Solvation Free Energy of Self-Assembled Complexes: Using Molecular Dynamics to Understand the Separation of ssDNA-Wrapped Single-Walled Carbon Nanotubes

Abstract

Molecular dynamics simulations were used to characterize the self-assembly of single-stranded DNA (ssDNA) on a (6,5) single-walled carbon nanotube (SWCNT) in aqueous solution for the purpose of gaining an improved theoretical understanding of separation strategies for SWCNTs using ssDNA as a dispersant. Four separate ssDNA sequences, ((TAT)₄, TTA(TAT)₂ATT, C₁₂, (GTC)₂GT), at various levels of loading, were chosen for study based on published experimental work showing selective extraction of particular SWCNT species based on the ssDNA dispersant sequence. We develop a unique workflow based on free energy perturbation (FEP) and use this to determine the relative solubility of these complexes due to the adsorption of the ssDNA on the SWCNT surface, and hence, rank the favorability of separations observed during experiments. Results qualitatively agree with experiments and indicate that the nucleobase sequence of the adsorbed ssDNA greatly affects the free energy of complex solvation which ultimately drives SWCNT separation. Further, to elucidate the underlying physics governing the ssDNA-SWCNT solubility rankings, we also present calculations for four structural characteristics of ssDNA adsorption. We demonstrate that a unique type of intra-strand hydrogen bonding is the most important factor contributing to the stability of the ssDNA-SWCNT complexes and show how these adsorption characteristics are coupled with the FEP results.

1. INTRODUCTION

Single-walled carbon nanotubes (SWCNTs) possess unique mechanical, electrical, and optical properties that make them attractive in a myriad of applications such as structural fillers in polymer nanocomposites (PNCs)^1,2, electronic components in molecular^3,4 and optical sensors^5,6, drug delivery agents in therapeutic dispersions^7,8, and as additives in PNC desalination membranes^9,10. However, difficulty in realizing widespread use of these materials stems from their manifold polydispersity in terms of size, chirality, and enantiomeric handedness due to the uncontrolled nature of current synthesis techniques. Numerous post-synthesis separation techniques have been developed so that specific nanotube species can be isolated for use in industry. These methodologies generally follow a two-step protocol. First, the SWCNTs are dispersed in aqueous media using a surfactant or polymer dispersant, e.g. ionic surfactants such as sodium dodecyl sulfate (SDS)¹¹, sodium deoxycholate (DOC) or other related bile salts^11–13, polymers containing aromatic moieties including aryleneethynylenes^14,15 or single-stranded DNA (ssDNA)^16,17. This is followed by separation to purify the dispersed SWCNTs based on their physiochemical properties using various techniques such as ultracentrifugation¹², ion-exchange chromatography (IEX)¹⁶, or aqueous two-phase extraction (ATPE).¹³

Of these different methodologies, separations based on ssDNA dispersed SWNCTs are perhaps the most interesting as they are able to select for specific nanotube chiralities based on the nucleobase (NB) sequence of the ssDNA oligomer^18,19. Refinement of these techniques have further demonstrated the ability to differentiate between the left- and right-handed enantiomers of single nanotube chiralities²⁰. The key to this is the observation termed sequence-specific hybridization: particular ssDNA sequences hybridize in such a way upon adsorption on specific SWCNT chiralities to the degree that isolation of single species from highly heterogeneous mixtures is possible. The challenge in taking advantage of this paradigm has been the identification of ssDNA-SWCNT sequence-chirality pairs that lead to this beneficial hybridization. Currently, the search is accomplished only via labor-intensive trial and error experimentation as the mechanism by which the separation is driven by ssDNA sequence is not yet understood. While the library of sequence-chirality pairs is increasing,^21,22 a robust method for predicting additional pairs based on theory and/or modeling efforts that explain the underlying binding mechanism is highly desirable, and this is the goal of the present work.

It is important to first discuss some preliminary factors that are understood from previous work in the literature. Adsorption of ssDNA to SWCNTs is driven by a number of mechanisms. First is the hydrophobic effect – the hydrophobic ssDNA bases are driven to the tube surface and engage in π-π stacking with the tube, while the polar phosphate groups are solvated by the water and the free ions. Driven by this mechanism, all ssDNA sequences undergo binding adsorption with SWCNTs to some extent irrespective of chirality. The second element is hybridization – as ssDNA strands adsorb, they tend to wrap helically along the tube axis and undergo two types of hydrogen bonding. The first type is intra-strand bonding where bases on the same strand hybridize with each other, the so-called “self-stitching”²³. The second type is inter-strand where the neighboring ends of adjacent strands coordinate and hybridize akin to a supramolecular polymerization. Both are important factors in stabilizing the adsorption. It is this second mechanism of differences in hydrogen bonding which is thought to differentiate free energy changes during adsorption based on a sequence-specific effect. A final important point is that while DNA naturally forms Watson-Crick hydrogen bonds during the formation of the double-helix conformation that exists in biology, many other non-Watson-Crick hydrogen bonding schemes arise when the nucleobases assume their adsorbed configurations on surfaces such as SWCNTs. This is because the short nucleotides behave as a random polymer coils in the aqueous environment surrounding the surface, allowing any available hydrogen bonds to form. Neither their biological function nor structure is preserved in this adsorbed state.

Because the separation in the ATPE framework is driven by differences in the solvation free energy, our starting hypothesis is that the selective partitioning observed experimentally occurs only when there are narrow, non-overlapping distributions of solvation free energy between different ssDNA-SWCNT complexes. Contrastingly, complexes which have overlapping solvation free energy distributions would not be able to be separated in such a scheme. This hypothesis is illustrated by the schematic in Figure 1.

Figure 1.

Hypothesized mechanism of separation made possible by non-overlapping distributions of the solvation free energy of the ssDNA-SWCNT complex in aqueous phases.

