Molecular Dynamics Studies of Ion Distributions for DNA Duplexes

The coordinates x and y are in units of angstroms, and the colored contours specify concentration in mol/L. .... condensation theory (76%).24 The latt...
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J. Phys. Chem. B 2006, 110, 2918-2926

Molecular Dynamics Studies of Ion Distributions for DNA Duplexes and DNA Clusters: Salt Effects and Connection to DNA Melting Hai Long, Alexander Kudlay, and George C. Schatz* Department of Chemistry, Northwestern UniVersity, EVanston, Illinois 60208-3113 ReceiVed: October 5, 2005; In Final Form: December 2, 2005

We present extensive molecular dynamics simulations of the ion distributions for DNA duplexes and DNA clusters using the Amber force field with implicit water. The distribution of ions and the electrostatic energy of ions around an isolated DNA duplex and clusters of DNA duplexes in different salt (NaCl) concentrations over the range 0.2-1.0 mol/L are determined on the basis of the simulation results. Using the electrostatic energy profile, we determine a local net charge fraction φ, which is found to increase with increasing of salt concentration. For DNA clusters containing two DNA duplexes (DNA pair) or four DNA duplexes, φ increases as the distance between the duplexes decreases. Combining this result with experimental results for the dependence of the DNA melting temperature on bulk salt concentration, we conclude that for a pair of DNA duplexes the melting temperature increases by 5-10 K for interaxis separations of 25-40 Å. For a cluster of four DNA duplexes, an even larger melting temperature increase should occur. We argue that this melting temperature increase in dense DNA clusters is responsible for the cooperative melting mechanism in DNAlinked nanoparticle aggregates and DNA-linked polymer aggregates.

I. Introduction Recent experiments have shown that gold nanoparticles can be used for DNA sensing applications wherein DNA hybridization produces aggregates containing hundreds to thousands of nanoparticles that are linked by the DNAs.1,2 The formation of aggregates can easily be detected by the color change that occurs upon aggregation due to sensitivity of the plasmon wavelength to interparticle distance.3 When these aggregates are heated, they undergo reversible melting within a relatively narrow (3 K) temperature range, which is to be compared with a width of 20 K for melting of the same DNA in bulk solution. The melting temperature of the aggregate is also higher by a few degrees.4 Single base pair mismatches, insertions, or deletions in the DNA result in a variation in the melting temperature of a few degrees, which because of the sharp melting is easily detected by colorimetric methods. This result is important for DNA detection,5 providing an advantage over DNA detection techniques based on fluorescence where the melting is much broader.6 One proposed mechanism for this sharper melting transition has been reported by Jin et al.4 They formulated a simple thermodynamic model for dehybridization as illustrated in Figure 1. In this model, the sharp melting arises from two effects: (a) multiple links between the particles and (b) cooperative interactions between closely packed DNA duplexes, i.e., DNA clusters. The latter effect arises because the close proximity of the DNA duplexes within a cluster leads to overlapping of the ion clouds around each DNA, and this stabilizes each DNA duplex, leading to an increase in the melting temperature for each DNA duplex by an amount that depends on DNA density. As a result, when melting begins, each single-strand DNA released leads to a reduction in local ion concentration due to counterion release, and this makes it easier to melt the remaining linkers in the aggregate. Since the lower ion concentration destabilizes the * Author to whom correspondence should be addressed. E-mail: [email protected]

Figure 1. Schematic of the DNA cooperative melting model.

DNAs,7 the melting temperatures for subsequent steps in the melting process decrease. This results in sharper melting curves for a pair of nearby DNA duplexes than those for isolated DNA duplex and even sharper melting for larger clusters. There is also a modest melting temperature increase that is also consistent with the experimental observations. By fitting the experimental melting data, it was found that only ∼2 linkers per nanoparticle pair are enough to achieve the observed reduction in the melting width. While the Jin et al. model is not the only mechanism for the sharp melting curves that has been proposed,8-10 it is the only one that explicitly factors in the influence of cooperative interactions between DNAs on the melting behavior. A key issue in the Jin et al. mechanism is whether the changes in ion clouds around the DNA clusters that occur when the nanoparticle aggregates form lead to a sufficiently large increase in melting temperature to be consistent with the cooperative melting mechanism. Although there have been plenty of studies of ion clouds around DNA using molecular dynamics (MD),11-17 Monte Carlo (MC) simulation methods,12,18-21 Poisson-Boltz-

10.1021/jp0556815 CCC: $33.50 © 2006 American Chemical Society Published on Web 01/25/2006

