Structure of Homopolymer DNAâ CNT Hybrids

J. Phys. Chem. C 2007, 111, 17835-17845

17835

Structure of Homopolymer DNA-CNT Hybrids† Suresh Manohar,‡ Tian Tang,§ and Anand Jagota*,‡ Department of Chemical Engineering, Lehigh UniVersity, Bethlehem, PennsylVania 18015, and Department of Mechanical Engineering, UniVersity of Alberta, Edmonton, AB T6G 2G8, Canada ReceiVed: February 15, 2007; In Final Form: May 22, 2007

Single-stranded DNA-carbon nanotube (CNT) hybrids have been used successfully for dispersion and structurebased sorting of CNTs. The structure of the hybrid determines its behavior in solution. Using scaling arguments and molecular dynamics simulations, we have studied various factors that contribute to the free energy of hybrid formation, including adhesion between DNA bases and the CNT, entropy of the DNA backbone, and electrostatic interactions between backbone charges. MD simulations show that a significant fraction of bases unstack from the CNT at room temperature, which reduces effective adhesion between the two per base. For homopolymer wrappings, we show that at low ionic strength, the dominant influences on the structure are adhesion between DNA and the CNT and electrostatic repulsion between backbone charges on the DNA. With a simple analytical model, we show that competition between these two can result in an optimal helical wrapping geometry.

1. Introduction Hybrids of single-stranded DNA (ssDNA) with single-wall carbon nanotubes (CNTs) have been used successfully for solution-based CNT manipulation. DNA forms a stable hybrid with CNTs by wrapping around them in a helical fashion.1,2 The hybrid is a negatively charged colloidal rod that can adhere to a positively charged surface. This adhesion is modulated by the electronic properties of the carbon nanotube, allowing their separation. The hybrid is useful for dispersion, sorting1,2 and patterned placement3 of nanotubes, for transportation of DNA into a cell, and for thermal ablation treatment.4 Strength of dispersion, ability to sort, and stability in the cellular environment all depend on the interaction between DNA and a CNT.5 From the point of view of carbon nanotube manipulation, DNA-CNT hybrids provide a model system to understand, more generally, how polymers wrap around carbon nanotubes. Because DNA forms a new kind of structure with CNTs, study of the hybrid also sheds light on the behavior of DNA. Several physical effects potentially contribute to the formation of DNA-CNT hybrids. Some of these are entropy loss due to the confinement of the DNA backbone, van der Waals and hydrophobic interactions between DNA bases and the CNT, electrostatic interactions between DNA charges, and CNT deformation. One of the goals of this paper is to sort out which of these contribute most strongly to the free energy of binding between the two components of the hybrid. For helical wrapping, one may ask whether there is an optimal pitch for a given sequence and CNT. Because the helical pitch directly controls the linear charge density of the hybrid, it strongly influences the electrostatic field near it and is predicted to be an important parameter controlling separation in ion-exchange chromatography.5 Several investigators have recently examined the interaction of flexible polymers with carbon nanotubes. McCarthy et al.6 †

Part of the special issue “Richard E. Smalley Memorial Issue”. * To whom correspondence should be addressed. E-mail: [email protected]. ‡ Lehigh University. § University of Alberta.

propose from a study of interactions of a conjugated polymer with a CNT that π-stacking objects align with the underlying structure or helicity of the CNT. Others have proposed that the structure is determined by competition between nonspecific contributions to the binding free energy such as adhesion and bending strain energy. Kusner and Srebnik7 model the polymer as a bead spring with either random walk or excluded volume (self-avoiding random walk) statistics on a smooth cylindrical surface. They predict that a large radius of curvature promotes helical conformations in neutral semiflexible chains, whereas a small radius of curvature promotes fully stretched linear adsorption. Coleman and Ferreira8 consider two competing factors, adhesion and elastic bending energy. Considering adhesion to be independent of conformation, ignoring entropic effects, and allowing multiple strands on the surface, they show that because of a quantized increase in the number of strands with increasing helical pitch, there are certain minima in the free energy. Wall and Ferreira9 focus on elastic and electrostatic contributions to the free energy. Kunze and Netz and Chertsvy and Winkler10-12 have considered the interaction of charged polymers with oppositely charged spheres or cylinders. Their work is based on interactions between bending and electrostatic potential energies, that is, where entropic contributions can be neglected. The remainder of this paper is organized as follows. In section 2, we consider several contributions to the free energy of binding of a ssDNA strand to a CNT. Section 3 describes the molecular dynamics (MD) methods used in this work. Section 4 uses results from MD simulations of simple DNA-CNT hybrids to explore its structure in greater detail and to validate some of the conclusions of section 2. Together, these two sections allow us to propose that for homopolymers at low ionic strength, the main contributions to the free energy are adhesion between bases and the nanotube and electrostatic interactions due to the charged phosphate groups. On this basis, in section 5, we present a model that predicts how the competition between adhesion and electrostatics can lead to helical wrapping with a preferred pitch.

