binding becomes weaker and weaker as the concentration of monovalent salt increases, which has already been demonstrated in our previous study.15 When the concentration of monovalent salt exceeds a limit value (for [Spd3+] ) 20 µM, [NaCl] > 200 mM), DNA molecules are gradually transformed into globules because of the weak adsorption on the surface (Figure 1b,c). This is the transition between adsorption and nonadsorption due to the binding competition between monovalent and multivalent counterions. At NaCl concentrations significantly larger than the transition limit, we do not observe a significant decrease in the number of DNA molecules on the surface. This indicates that DNA molecules still interact with the surface, even if this interaction is weak.
3. Results 3.1. Phase Diagrams. The phase diagrams of DNA molecules on mica have two common features, whatever the counterion charge or affinity with DNA or mica (see Figures 3 and 4). First, at concentrations of multivalent cations larger than a critical value C*, DNA molecules are released from the surface independently of the concentration of monovalent salt. C*, the nonadsorption threshold, depends on the valence and on the nature of the counterions. This is characteristic of an overcharging effect by multivalent counterions which, as an example, could lead to DNA redissolution by polyamines38,39 (see also section 3.2). Second, for a concentration of multivalent cations lower than C*, an adsorption/nonadsorption transition is also observed, which comes from the binding competition between monovalent and multivalent counterions. Indeed, monovalent counterions cannot usually induce DNA adsorption by counterion correlations. The presence of a nonadsorption limit C* could be evidence that counterion correlations are involved in the adsorption mechanism. The main feature of C* is that it does not depend on the concentration of monovalent salt up to 300 mM. For Mg2+ cations, C* is about 800 mM. For spermidine and cobalt hexamine, C* is about 0.8 mM and 4 mM, respectively. Nonadsorption occurs at a larger concentration of multivalent cations when they interact specifically with DNA and (or) mica. Indeed, there is an energy benefit in DNA adsorption due to the specific interactions that increases the value of C*. It should be noted that C* is closely related to the redissolution limit observed in DNA condensation by multivalent cations,23 which also does not depend on the monovalent salt concentration. One point of comparing DNA condensation and DNA adsorption is that the polyamine concentration at the redissolution limit for DNA condensation (50 mM for spermidine and 220 mM for cobalt hexamine) is significantly larger than C* (0.8 mM for spermidine and 4 mM for cobalt hexamine). DNA in solution is therefore negatively charged when nonadsorption occurs. It is then intriguing that a negatively charged molecule can be repelled from the mica surface because of the attraction mediated by multivalent counterion correlations. A more detailed description of the phenomena that induces the release of DNA from the mica surface is necessary to answer this question (see last section). It could also be noted that DNA condensation by divalent counterions does not occur in physiological conditions, whereas DNA adsorption does. We shall see in the next sections that the attraction induced by counterion correlations is stronger between mica and DNA than it is between two DNA segments. In addition, (38) Raspaud, E.; Olvera de la Cruz, M.; Sikorav, J. L.; Livolant, F. Biophys. J. 1998, 74, 381-393. (39) Nguyen, T. T.; Rouzina, I.; Shklovskii, B. I. J. Chem. Phys. 2000, 112, 2562-2568.