Ion Distributions for DNA Duplexes and Clusters TABLE 1: Simulation Sets simulation set 1 2 3 4 5 6 7

counterions 20Na+ 20Na+ 20Na+ 20Na+ 20Na+ 20Na+ 20Na+

extra salt

Isolated DNA Duplexes 20Na+ 20Cl30Na+ 30Cl40Na+ 40Cl50Na+ 50Cl60Na+ 60Cl70Na+ 70Cl80Na+ 80Cl-

extra salt concentration (mol/L) 0.244 0.366 0.487 0.609 0.731 0.853 0.975

DNA Pairs: Separation Distance D ) 25, 30, 35, 40, 45, and 50 Å 8-13 40Na+ 40Na+ 40Cl0.244 60Na+ 60Cl0.366 14-19 40Na+ 20-25 40Na+ 80Na+ 80Cl0.487 Four DNA Clusters: Separation Distance D ) 25, 30, 35, 40, 45, and 50 Å 26-31 80Na+ 80Na+ 80Cl0.244

mann (PB) solutions,20,22,23 and Manning’s counterion condensation theory,24-30 most of these studies were focused on isolated DNA duplexes. The changes in the counterion clouds when DNA clusters are formed are still poorly understood. In this paper, we first use MD simulations to study the ion clouds around isolated DNA duplexes and around DNA clusters containing two DNA duplexes (DNA pair) and four DNA duplexes for a range of salt concentrations. From the MD simulation results, the ion distribution and the electrostatic energy of ions around DNA in the presence of added salt have been computed. We then connect the DNA melting temperature to the ion distribution based on the properties of a local net charge fraction (φ) that is calculated from the ion distribution. Here φ is defined as the ratio of local net ion charge around a DNA duplex (or cluster) to the total charge on the DNA duplex (or cluster). This local net ion charge is determined from an electrostatic energy criterion. The major advantage of the electrostatic energy criterion over other criteria such as the condensation radius is that it does not require cylindrical symmetry, so it can be applied to DNA clusters of arbitrary structure. Combining the experimentally known dependence of DNA melting temperature on salt concentration with the dependence of φ on salt concentration found from the simulation, we obtain a melting temperature increase (∆T) for the DNA clusters as a function of salt concentration and DNA separation distance. The results corroborate the cooperative melting model of DNA gold nanoparticle aggregates. II. Details of Simulations The simulations were performed using the Amber molecular dynamics program package31 with the Amber force field and Amber partial charges on the DNA residue atoms.32 To focus on the ion distribution, a 10 base pair poly(A-T) DNA duplex (i.e., one turn of the helix) in the classical B-type conformation33 was used with periodic boundary conditions (PBC) applied along the helical axis to mimic an infinite length duplex. The total charge on the 10 base pair DNA is -20e. We have run seven simulations for this DNA duplex with NaCl concentrations ranging from 0.2 to 1 mol/L (Table 1). The simulations used an NVT ensemble and PBC in a 65.0 Å × 65.0 Å × 33.8 Å simulation box. The initial positions of Na+ and Cl- ions were randomly located around the DNA duplex. (In this paper, we consider only NaCl salt.) The box dimension in directions perpendicular to the helix (the x- and y-directions) is 65 Å, which is large enough for the electrostatic interaction between the DNA and its periodic replicas to be screened out even for the lowest salt concentration, [NaCl] ) 0.244 mol/L (see discussion below),

J. Phys. Chem. B, Vol. 110, No. 6, 2006 2919 we considered. Water was treated implicitly by setting the dielectric constant  ) 78.5. This approximation is essential for computational feasibility, as it allows the ions to diffuse quickly to give equilibrium ion distributions. The difference between implicit and explicit water has been studied in the past12 and from the perspective of the present study is of minor significance. The particle mesh Ewald method was employed to calculate the electrostatic interactions. During the simulation, all DNA atoms were fixed, and only the ions were allowed to move. This enables us to use a simulation time step of 4 fs. For each simulation run, a 1000 step energy minimization using the steepest descent method was first applied to remove any unfavored close contacts between atoms. After the minimization, a 2 ns equilibrating MD simulation was started with an initial temperature of 100 K. The temperature was gradually raised to 300 K within 20 ps, and then the temperature was fixed at 300 K using the Berensten temperature coupling algorithm.34 After the 2 ns equilibrating MD simulation, an 8 ns production simulation was run with the temperature coupled to 300 K. During the production runs, the positions of all ions were saved every 10 steps for the ion distribution calculation. The simulations of DNA clusters considered two or four DNA duplexes, all chosen to have parallel helical axes and with different axis-axis separation distances (D ) 25, 30, 35, 40, 45, and 50 Å). In both the two and four DNA cluster simulations, the same simulation methods were used as those used for the isolated DNA duplex simulations except that we used a larger simulation box. For the DNA pair case, the DNA central axes were located at ((D/2, 0), perpendicular to the xyplane in a simulation box of 130.0 Å × 65.0 Å × 33.8 Å. For the four DNA cluster case, the DNA axes were located at ((D/ 2, (D/2) with a simulation box of 130.0 Å × 130.0 Å × 33.8 Å. For DNA pairs, three sets of simulations with salt concentrations of 0.244, 0.366, and 0.487 mol/L were performed and with D ranging from 25 to 50 Å in 5 Å increments For the four DNA clusters, only one series of simulations at [NaCl] ) 0.244 mol/L was done. The simulation sets are summarized in Table 1. III. Results A. Ion Distribution around an Isolated DNA Duplex. Figure 2 presents a two-dimensional plot of (a) the net charge concentration (CNet ) [Na+] - [Cl-]) distributions averaged along the DNA axis, (b) the averaged net charge concentration distributions around one DNA base pair, and (c) the averaged net charge concentration distribution along the cross section of DNA for [NaCl] ) 0.487 mol/L. The two-dimensional averaged net charge distribution has a near-cylindrical symmetry (Figure 2a); therefore, it can be well characterized using a radial net charge distribution. Figure 3a presents the radial distribution for the Na+ and Cl- ions and the net charge distribution (averaged along the DNA axis) for [NaCl] ) 0.487 M, where the radius is measured from the DNA axis. The peak at 5 Å corresponds to Na+ in the DNA major and minor grooves as shown in Figures 2b and 2c. The shoulder peak at 11-12 Å is due to Na+ territorially bound to the DNA phosphate groups. At around 15 Å, the Na+ curve begins dropping sharply due to the electrostatic screening of the DNA charges (Figure 3). Although the most charged parts of DNA are located on the phosphate group, locations deep in the DNA grooves that are between the two phosphates and close to some electronegative atoms on the DNA bases have the lowest electrostatic potentials and give the highest local Na+ concentrations. These results match with the findings of other groups.12,13,16 There are very