The second part of our hypothesis here is that it is the stability of the adsorption due to hybridization which differentiates successful ssDNA sequences from unsuccessful ssDNA sequences by resulting in narrow or broad distributions in the solvation free energy. Stable packings with relatively few conformations should behave more uniformly in solution when compared to less stable packings featuring many molecular orientations or arrangements. Heretofore, there has been no work to quantify this phenomenon. Thus, one aspect of our analysis is to develop a method capable of calculating and comparing the relative solubilities of these different molecular packings. A further and equally important question to answer in the analysis is: What drives the stability? As mentioned above, two types of hybridization contribute to ssDNA adsorption on a SWCNT, and these factors, plus the chirality of the tube, must somehow cooperate to coordinate stable packing. This leads us to also focus analysis on these two differing elements of hybridization and develop metrics to ascertain their individual roles in the process. The problem of computational calculation of solubility for complex self-assembling systems is a challenging and evolving field of study. Molecular simulation provides a valuable tool by allowing for the estimation of energetic and structural behavior on a molecular level and has been applied to many problems in the fields of biomolecular modeling²⁴ and materials science²⁵. The application of this tool toward the calculation of solvation free energies is widespread^26,27, especially in the field of computer-aided drug design (CADD). In such a framework, the goal is to reach a better understanding of the behavior of drug molecules in different environments and to compare the binding energies of drug ligands at protein sites to help determine/predict their activity in the body^28–35. These methodologies have mostly been limited to small molecules because the predictions become more difficult as the conformational phase space of the molecule increases. Group contribution methods have also been used to predict solvation free energies^36,37. However, it has been shown that solubilities calculated via explicit-solvent MD simulations for a sub-class of amino acid molecules are considerably different from those obtained via additive group contribution³⁸. Moreover, the theory behind these methods does not yet account for free energy changes associated with self-assembly. Subsequently, to our knowledge no systematic method exists for the prediction of solvation free energies of large, self-assembled complexes.

In this work, we discuss that a measurement of relative free energies can be made since the self-assembly decreases the conformational degrees of freedom as the molecules assume specific, co-operational arrangements for hybridization. To this end, we propose a free energy pathway to estimate relative solvation free energies of ssDNA-SWCNT complexes based on the hypothesis that the number of adsorbed ssDNA conformations is reduced and stabilized. The relative free energy pathway proposed in this study is shown in Figure 2 and allows for comparison between different ssDNA sequences or conformations wrapped on the same SWCNT by applying Eq. (1).

$$\Delta \Delta G_{solv}=\left(\Delta G_1+\Delta G_2\right)-\left(\Delta G_3+\Delta G_4\right)$$ (1)

Here, ΔΔG_solv is the relative free energy difference between the same chirality SWCNT bound to differing ssDNA sequences, ΔG₁ and ΔG₃ are the change in free energy associated with decoupling the electrostatic interactions of the ssDNA oligomers, and ΔG₂ and ΔG₄ represent the decoupling of the remaining van der Waals interactions. Further details of the approach, which is based on free energy perturbation (FEP),^39,40 and all other computational details are explained in Methods.

Figure 2.

Schematic for the relative free energy change (ΔΔG_solv) between different ssDNA sequences bound to the same chirality tube. For each ssDNA conformation or sequence to be examined, the oligonucleotides are decoupled from the surrounding environment to reach the bare SWCNT reference state. The red and green ssDNA molecules represent different sequences adsorbed to the same chirality nanotube undergoing full molecular interactions. The white colored molecules in the intermediate state, corresponding to transitions ΔG₁ and ΔG₃, represent sequences in which the electrostatic interactions have been decoupled from the surroundings. Finally, to reach the bare SWCNT reference state, corresponding to transitions ΔG₂ and ΔG₄, the van der Waals forces are additionally decoupled from the surroundings.

The paper is organized as follows. First is a description of our newly developed technique to determine the free energy differences of assembled ssDNA-SWCNT complexes. This is followed by the results of our FEP technique for both the adsorption of (TAT)₄ on a (6,5) SWCNT as a function of increased loading and as a comparison of four different ssDNA sequences at maximum loading. The ssDNA sequences are strategically chosen to allow comparison with experimental results for binding and non-binding pairs. We then present calculations of four structural metrics to characterize the adsorption and provide insight into the physics driving the observed free energy differences. We show that our FEP results are consistent with experimental results for both selecting and non-selecting sequence-chirality pairs and that measurable, quantitative differences in intra-strand hydrogen binding are the main the driver of free energy differences. Specifically, we define criteria which show that more favorable free energies occur when long range, intra-strand self-stitching is coordinated with a high degree of chain wrapping for the ssDNA/SWCNT pair.

2. METHODS

2.1. Simulation Details and System Description

All simulations were carried out using GROMACS (ver. 5.1.2)^41–43 applying the CHARMM36 forcefield⁴⁴ for the ssDNA and using SPC/E water⁴⁵. All carbon atoms in the SWCNT were treated identically as uncharged, aromatic sp² carbon as in our previous work,⁴⁶ as well as numerous previous studies of ssDNA/SWCNT systems.^23,47–49 While parameters for alternative forcefields exist that focus specifically on interfacial interactions,⁵⁰ they have only recently incorporated graphene-like surfaces (through the introduction of complex virtual-atom sites⁵¹) and have not yet been tested and verified for the nucleotides of interest in this work. This might be a consideration for future work.

Due to the nature of the ssDNA molecule and the large number of conformational possibilities that exist when such a molecule adsorbs onto a surface, obtaining equilibrated structures requires the use of replica-exchange molecular dynamics (REMD)^52,53. This enhanced sampling technique helps overcome energy barriers that may otherwise present a problem and cause the adsorbed ssDNA to get “stuck” in a certain packing motif that is not the most energetically favorable. Previous studies on ssDNA-SWCNT complexation have also had success using this technique^23,47,48 to probe favorable conformations of the assembled complex. The protocol used to construct an equilibrated ssDNA-SWCNT complex is as follows: The ssDNA molecules are constructed using the 3D-DART⁵⁴ interface and then placed adjacent to the SWCNT in their natural helix configuration (Figure S3). Following solvation, the energy of the system is minimized and then run through a 200 ps NVT equilibration using a velocity rescaling thermostat⁵⁵ during which positional restraints are applied to the SWCNT and the ssDNA. The LINCS⁵⁶ algorithm was used to maintain the correct bond lengths of all hydrogen-containing bonds allowing a timestep of 2 fs. The positional restraints on the ssDNA are then lifted followed by an additional 200 ps in the NVT ensemble and 200 ps of NPT equilibration applying the Parrinello-Rahman barostat⁵⁷ in a semi-isotropic fashion to maintain the correct box-size in the axial dimension corresponding to the length of the periodic SWCNT unit cell.

Following these equilibration steps, the REMD scheme is begun using 40 replicas spanning temperatures from 300 K to 600 K in an exponential profile and attempting exchanges every 2 ps. This choice of replicas results in a uniform exchange acceptance ratio of roughly ≈20% across all replicas. Simulation times on the order of ≈200 ns are necessary for sufficient equilibration which is monitored by the solvent accessible surface area (SASA) of the SWCNT. Only the bottom temperature replica (300 K) is used for all subsequent analysis.