10.1021/jp071316x CCC: $37.00 © 2007 American Chemical Society Published on Web 07/20/2007

17836 J. Phys. Chem. C, Vol. 111, No. 48, 2007 2. Estimates of Contributions to the Binding Energy The main feature of the DNA-CNT structure is that the DNA bases, which in B-DNA lie approximately in a plane orthogonal to its helical axis, are here usually stacked onto the surface of the carbon nanotube.1,2 When referring to a DNA base near the CNT surface, the terms stack and unstack refer to the adsorbed and desorbed states of the base, respectively. In the adsorbed state, the normal to the plane of the base is nearly aligned with the normal exiting the CNT surface. The DNA backbone remains exposed to the aqueous environment, allowing its easy hydration. Formation of the hybrid is driven by adhesion between the bases and the CNT. Entropy loss upon adsorption, increased electrostatic interactions, and possible elastic strain energies of the DNA and nanotube are among the other potentially important contributions to the free-energy difference between a hybrid and its separate constituents. 2.1. Adhesion between a Base and the Carbon Nanotube. Adhesion is driven primarily by favorable interaction between the DNA bases and the CNT due to van der Waals and hydrophobic interactions. While reliable values for CNT-base interactions are not available, Sowerby and coauthors13-21 have examined the binding of nucleic acid bases to graphite. A comparative study in ref 13 shows that the binding energy decreases in the sequence G > A > T > C, with a value of about 20 kJ/mol for adenine, or about 8 kBT per base14 (where kB is Boltzmann constant and T is room temperature, 300 K). Since the separation between neighboring phosphorus atoms on the DNA backbone is 6-7 Å, while the Kuhn length for ssDNA is 1.6 nm at a 150 mM salt concentration,22 this implies that the adhesion energy is considerably greater than kBT per Kuhn length. Taking the distance between phosphorus groups to be 6.5 Å, for a range of wrapping angles (0-3π/8), the adhesion binding energy for poly-d(A) would be between about -12.5 and -32 kBT per nm of the CNT if all bases remained stacked on the CNT. The question of whether this value is attenuated because only a fraction of the bases stack onto the CNT at any one time is addressed later using molecular simulations. Differences between bases are primarily due to the different sizes of purines and pyrimidines and due to differences in hydrophobicity. There is also believed to be a significant contribution to the binding energy from hydrogen bonding since, in several cases, bases are known to form ordered monolayers stabilized by interbase hydrogen bonding. Interestingly, the efficiency of homopolymeric ssDNA as a dispersant for CNTs does not follow this series; rather, it is T > C > A,1 guanine not being accessible experimentally since poly-dG is insoluble in water. We propose that the difference between the behavior of single bases and of homopolymers is partly due to the difference in how many bases remain stacked to the nanotube. Later in this paper, we will present results of molecular simulations that are consistent with this interpretation. Shi et al.23 have studied the peeling of ssDNA from a graphitic sheet using MD simulations (CHARMM) and quote base binding energies in vacuum for G/A/T/C to be 23/21/19/17 kcal/ mol for the base alone and 30/27/26/24 kcal/mol for deoxyribose. These are consistent with the values quoted by Edelwirth et al.,17 also in vacuum, but are much higher than the reported experimental values in solution. 2.2. Entropy Decrease due to Confinement of DNA Backbone. To establish the order of magnitude of entropic loss due to restriction of backbone degrees of freedom, consider the free-energy expenditure required to stretch the ends of a freely

Manohar et al. jointed chain with Kuhn length bk and number of segments N to a distance R24

3 R2 U e ) kB T 2 Nb 2

(1)

k

While this expression is accurate only for a Gaussian chain (small stretches compared to Nbk), it can be used to obtain an order-of-magnitude estimate of the entropic free energy by replacing R by Nbk

N 2bk2 3 3 ) NkBT U e ) kB T 2 Nb 2 2

(2)

k

that is, about kBT per Kuhn length. The Kuhn length of ssDNA is about 1.6 nm at 150 mM monovalent salt concentration.22 We are often concerned with a considerably smaller ionic strength, and under these conditions, the Kuhn length can be significantly larger.25 Therefore, in comparison with the adhesion energy, the free-energy increase because of entropy loss due to confinement of the backbone is small. This conclusion is consistent with experimental measurements that show that ssDNA requires only about 5 pN under 150 mM conditions to be stretched halfway to full length.22,25,26 For a wrapping angle of 45° and Kuhn lengths in the range of 16-50 Å, this estimate of entropy loss is in the range of 0.4-1.3 kBT per nm of the nanotube. 2.3. Electrostatic Energy. Especially at small ionic strength, electrostatic interactions between the backbone charges, ions, and between them and the nanotube play a very important role. We will present a more detailed analysis of these interactions in section 5. Here, to establish the scale of this interaction, we calculate the electrostatic energy of interaction, Uel, using Manning’s counterion condensation theory27,28 for a line of charges