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Figure 3. (a) Radial ion distribution plots for Na+ (black) and Cl(red) and net charge concentration (green) for 0.487 mol/L extra salt in a 65.0 Å × 65.0 Å × 33.8 Å simulation box. (b) Radial net charge concentration plot for 0.487 mol/L extra salt in a 65.0 Å × 65.0 Å × 33.8 Å simulation box (black), in a 80.0 Å × 80.0 Å × 33.8 Å simulation box (red), and in a 65.0 Å × 65.0 Å × 67.6 Å simulation box (green). The concentrations within the DNA grooves are calculated using the ion-accessible volume.

Figure 2. (a) Net charge concentration distribution averaged along the DNA axis, (b) averaged net charge concentration distribution around one base pair, and (c) averaged net charge concentration distribution along the DNA cross section for [NaCl] ) 0.487 mol/L. The concentrations within the DNA grooves are calculated using the ionaccessible volume. The coordinates x and y are in units of angstroms, and the colored contours specify concentration in mol/L.

few Cl- ions close to DNA, and the Cl- concentration increases with increasing radius. The falloff portion of the net charge

distribution curve can be fitted by an exponentially decaying function, giving a Debye screening length κ-1 ) 3.25 Å. If Debye-Hu¨ckel theory were valid for these salt concentrations, then this would correspond to an averaged local salt concentration of 0.89 mol/L in the vicinity of the DNA. Similarly, for the lowest salt concentration (0.244 mol/L), the Debye length κ-1 ) 4.48 Å corresponds to a 0.46 mol/L local salt concentration. However, Debye-Hu¨ckel theory is not appropriate here, so we will not use these results in what follows. As can be seen from the plot in Figure 3a, the net charge distribution drops to zero (except for thermal noise) at around 30 Å from the DNA axis. This is still within the simulation box, showing that our box is large enough to screen out the interaction between the DNA and its PBC replicas. To check the influence of the box size, we have performed two other independent simulations with larger boxes. One simulation is performed in a wider 80.0 Å × 80.0 Å × 33.8 Å box in 0.487 mol/L salt solution (Figure 3b, red curve). The other simulation checks the periodicity in the z-direction; it is performed in a 65.0 Å × 65.0 Å × 67.6 Å box with 20 base pairs of DNA duplex (two helical turns) in 0.487 mol/L salt solution (Figure 3b, green curve). The radial net charge distributions from these