2.2. Free Energy Calculation Details

The change in Gibbs free energies of solvation, ΔG_solv, is often calculated using free energy perturbation by adding a coupling parameter, λ, to the Hamiltonian. By gradually changing this parameter between 0 and 1, the molecule can be “grown into” or “faded out of’ the solution. In this case, λ = 1 corresponds to a state in which the solute molecule of interest does not see the solvent and effectively behaves as if in vacuum and λ = 0 corresponds to the state of normal molecular/ solvent interaction. Values of λ between zero and unity correspond to intermediate “ghost states” in which a soft-core Lennard-Jones type potential is used to avoid singularities. This technique requires that separate equilibrium simulations are run for each value of λ and the average derivative of the parametrized system Hamiltonian, ⟨∂H/∂λ⟩, be calculated. The solvation free energy can then be found using acceptance ratios to estimate the statistical error^58–60. Such a method was implemented via the g_bar module within GROMACS and is identical to the scheme we used previously when studying the solvation of bare carbon nanoparticles⁴⁶.

2.3. Defining a Relative Free Energy Pathway for Complex Solvation

Generally, the computational approaches for calculating solvation free energies take advantage of the fact that the free energy difference between two states is path-independent by exploring intermediate states that exist solely in silico. The proposed scheme for calculating the relative solvation free energy of the self-assembled complexes will similarly use the FEP technique, applying the λ coupling parameters in a multi-step pathway to deconstruct the ssDNA wrapped SWCNT to a reference state defined as the bare SWCNT in water (Figure 2) which we have previously studied extensively⁴⁶.

The relative free energy pathway proposed in this study allows for comparison between different ssDNA sequences or conformations wrapping on the same SWCNT by applying Eq. (1) (see Introduction). Multi-step free energy pathways such as this are quite common in the field of computer-aided drug design (CADD) when calculating binding free energies of small ligand molecules to larger proteins or relative binding energies of different ligands. This technique is often referred to as the “double annihilation method” or “double decoupling method”^61–64 as the ligand is decoupled from both its bound state in the protein binding pocket and also from a purely solvated state to elucidate the free energy difference, ΔG_bind, that arises during the localization of the drug to the binding site. Many additional difficulties arise during FEP application to these systems. Because these complexes are not stabilized by any sort of covalent bonds between the SSDNA and SWCNT, the ssDNA oligomers begin to diffuse off of the carbon surface during the unmodified FEP process (Figure S1). In order to maintain the assembled structure, further restraints must be added to the system (Figure S2). See the Supplementary Information for discussion of these intricacies of applying the FEP technique to the assembled ssDNA-SWCNT complexes.

3. RESULTS AND DISCUSSION

3.1. Study of Two Cases: Effect of ssDNA Loading and the Effect of Nucleobase Sequence

The results presented here fall into two distinct groupings: (1) the effect of increased ssDNA loading on a fixed SWCNT chirality; and (2) the effect of differing nucleobase sequence at conditions of maximum loading. The same (6,5) SWNCT was used in all simulations to isolate the effect of ssDNA sequence. For the first grouping, because the ssDNA acts as a surfactant by facilitating nanoparticle dispersion, it is expected that increasing the number of ssDNA molecules will increase the solubility of the self-assembled complex. In order to verify this behavior, and thus our FEP method, we assembled complexes consisting of a (6,5) SWCNT, two unit cells in length (8.1127 nm), with one, two, three, and four ssDNA molecules of the sequence (TAT)₄. This sequence-chirality pair was chosen as it has been shown experimentally to yield a higher single chirality enrichment than others¹⁸. We first present calculations of the relative free energy in this increased loading case. Additionally, we note that the length of the SWCNT was chosen for convenience and at loadings beyond this range, finite size effects are expected to impact the observed packing motifs. This study is intended to develop a method to detect free energy differences between various dispersants and the effect of the ratio between the ssDNA strand length and the SWCNT length will be investigated in future work.

We also compare ssDNA adsorption on a (6,5) SWCNT for different nucleobase sequences at conditions of maximum loading. The purpose of this is to determine whether our FEP method can discriminate sequence-specific binding effects. Complexes assembled from three (3) additional ssDNA sequences are compared with the (TAT)₄ we examine in the loading section: TTA(TAT)₂ATT, C₁₂, and (GTC)₂GT (see Table 1). There are several rationales for these choices. The TTA(TAT)₂ATT is somewhat similar to the (TAT)₄ sequence, but more importantly, it has also proven experimentally to select for (6,5) SWCNT_s²⁰. This allows us to compare the adsorption metrics of two sequences which both select for the same tube. The C₁₂ sequence is chosen because it is known from experiment to not select for the (6,5) SWCNT (although it does select for others, such as (11,0)²⁰). This gives us contrast between the metrics of selectors and non-selectors. The (GTC)₂GT is chosen because it also does not select for the (6,5) SWCNT but it does select for the (9,1) SWCNT which has an identical diameter to the (6,5) SWCNT.

Table 1.

Comparison of different NB sequences studied for conditions of maximum loading on a (6,5) SWCNT.

ssDNA Sequence	(6,5) Selective?	Significance
(TAT)4	Yes	Experimentally observed selector for (6,5).
TTA(TAT)2ATT	Yes	Experimentally observed selector for (6,5). Enables comparison of two selectors.
C12	No	Selects for many chiralities but not (6,5). Enables comparison of selectors and non-selectors.
(GTC)2GT	No	Selects for (9,1), same diameter as (6,5). Enables discrimination between chemical and geometric effects.

This gives us additional contrast between selectors and non-selectors and indirectly allows us to discriminate between geometry and chemistry during adsorption. The number of ssDNA oligomers was chosen such that the number of nucleobase residues was equal across all systems: four (4) strands of (TAT)₄, TTA(TAT)₂ATT, and C₁₂, and six (6) strands of (GTC)₂GT, resulting in a total of 48 nucleobases in each case. We hope by these comparisons to validate our FEP scheme and determine which of these metrics is predictive for explaining the different partitioning behavior observed in experiments.