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the entropy penalty induced by DNA condensation could be larger than that for DNA adsorption. Indeed DNA molecules adsorbed on mica are mobile on the surface.36 For multivalent counterion concentrations lower than C*, the adsorption is controlled by the binding competition between monovalent and multivalent counterions. We have shown in a previous study that the attraction force between DNA and mica is constant for a given ratio, [MZ+]/[NaCl]Z, where Z is the cation charge.15 Consequently, in a log-log plot of multivalent cation concentration and NaCl concentration, the transition between adsorption and nonadsorption due to the binding competition should be a straight line, the slope of which is the multivalent counterion charge Z. This is valid for Mg2+ and Spd3+ ions (see Figure 3), for which the slopes are about 1.9 and 2.6, respectively. The small discrepancy between the value of the slope and Z could come from their sizes and geometries. Concerning the phase diagram of CoHex3+, the adsorption/nonadsorption transition can also be observed, but the slope of the transition line in a log-log plot is about 1.60. CoHex3+ ions have a high affinity toward DNA and mica surfaces because of specific interactions. Consequently, the transition between nonadsorption and adsorption in these diagrams is weakly dependent on the monovalent salt concentration. A simple electrostatic model is not appropriate to describe DNA adsorption by CoHex3+. DNA condensation by multivalent counterions has been the subject of many AFM studies in which toroids or rods, the typical forms adopted by condensed DNA in diluted solution, were observed.40-43 The analysis of our AFM images also reveals the presence of toroids or DNA aggregates with cobalt hexamine for low concentrations of NaCl (see black squares in Figure 4). The dashed gray line in Figure 4 shows the critical concentration of cobalt hexamine required to collapse DNA versus the NaCl concentration obtained in solution in a previous study.21 These critical concentrations are in good agreement with these obtained from the AFM analysis. However, only very few of these structures maintain their typical toroidal forms, even for CoHex3+ concentrations significantly larger than the critical values required for DNA condensation (Figure 5). Indeed single condensed DNA molecules or DNA aggregates by spermidine or cobalt hexamine very often spread upon contact with the mica surface by adopting a flowerlike structure (Figure 5). We will address this issue in the last section of the paper. For DNA condensation with spermidine, toroids and rods are not adsorbed on the mica, whatever spermidine concentration is used. However, we observe multiple crossovers between DNA segments, leading to flowerlike structures as observed in a previous AFM study.40 In fact, it is difficult to adsorb toroids or rods on mica with spermidine since C* (about 0.8 mM) is generally lower than the concentration necessary to trigger DNA condensation. 3.2. Model of DNA Adsorption. The attraction between two like-charged surfaces by counterion correlations depends on their surface charge densities and on the multivalent counterion valence Z. Since DNA and mica are both highly charged (see below), the attraction force can be strong. In addition, a monovalent ion cannot induce a significant attraction between DNA and mica, which means that the attraction is closely related to the binding competition between monovalent and multivalent counterions.15,44 (40) Fang, Y.; Hoh, J. H. J. Am. Chem. Soc. 1999, 120, 8903-8909. (41) Golan, R.; Pietrasanta, L. I.; Hsieh, W.; Hansma, H. G. Biochemistry 1999, 38, 14069-14076. (42) Hansma, H. G.; Golan, R.; Hsieh, W.; Lollo, C. P.; Mullen-Ley, P.; Kwoh, D. Nucleic Acids Res. 1998, 26, 2481-2487. (43) Liu, D.; Wang, C.; Li, J.; Lin, Z.; Tan, Z.; Bai, C. J. Biomol. Struct. Dyn. 2000, 18, 1-9. (44) Burak, Y.; Ariel, G.; Andelman, D. Biophys. J. 2003, 85, 2100-2110.
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molecules should be taken into account, but planar approximation can still provide interesting information for a qualitative description of the forces acting between DNA and mica. Thus, we consider DNA as a plane surface, with a surface charge density of σ ≈ 1 e-/nm2.32 The mica surface charge density in solution is σ ≈ 2 e-/nm2.48,49 It results from the release of potassium ions on the mica surface, without taking into account the neutralization by the adsorbed cations. Indeed, under typical solution conditions, most of the negative lattice sites are neutralized with condensed cations (Na+, Mg2+, etc.).48,50 Let us describe the free energy associated with the counterions that are adsorbed on mica and DNA surfaces. Since mica and DNA are both highly negatively charged, multivalent counterions get attracted toward the surface by a strong uniform electric field, Eelec:
Eelec ) Figure 5. AFM images of DNA condensation (Tris 10 mM, pH 7.5, NaCl 10 mM, cobalt hexamine 0.5 mM). We observe large DNA aggregates that spread on the surface by adopting a flowerlike form. Few toroids (see arrow) are adsorbed on the surface, whereas it is the dominant form in solution. The spreading of the DNA molecules and the presence of uncondensed DNA molecules could be induced by the surface interaction with the DNA molecules. Scan area: 4 × 4 µm2; z range: 5 nm; scan frequency: 1.5 Hz.