Ion Distributions for DNA Duplexes and Clusters box sizes are almost identical with the results of a 65.0 Å × 65.0 Å × 33.8 Å box (Figure 3b, black curve), which shows that the effect of box size is negligible. B. Electrostatic Energy of Ions around an Isolated DNA Duplex. The electrostatic energy of ions around the DNA duplex was calculated as follows: First, the whole simulation box was divided into 1.0 Å × 1.0 Å × 1.0 Å grid cells. Next, in each snapshot of the MD calculation, when an ion was located inside a certain grid cell, the electrostatic energy at this grid cell was set equal to the electrostatic interaction energy between this ion and all the other ions and the DNA duplex as well as periodic replicas of this snapshot. Ewald summation was used to compute the electrostatic interaction between this ion and the ions and DNA in the periodic replicas. Finally, by averaging over the whole 8 ns MD simulation, the averaged electrostatic energy at each grid point was obtained. Note that the resulting electrostatic energy corresponds to the electrostatic energy of a unit charge located at the grid point rather than to the potential energy obtained by averaging all configurations, i.e., including those where the ion is not in the cell. The averaged net charge concentration for each grid cell is obtained in the same calculation. Figure 4 presents a two-dimensional plot of (a) the electrostatic energy of ions averaged along the DNA axis, (b) the averaged electrostatic energy of ions around one base pair, and (c) the averaged electrostatic energy of ions along the cross section of DNA for [NaCl] ) 0.487 mol/L. The lowest electrostatic energy regions around DNA are in the major and minor groove as well as in the vicinity of the phosphate group. These lower electrostatic energies of course correspond to high counterion concentrations in these regions (Figure 2). The radial electrostatic energy curves for three different salt concentrations (0.244, 0.487, and 0.975 mol/L) are shown in Figure 5. This shows that the electrostatic energy of ions in the vicinity of DNA (closer than roughly 15 Å from the DNA axis) is much lower than the energy in bulk solution (asymptote). The bulk energy can be calculated by fitting the exponential falloff part of the curve and extracting the asymptote. The finite negative value of the asymptote (instead of zero) results from the specific way that we have calculated the electrostatic energy: To obtain the electrostatic energy at a certain point, we averaged only over the configurations in which an ion is found inside this grid cell. Thus our electrostatic energy is in fact the potential energy of the ion at this point. This electrostatic energy contains a local correlation part, which is due to correlations between the ion and the ion cloud around it. The electrostatic energy of ions is different from the “unperturbed” electrostatic potential such as the one in the Poisson-Boltzmann equation, which would be obtained if we average over all configurations, i.e., assuming that the “test charge” does not influence the system. This “unperturbed” potential should fall off to zero, which corresponds to electro-neutrality in the system. However, calculating the “unperturbed” potential is numerically inconvenient, due to the singularity at points where the ions are located. The minima at around 5 and 3 Å in Figure 5 are from the DNA major and minor grooves, respectively, corresponding to the high peak at 3-5 Å in Figure 3a. The kink at around 1112 Å is from the phosphate groups where most of the DNA charges are located and corresponds to the shoulder peak at 1112 Å in Figure 3. At radii larger than 12 Å, the absolute value of the electrostatic energy decreases exponentially and approaches a finite negative constant, implying that the charges on the DNA have been screened.

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Figure 4. Two-dimensional plot of (a) the electrostatic energy of ions averaged along the DNA axis, (b) the averaged electrostatic energy of ions around one base pair, and (c) the averaged electrostatic energy of ions along the DNA cross section for [NaCl] ) 0.487 mol/L. The coordinates x and y are in units of angstroms, and the colored contours specify the electrostatic energy as multiples of kT (T ) 300 K).

Figure 5 shows that as the salt concentration increase, the electrostatic energy of ions in the bulk solution decreases. This

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Figure 5. Radial plots of electrostatic energy of ions for 0.244 mol/L salt (solid), 0.487 mol/L salt (dotted), and 0.975 mol/L salt (dash dot).

lower electrostatic energy for bulk solution in a high salt concentration solution arises from stronger correlations between the ions. However, in the vicinity of the DNA duplex, the electrostatic energy is less negative for a high salt concentration. This phenomenon is due to better screening of the DNA negative phosphate charges. Thus, the space around the DNA can be divided into two regions: one corresponding to the bulk, where bulk ion-ion interactions are dominant, and the other a “local” region, where screened DNA phosphate charges are dominant. A simple way to define the outer boundary of the “local” region is to require that the electrostatic energy be lower in the local region than the average of the asymptotic and short range (say, at the surface of DNA) electrostatic energies. Thus we require that the electrostatic energy U in the local region satisfies

Ue

U b + Up 2

(1)

where Ub is the bulk electrostatic energy of the ions and Up is the electrostatic energy of the ions around the phosphate group (the kink at 11-12 Å from the DNA axis in Figure 5). Other definitions are possible, but this one has the advantage that it can be applied to arbitrary DNA clusters that have no radial symmetry. Since the ion-ion correlations become stronger at higher salt concentrations where Ub is already close to -kT (Figure 5), we also cannot adopt the previously suggested definition of U < -kT.20 By applyication of this criterion, the total net charge in the local region, Nlocal, can be calculated from the ion concentration distribution. The local net charge fraction, φ, is defined by

Nlocal | Ntotal

φ)|

(2)

where Ntotal is the total charge on the DNA. For a salt concentration of 0.244 mol/L, the local net charge fraction is found to be φ ) 68.8% ( 0.2%. In addition, we find that φ increases with increasing salt concentration, showing a nearly linear relationship with the logarithm of the salt concentration (Figure 6). The solid curve in the figure shows a fit based on eq 3 for which we find a correlation coefficient R ) 0.9990.