Furthermore, four separate metrics are used to characterize the geometric arrangement of the adsorbed oligonucleotides in order to interpret the underlying physics that drive the observed free-energy changes:

1. Solvent accessible surface area (SASA) of the SWCNT

2. Number of nucleobases π – stacked with the SWCNT lattice

3. Histograms of the angle between the ssDNA-backbone and the SWCNT axis during wrapping

4. Number and character of nucleobase-nucleobase hydrogen bonds

The measurement of each of these metrics is discussed below along with the method by which they effect the free energy of the assembled complex. Values calculated are an average over the last 100 ns of the 300 K trajectory ensemble produced by replica-exchange molecular dynamics (REMD) simulations (see METHODS).

3.2. Relative Solvation Free Energy

3.2.1. Relative Solvation Free Energy with Increased ssDNA Loading

The FEP analysis described in the METHODS section was carried out on three (3) representative structures from each complex. These structures were selected via a clustering algorithm (see the Supplementary Information for additional details about this approach). In terms of free energy differences, the addition of more dispersing surfactant (i.e., higher ssDNA loading) means that a complex comprised of a SWCNT bound by N ssDNA oligomers will have a free energy that is more negative (i.e., larger magnitude) than a complex of the same SWCNT bound by N-l ssDNA oligomers. Because we are focused on the relative differences between the various ssDNA loadings and not the absolute values of a single wrapping, the single strand ssDNA-SWCNT complex is used as a basis for normalization. Results are presented in Table 2 where we show relative (column 2) and excess (column 3) free-energies. Absolute numbers obtained are tabulated in the Supplementary Information, Table S1.

Table 2.

Relative and excess free energies for (TAT)₄/ (6,5)-SWCNT with increasing ssDNA loading with respect to ΔG_0→1. Uncertainties represent the standard deviation of N = 3 different configurations.

Number ssDNA, i	$\frac{\Delta G_{0\to i}}{\Delta G_{0\to 1}}$	$\frac{\Delta G_{ex,i}}{\Delta G_{0\to 1}}$
1	1 (definition) ± 0.02	0 (definition) ± 0.02
2	2.01 ± 0.02	0.01 ± 0.02
3	3.06 ± 0.03	0.06 ± 0.03
4	4.15 ± 0.02	0.15 ± 0.02

The relative free-energy values shown in Table 2 (column 2) show a steady increase with increased ssDNA loading. This result demonstrates that our FEP method can reproduce the intuitive phenomenon stated previously; that the addition of increasing amounts of dispersant molecules serve to make the nanoparticle/water interactions more energetically favorable. It is also apparent that the increase is not perfectly linear but that there is a cooperative effect with increased loading. We show later by means of our metric analyses that this can be explained by various effects related to hydrogen bonding (coordinated wrapping, increase in inter-strand HBs, and the presence of stabilizing intra-strand HBs).

To better quantify this cooperative effect, we define the “excess free energy”, ΔG_ex,i as in Eq. (2) by subtracting multiples of the single strand free energy change, ΔG_0→1 , from the free energy change observed for i ssDNA molecules:

$$\Delta G_{ex,i}=\Delta G_{0\to i}-i\Delta G_{0\to 1}$$ (2)

The numbers are shown in Table 2 (column 3) and indicate that the cooperative effect is negligible upon addition of the second strand, but grows non-linearly with an increase to 15% of the single strand free energy difference at a loading of four oligomers per 8.2117 nm. The free energy differences across different loadings are significant in terms of the SWCNT partitioning because as the sequence of the ssDNA is changed the loading may also vary ultimately leading to a complex that is more or less stable in the aqueous environment. The behavior of the excess free energy with increasing system size is also of interest but this effect will be pursued in a future study. The results observed here validate that our FEP method behaves as expected and we next apply this to ssDNA oligomers of differing sequence.

3.2.2. Relative Solvation Free Energy of Different ssDNA Sequences (at Maximum Loading)

The FEP scheme was applied to the three additional ssDNA sequences described above and the relative free energies for each binding pair at maximum loading were computed and then compared with our original calculations for the (6,5) SWCNT with (TAT)₄. It was our hypothesis in constructing the scheme, that because changes in the nucleobase sequences has proven experimentally to result in different partitioning behavior within the ATPE scheme, this should be reflected by discemable free energy differences. The outcome of this calculation, therefore, is a prediction of which complexes formed with the (6,5) SWCNT are more energetically favorable relative to one another. To facilitate ease of comparison, the same reference state was used as in the previous section (the adsorption of one strand of (TAT)₄ on the (6,5) SWCNT). Results are presented in Table 3 where we show relative free energies for each sequence at the condition of maximum loading. For the computations with the new surfactants, we only studied the case of maximum loading, so excess free energies are not presented as in the previous section.

Table 3.

Relative free energy changes obtained using the FEP approach outlined in the Methods section. All values are normalized by the free energy change upon adding a single strand of (TAT)₄ to a bare (6,5) SWCNT Uncertainties represent the standard deviation of N = 3 different configurations.

Sequence, i	$\frac{\Delta G_{0\to i}}{\Delta G_{0\to 1\times {(TAT)}_4}}$
4 × (TAT)4	4.15 ± 0.02
4 × TTA(TAT)2ATT	4.06 ± 0.02
4 × C12	3.66 ± 0.02
6 × (GTC)2GT	3.02 ± 0.03

The results show that the computations qualitatively agree with those obtained in the experimental partitioning via the ATPE method (we say qualitatively because those experiments do not quantify free energy differences but are simply related to such a quantity). The computations for the (TAT)₄ and TTA(TAT)₂ATT sequences predict relative free energy changes that are higher and therefore more favorable, and these have shown the highest levels of selection for the (6,5) SWCNT in experiment^18,20. In contrast, the sequences which are known non-selectors for the (6,5) SWCNT have demonstrably lower free energies. Decreases of ≈9% and ≈25% (relative to the best selector (TAT)₄) are observed for the C₁₂ and (GTC)₂GT sequences, respectively. These are significant changes as a 25% decrease in this setup is equivalent to removing an entire ssDNA strand from the surface. We believe that these differences are driven by changes in the geometric arrangement of the ssDNA on the SWCNT surface that arise due to disparities in hydrogen bonding. We discuss this in detail below in the Hydrogen Bonding analysis.