Uel ) -

1 1 1 1 2log(1 - exp(-κb)) - + 2 z zξ z zξ

(

)

(3)

where z is the valence of the counterions, ξ ) lB/b ) q2/(4πokBTb) is the ratio of the Bjerrum length lB to the distance between adjacent charges along the line, b. Constants and o are the dielectric constant of water and permittivity of free space, respectively, q is the magnitude of the charge of an electron, and κ is the reciprocal of the Debye screening length. For 100 µM monovalent salt at room temperature (T ) 300 K) and for a range of wrapping angles (0-3π/8), Uel is in the range of 1.8-3.8 kBT per nm. Interaction with the nanotube and the helical structure of the ssDNA will further increase the electrostatic energy. We can conclude that with decreasing ionic strength, the electrostatic contribution will dominate over entropic terms. 2.4. Enthalpy Change due to Bending, Stretching, and Torsion of ssDNA and the CNT. Because the extensional stiffness of DNA22 plays a role only at stretches approaching the contour length and because there are no constraints on the backbone length of ssDNA on a CNT, it is very unlikely that this stiffness plays a significant role. However, it is germane to consider whether the enthalpy of bending and torsion of the DNA backbone matters. Indeed, several studies of wrapping polymers around cylindrical objects assign an energy to bending and torsion.6,8-12 If we view ssDNA as an object with a Kuhn length of 1.6 nm or larger, we might expect elastic bending energies to be important in wrapping it around a cylinder with

Structure of Homopolymer DNA-CNT Hybrids

J. Phys. Chem. C, Vol. 111, No. 48, 2007 17837

Figure 1. Total system energy versus time during the production phase for poly-dT(12) with helical pitches of 4.1, 17.7, and 32.7 nm.

TABLE 1: Estimates of Contributions to the Binding Energy of SsDNA and a CNT

contributing term

estimate (kBT/nm for a 1 nm tube and wrapping angle between 0 and 3π/8)

1

base-CNT adhesion

13-35 (based on base-graphite adsorption data)

2

entropic free-energy increase due to backbone confinement

0.4-1.3

3

electrostatics

1.8-3.8 (100 µM salt)

4

enthalpy increase due to DNA/CNT deformation

negligible for DNA and for CNTs < 1 nm in diameter

5

base-base stacking

order of base-CNT adhesion, absorbed into it

6

hydrogen bonding

potentially dominant, sequence dependent (28-11 kBT for GC/AT); negligible for cases studied here

a diameter on the order of 1 nm. On the other hand, if the value of the Kuhn length is primarily due to electrostatic stretching and the underlying “null” ssDNA is much more flexible, then there may not be significant enthalpy of bending and torsion. Results of molecular simulations (section 4) show that the latter is indeed the case. By accounting for electrostatic effects directly, we therefore are able to neglect the intrinsic bending and torsional stiffness of ssDNA. Beyond a critical diameter, adhesion between CNTs is strongly influenced by the ability of their sidewalls to deform further to increase the area of contact. The critical diameter for adhesion-induced deformation is about 10 Å.29 Therefore, for larger diameter CNTs, it is likely that DNA adhesion will cause significant deformation. In this work. we are restricting our attention to smaller diameter nanotubes and hence neglect this effect. 2.5. Stacking and Hydrogen Bonding. Stacking of neighboring bases competes with stacking onto the CNT surface, and its energy is of similar magnitude.30 It contributes, for example, to the fact that a certain fraction of bases in an overall bound chain are released from the CNT surface and stack with neighboring ones. We use MD simulations, presented in the

Figure 2. Snapshot of the final structure at the end of a 400 ps production run for poly-dT with helical pitches of (a) 4.1, (b) 17.7, and (c) 32.7 nm. Grey, CNT; yellow, DNA backbone; red, DNA bases; cyan, sodium ions. Water has been removed for clarity.

next section, to assess the importance of this factor and find that the fraction of unstacked bases is substantially independent of the helical wrapping angle. While it remains a significant effect, we are therefore able to absorb it into a reduction of the effective adhesion energy per unit length. Hydrogen bonding between bases can be significantly stronger than the adhesion energy. Watson-Crick GC and AT base-pair free energies are estimated in vacuum to be about -28 and -11 kBT, respectively.30 In water, they have beeen estimated to be about -9.7 and -7.2 kBT, respectively.39 Several other hydrogen-bonding possibilities that are normally not stable in fully solvated chains can be stabilized by the presence of a low-dielectric CNT. Indeed, it has been suggested that this contributes to the stability of poly-d(GT) structures, which is important for their efficacy for structure-based sorting of CNTs.2 Consideration of multiple chains will mean that one must also account for effects such as electrostatic repulsion and additional entropic confinement. In this work, we restrict our attention to CNTs singly wrapped by homopolymers without any internal hydrogen bonding. Table 1 summarizes these estimates of the main contributions to the binding energy between ssDNA and a CNT. The following picture emerges. For homopolymers in the limit of low salt concentration, binding energy is dominated by adhesion and electrostatics. Because adhesion favors a tighter wrap whereas electrostatic repulsion prefers the converse, there may be an optimal wrapping geometry. We present a model for this possibility in section 5. In the limit of high salt concentration, we expect entropic effects to be significant, and it is not clear that well-defined optimal geometries exist. We leave aside the

17838 J. Phys. Chem. C, Vol. 111, No. 48, 2007

Manohar et al.