In this model, we will only consider Spd3+ and Mg2+counterions because they interact nonspecifically with DNA and mica. Additionally, correlations between multivalent counterions can also lead to an overcharging of a charged surface.31,45,46 The multivalent counterion correlations favor an excess of multivalent counterions on the surface and therefore an inversion of the net surface charge. Examples of charge inversion include repulsion between an AFM tip and a charged surface45 or reversal of the sign of the electrophoresis mobility of charged colloids.46 DNA redissolution by polyamines could be also considered an example of DNA overcharging,39 but it was recently reported that DNA redissolution could also be induced by the association of co-ions (Cl-) with high spermidine concentrations (beyond 1 mM).47 However, the transition from DNA adsorption to nonadsorption on mica arises at a SpdCl3 concentration of about 0.8 mM. Therefore, for [SpdCl3] ) 0.8 mM, the concentration of Spd-Cl2+ ions is significantly lower than the concentration of Spd3+ ions, since the estimated dissociation constant for Spd-Cl2+ in NaCl solution is about 150 mM.47 In addition, even if a small amount of divalent counterions was present in solution at spermidine concentrations significantly larger than 1 mM, it would favor DNA adsorption rather than nonadsorption. 3.2.1.General Criterion for Polyelectrolytes Adsorption on a Like-Charged Surface in Absence of MonoValent Counterions. Highly charged polyions such as DNA can be treated as charged plane surfaces, provided that the ionic strength is higher than 100 mM and is lower than 1 M.32 For ionic strengths between 10 mM and 100 mM, the cylindrical geometry of the DNA (45) Besteman, K.; Zevenbergen, M. A.; Heering, H. A.; Lemay, S. G. Phys. ReV. Lett. 2004, 93, 170802. (46) Lozada-Cassou, M.; Gonzalez-Tovar, E.; Olivares, W. Phys. ReV. E. 1999, 60, 17-20. (47) Yang, J.; Rau, D. C. Biophys. J. 2005, 89, 1932-1940.
4πσ*
(1)
where σ* is the net surface charge density (σ* ) σ + Zne), is the water dielectric constant, and n is the surface density of multivalent counterions. Then, the electrostatic energy Welec (in kBT units) associated with the electric attraction of one multivalent counterion (or repulsion when overcharging takes place, since, in this case, σ* > 0) is
lb Welec Zeψ(o) ) ≈ 4πσ*d kBT kBT e
(2)
where lb is the Bjerrum length, and d is the hydrated radius of the cation. It is assumed that the potential ψ is ∼0 at a distance d from the surface, since a large part of the surface charge is neutralized by adsorbed counterions within a distance d. Near the nonadsorption limit C*, the presence of monovalent counterions on the surface can be neglected since charged surfaces are preferentially neutralized by multivalent counterions. Multivalent counterions condensed on the surface form a twodimensional liquid with a lattice constant A. From the model developed by Shklovskii,30 the chemical potential µ of the adsorbed ions of valence Z is equal to
µ ) µ0 + µWC
(3)
where µ0 is the chemical potential of a system of neutral hard disks of diameter d, and µWC is the chemical potential due to the counterion correlations. The adsorbed layer of counterions can be considered a metal surface. Indeed a new multivalent counterion repels the adsorbed ones, which creates an image charge of opposite sign, such as that in a metal surface. The value of µWC is therefore expressed by30
µWC(n) ) - kBT(1.65Γ(n) - 2.61Γ1/4(n) + 0.26 ln(Γ(n)) + 1.95) (4) where (48) Pashley, R. M.; Quirk, J. P. Colloids Surf. 1984, 9, 1-17. (49) Pashley, R. M.; Israelachvili, J. N. J. Colloid Interface Sci. 1984, 97, 446-455. (50) Nishimura, S.; Scales, P. J.; Tateyama, H.; Tsunematsu, K.; Healy, T. W. Langmuir 1995, 11, 291-295.
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Γ(n) ) -
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Z2e2 xπn kBT
The expression µ0(n) is obtained by assuming an ideal gas of counterions:51
µ0(n) ) kBT ln(csν0)
(5)
where V0 is the molecular size of the counterion, and cs is the surface concentration of multivalent counterions:
cs )
n λ
(6)
with λ being the Gouy-Chapman length, which defines the sheath near the surface within which the counterions are confined. In eq 5, the sterical hindrance between counterions is neglected. It plays a significant role only when the distance between two nearest counterions on the surface approaches d, the hydrated diameter of the counterion (see Appendix A for details). For studying the transition between DNA adsorption and nonadsorption, it is necessary to describe the free energy of a DNA chain adsorbed on the surface and the free energy of a DNA chain released from the surface. Let us start with the first configuration, which is a DNA molecule adsorbed on mica as a result of multivalent counterions condensed or localized at the interface. The equivalent charge density of the interface σi is the sum of the DNA surface charge density σs and the mica surface charge density σp. Therefore the net “surface” charge of this system is
σ/i ) σi - niZe ) (σs + σp) - niZe,
(7)
where ni is the number of multivalent counterions per area unit at the polyelectrolyte/surface interface (DNA/mica). This hypothesis seems sound since the system is symmetric with regard to a parallel plane situated at an equal distance from each surface. The free energy associated with N counterions in a system of area S is then
∆Fadsorption )
∫0N [µ0(N) + µWC(N) + Welec(N)]dN
(8)
The second configuration describes the polyelectrolyte released from the charged plane. The surface considered here is the same as that of the first configuration. We find
Figure 6. Plot of d(∆F)/dN in kBT units versus the distance between two nearest counterions. 2A/2 and 2A/3 represent the separation distances at the transition limits between adsorption and nonadsorption for Z ) 2 and Z ) 3, respectively.