φ(%) ) 9.6 log[NaCl] + 74.65

(3)

The slope in this curve is 9.6 ( 0.19, and the intercept is 74.65 ( 0.08. When the salt concentration is 1 mol/L, φ is 74.65% ( 0.4%, which is close to the condensed ion fraction from Manning’s counterion condensation theory (76%).24 The

Figure 6. Local net charge fraction φ as a function of log[NaCl]. The black straight line is the least-squares fit.

latter result is inferred at a very low salt concentration, and it often is assumed to be weakly dependent on concentration, although clearly the concentrations considered here are higher than are appropriate for Manning theory to be strictly valid. Of course φ is not intended to match the condensed ion fraction, but we see that the two quantities have similar behaviors. C. Ion Distributions and Electrostatic Energy of Ions around DNA Clusters. The ion distributions around DNA duplex pairs are calculated using the same methods as those used for the isolated DNA duplex calculation. The axes of the DNA duplexes are taken to be separated by 25, 30, 35, 40, 45, and 50 Å and with extra salt concentrations of 0.244, 0.366, and 0.487 mol/L, respectively. Figure 7a shows the net charge distribution around a DNA pair with D ) 30 Å in 0.244 mol/L salt. As we can see, the local net charge concentration in the region between the DNAs is higher than that in bulk solution, which leads us to expect that φ will also be increased for DNA pairs at small separations. The total electrostatic potential energies calculated from the MD simulations at different separation distances and [NaCl] ) 0.244 mol/L are given in Table 2. The fluctuations did not decrease significantly when simulations were extended. When the DNA duplexes are close to their smallest possible separation, the potential energy drops a little bit (∼2 kcal/mol) compared to larger separations but with fluctuations on the order of 4 kcal/ mol. This tentatively implies a very weak attraction (a few kcal/ mol) between the 10 base pair DNA duplexes in the DNA pair. This conclusion is in agreement with other theoretical predictions.25,30 Previous simulations of isolated DNA duplexes suggest attractions between DNAs for monovalent salt concentrations on the order of 1 M.35 From the MD simulation results and through the use of the same method used in the isolated DNA duplex calculations, the electrostatic energy of the ions around the DNA pairs can be obtained. For D ) 30 Å, the electrostatic energy of the ions in the region between the DNA duplexes is slightly lower due to ion cloud overlap (Figure 7b). From the electrostatic energy results, a local region around the DNA pair can be defined using the criterion given by eq 1, where Up is the averaged electrostatic energy around phosphate groups in the DNA pairs. Then, similar to the isolated duplex calculations, the local net charge fraction φ of the DNA pairs can be computed. Figure 8 shows a plot of φ for the DNA pair as a function of duplex separation distance at [NaCl] ) 0.244 mol/L. Due to overlapping ion clouds in the DNA pairs, a local region appears between the DNAs as well as in the region around each DNA, and φ of the DNA pair is

Ion Distributions for DNA Duplexes and Clusters

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Figure 7. (a) Net charge distribution and (b) electrostatic energy of ions (local region only) around a DNA pair for an axis separation D ) 30 Å and [NaCl] ) 0.244 mol/L. The coordinates x and y are in units of angstroms, and the colored contours specify mol/L for part a and kT (T ) 300 K) for part b.

higher than φ of an isolated DNA duplex for the same salt concentration. At D ) 25 Å, φ in the DNA pair for [NaCl] ) 0.244 mol/L accounts for 76.6% ( 0.2% of the total charges on DNA. When the DNA pair separation distance increases, φ decreases due to less overlap of the ion clouds; however, even for D ) 50 Å with [NaCl] ) 0.244 mol/L, the φ value is 69.5% ( 0.3%, still slightly higher than φ of an isolated DNA duplex at the same salt concentration (68.8% ( 0.2%). According to eq 3, for an isolated DNA duplex φ is linearly dependent on log[NaCl]; therefore, a higher φ in the DNA pair is equivalent to a DNA duplex in a much higher effective or “virtual” salt concentration than the bulk salt concentration. We introduce a “virtual equivalent salt concentration”, [NaCl]virtual,

for DNA clusters as the concentration for which an isolated DNA duplex has the same φ as the φ of the DNA in the cluster, that is

φduplex([NaCl]) ) φcluster([NaCl]virtual)

(4)

Clearly, [NaCl]virtual depends on separation distance D. Using the results in Figure 8 and eq 2, we obtain the dependence of [NaCl]virtual on D that is depicted in Figure 9. From this result, we can further obtain the melting temperature increase (∆T) due to the formation of a DNA cluster. From experiment, it is known that the melting temperature of an isolated DNA duplex

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TABLE 2: Total Electrostatic Potential Energies for DNA Pairs at Different Separation Distances on the Basis of the MD Simulationsa separation distance D (Å)

total potential energy (kcal/mol)

25 30 35 40 45 50 ∞

-204.8 ( 4.2 -203.8 ( 4.2 -202.6 ( 4.1 -202.5 ( 4.3 -202.7 ( 4.4 -202.4 ( 4.0 -202.8 ( 3.9

a The salt concentration is 0.244 mol/L. The value at infinite separation is deduced by doubling the value of the isolated DNA duplex simulation at same salt concentration.