3.3. Solvent Accessible Surface Area (SASA)

The SASA of proteins is a common metric in the simulation of biomolecules^65–67 and has been similarly applied to ssDNA-SWCNT systems.⁴⁶ It is an important measure to quantify when characterizing the solvation of the entire complex. Previous work shows that the SASA for a SWCNT decreases as successive ssDNA strands adsorb onto the surface (a result which may be expected). We look to determine whether this metric is related to our relative free energy results. We calculated the SASA using the g_sasa tool within GROMACS which uses the algorithm from Eisenhaber et al⁶⁸. However, to characterize and compare the assembly of the ssDNA strands, it is more convenient to define the hybridized surface area (HSA) which quantifies the portion of the hydrophobic SWCNT surface shielded by these adsorbed molecules. The HSA is calculated via Eq. (3) by subtracting the SASA of the wrapped SWCNT from the total SASA of the same bare nanotube:

$$HSA=SAS{A}_{bare}-SASA$$ (3)

3.3.1. HSA with Increased ssDNA Loading

The total HSA and the HSA per nucleobase are shown in Figure 3a and Figure 3b, respectively. Figure 3a shows that the HSA grows steadily with increased loading but in a nonlinear fashion. The dashed line in Figure 3a denotes the value of SASA_bare. This shows that even with four strands of ssDNA, the tube does not achieve complete surface coverage. The normalized data in Figure 3b shows a slight maximum at two strands. This suggests that for the case of this sequence-chirality pair, the most efficient wrapping occurs at an ssDNA loading of two strands (one strand per unit cell). Beyond this value, the surface becomes increasingly crowded and the individual nucleobases cannot fully hybridize with the carbon lattice. It is probable that tube periodicity plays a role in this result – in particular, the ratio of the ssDNA strand length to unit cell size – and this will be assessed in future work.

Figure 3.

(a) Total and (b) Normalized hybridized surface area in the adsorption of (TAT)₄ on the (6,5) SWCNT for different levels of loading. (c) Total hybridized surface area for the adsorption of 4 different ssDNA sequences on the (6,5) SWCNT. The dashed line denotes the SASA of the bare SWCNT. Error bars represent one standard deviation.

3.3.2. HSA of Different ssDNA Sequences

Results of the HSA analysis for the four different ssDNA sequences are shown in Figure 3c. These show no clear pattern that relates to the free energy differences. The TTA(TAT)₂ATT and C₁₂ sequences show decreases of 9% and 12% with respect to (TAT)₄, respectively. Curiously, the (GTC)₂GT sequence does not result in a similar decrease even though it does not select for the (6,5) SWCNT in experiments. This is most likely due to the shorter chain length, which allows for different packing motifs and inter-strand interactions. These differences could have a significant impact on the behavior of the complex in solution due to the hydrophobic nature of the SWCNT surface. The greatest significance of these results is that the HSA does not seem to be a meaningful predictor of sequence selectivity and is simply an emergent quantity. This is in keeping with our hypothesis that binding stability is the primary driver.

3.4. π – Stacking between ssDNA and SWCNT

It is commonly understood that molecules containing aromatic rings can undergo π-stacking^69–72, In addition to hydrophobic interactions, this phenomenon helps drive the adsorption of ssDNA on the SWCNT lattice^23,47. We characterize this quantity here to determine if it relates to and is a driver of free energy differences between the different ssDNA-SWCNT complexes (which has been speculated). Simple geometric relationships between the constituent atoms define π – π stacking making it straightforward to quantify at each snapshot of the simulation. We use the criteria⁷³ that two aromatic rings (in this case one within the SWCNT lattice and one in the NB) are said to be stacked if the distance between their centers-of-mass, d, are within 4 Å and the angle between their two normal vectors, ϕ, is less than 20° (see Figure 4a).

Figure 4.

(a) Geometric definition of π – π stacking; d < 4 Å and ϕ < 20°. (b) Fraction of (TAT)₄ nucleobases stacked on (6,5) SWCNT surface at increasing ssDNA loading. (c) Fraction of nucleobases stacked on the SWCNT surface for the 4 different ssDNA sequences. Error bars denote on standard deviation.

3.4.1. Effect of ssDNA Loading on π – Stacking

In Figure 4b, we show the fraction of stacked bases (i.e., the total number of stacked bases divided by the total number of bases) vs. loading. Overall, the results are flat and insensitive although in a manner similar to the fractional HSA, there is a slight decrease at the upper loading value. This decrease is consistent with the HSA analysis which showed that at full ssDNA loading (four total strands, two strands per unit cell), steric crowding prevents full hybridization due to periodicity effects for this size and number of unit cells.

3.4.2. Effect of ssDNA Sequence on π – Stacking

The fraction of NB’s undergoing π – π stacking with the SWCNT surface for the four different ssDNA sequences are shown in Figure 4c. For the same reasons previously discussed for this metric in the loading study, the nucleobase sequence does not seem to have an effect on the amount of stacking between the SWCNT surface and the adsorbed molecules. The results are flat and the small differences that are observed are within the measured uncertainty and cannot be claimed to be statistically significant. Hence, this measure, while a driver for adsorption itself, is also not a meaningful predictor of sequence selectivity.

3.5. Wrapping Angle of the ssDNA Backbone

The angle assumed by the ssDNA backbone with respect to the SWCNT axis can also help characterize the adsorption. The side and top views of a model system in Figure 5a define the geometry of this measurement. Three vectors are defined: First, ν_P→P, which extends between successive phosphorous atoms (P_i → P_i+1); Second, ν_A→P, which has its origin on the SWCNT axis and extends to the first of the two phosphorus atoms (P_i) in a fashion that is orthogonal to the SWCNT axis; and third, ν_axis, which runs along the SWCNT axis and shares its origin with ν_A→P. The wrapping angle, ω, can then be found as the dihedral angle between the three vectors. A value of 0° or 180° indicates that that particular portion of the backbone runs parallel or anti-parallel to the SWCNT axis while a value of 90° or 270° represents wrapping perpendicular to the axis. The various quadrants can be used to define if the wrapping occurs in a left- or right-handed manner. Quadrants I and III (shaded green) signify left-handed wrapping and quadrants II and IV (shaded white) signify right-handed wrapping. Normalized histograms of these wrapping angles can then be constructed to observe the average behavior of the backbone and to allow a more direct comparison.

Figure 5.

(a) Side and top view of the geometric definition of ssDNA wrapping angle, ω (Not to scale). Red circles denote phosphorous atoms, the grey rectangle with the red dashed line represents the SWCNT and its axis. (b) Histograms of angles for increasing (TAT)₄ loading on (6,5) SWCNT. (c) Histograms of angles for the 4 different ssDNA sequences on (6,5) SWCNT.