the chemical potential associated with the two configurations should be equal at the transition limit:
d(∆Fnonadsorption) d(∆Fadsorption) ) dN dN
(11a)
d(∆F) d(∆Fnonadsorption - ∆Fadsorption) ) )0 dN dN
(11b)
The solution of this equation is obtained when the concentration of multivalent counterions is equal to C*. There is also another transition in the presence of monovalent salt, but the problem is then more complicated (see next section). Near the transition limit C*, both DNA and mica surfaces are nearly fully neutralized or partially overcharged by multivalent counterions. By slightly increasing the concentration of multivalent counterions to a concentration larger than C*, the counterions that were initially localized at the interface between DNA and mica during adsorption (configuration 1) divide among the DNA and mica surfaces after dissociation (configuration 2). The distribution of counterions depends on the surface charge density of each surface. We assume here that the overcharging of the charged surfaces is weak. We shall see that this assumption is valid for the range of concentration studied here (see next section). We find
(9)
σs Ns ≈ N σp + σs
(12a)
where ∆Fs is the free energy associated with Ns counterions on the polyelectrolyte of surface S with a charge density σs, and ∆Fp is the free energy associated with Np counterions on the charged plane of charge density σp:
σp Np ≈ N σp + σ s
(12b)
∆Fnonadsorption ) ∆Fp + ∆Fs
∆Fs ) ∆Fp )
∫0N [µ0(Ns) + µWC(Ns) + Welec(Ns)]dNs
∫0N
s
p
(10a)
[µ0(Np) + µWC(Np) + Welec(Np)]dNp (10b)
To find the transition between nonadsorption and adsorption, we adopt the same procedure that has been used to study the redissolution limit of DNA condensation by polyamines.39 The transition between the two configurations (adsorption S nonadsorption) occurs when the two phases coexist. This means that (51) Nguyen, T. T.; Shklovskii, B. I. Phys. ReV. E 2001, 64, 4140741415.
These equations are used to perform the derivation of eq 11b. Figure 6 represents d(∆F)/dN versus the distance (2A) between multivalent counterions at the DNA/mica interface. For Z ) 3, when the surface density of multivalent counterions increases, the surface potential becomes positive and induces an electrostatic repulsion of the cations. When d(∆F)/dN ) 0, the electrostatic repulsion due to the overcharging of the DNA/mica interface compensates the attractive potential due to the correlations between multivalent counterions. The level of overcharging at the depletion limit can then be calculated since there is a relation between the distance of separation of two nearest counterions at the DNA/mica interface, 2A* (see Figure 6), and their surface
DNA Adsorption on Mica by MultiValent Cations
Langmuir, Vol. 22, No. 15, 2006 6657
density, n/i ) 1/2x3A*2. We then obtain that the level of overcharging at the DNA/mica interface required to initiate the depletion is about 15% for spermidine and 5% for Mg2+. However, for divalent cations, we see that overcharging may not take place because of the sterical hindrance between adsorbed divalent counterions (see discussion below). By solving eq 11b, one can estimate the bulk concentration of multivalent counterions (C*) required for DNA nonadsorption (see Appendix B):
C* ≈
(
)
µWC(n/i ) + Welec(n/i ) σ exp Ze‚λ kBT
(13)
This expression is an extension of that derived by Shklovskii and co-workers31 for the concentration of multivalent counterions at the neutral point (n/i ) {σi/Ze}). For spermidine and magnesium, C* is about 4 mM and 0.15 M, respectively. Let us compare these estimations with the experimental results presented in Figure 3. From the diagrams, C* is about 0.8 mM for spermidine. This value of the depletion limit roughly agrees with the model, even though the theoretical value (4 mM) is larger than the experimental one. This discrepancy could be explained since the adsorption/ nonadsorption transition, as measured by AFM and defined in section 2.3, occurs when DNA molecules are loosely adsorbed on the surface so that DNA molecules are still weakly bound on mica. The surface depletion occurs at a higher concentration of multivalent salt. For Mg2+ ions, DNA nonadsorption takes place at a concentration of about 600-800 mM. The predicted value (150 mM) underestimates the experimental one. Indeed, the spacing between divalent counterions at the depletion limit is about 0.88 nm, which is close to the diameter of the hydrated Mg2+ ion (about 1 nm). The DNA/mica interface is then crowded with divalent counterions since 2A tends to d near the depletion limit (see Appendix A for details). If a multivalent counterion approaches the interface, it cannot create its correlation hole in the strongly correlated cloud of counterions to be adsorbed on the surface. Therefore, overcharging could only take place if a second layer of divalent counterions is formed.51 Let us note that, for spermidine (Z ) 3), the spacing between multivalent counterions is about 1 nm on mica at the redissolution limit. Since its radius of gyration is about 0.35 nm, the sterical hindrance is certainly of less importance. 