Figure 3. (a) Location of P atoms (yellow, starting structure; red, final structure) for DNA with a helical pitch of 61.5 nm. (b) Solvated P atoms. Blue, P atoms; green, water.

performed with a (10,0) CNT. For all of the results reported here, CNT atoms were constrained, and their only interaction with other atoms was via a (12-6) van der Waals potential with a Lennard-Jones (LJ) well depth of -0.07 kcal/mol and minimum interaction radius of 1.99 Å.32 For a sufficiently large band gap, a semiconducting substrate responds to external fields like a dielectric.34,35 In separate computations, we have found that charge generation on semiconducting nanotubes about 1 nm in diameter due to the phosphate charge in solution is considerably smaller than the DNA charge itself and that the semiconducting tube behaves essentially like a dielectric. Therefore, one may view the molecular simulations as relevant to the case of a high band gap semiconducting CNT with a dielectric constant much less than that of the surrounding medium. Each phosphate group carries a negative charge that is neutralized by a Na+ ion. For the size of simulations performed (described below), this corresponded to a concentration of 160 mM. The corresponding Debye screening length is κ-1 )

xo kBT /q 2F ) 1.1 nm at T+) 300 K,36 where F is the number

Figure 4. (a) Base stacked onto the CNT surface. (b) Base unstacked from the CNT (green, water.).

potentially important role that could be played by sequencespecific hydrogen bonding interactions in such a case. We turn next to molecular simulations to study some aspects of the structure in greater detail. 3. MD Simulation Methodology We have studied the structure of DNA-CNT hybrids using MD simulations employing the CHARMM31,32 program and force field. Carbon atoms in the CNT were assigned benzenelike parameters. These were validated by simulating the bending and stretching of CNTs and comparing computed moduli with those reported in the literature.33 The value of the bending stiffness obtained from simulations was 3 × 10-19 J, the effective Young’s modulus was 3.06 TPa, and the effective thickness of the CNT was 1.04 Å. All simulations were

density of counterions (Na ). Note that the simulations were performed at a concentration where electrostatic repulsion is strongly screened. In all of the MD simulations, electrostatic interactions were treated with the particle mesh Ewald (PME) method.31,32,38 PME calculations were performed using a real-space cutoff of 10 Å, a convergence parameter of 0.36 Å-1, and a sixth degree B-spline interpolation. A cutoff of 12 Å was used for van der Waals interactions. Simulations were performed with homopolymer ssDNA strands 12 bases long with constrained ends. A few simulations were performed with DNA free on the CNT and in solution. Starting structures with ssDNA wrapped helically on the CNT surface were generated in the commercial program Materials Studio (MS), as described previously.1 A parent chain was transformed into other DNA-CNT structures with a varying helical pitch. Sodium ions were placed initially at a distance of 3.5 Å from the phosphorus atom along the bisector of the angle defined by the phosphorus atom and nonbridging oxygen atoms. A pre-equilibrated water box of dimension 102 × 39 × 33 Å3 was used. The solute (DNA + CNT + ions) was placed at the center of the water box, and the water molecules overlapping the solute were removed. Periodic boundary conditions were employed using the CRYSTAL command in CHARMM. The starting structure was first minimized for 500 steps with the Adopted Basis Newton Raphson (ABNR) method. Equilibration of the structure was done in stages so that the most strained parts of system could adjust without introducing artifacts. With DNA and CNT constrained, the structure was heated to 300 K in 2 ps, allowing the water and ions to equilibrate around the hybrid. The constraints on the DNA, excluding the end oxygen



Figure 5. (a) Definition of base orientation parameter, Borien, and orientation angle, R. (b) Base orientation parameter Borien for poly-dT with a helical pitch of 17.7 nm at various stages during a simulation. Twelve bases are numbered from 1 (O5′) to 12 (O3′).

