Energetic Basis of Single-Wall Carbon Nanotube Enantiomer

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Energetic Basis of Single Wall Carbon Nanotube Enantiomer Recognition by Single Stranded DNA Akshaya Shankar, Ming Zheng, and Anand Jagota J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b05168 • Publication Date (Web): 18 Jul 2017 Downloaded from http://pubs.acs.org on July 18, 2017

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The Journal of Physical Chemistry

Energetic Basis of Single Wall Carbon Nanotube Enantiomer Recognition by Single Stranded DNA Akshaya Shankar†, Ming Zheng‡, Anand Jagota⁋* †

Department of Chemical & Biomolecular Engineering, Lehigh University

‡

Materials Science and Engineering Division and Semiconductor and Dimensional Metrology

Division, National Institute of Standards and Technology, 100 Bureau Drive, Gaithersburg, Maryland 20899, United States ⁋

Department of Chemical & Biomolecular Engineering and Bioengineering Program, Lehigh

University E-mail: [email protected] Abstract Hybrids of single stranded DNA and single walled carbon nanotubes (SWCNTs) have proven very successful in separating various chiralities and, recently, enantiomers of carbon nanotubes using aqueous two-phase separation. This technique sorts objects based on small differences in hydration energy, which is related to corresponding small differences in structure. Separation by handedness requires that a given ssDNA sequence adopt different structures on the two SWCNT enantiomers. Here we study the physical basis of such selectivity using a coarse grained model to compute the energetics of ssDNA wrapped around an SWCNT. Our model suggests that difference by handedness of the SWCNT requires spontaneous twist of the ssDNA backbone. We also show that differences depend sensitively on the choice of DNA sequence.

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1. Introduction Single walled carbon nanotubes (SWCNTs) are low dimensional tubular structures of a single layer of sp2 hybridized carbon atoms bonded in a hexagonal lattice except at their ends.1 SWCNTs can be chiral or achiral and their structure is related directly to their electronic and optical properties.2–6 Like other chiral molecules, chiral SWCNTs can exist as right handed and left handed enantiomers. Due to their extraordinary mechanical, electronic and optical properties7,8, SWCNTs have been demonstrated to be useful in a number of applications such as thin film transistors9,10, organic photovoltaics11, various types of biosensors12–17 and targeted drug delivery18,19. Many potential biomedical and sensing applications require separation of SWCNTs according to chirality and handedness such as optical transition based applications20–23, single photon light sources24, molecular sensing of amino acids25, penetration of SWCNTs into cell membranes26 and chirality based effects.27,28 For this reason, significant work has been done to develop separation techniques for sorting SWCNTs using techniques such as density gradient centrifugation29,30, chiral nanotweezers31,32,

gel

column

chromatography33,34,

chiral

molecules35,

DNA-assisted

separation36,37 and aqueous two phase separation38–40. Few of these techniques can separate SWCNT according to their handedness. For example, some previous work has been done in which SWCNT chiral enantiomers have successfully been separated using density gradient ultracentrifugation30,41 block co-polymers42 and most recently DNA-assisted separation using polymer aqueous two phase systems43. The structure of DNA on SWCNT has been probed in many ways including molecular dynamics (MD) simulations44–47, measuring activation energy of displacement of DNA by a surfactant48–50, AFM studies51–53 and aqueous two phase separations40. MD simulations have provided a set of structural motifs for how DNA adsorbs onto the SWCNT surface. Surfactant exchange studies, ATP studies and AFM studies provide information about activation barriers, differences in solvation energy, and binding free energy, respectively, for different DNA sequences on different SWCNT chiralities. In particular, there is strong correlation between recognition DNA/SWCNT pairs and the activation energy for separation of the two using a surfactant.48,49 The same recognition sequence/SWCNT pairs are also useful for separation of nanotube chiralities using the 2 ACS Paragon Plus Environment

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aqueous two phase technique40. Thus, differences in strength of binding between DNA and SWCNTs correlate with separability. DNA assisted methods of separation usually exploit the differences in the DNA structure on various SWCNT types. The purpose of this paper is to develop a model based on the energetics of DNA binding to SWCNT in order to understand the physical and structural basis for enantiomer separation. Based on all the previous work, we understand that separation relies on small differences in solvation free energy that are reflected in small differences in structure, for which we have as proxy small differences in binding energy of the hybrid. Accounting for the various contributions to this energy, such as adhesion between bases and the SWCNT, hydrogen bonding between bases, bending and torsion of ssDNA, we ask what it is that allows enantiomeric recognition to occur. 2. Methods 2.1. Representation and Structure of the Hybrid SWCNTs are usually labeled as (m,n), representing a roll-up vector comprising the sum of m and n basis vectors.54 The coordinates for the carbon atoms in each SWCNT of length ~ 40 nm were generated using the “Nanotube builder” in the VMD package.55 We used a coarse grained model for ssDNA in which each ‘mer’ is represented by two-beads. This is the simplest coarsegrained model that allows us to incorporate base-specificity and surface adsorbed structures with bases that alternate on each side.56 One bead represents the backbone (consisting of the sugar and phosphate) and the second bead is for the base. The co-ordinates for each bead were generated using code written in MATLAB®. Based on structural motifs observed in all atom molecular dynamics simulations, the DNA strand backbone is made to adopt a helical configuration and the DNA bases alternate on each side of the backbone.44,45,57–59 We considered single stranded 20 bases long (20-mer) DNA strands for this study, as sufficiently long to yield results that are essentially independent of strand length.” The SWCNT is placed along the Z-axis and the DNA backbone beads are arranged along a helical path which is co-axial with the SWCNT. The bond between the backbone bead and the base bead is perpendicular to the local tangent to the backbone, and alternates on either side of it.57 Given a starting point of the helical backbone, all bead positions are therefore specified once the helical angle or pitch is given. In case of a single DNA strand on the SWCNT, the position of the 3 ACS Paragon Plus Environment