Figure 10. Melting temperature increase (∆T) for DNA clusters for different salt concentrations and separation distances. Squares, DNA pairs in 0.244 mol/L salt; circles, DNA pairs in 0.366 mol/L salt; uptriangles, DNA pairs in 0.487 mol/L salt; down-triangles, four DNA clusters in 0.244 mol/L salt. Error bars are not shown for clarity. The standard deviations are on the order of 0.5-1 K.

Figure 8. φ for a DNA pair as a function of the DNA axis separation distance D for [NaCl] ) 0.244 mol/L.

be attributed to the changes in ∆S;36 this scheme corresponds to assuming that the positive entropic correlation due to the proximity of neighboring duplexes results from a local salt environment that can be characterized by φ. The ∆T of DNA pairs in different salt concentrations and separation distances are shown in Figure 10. With D ) 25 Å and 0.244 mol/L bulk salt concentration, ∆T can be as large as 12 K. Note that ∆T is not very sensitive to salt concentration. When the bulk salt concentration increases from 0.244 to 0.467 mol/L, ∆T only decreases by ∼3 K. We can generalize these results to the case of clusters containing four DNA duplexes in a square arrangement. Figure 11 presents the ion distribution and electrostatic energy of ions for the four DNA cluster with D ) 35 Å at [NaCl] ) 0.244mol/ L. Because of greater overlap between the ion clouds, the four DNA clusters have a much higher φ than that either an isolated DNA duplex or a DNA pair under the same conditions. Consequently, the four-DNA clusters have a higher ∆T than that of the DNA pairs (the down-triangle curve in Figure 10). IV. Discussion

Figure 9. Virtual equivalent salt concentration as a function of the DNA axis separation distance D for [NaCl] ) 0.244 mol/L.

increases with the salt concentration as follows4

∆T (K) ) 15.8 log

C1 C2

(5)

Given C1 ) 0.244 mol/L, which is the real bulk salt concentration for the DNA pair, and C2 is [NaCl]virtual for the DNA pair at the same bulk concentration, the melting temperature increase (∆T) for the DNA pair compared to the melting temperature of the isolated DNA duplex can be calculated from eq 5. Since the ∆H of DNA melting is largely independent of salt concentration, the influence of salt on DNA melting can

In the DNA gold nanoparticle aggregates, the DNA chains are terminally attached to the nanoparticle surface by thiol bonds so that the DNAs can form a variety of cluster configurations depending on coverage, nanoparticle size, and hybridization efficiency. According to DNA coverage measurements, the surface coverage density of DNA on the gold nanoparticle surface is in the range from 15 to 34 pmol/cm2,37 which corresponds to average axis separations from 25 to 40 Å. The proximity of the DNAs results in overlapping ion clouds thereby increasing the local charge fraction φ. As a result, the DNA duplexes in nanoparticle aggregates have a higher melting temperature than isolated DNA duplexes in bulk solution. The higher melting temperature of DNA in the pairs or clusters leads to a cooperative mechanism of melting in these aggregates. In this mechanism, melting only occurs when the temperature is higher than the melting temperature of the DNA cluster, but once melting starts, the released single-strand DNAs migrate out of the aggregates with their complement of counterions, decreasing the stability of the remaining DNA duplexes so that all the DNAs in the cluster melt in a very narrow temperature range.

Ion Distributions for DNA Duplexes and Clusters

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1 QSn Q2S2n QNSnN ) + + ‚‚‚ + K K1 K1K2 K1K2 ‚‚‚ KN

(8a)

Since the melting temperature for each step decreases as we have demonstrated in the previous section, the first step has the highest temperature and therefore the smallest equilibrium constant, i.e., KN < KN-1 < ‚‚‚ < K1. In this case the last term in eq 8a is dominant, and we have

1 QNSnN ≈ K K1K2 ‚‚‚ KN

(8b)

Combining eq 8b with the van’t Hoff equation, the melting curve can be expressed by

f)

1 ∆Htot 1 1 1 + exp R T Tm

( [

])

(9)

where

∆Htot )

∑i ∆Hi

(10a)

∆Htot

Tm )

(10b)

∆Hi

∑i T

m,i

If we assume that

∆Htot )

Tm ) Figure 11. (a) Net charge distribution and (b) electrostatic energy of ions (local region only) around a four DNA cluster for D ) 35 Å and [NaCl] ) 0.244 mol/L. The coordinates x and y are in units of angstroms, and the colored contours specify mol/L for part a and kT (T ) 300 K) for part b.