atoms (O3′ and O5′), were removed, and the system was again slowly heated to 300 K in 2 ps. The heated system was equilibrated for 20 ps in the CPT ensemble, allowing the box dimensions to change only in the x direction (along the axis of the CNT), with pressure maintained at 1.0 atm. The resulting equilibrated structure was used for the production phase, which was performed for 400 ps. The box dimensions were fixed during the production phase, and data was collected during this period. This procedure was followed for structures with varying helical pitch in the case of poly-dT and for a single pitch for other homopolymers. Figure 1 shows examples of total system energy as a function of time. In all cases, after 250 ps, the variation in total energy was small, indicating the system had reached dynamic equilibrium. Because of the procedure followed in constructing the model, the number of water molecules in each run was slightly different. To compare the energies obtained from different runs, we have corrected for this difference. For this purpose, we performed simulations on water boxes with different numbers of water molecules and computed the total energy change for the addition of a single water molecule at 300 K to be -9.64 kcal/mol and the potential energy

to be -7.85 kcal/mol. Figure 2 shows representative structures of poly-dT with different helical pitches. The ionic strength in our simulations was sufficiently high that, as expected by the arguments in section 2 and observed in the simulations, conformations of a single unconstrained ssDNA strand on the CNT surface fluctuated considerably. The series of simulations with fixed ends allowed us systematically to study the structure, with control over the effective helical pitch. The entropic and enthalpic change in free energy due to this constraint was small because of the flexibility of null ssDNA and the relatively large Kuhn length. This is also confirmed by measurements of the average constraining force at the ends, which is only a few pN. 4. MD Results 4.1. Basic Structural Features. As noted previously,1,37 the principal feature of the family of helically wrapped DNA-CNT structures in comparison with the B-DNA double helix is that bases stack with the nanotube surface rather than with each other. For minimized poly-dT structures, this is accommodated


Manohar et al.

Figure 6. Mean base orientation parameter in simulations of poly-dT on a (10,0) CNT as a function of pitch (blue, starting minimized structure; red, mean taken during the production phase).

by a particular set of DNA backbone torsion angles, and nearly all bases are stacked onto the CNT surface [see ref 37 and current results]. Figure 3a shows the location of phosphorus (P) atoms at the start and end of the simulation for a structure with a large DNA helical pitch of 61.5 nm. On average, P atoms move slightly away from the CNT and remain solvated (Figure 3b). They are located at a distance of 9.8 ( 0.5 Å from the center of the CNT. Figure 3b also shows a base stacked onto the CNT surface. Upon equilibration in water at 300 K, we find that a significant fraction of bases unstack. Also, backbone angles fluctuate considerably and are no longer within specific ranges observed in the minimized structure.37 Figure 4a shows part of the model where a base (T) is stacked onto the carbon nanotube surface; Figure 4b shows one that has unstacked. It is clear that there is no water between the stacked base and the CNT. The stacking distance between the thymine base and CNT surface is 3.45 ( 0.05 Å, and the water envelope starts at a radial distance of 6.8 ( 0.5 Å from the CNT axis. In the case of stacked bases, we observed that the sugar ring plane is usually nearly parallel to the plane of the base, while for unstacked bases, the sugar ring plane is usually perpendicular to the base. The latter configuration is lower in torsional energy by about 0.6 kcal/mol (∼kBT). To quantify the tendency of bases to unstack from the CNT, we track a base orientation parameter Borien ) cos R, which is computed as the inner product between the unit normal to the base and a unit vector oriented along the line drawn from the center of the base perpendicular to the CNT axis, where R is the base orientation angle (Figure 5a). The base orientation parameter, Borien, approaches unity if the base normal is oriented radially toward the CNT axis and reduces in magnitude for other orientations. Figure 5b shows how the base orientation parameter for each base in a poly-dT wrap of a helical pitch of 17.7 nm varies during the simulation. Several bases stack to and unstack from the CNT during the simulation. We have also observed that sometimes adjacent bases unstack from the CNT in pairs and stack with each other. Figure 6 shows the mean value of Borien during the production phase as a function of helical pitch. With the exception of the smallest and largest helical pitch, the degree of unstacking from the CNT is

systematically larger during the production phase than that in the minimized structure. There is some variability in Borien as a function of pitch because particular runs can be trapped in metastable states over the time scale of the simulation. However, since there is not a systematic dependence on helical pitch, we will assume in the following section that the propensity to unstack from the CNT is independent of pitch. Therefore, the effective adhesion energy between DNA and the CNT will be reduced and will depend only on the length of the DNA strand. In section 5, where we present a model for competition between adhesion and electrostatics, we will accordingly allow for a reduction in the average adhesion energy between DNA and the CNT. In Figure 7, we plot the probability distribution of the orientation angle R for the four homopolymers. Also shown in the figure is the van der Waals interaction energy between a single nucleotide and the CNT for the corresponding orientation angles. These values were obtained by first deleting all atoms except for a single nucleotide and the carbon nanotube from the model and then computing the potential energy of interaction between the remaining nucleotide and CNT. Observe that the distribution of base orientation angles is bimodal. Comparing it to the plot for van der Waals interaction energy, we define the stacked (bound) state to be R e 35° and the unstacked (unbound) state to be R g 35°. Let us take as a measure of a single base binding energy the difference between the van der Waals energy for R(0) - R(π/2). From Figure 7, we estimate this to be -7.8 kcal/mol for adenine, -7.27 kcal/mol for guanine, -6.46 kcal/mol for cytosine, and -6.3 kcal/mol for thymine, that is, A > G > C > T. The effective base binding energy, calculated by averaging the van der Waals energy weighted by the probability, is -2.32, -2.36, -2.57, and -3.3 kcal/mol for poly-dC, poly-dA, poly-dG, and poly-dT, respectively, that is, poly-dT > poly-dG > poly-dA > poly-dC. This finding suggests that the difference in the propensity to unstack can significantly modify the effective binding energy per nucleotide. To compare the overall binding efficiency, one would need to calculate the free energy of ssDNA in solution, which we have not done. However, it is interesting to note that, while the single-base binding energy to graphite is greater for