3.5.1. Effect of ssDNA Loading on the Wrapping Angle

Normalized histograms of the ssDNA backbone wrapping angle are shown in Figure 5b. These histograms display how the wrapping, and subsequently the inter-strand packing, changes as the ssDNA loading is increased. All ssDNA strands at all four levels of loading prefer to wrap in a left-handed manner (peaks in green regions). While this is opposite the observation reported by Roxbury et al.⁴⁸ it should be noted that the initial conditions are different for the two studies. In the case of the Roxbury study, the system was constructed with the ssDNA already partially wrapped around the SWCNT in a right-handed manner (see Figure 4 in the cited work). Even though the nucleobases are not in contact with the SWCNT surface, this initial configuration suggests the final configuration. Our initial configuration (see Supplementary Information, Figure S3) consists of the ssDNA placed beside the SWCNT in a right-handed helix with no “prewrapping”. This gives us further confidence that our system is at equilibrium as the ssDNA configuration had to change handedness upon adsorption. We also note that the data reported in Figure 5 is a histogram of the orientation of single bonds, and that some right-handed configurations are still present (this is not surprising as the trajectory from the REMD simulation should not be thought of as a times series as in standard MD, but more like an ensemble of configurations as obtained in various Monte Carlo approaches).

A single ssDNA oligomer is equally likely to wrap in a parallel or anti-parallel manner, as signified by the two nearly equal peaks at roughly 20° and 200°. As more strands are added, the peak in quadrant IV diminishes indicating a preference that all ssDNA strands wrap in the same direction with respect to one another. The left-handed peak also shifts to around 60° representing that the oligomers wrap in a more elongated fashion and assume a shallow angle with respect to the tube axis. Both of these effects again suggest an increased coordination between the ssDNA strands during the hybridization with the SWCNT surface as loading increases. Also of note is the observation of what appears to be a transition state of the wrapping angle for three ssDNA strands. This is another indication that this level of loading appears to be a threshold between loosely and highly coordinated dispersant interactions.

3.5.2. Effect of ssDNA Sequence on the Wrapping Angle

Histograms for the backbone wrapping angles for the four different ssDNA sequences are shown in Figure 5c. As in the loading study for this metric (Figure 5b), it is apparent that all four sequences prefer to wrap in conformations that are left-handed (peaks in green regions). However, the results indicate that while this measure gives us a better picture of how the ssDNA oligomers arrange in a macromolecular sense, it does not give us insight into sequence-selectivity. To that point, the overall results for (TAT)₄ and C₁₂ (which are sequence selective and non-selective, respectively) are somewhat similar. Likewise, the overall results for TTA(TAT)₂ATT and (GTC)₂GT (also sequence selective and non-selective, respectively) are surprisingly similar. Thus, the results for every case can be thought of as individual responses (with some similarities and some differences) that are functions of the stiffness and hydrogen bonding characteristics of the specific oligomers. There are some individual responses of note. The shorter (GTC)₂GT sequence displays a slightly larger tendency for wrapping in an anti-parallel manner, likely due to the shorter chain-lengths of the six strands of eight nucleobases. Additionally, the C₁₂ sequence wraps in a manner that is more elongated than any of the other sequences (large peak around 70°). These differences can be ascribed to the way in which the specific nucleobases interact and adsorb onto the carbon lattice which is then imparted on the arrangement of the phosphate backbone.

3.6. Hydrogen Bonding

Another stabilizing factor within the complex is the formation of hydrogen bonds between nucleobases adsorbed on the SWCNT surface. This effect has been examined previously^23,48,49 and it is believed to demonstrate the cooperative nature that occurs during the assembly of multiple ssDNA oligomers. While DNA naturally forms Watson-Crick hydrogen bonds during the formation of the double-helix conformation that exists in biology, other non-Watson-Crick hydrogen bonding schemes arise when the nucleobases assume their adsorbed configurations on surfaces such as SWCNTs. For example, in a (TAT)₄ system, this allows for A-A and T-T hydrogen bonds to form during assembly in addition to the common biological A-T association.^23,74

We monitored the presence of hydrogen bonds throughout the simulation via simple geometric relationships. In molecular simulations, hydrogen bonds are commonly defined⁷⁵ as in Figure 6a, where the Donor–Acceptor distance, r, is less than 3.5 Å and the Hydrogen–Donor–Acceptor angle, θ, is less than 30°. Additionally, we differentiate the total into contributions from intra- and inter-strand varieties (using the GROMACS g_hbond utility) which yields a more complete understanding of the cooperative effect of hybridization. These values are then normalized by the number of nucleobases present to facilitate the comparison across different levels of loading.

Figure 6.

Definitions used in the analysis of hydrogen bonds. (a) Geometric definition of hydrogen bond; D ≡ Donor, A ≡ Acceptor, r < 3.5 Å, θ < 30°. (b) Various degrees of ssDNA wrapping described by the COM-distance, d_COM. The SWCNT and its axis are viewed on end and are denoted by the black circle and dot. The red line represents the ssDNA strand and the blue dot its center-of-mass. The dashed line is the measure d_COM for the relevant conformation. Small values of d_COM (less than the tube radius) are a necessary measure in order to characterize an intra-strand HB as stabilizing.

In order to further quantify how hydrogen bonding affects the stability of the ssDNA-SWCNT complex, we have developed a set of criteria which characterizes the intra-strand HBs as either stabilizing or non-stabilizing. Non-stabilizing HBs are defined as occurring between successive or nearly-successive nucleobases when the relevant section of ssDNA is adsorbed on the carbon surface. This contrasts with stabilizing HBs which are defined as occurring between nucleobases far enough apart along the backbone to allow the ssDNA strand to completely wrap around the SWCNT – this has previously been termed “self-stitching”.²³ This definition leads to two parameters that can aid in the differentiation of these intra-strand HBs. The first is the “distance” between nucleobase residues along the ssDNA backbone (Eq. (4)).

$$d_{RES}=\left|ResI{D}_1-ResI{D}_2\right|$$ (4)

Here d_RES is defined as the number of nucleobases between the two residues, ResID_i, participating in a hydrogen bond. If d_RES, is small (e.g. between 1 and ≈5) then the hydrogen bond must be non-stabilizing because there is not a large enough amount of ssDNA to allow for wrapping of the SWCNT. A larger d_RES (e.g. ≈8 to 11) signifies that the hydrogen bond may be stabilizing, but only if the relevant conformation wraps around the nanotube (this is similar to the “self-stitching” observed by Roxbury et al.²³). A such, it is necessary to introduce a second parameter for HB characterization – the distance of the ssDNA strand center-of-mass from the nanotube axis in the xy-plane. This measurement, d_COM, helps describe whether the ssDNA in the current conformation is wrapped around the nanotube or simply adsorbed onto the surface. A d_COM that is less than the radius of the nanotube signifies a high degree of ssDNA wrapping versus a d_COM greater than the SWCNT radius which describes an ssDNA strand adsorbed on the surface. This is described pictorially in Figure 6b.