3.2.2. Adsorption and Nonadsorption in the Presence of MonoValent Salt for MultiValent Cation Concentrations Lower than C*. The analysis of the transition between adsorption and nonadsorption first requires one to take into account the competition between monovalent and multivalent counterions so that their surface densities can be estimated. For this purpose, we use the Poisson-Boltzmann (P-B) equation (see Appendix C), which is valid as long as the contribution of the counterion correlations is negligible compared with the electrostatic potential. P-B equations have already been successfully used to describe the binding competition between monovalent and multivalent counterions for both mica52 and DNA surfaces.32 In addition, we consider here that the correlations between monovalent counterions are ineffective in DNA adsorption.15,18 The two main contributions to DNA adsorption are the free-energy variation ∆FWC due to the counterion correlations, (52) Pashley, R. M. J. Colloid Interface Sci. 1984, 102, 23-35.
Figure 7. Plot of d(∆F)/dN in kBT units versus the concentration of spermidine for two NaCl concentrations (200 mM and 400 mM).
and the energy gain ∆F0 in releasing monovalent counterions:
∆F ) ∆FWC + ∆F0
(14)
However, because the P-B equation is considered to be valid, the entropy penalty of adsorbing one multivalent counterion is nearly equal to the entropy gain in releasing Z monovalent counterions (see Appendix C). ∆F0 is therefore neglected. ∆FWC depends on the fractional occupancies of multivalent counterions on DNA, mica, and the DNA/mica interface. Since the concentration of multivalent counterions is larger at the DNA/ mica interface than on DNA and mica surfaces, the energy benefit due to the localization of one counterion at the interface is
ns np d(∆FWC) ≈ µWC(ni) - µWC(ns) - µWC(np) dN ni ni
(15)
where ni, ns, and np are the surface densities of multivalent counterions for the DNA/mica interface, DNA, and mica surfaces,respectively:
nˆ i(σp + σs) Ze
(16a)
ns )
nˆ sσs Ze
(16b)
np )
nˆ pσp Ze
(16c)
ni )
where nˆ i, nˆ s, and nˆ p are the fractional surface charge densities of the multivalent cations adsorbed on the DNA/mica interface, DNA, and mica, respectively (see Appendix C for details). Figure 7 represents the plot of {d(∆F)}/{dN} versus the multivalent salt concentration for Z ) 3, [NaCl] ) 200 mM, and [NaCl] ) 400 mM. It is worth noting that this function is negative for the full range of concentrations of multivalent counterions ([Spd3+] < C*). This indicates that multivalent counterions prefer to be localized at the DNA/mica interface rather than on mica or DNA surfaces. The gain in correlation energy per counterion due to the binding of one multivalent counterion at the interface increases as the multivalent cation concentration increases. The reason is that monovalent counterions are gradually replaced by multivalent counterions at the DNA/mica interface. Then, at higher spermidine concentrations ([Spd3+] g 20 µM for [NaCl] ) 200 mM), {d(∆F)}/{dN} increases, which indicates that the replace-
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ment of monovalent counterions arises at a larger concentration of multivalent counterions for mica and DNA surfaces relative to that for the DNA/mica interface. By using the experimental data provided by the diagrams, we can calculate the lowest percentage of DNA neutralization by multivalent counterions required to trigger DNA adsorption. At the adsorption/nonadsorption limit (∼20 µM of spermidine for NaCl ) 200 mM; ∼15 mM of Mg2+ for NaCl ) 200 mM), 15% of the DNA backbone is neutralized by spermidine, whereas 50% of the DNA backbone is neutralized by magnesium. A significantly higher fractional occupancy of multivalent counterions is then required to adsorb DNA on mica when using magnesium compared to that required when using spermidine. Let us now estimate the transition limit between adsorption and nonadsorption. We consider that adsorption takes place when an energy barrier ∆Ea is overcome:
1 S
dN e ∫d∆F dN
∆Ea S
(17)
Figure 8. Free-energy benefit in DNA adsorption versus multivalent salt concentrations for two counterions (Mg2+ and Spd3+). Line curve: [NaCl] ) 200 mM; dashed curve: [NaCl] ) 400 mM. The open circles show the adsorption/nonadsorption transition due to the binding competition between monovalent and spermidine counterions obtained from AFM analysis. The gray band represents the estimation of ∆Ea/S, the energy barrier per area unit.