Figure 7. (a) Probability distribution (shown by lines) and (b) van der Waals interaction energy (shown by circles) as a function of the base orientation angle, R. The data have been shifted so that energy is zero for the largest angle.

A than that for T,13 the dispersion efficacy of poly-dT is greater than that of poly-dA.1 4.2. Null DNA Elasticity. We suggested in section 2.4 that the null, uncharged ssDNA chain has negligible elasticity. For poly-dT-CNT hybrids with varying helical pitch, 2πc, we have plotted in Figure 8 the potential energy of the ssDNA chain (with and without the backbone charges) as a function of 1/c, If a is the radius of the helix, the Cartesian coordinates of the charges along it, x, y, and z, are

x ) a cos θ

y ) a sin θ

z ) cθ

(4)

where the difference in θ between the neighboring charges is ∆θ, that is, θ ) n∆θ (n ) 0, 1, ..., N). Then, the distance between every two charges on the helix, rmn, is

rmn )

x

(a cos n∆θ - a cos m∆θ)2 + (5) (a sin n∆θ -a sin m∆θ)2 + (cn∆θ - cm∆θ)2

The electrostatic energy in vacuum of a collection of charges arranged on a helix can then be calculated directly by the sum N-1

N

∑ ∑

n)0 m)n+1

q2 4πormn

(6)

which is also shown in Figure 8. It is clear that the increase in potential energy of the charged helix with decreasing helical pitch (increasing 1/c) is primarily due to electrostatic repulsion between backbone charges. Consistent with this, we find that there is little change with varying (1/c) of the potential energy of the null or uncharged ssDNA. Because in our model we account for electrostatic energy explicitly, we can neglect backbone elasticity. 4.3. Lateral Mobility of Bases. We now examine the question of lateral mobility of bases on the CNT surface to understand whether specific registration between DNA bases and the CNT surface affects the structure. Some researchers have argued that base-stacking moieties find particular orientations on the surface,6 and studies of adenine monolayers indicate a well-organized structure in which bases straddle carbon atoms.17 Others have argued that barriers for lateral sliding are negligible.8,9,23 To study this question, we track the location of the centroid of each base relative to the location of carbon atoms on the CNT. In particular, for every base, we project the CNT atom closest to the centroid of the base onto the plane of the base along the base normal. Figure 9a shows four locations for a base at different times during a simulation, an example where the base remains tied to a particular site over this duration. Figure 9b shows an example where the base samples several sites over this time period; this is the more common occurrence. In Figure 10a, we show the distribution of the location of the nearest CNT carbon atom projected onto the plane of the thymine base, rcnt, considering only stacked bases (bases are considered stacked if R e 35°). Using these results, we calculated the probability density (per unit area) of finding a CNT carbon atom at a distance of |rcnt - rc|, where these vectors are in the plane of the base (Figure 10b). It shows the expected tendency for the base to place its centroid rc near the closest CNT carbon atom, a favored state. If the base and CNT hexagons stack in commensurate fashion, the distance |rcnt rc| equals the C-C distance (1.39 Å), and this is an unfavorable state. We estimate the free-energy difference between these two states to be kBT log (P(0)/P(1.39)) ∼ 2kBT, where P(x) is the probability distribution function. While this value is small enough to allow bases to readily sample several sites during our simulations, it is significant enough to play a role in rapid nonequilibrium events such as peeling DNA from CNT. Because our interest here is in equilibrium DNA-CNT structures, the model presented in the next section will neglect site-specific registration effects. 4.4. Kuhn Length of DNA on the CNT. We argued in section 2.2 that the persistence length of the ssDNA is large enough that entropic free-energy increase due to confinement of the backbone to the CNT surface will be less than kBT per base. This argument used measured values for the Kuhn length (16 Å in ∼ 150 mM monovalent salt22). To further test this argument, we have estimated the Kuhn length of DNA on the CNT. Configurations of a poly-dT-CNT hybrid (pitch )12.67 nm) were computed with the constraint on the ends of DNA removed after the second stage of heating and equilibration (Figure 11). The average P-P distance was 6.57 Å. The coordinates of P atoms were monitored every 10 ps during the production phase. We define a series of vectors ri that connect adjacent P atoms and compute 〈ri ‚ rj〉 as a function of arc length s along the backbone, as shown in Figure 12. Fitting the initial decay to 〈ri ‚ rj〉 ) exp(-2s/lK) yields a Kuhn length lK of about 50 Å. Since entropic free-energy gain is on the order of kBT per Kuhn length and the latter will increase with increasing


Manohar et al.