These two parameters are measured for every instance of an intra-strand HB during the replica-exchange simulation and then binned into a two-dimensional histogram and normalized by the volume of the cylindrical shell containing the ssDNA COM (Eq. (5)):

$$h\left(d_{RES,i},d_{COM,j}\right)=\frac{N\left(d_{RES,i},d_{COM,j}\right)}{V_{shell}\left(d_{COM,j}\right)}$$ (5)

This histogram can then be transformed into a relative free energy surface (FES) using Eq. (6):

$$\frac{\Delta G\left(d_{RES,i},d_{COM,j}\right)}{kT}=-ln\left(\frac{h\left(d_{RES,i},d_{COM,j}\right)}{h_{\mathit{max}}}\right)$$ (6)

This new method allows for quick visualization of the character and probability of all intra-strand hydrogen bonds that form during the entire REMD trajectory and therefore facilitates comparisons between multiple systems.

3.6.1. Effect of ssDNA Loading on Hydrogen Bonding

The total number of hydrogen bonds per NB vs. loading is shown in Figure 7a. The figure shows that the total number of hydrogen bonds per NB increases with increasing ssDNA coverage. However, the breakdown of these values shows that this increase is solely due to the formation of inter-strand hydrogen bonds. The fractional number of intra-strand HBs remains relatively constant across all levels of ssDNA loading.

Figure 7.

Hydrogen bond statistics for (a) increased loading of (TAT)₄ on a (6,5) SWCNT, and (b) the 4 different ssDNA sequences on a (6,5) SWCNT Total number of hydrogen bonds per ssDNA NB broken down into the intra- (red) and inter-strand (green) contributions. ssDNA Sequence Index: 1 = (TAT)₄; 2 = TTA(TAT)₂ATT; 3 = C₁₂; 4 = (GTC)₂GT. Error bars represent one standard deviation.

The free energy surfaces (FESs) corresponding to an increasing number of ssDNA strands of sequence (TAT)₄ are presented in Figure 8a – 8d and display the character and relative probability of the various HBs. The index between hydrogen bonding residues, d_RES, is along the x-axis and d_COM is on they y-axis from 0 to 1.5 nm (top to bottom). The HBs that exist in the upper right-hand region (indicated by the blue circles) can be said to be stabilizing as defined previously. They occur between nucleobase residues that are not adjacent (d_RES = ≈8 to 11) and when the ssDNA strand is wrapped around the nanotube (as indicated by the low d_COM values). While there are slight variations in the free energy surface presented in Figure 8a – 8d, they all have the same general shape and magnitude. This agrees with our previous observation that the number of intra-strand HBs per nucleobase remains constant as increasing strands are added to the nanotube. We make greater use of this metric in the following section to demonstrate how it helps explain free energy differences for different ssDNA sequences at maximum loading.

Figure 8.

Free energy surface (FES) characterization of the intra-strand hydrogen bonds for ssDNA on a (6,5) SWCNT: (a) One strand (TAT)₄ (b) Two strands (TAT)₄ (c) Three strands (TAT)₄ (d) Four strands (TAT)₄ (f) Four strands TTA(TAT)₂ATT (g) Four strands C₁₂ (h) Six strands (GTC)₂GT. The dashed line denotes the center of the nanotube wall. The presence of HBs in the circled regions indicate HBs that have stabilizing character.

3.6.2. Effect of ssDNA Sequence on Hydrogen Bonding

The total number of hydrogen bonds per NB for the four different ssDNA sequences are shown in Figure 7b. In contrast to the other geometric characteristics that we have examined, the sequence plays a large role in the number of hydrogen bonds that form within the adsorbed molecular network. In terms of the total number of HBs formed, the C₁₂ and (GTC)₂GT sequences display a 20% and 60% increase respectively over (TAT)₄. However, since these are poor selectors for this particular SWCNT, this is the first indication that more hydrogen bonds are not necessarily better. We discuss this point in detail below. Another interesting observation is the fraction of intra- and inter-strand hydrogen bonding for the different sequences. The decrease in HBs for the TTA(TAT)₂ATT sequence is solely due to a decrease in inter-strand bonding. In the case of C₁₂, there is a disproportionate increase in the amount of intra-strand bonding (so large that it actually surpasses the number of inter-strand HBs). Contrastingly, the increase for (GTC)₂GT is largely due to the formation of more inter-strand bonds.

Figure 9a – 9d present the free-energy surfaces (FESs) for the intra-strand HBs that form in these complexes. Recall that the favorable indicator of stabilizing hydrogen bonds is a local energy minimum in the high d_RES, low d_COM region, which is the upper right-hand corner of these plots (indicated by the circles). Clearly the stabilizing region displays this local minimum for the cases of Figure 9a ((TAT)₄) and Figure 9b (TTA(TAT)₂ATT), which are the sequences shown to select for this SWCNT in experiments and which have the most favorable relative free energies computed in this study. In contrast, the stabilizing region is depleted for the cases of Figure 9c (C₁₂) and Figure 9d ((GTC)₂GT), which are the experimental non-selectors and show significantly less favorable relative free energies via our computational approach. The relationship between good experimental selection, more favorable computed relative free energy, and stabilizing hydrogen bonds is clear from these results.

Figure 9.

Free energy surface (FES) characterization of the intra-strand hydrogen bonds for ssDNA on a (6,5) SWCNT: (a) Four strands (TAT)₄ (b) Four strands TTA(TAT)₂ATT (c) Four strands C₁₂ (d) Six strands (GTC)₂GT. The dashed line denotes the center of the nanotube wall. The presence of HBs in the circled regions indicate HBs that have stabilizing character.

These phenomena confirm that it is not just the number of hydrogen bonds formed that is important for the stabilization of the adsorbed ssDNA, but also the quality of these hydrogen bonds. The upper right-hand region of the FES that signifies stabilizing HBs is very sparsely populated for C₁₂ and nearly empty for (GTC)₂GT. This means that for these sequences, most intra-strand HBs are formed between nearly-adjacent nucleobases and do not occur in wrapped conformations. We call these non-stabilizing HBs as opposed to destabilizing because their effect is more neutral and does not actively work against the formation of these complexes. This type of intra-strand bonding does not lead to any degree of stabilization because they occur between nucleobases that are in close proximity with no significant wrapping of the backbone around the SWCNT between them. Thus, unless the ssDNA backbone wraps around the SWCNT and a hydrogen bond forms between the two ends (i.e., the previously observed self-stitching²³) intra-strand HBs can be predominately superficial and it is a small population of stabilizing HBs that are responsible for better adsorption. We note that inter-strand HBs are also important for the stabilization of the entire complex, but we make no observation that directly corresponds the differences in free energy observed with FEP.