To model the melting process, we consider a cluster DN in which there are N DNA duplexes. Following the development of Jin et al.,4 we imagine that the melting process involves the sequence of steps

DN ) DN-1 + Q + nS DN-1 ) DN-2 + Q + nS ‚‚‚ D1 ) D0 + Q + nS

(6)

where Q is the target oligonucleotide (released from the cluster during each melting step) and nS stands for the n counterions (S) that are released in each step. The overall melting process can be written

DN ) D0 + NQ + nNS

(7)

and the equilibrium constant K for this reaction can be related to the equilibrium constants Ki for each step via4

∑i ∆Hi ≈ N∆H1

∆Htot ∆Hi

∑i T

m,i

(11a)

N∆H1

≈

(12b)

∆H1

∑i T

m,i

A Taylor expansion of eq 12b gives

Tm ) Tm,1 + ∆T ≈ Tm,1 +

∑i ∆Ti N

(13)

According to Jin et al.,4 by fitting the melting curve of the DNA gold nanoparticle aggregates, an average number N of 1.6 can be obtained from eq 8a, which corresponds approximately to a pair of DNA duplexes as the initial cluster. The experimental ∆T is around 3-6 K,4 which gives ∆T2 ) 6-10 K on the basis of eq 13. This value corresponds to a separation distance of around 25-40 Å according to Figure 10, which is in agreement with the results of DNA surface coverage measurements. As a second application we consider the results of Gibbs et al.,38 who reported that in DNA-polymer aggregates the melting curves are even sharper than those of DNA-gold nanoparticle aggregates. The measured value of N for their DNA-polymer aggregates is equal to 4, which suggests a four DNA cluster melting process. The measured ∆T in the DNA-polymer aggregate is more than 10 K. This result can also be explained by our theory: for a separation distance of around 25-40 Å,

2926 J. Phys. Chem. B, Vol. 110, No. 6, 2006 the four DNA cluster gives a ∆T4 as large as 6-20 K, which is much larger than what we obtained for the DNA pair. V. Conclusion In this paper, we employed MD simulations to study the ion atmosphere in isolated DNA duplexes and DNA clusters. We have proposed a new criterion to define a local region around each DNA on the basis of the electrostatic energy of the ion. This criterion has the advantage that it does not require radial symmetry and can be applied to DNA clusters. For isolated DNA duplexes, we have demonstrated that the local net charge fraction φ increases as salt concentration increases, although with a relatively small slope. When the salt concentration changes from 0.2 to 1.0 mol/L, there is only a 5% increase in φ. Although φ is not intended to be the same as the condensation fraction in Manning’s theory, the weak dependence of φ on salt concentration is similar to what would be expected from that theory. For DNA clusters containing two or four DNA duplexes, we found that φ is higher than φ of the isolated DNA duplex at the same salt concentration. In addition, φ was found to increase with decreasing distance between DNA duplexes. To calculate the melting temperature increase ∆T, we introduced the virtual equivalent salt concentration [NaCl]virtual and found the dependence of [NaCl]virtual on the interduplex distance D in the DNA clusters. Taking into account the experimentally known dependence of the melting temperature on [NaCl],4 finally we were able to estimate ∆T for the DNA clusters. The value of ∆T increases with decreasing D and is largest for the four DNA cluster case, where the ion overlap is most significant. For the four DNA cluster with D ) 25 Å, we obtained a rather significant ∆T of 20 K. Applying this result to the cooperative melting model proposed earlier for the melting of DNA-gold nanoparticle aggregates,4 we found that the increase in the melting temperature of a DNA pair with separation of 25-40 Å is sufficient to explain the sharp melting transition that is experimentally observed in the DNA-linked gold nanoparticle aggregates. In addition, results for a four DNA cluster can be used to explain the higher and sharper melting in DNA-polymer aggregates. Acknowledgment. This research was supported by the National Science Foundation (DMR-0076097) through the Nanoscale Science and Engineering Initiative and by the Air Force DURINT program. References and Notes (1) Elghanian, R.; Storhoff, J. J.; Mucic, R. C.; Letsinger, R. L.; Mirkin, C. A. Science 1997, 277, 1078. (2) Mirkin, C. A.; Letsinger, R. L.; Mucic, R. C.; Storhoff, J. J. Nature 1996, 382, 607.