With eq 15, we find
∫0
ni
µWC(ni)dni -
∫0
ns
ns µ (n )dn ni WC s s ∆Ea np np (18) µ (n )dnp e 0 n WC p S i
∫
in which ∆Ea/S is a priori unknown and includes all the energetic terms able to hinder DNA adsorption. It has three main components: 1. ∆Ethermal, the energy barrier due to the interpenetration of the clouds of DNA and mica counterions. If we only consider the thermal pressure due to the monovalent counterions, the entropic penalty (in kBT units) for adsorbing one cation on a charged surface is ln(csνo). Then the entropic cost in adsorbing DNA is the sum of the energies required to insert the DNA counterions (σs/e) and the mica counterions (σp/e) into the DNA/ mica interface. The thermal pressure of the interpenetrating clouds of counterions is then
()() ()()
σs csi σp csi ∆Ethermal ≈ln + ln S‚kBT e css e csp
(19)
where csi,css, and csp are the counterion surface concentrations of the DNA/mica interface, the mica surface, and the DNA surface, respectively (see Appendix C). We obtain ∆Ethermal/S ≈ -3.8 kBT/nm2. This is one of the major contributions to the energy barrier that prevents DNA adsorption. 2. ∆Eel, the electrostatic energy induced by the repulsion between negatively charged DNA and mica surfaces. From different models,32,53 we know that about 75% of the DNA surface charge is neutralized by monovalent counterions, whereas more than 85% of the mica surface charge is neutralized at distances larger than lb.33 The free-energy penalty due to this electrostatic repulsion scales as
( ) (
)
σs σp ∆Eel ≈ - 0.25 4π 0.15 d‚lb S‚kBT e e
(20)
Therefore, ∆Eel/S ≈ -0.66 (kBT/nm2) for d ) 1 nm. We note that ∆Eel/S is significantly weaker than ∆Ethermal/S. (53) Manning, G. S. J. Chem. Phys. 1969, 51, 924-933.
3. ∆Econf, the entropy cost in DNA confinement. To estimate ∆Econf, Odijk introduced the deflection length λC, which is the average distance between two contact points:54
λC ≈ C2/3lp1/3
(21)
where lp is the DNA persistence length (50 nm), and C can be thought of as the thickness of the adsorption layer. The energy due to the entropic repulsion is then obtained by roughly assuming that the polymer loses kBT for each contact point. More accurately, when L g lp and lp > λC, we find54
∆Econf 2 L ≈ - ‚ 2/3 kBT 3C l
1/3
ln(lp/C)
(22)
p
where L is the DNA segment length, which contains N multivalent counterions at the interface between DNA and mica. Obviously, the entropy loss is larger for a thin adsorption layer. For DNA adsorption on mica, the adsorbed layer is expected to be thin because the attraction is short ranged.15 With L ) 100 nm and C ≈ 4 nm, we obtain ∆Econf ∼ -18 kBT for a nearly complete immobilization. Let us note that ∆Econf is certainly overestimated since DNA diffusion on mica is still possible, as observed by AFM in liquid.36 To compare with the other energy terms, we should consider ∆Econf/S, where S is the area of the adsorbed DNA surface. If we assume that, between 3 and 30% of the DNA surface is adsorbed on the mica surface as a result of the likely DNA deformation, ∆Econf/S should be contained between -1 and -0.1 kBT/nm2. Since the transition from adsorption to nonadsorption should occur when the free energy decreases to a critical value ∆Ea, the plot of the free energy gain in DNA adsorption per area unit can provide an estimation of ∆Ea/S. Figure 8 represents 1/S ∫(d(∆F)/dN)dN versus the multivalent counterion concentrations (Mg2+and Spd3+) for two monovalent salt concentrations ([NaCl] ) 200 mM and 400 mM). This function decreases as the multivalent cation concentration increases, which indicates an energy benefit in DNA adsorption. Since ∆Ea/S should be equal to the value of 1/S∫(d(∆F)/dN)dN at the transition limits, experimental data points corresponding to the transitions from adsorption to nonadsorption offer an estimation of ∆Ea/S (see circles, Figure 8). For both Mg2+ and Spd3+ ions, we see that (54) Odijk, T. Macromolecules 1983, 16, 1340-1344.