Another important factor that the FES points to is stability. The FES is a statistical sampling over a large number of conformations. Recall, our initial hypothesis was that sequence-specific pairs have narrow distributions due to stable packings. The FES for C₁₂ and (GTC)₂GT show evidence of statistical fluctuations spread over the complete range of d_COM values, with almost no high d_RES values at any value of d_COM. This evidence of higher fluctuations shows that they are less stable in this environment (quite possibly due at least in part to the inability to form stabilizing intra-strand bonds). In contrast, the (TAT)₄ sequence, which yields the most favorable free energy, shows a more compact overall distribution (less fluctuation) paired with the likelihood of forming stabilizing HBs. The case of TTA(TAT)₂ATT is somewhere in the middle, displaying a noncompact distribution but with a good likelihood of forming stabilizing HBs. This indicates that even if the statistical fluctuations are large, if the assembled ssDNA is able to form these stabilizing HBs during the sampling period, it may be enough to drive the system towards more favorable free energy conditions.

4. CONCLUSIONS

We have used molecular dynamics simulations to study ssDNA adsorption onto SWCNTs in aqueous solutions with the goal of developing protocols which enable us to rank ssDNA-SWCNT sequence-chirality pairs that are more favorable in SWCNT purification and separation operations in which ssDNA is used as a dispersant. A significant result is the development of a workflow based on free energy perturbation (FEP) that enables us to rank the relative free-energy of ssDNA-SWCNT binding pairs for differing ssDNA sequences on the same SWCNT. The results for increased loading of (TAT)₄ on the (6,5) SWCNT show that the tube becomes increasingly soluble as dispersant concentration increases as measured by the growth in the relative free-energy. These results also show that there is an excess free-energy contribution with increasing number of strands which we attribute to the cooperative effect of increased inter-strand binding. This effect must be further studied in larger scale simulations to quantify the growth of the excess contribution with additional loading. Our results for the ranking of four differing ssDNA sequences on a (6,5) SWCNT at maximum loading were consistent with experimental results – the sequences that are known from experiment to select for this chirality show the highest relative free-energy, while two known non-selectors showed a significantly lower relative free-energy change. We did not characterize the excess free-energy with increased loading for all the selectors and non-selectors, but it is possible that such differences based on sequence could also play a role in sequence selectivity and will be fully explored in future work.

The FEP scheme proposed here does not yet allow direct comparisons between different nanotubes bound by different sequences. This requires the addition of another step in the free energy pathway depicted in Figure 2 to measure the free energy difference between the various reference states for each tube. This should not prove difficult to implement but was beyond the scope of the current work and will also be the subject of future investigations.

The FEP results enable the ranking of the binding pairs, but do not provide insight into the physics driving such differences. To explore this, we performed calculations of four different metrics related to the adsorption in an effort to determine the driving forces which explain the subtle differences in the relative free energy. Solvent accessible surface area (SASA) and the degree of π-stacking of the NBs with the SWCNT surface – two metrics which have been oft used in the past in the analysis of ssDNA-SWCNT adsorption – are shown for both the case of increased loading and the sequence study at maximum loading not to be meaningful indicators of sequence selectivity. We also introduced a new and original metric, the ssDNA-SWCNT backbone wrapping angle. This did provide us some unique insights into the physics of wrapping, however, the differences in behavior do not correlate with sequence-selectivity. The SASA, π-stacking and backbone wrapping angles all appear to be emergent quantities and not drivers in the adsorption.

The final metric of hydrogen bonding we believe is the driver of the observed free-energy differences, and hence, the origin of sequence selectivity. The results for the loading of (TAT)₄ on the (6,5) SWCNT show the intra-strand contributions per NB remain at a constant value while the inter-strand bonds per NB increase somewhat linearly. Recalling the growth in the excess free-energy with increased wrapping and comparing with this increase in inter-strand bonding shows that the inter-strand bonds are stabilizing. Intra-strand bonding therefore remains as the sole determining factor governing sequence selectivity. We have developed a classification dividing these intra-strand bonds into two categories, stabilizing and non-stabilizing (as opposed to destabilizing as discussed). We first deduced these two categories from conformation visualization in which it was noticed that intra-strand hydrogen bonds formed and broke apart between NBs that were relatively adjacent to each other and that this drove distortions of the conformation. We have quantified this by means of the intra-strand FES plots for both the analysis of increased loading and the comparison of maximum loadings. The heat map comparisons for the case of maximum loading show that sequences which are strong selectors contain many more hydrogen bonds in the stabilizing region, while the non-selectors not only have fewer bonds in the stabilizing region but few at all in which the ssDNA center of mass lies inside the tube radius. While both selectors and non-selectors contain intra-strand hydrogen bonds outside the stabilizing region, the non-selectors are dominated by this type and this prevents the strands from achieving more stable conformations which wrap the tube. We also observe the compactness of the free energy surface of the (TAT)₄ sequence in comparison to that of the other sequences. This suggests the presence of less fluctuations and a more narrow energy distribution that allows for improved separability as presented in our hypothesis. What drives these differences in hydrogen bonding is fundamental to understanding what makes some sequences good selectors for some tube chiralities but not for others, and the present work has given us a more focused metric to make use of in further studies.

While the relative free-energy measure we have developed does not answer the question of what is the optimal binding sequence for a given SWCNT chirality, it does enable us to now rank candidate sequences and this is great progress. We also note that the method could be used as a fitness function in a genetic evolution algorithm to optimize sequences although this may not be robust in the current state. A workflow for the more general problem of comparing the relative free-energy differences between different ssDNA-SWCNT sequence-chirality pairs is something that can easily be constructed based on these results, and we plan to investigate this at some point in continued work. This work is the first of its kind in applying FEP to multi-component, self-assembling systems. It has broader application to the general problem of solubility in systems which feature self-assembly and adsorption. Such systems have been difficult to characterize and are not amenable to analysis by methods used to analyze homogeneous dissolution such as Hansen Solubility Parameters.