Figure 8. Energy of null and charged DNA as a function of 1/c. The energy data have been shifted in both cases for clarity.

screening length, this confirms that the scale of entropic free energy is considerably smaller than that of adhesion and electrostatic energies (for small ionic strength). 5. Model for Optimal Wrapping Geometry From the previous sections, we conclude that, for small ionic strength, electrostatic and adhesive interactions between the DNA and CNT are the two important contributions to the free energy of binding. Since the effective adhesive interaction energy is substantially independent of the helical pitch, in excess of DNA in solution, a very tight wrapping is expected to achieve maximum coverage of DNA per unit length of the CNT. Electrostatic repulsion, on the other hand, tends to stretch the DNA to form a straight line. Therefore, the competition between adhesive and electrostatic interactions may result in an optimal wrapping geometry. In this section, we present a model that allows us to investigate this possibility. Figure 13 shows a helix of discrete charges wrapped around a cylinder. The helix has radius a and pitch 2πc. The discrete charges q on the helix represent the phosphate group on the DNA backbone and are at a distance d from the surface of the cylinder. The cylinder represents the space inside the DNA occupied by the CNT and vacuum and has a much lower dielectric constant 2 than that of the outer medium (water, 1). The charge is located at the phosphorus atom which, from molecular simulations, is at a distance of 9 Å from the center of the CNT. The distance from the location of the charge to the interface between water and free space is about 2 Å. Therefore, we take a ) 9 Å and d ) 2 Å. The arc length between adjacent charges on the helix is δ, which is about 7 Å. The projection of δ onto the axis of the cylinder is denoted by b, and it varies with the helical pitch. Clearly

b)

cδ

xa

2

+ c2

(7)

Figure 9. Location of a single base at four different times during the production phase in relation to the underlying CNT structure for (a) an example where a base is bound to a single site (orange, 150 ps; green, 230 ps; blue, 320 ps; yellow, 400 ps) and (b) a more common example where the base samples several sites (orange, 150 ps; green, 210 ps; blue, 260 ps; yellow, 380 ps).



Figure 11. Snapshot of unconstrained DNA of a helical pitch of 12.7 nm. (a) Starting minimized structure. (b) Final structure at the end of a 400 ps production phase. Water has been removed for clarity.

Figure 10. (a) Location of the nearest carbon atom on the CNT projected onto the plane of the base. The mean centroid-to-corner distance for the T base is 1.39 A°. (b) Probability density (per unit area) as a function of projected distance.

In the following, we calculate the total free-energy density (energy per unit length of the CNT) as a function of the helical pitch and ask if there is an optimal wrapping geometry by looking for an energy minimum. The contribution to the free energy from the adhesive interaction is given by

gad ) -lγ

(8)

where γ is the absolute value of the adhesion energy per unit arc length of the DNA and is a constant independent of the pitch, and l is the arc length of the DNA per unit length of the CNT, which is given by

l)

xa

2

+c c

Figure 12. Correlation plot to calculate persistence length.

2

(9)

To calculate the electrostatic free-energy density, we consider the electrostatic interaction between pairs of charges while accounting for the effect of a low dielectric medium inside the helix. In a previous work,33,34 we have demonstrated, using the theory of counterion condensation,26,27 that the existence of a low dielectric medium near a straight line of charges strengthens the electric field around the line, resulting in increased counterion condensation. Here, we apply this solution to the case of a helix of charges by making the following assumptions. (1) The electrostatics is governed by the theory of counterion condensation. That is, with an effective charge density of the helix, the electric potential can be solved from the linearized Poisson-Boltzmann or Debye-Huckel equation.35 (2) When

calculating the interaction between a charge and the cylinder, the cylinder is treated locally as a half space. (3) In addition to 2, when calculating the electrostatic energy between any two charges, the charges are taken to be on the same plane above the half space. With these assumptions, the total electrostatic energy can be calculated by superimposing the interactions between any two DNA charges along the helix. Detailed calculations will be published elsewhere. Here, we only give the result for the electrostatic energy density gel

gel )

kBT b

{[

1-

( )]( ) [ ( )]}

2 1 1+ 4ξ 1

1+

2 (f + h) 1 1-

2 1 1+ 2ξ 1

(10)


Manohar et al.