DNA Adsorption on Mica by MultiValent Cations
∆Ea/S has nearly the same value, whatever the concentration of NaCl. It adds credit to the model since ∆Ea/S should weakly depend on the ion valence and on the monovalent salt concentration. In addition, we note that ∆Ea/S is contained between -4.5 and -3.5 kBT/nm2 (see Figure 8), which is in agreement with the theoretical estimation of the energy barrier (-5.4 kBT/nm2 < (∆Ethermal + ∆Econd + ∆Eel)/S < -4.5 kBT/nm2; see eqs 1922).
4. Discussion One of the major issues for AFM investigations of DNA/ ligand complexes is the adsorption of DNA on mica in physiological conditions, that is, at high ionic strength. For a large range of monovalent salt concentrations, the concentration of multivalent salt required for adsorbing DNA on mica can now be easily determined by using the phase diagrams. However, the difficulty remains in studying DNA by AFM at high monovalent salt concentrations. Indeed, because the critical concentration of multivalent counterions required to adsorb DNA is proportional to {cbZ}/{cb1Z}, a high concentration of divalent salt must then be used for high monovalent salt concentrations ([NaCl] > 0.1 M). This is not adequate for studies of biological importance. This is why a low ionic strength is generally preferred when Mg2+ ions are employed for AFM studies. The experiments presented in this work show that spermidine seems to be a better choice, since a low spermidine concentration is sufficient to induce DNA adsorption. Indeed, 20 µM of spermidine is enough to adsorb DNA in the presence of 200 mM NaCl, which is far below the concentration of spermidine required for DNA condensation. Concerning the adsorption of DNA molecules by using CoHex3+, because it interacts specifically with DNA and/or mica, its use is not advised to study DNA/ ligand complexes. Another point of this work is that the experimental and theoretical results give further insight into the mechanism of DNA adsorption on mica. The presence of a nonadsorption transition at high multivalent counterion concentrations bears evidence that DNA adsorption is mainly induced by the correlations between multivalent counterions at the DNA/mica interface. In that respect, the adsorption process of an anionic polyelectrolyte chain such as DNA onto a highly negatively charged surface obeys the same rules as that apply to DNA condensation. In addition, we show that there is an energy barrier that hinders DNA adsorption. This barrier has three main components: the entropy cost in DNA confinement on a thin adsorbed layer, the thermal repulsion due to the interpenetration of the counterion clouds, and the electrostatic repulsion between DNA and mica, which are only partly neutralized. Figure 8 provides some interesting information about the choice of the counterion valence for polymer adsorption. For divalent counterions, it is worth noting that the correlation energy reaches an energy plateau at about -4.5 kBT/nm2 (see Figure 8). This indicates that the adsorption energy could not be increased, even though the divalent counterion concentration is increased. If the energy barrier preventing DNA adsorption is lower than -4.5 kBT/nm2, DNA can not be adsorbed on mica by using divalent counterions. However, the adsorption force can be enhanced by using trivalent counterions, since the energy plateau is lower for spermidine. Moreover, the fractional occupancy of Spd3+ ions on DNA surfaces at the transition limit is significantly lower than the fractional occupancy of Mg2+ ions, which is a good point for AFM investigations of DNA/ligand complexes. Indeed,
Langmuir, Vol. 22, No. 15, 2006 6659
high surface concentrations of Mg2+ can limit DNA accessibility to the ligand.55 Let us also address the issue related to the observation by AFM of stable DNA toroids or rods induced by multivalent counterions on mica. The surface does not alter the threshold of DNA condensation observed in solution. However, it strongly influences the condensed forms of DNA. Toroids and rods are typical forms adopted by DNA molecules in their condensed states, as visualized by AFM40,42 and electron microscopy.56,57 These DNA forms are stable in solution as a result of the multivalent ion correlations, which maintain the attraction between DNA segments. Nevertheless, DNA molecules adsorbed on mica in their condensed state generally appear as a flowerlike structure or as a large aggregate that spreads onto the surface, as shown in Figure 5. Indeed, very few toroids or rods are observed. So, we think that adsorbed toroids or rods are not stable on the surface and are transformed into flowerlike structures. The model developed in this study also suggests that multivalent counterions prefer to be localized at the DNA/mica interface rather than at the DNA/DNA interface. The reason is that the DNA surface is less charged than that of mica so that the multivalent counterions, which participate in the DNA adsorption, cannot simultaneously participate in DNA/DNA self-attraction. It results in the spreading of large aggregates onto the surface, which then adopt a flowerlike form. Herein is an example of the use of the model for explaining the influence of the surface.