Long et al. (3) Storhoff, J. J.; Lazarides, A. A.; Mucic, R. C.; Mirkin, C. A.; Letsinger, R. L.; Schatz, G. C. J. Am. Chem. Soc. 2000, 122, 4640. (4) Jin, R.; Wu, G.; Li, Z.; Mirkin, C. A.; Schatz, G. C. J. Am. Chem. Soc. 2003, 125, 1643. (5) Storhoff, J. J.; Elghanian, R.; Mucic, R. C.; Mirkin, C. A.; Letsinger, R. L. J. Am. Chem. Soc. 1998, 120, 1959. (6) Taton, T. A.; Mirkin, C. A.; Letsinger, R. L. Science 2000, 289, 1757. (7) Owczarzy, R.; You, Y.; Moreira, B. G.; Manthey, J. A.; Huang, L.; Behlke, M. A.; Walder, J. A. Biochemistry 2004, 43, 3537. (8) Park, S. Y.; Stroud, D. Phys. ReV. B 2003, 67, 212202. (9) Lukatsky, D. B.; Frenkel, D. Phys. ReV. Lett. 2004, 92, 068302. (10) Lukatsky, D. B.; Frenkel, D. J. Chem. Phys. 2005, 122, 214904. (11) Young, M.; Ravishanker, G.; Beveridge, D. L. Biophys. J. 1997, 73, 2313. (12) Young, M. A.; Jayaram, B.; Beveridge, D. L. J. Am. Chem. Soc. 1997, 119, 59. (13) Feig, M.; Pettitt, B. M. Biophys. J. 1999, 77, 1769. (14) Bonvin, A. M. Eur. Biophys. J. 2000, 29, 57. (15) Varnai, P.; Zakrzewska, K. Nucleic Acids Res. 2004, 32, 4269. (16) Rueda, M.; Cubero, E.; Laughton, C. A.; Orozco, M. Biophys. J. 2004, 87, 800. (17) Ponomarev, S. Y.; Thayer, K. M.; Beveridge, D. L. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 14771. (18) Korolev, N.; Lyubartsev, A. P.; Rupprecht, A.; Nordenskiold, L. J. Phys. Chem. B 1999, 103, 9008. (19) Korolev, N.; Lyubartsev, A. P.; Rupprecht, A.; Nordenskiold, L. Biophys. J. 1999, 77, 2736. (20) Lamm, G.; Wong, L.; Pack, G. R. Biopolymers 1994, 34, 227. (21) Washizu, H.; Kikuchi, K. J. Phys. Chem. B 2002, 11329, 11329. (22) Pack, G. R.; Wong, L.; Lamm, G. Biopolymers 1999, 49, 575. (23) Wagner, K.; Keyes, E.; Kephart, T. W.; Edwards, G. Biophys. J. 1997, 73, 21. (24) Manning, G. S. Q. ReV. Biophys. 1978, 11, 179. (25) Ray, J.; Manning, G. S. Langmuir 1994, 10, 2450. (26) Manning, G. S. Macromolecules 2001, 34, 4650. (27) Record, M. T., Jr; Anderson, C. F.; Lohman, T. M. Q. ReV. Biophys. 1978, 11, 103. (28) Record, M. T., Jr; Lohman, T. M.; Haseth, P. D. J. Mol. Biol. 1976, 107, 145. (29) Bond, J. P.; Anderson, C. F.; Record, M. T., Jr. Biophys. J. 1994, 67, 825. (30) Ray, J.; Manning, G. S. Macromolecules 2000, 33, 2901. (31) Case, D.; Pearlman, D.; Caldwell, J.; Cheatham, T., III; Wang, J.; Ross, W.; Simmerling, C.; Darden, T.; Merz, K.; Stanton, R.; Cheng, A.; Vincent, J.; Crowley, M.; Tsui, V.; Gohlke, H.; Radmer, R.; Duan, Y.; Pitera, J.; Massova, I.; Seibel, G.; Singh, U.; Weiner, P.; Kollman, P. AMBER 7; University of California: San Francisco, 2002. (32) Cornell, W. D.; Cieplak, P.; Bayly, C. I.; Gould, I. R.; Merz, K. M.; Ferguson, D. M.; Spellmeyer, D. C.; Fox, T.; Caldwell, J. W.; Kollman, P. A. J. Am. Chem. Soc. 1995, 117, 5179. (33) Arnott, S.; Campbell-Smith, P. J.; Chandrasekaran, R. In Handbook of Biochemistry and Molecular Biology; 3rd ed.; CRC Press: Cleveland, OH, 1976; pp 411-422. (34) Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; DiNola, A.; Haak, J. R. J. Chem. Phys. 1984, 81, 3684. (35) Abascal, J. L. F.; Montoro, J. C. G. J. Chem. Phys. 2001, 114, 4277. (36) SantaLucia, J., Jr. Proc. Natl. Acad. Sci. U.S.A. 1998, 95, 1460. (37) Demers, L. M.; Mirkin, C. A.; Mucic, R. C.; Reynolds, R. A., III; Letsinger, R. L.; Elghanian, R.; Viswanadham, G. Anal. Chem. 2000, 72, 5535. (38) Gibbs, J. M.; Park, S.; Anderson, D. R.; Watson, K. J.; Mirkin, C. A.; Nguyen, S. T. J. Am. Chem. Soc. 2005, 127, 1170.