Figure 13. Model system to study the optimal wrapping geometry of ssDNA on a CNT. Electrostatic repulsion between charges favors a large helical pitch, while adhesion between DNA and the CNT favors a smaller pitch. Competition between these two tendencies can lead to an optimal pitch. (a) Side view; (b) front view.

where kB is the Boltzmann constant, T is temperature, ξ ) lB/b ) q2/(4πo1kBTb) is the ratio of the Bjerrum length to b, and f and h are, respectively, given by ∞

f≡

∑ n)1

[

e

x

-κb

xn

2

xn +2a /b [1-cos(nb/c)]+4η

∞

2

2

2

2

+ 2a /b [1 - cos(nb/c)] + 4η

h≡

∑ ∫0 n)0

2

e-κb

xn 2

-

+ 2a /b [1 - cos(nb/c)]

2

2

∞

n2+2a2/b2[1-cos(nb/c)]

Fje-2η

2

2

xFj +(κb) 2

2

2

]

-

e-2κbη

x

2η

(11)

xFj

dFj

+ (κb) + (2/1)Fj 2

(12)

In eqs 11 and 12, κ is the reciprocal of the Debye length, η is the dimensionless ratio d/b, and Jo(x) is the 0th order Bessel function of the first kind. Combining eqs 8 and 10, the total free-energy density is

g)

kBT b

{[

1-

( )]( ) [ ( )]}

2 1 1+ 4ξ 1

2 (f + h) 1 2 1 11+ 2ξ 1

increases the electrostatic repulsion, therefore resulting in larger preferred wrapping geometry. It should be noted that g in Figure 14b and c is on the order of kBT/nm because it is the difference between the electrostatic and adhesion energy. Energies gad and gel alone can be as large as 2 orders of magnitude of kBT/nm, consistent with our previous argument that these are the dominant interactions. 6. Discussion and Summary

Jo(Fj n2 + 2a2/b2[1 - cos(nb/c)]) 2

Figure 14. Free energy per unit length of the CNT as a function of the helical pitch for a fixed Debye length ) 10 nm and different adhesion energy. Figures a and d do not have an energy minimum. The energy minimum in b and c has been indicated by the arrow.

1+

- lγ (13)

Taking a ) 9 Å, d ) 2 Å, o ) 8.85 × 10-12 C2/Nm2, 1 ) 80, 2 ) 1, T ) 300 K, and using eqs 7 and 9, g is only a function of c given any set of parameters (κ,γ). Figure 14 shows the total free-energy density as a function of helical pitch for four different values of γ with κ fixed at 10 nm. It can be seen that for very weak adhesion (case a), electrostatic repulsion dominates, and the free-energy density decays as the pitch increases, that is, a straight configuration is preferred. As the adhesion increases (case b), an energy minimum appears, corresponding to an optimal helical pitch. This optimal helical pitch decreases as the adhesion becomes stronger (from b to c), indicating tighter wrapping. In the limit of very strong adhesion (case d), the energy decreases monotonically with decreasing pitch, and again, optimal wrapping disappears. Similar behavior can be observed for fixed adhesion energy with varying Debye length. Specifically, increasing the Debye length

We have used a combination of scaling analyses, molecular simulation, and an analytical model to study the structure of single-stranded DNA adsorbed onto a carbon nanotube. In the limit of low ionic strength, the main contributors to the free energy of the system are adhesive interactions between the DNA and the CNT and electrostatic repulsion between charges on the DNA backbone (modulated by the presence of the CNT). We have shown with an analytical model that competition between these two can result in an optimal helical wrapping geometry for small ionic strength. Molecular simulations show that a significant fraction of bases unstack from the CNT. This unstacking reduces effective adhesion and can alter the relative strength of binding of homopolymers. At room temperature, the barrier between adjacent sites for registration of thymine bases is about 2 kBT, small enough that bases usually sample several sites during the simulation. On the basis of experimental values and molecular simulations, we can conclude that entropic freeenergy gain due to confinement of the backbone is considerably smaller than adhesive free-energy reduction. We have considered only single strands of homopolymers on the CNT. Especially at high ionic strength, it is likely that sequence-specific hydrogen bonding interactions between multiple chains play an important role in the formation of an organized structure around the CNT.2 This aspect of the structure will be studied elsewhere. Acknowledgment. This work was supported, in part, by the National Science Foundation under Grant CMS-0609050 and by NASA under Award No. NNX06AD01A for the Lehigh University/Mid-Atlantic Partnership for NASA Nanomaterials. References and Notes (1) Zheng, M.; Jagota, A.; Semke, E. D.; Diner, B. A.; McLean, R. S.; Lustig, S. R.; Richardson, R. E.; Tassi, N. G. Nat. Mater. 2003, 2, 338. (2) Zheng, M.; Jagota, A.; Strano, M. S.; Santos, A. P.; Barone, P.; Chou, S. G.; Diner, B. A.; Dresselhaus, M. S.; McLean, R. S.; Onoa, G. B.; Samsonidze, G. G.; Semke, E. D.; Usrey, M.; Walls, D. J. Science 2003, 302, 1545.

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