Appendix A: Potential of a System of Neutral Hard Disks Adsorbed on a Surface The expression µ0(n) is obtained by assuming an ideal gas of counterions, as described by Nguyen et al.51 The surface concentration Cs of the multivalent counterions is
Cs )
n λ
(A1)
with λ being the Gouy-Chapman length,
λ)
1 2πZσlb
(A2)
where lb is the Bjerrum length (about 0.7 nm in water). The chemical potential µ0(n) of an ideal gas of concentration Cs is equal to51
µ0(n) ≈ [ln(Csν0) - 1] + kBT
A A-
d 2
(A3)
where d is the diameter of the hydrated counterions by assuming a hard-sphere model, and ν0 is the molecular size of the counterion. Its exact value is not important because it vanishes from the final results. When d , 2A, the effect of the excluded volume is weak. We obtain
µ0(n) ) kBT[ln(Csν0)]
(A4)
For the results presented in this paper, eq A4 was used. The influence of the sterical hindrance is discussed when it (55) Pie´trement, O.; Pastre´, D.; Landousy, F.; David, M. O.; Fusil, S.; Hamon, L.; Zozime, A.; Le Cam, E. Eur. Biophys. J. 2005, 34, 200-207. (56) Midoux, P.; Le Cam, E.; Coulaud, D.; Delain, E.; Pichon, C. Somatic Cell Mol. Genet. 2002, 27, 27-47. (57) Hud, N. V.; Downing, K. H. Proc. Natl. Acad. Sci. U.S.A. 2001, 98, 14925-14930.
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plays a significant role, that is, near the close packing limit. Indeed, when 2A approaches d due to a high concentration of multivalent counterions on the surface, the second term of eq A3 is dominant and tends toward infinity. It indicates that no more additional multivalent counterions can be adsorbed on the surface.
Appendix C: Binding Competition between Multivalent and Monovalent Cations When P-B law is still valid, the surface concentrations of multivalent counterions csZ and monovalent counterions cs1 are
( ) ( )
Appendix B: Thermodynamic Equilibrium Equilibrium models of chemical potential require that the chemical potential of adsorbed cations be equal to the counterparts in solution. For the first configuration, multivalent counterions are adsorbed at the DNA/mica interface. We find
µ0(N) + µWC(N) + Welec(N) ) µbulk
(B1)
µbulk ) kBT ln(cbulkν0)
(B2)
with
where cbulk is the concentration of multivalent counterions in solution. For the second configuration, multivalent counterions are adsorbed on the mica and DNA surfaces:
(
)
µWC(N) + Welec(N) σ Cbulk ≈ exp Ze‚λ kBT
(B5)
(C1)
csZ ) cbZ exp
ZeΨ(0) kBT
(C2) (C3)
where cb1 and cbZ are the bulk concentrations of monovalent and multivalent counterions respectively, and Ψ(0) is the surface potential of the bare surface (mica, DNA, or the DNA/mica interface). The fractional surface charge density of the multivalent cations nˆ sZ versus the bulk concentrations of monovalentand multivalent cations is then obtained by using the P-B equation: 32
nˆ sZ (1 - nˆ sZ)Z
)Y
(C4)
with
Y)
cbZ‚csZ-1 cb1Z
(C5)
Let us recall that
[µWC(N) + kBT ln(cSν0) + Welec(N)] ) kBT ln(cbulkν0) (B4) Then,
eΨ(0) kBT
cs ) cs1 + csZ
[µ0(NP) + µWC(NP) + Welec(NP)]mica ) [µ0(NS) + µWC(NS) + Welec(NS)]DNA ) µbulk (B3) At the transition limit between adsorption and nonadsorption, both eqs B1 and B3 should be equated. The solution is found by considering that the counterions, which were initially localized at the DNA/interface, are divided among the DNA and mica surfaces after dissociation, thus a relation between N, NP, and NS is required to solve this problem. To obtain the concentration of multivalent counterions at the adsorption/nonadsorption transition, we use eq B1. We find
cs1 ) cb1 exp
cs )
σ Zeλ
Therefore, csi ∼ 60 M for the DNA/mica interface, css ∼ 6.6 M for DNA, and csp ∼ 26 M for mica. LA053387Y