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first backbone bead is sampled over the nanotube surface in the angular and axial direction. A representative figure of one DNA strand on nanotube model is shown in Figure 1, with the carbon nanotube atoms in black, DNA backbone and beads in blue, and DNA base beads in pink. The pitch angle ‘’is as defined in the Figure 1 inset, where ‘a’ is the helix diameter and 2c is the pitch of the helix. When the helix is left/right handed, ‘c’ and ‘’ are both negative/positive. When ‘’ is ± /2, the helix becomes a circle and when it is zero, the helix is a straight-line parallel to the helix axis (in this case the z-axis). Table 1 provides values of the several parameters used to construct the model.

Figure 1 DNA-SWCNT coarse grained model: conformation of an (AT)10 ssDNA sequence wrapped helically around a (9,1) SWCNT. SWCNT carbon atoms are in black, the DNA backbone beads (phosphate + sugar) are in blue, the blue line shows the helical backbone, and the DNA base beads are in pink.


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Table 1 Some parameters for DNA-SWCNT model co-ordinates Value used

Source

Radius of SWCNT i)

(9,1)

i)

0.373 nm

SWCNT Diameter

ii)

(1,9)

ii)

0.373 nm

= (n2 + m2 + nm)

iii)

(10,0)

iii)

0.392 nm

0.0783 nm 60

Distance between SWCNT

0.592 nm

Ref. 57

0.365 nm

Ref. 57

1/2

*

surface and ssDNA backbone (dcp) Distance between SWCNT surface and base (dcb) Radius of helix i)

(9,1)

i)

0.965 nm

SWCNT radius + distance

ii)

(1,9)

ii)

0.965 nm

from SWCNT to backbone

iii)

(10,0)

iii)

0.984 nm

i)

0.738 nm

SWCNT radius + Distance from SWCNT to base

Distance of base from SWCNT axis i)

(9,1)

ii)

0.738 nm

ii)

(1,9)

iii)

0.757 nm

iii)

(10,0)

Distance from backbone to base

0.64 - 0.8 nm

Ref. 61

0.49 - 0.6 nm

Ref. 61

0.65 nm

Ref. 57

(Adenine) (dpb-Ade) Distance from backbone to base (Thymine) (dpb-Thy) Distance

between

two

consecutive backbone beads (dpp)



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2.2. Energy Potentials Our goal is to calculate the energy of a DNA-SWCNT hybrid as a function of helical pitch and number of strands. We treat the SWCNT as rigid and so its carbon atoms are fixed and there is no need to specify or compute their internal interactions. The variable contributions that are summed to obtain the total energy are •

Adhesive interaction between DNA bases and SWCNT atoms,

•

Hydrogen bonding between DNA bases,

•

Bending and torsional energy of the DNA backbone

•

Base-backbone repulsion

•

Exclusion energy to prevent overlap of beads

It is assumed that electrostatic interactions are subsumed into the bending and torsional energies.

2.2.1. Adhesion energy between bases and SWCNT We represent the adhesive interaction between the base beads and the SWCNT by a 12-6 LennardJones potential 𝜎𝑎𝑑ℎ𝑒 12

𝐸𝑎𝑑ℎ𝑒 = ∑ 4 ∈𝑎𝑑ℎ𝑒 [(

𝑟𝑐𝑏

)

−(

𝜎𝑎𝑑ℎ𝑒 6 𝑟𝑐𝑏

) ]

(1)

Where rcb is the distance between the base bead and a SWCNT carbon atom. The values of σ adhe and εadhe for each type of base are obtained by calibrating equation (1) against experimental data reported by Iliafar et al. for adhesion energy per base for each DNA base on graphite

51

and

SWCNTs52. We set σadhe as 0.34 nm and found the values of εadhe for which Eadhe equaled the experimentally obtained values for the particular base and substrate in the above mentioned works on graphite and SWCNT. These values of εadhe and σadhe were then used in equation (1) to obtain adhesion energies between base beads and SWCNT carbon atoms. The values we found and used in our model are εadhe = 0.636 and 0.726 kT for A and T on graphite, respectively to get Eadhe ~ 9.9 kT per Ade on graphite and Eadhe ~ 11.3 kT per Thy on graphite. Here and elsewhere in this


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manuscript kT refers to the product of Boltzmann’s constant and room temperature of 300 K. Based on measurements of adhesion energy per base on SWCNTs, we obtain εadhe = 3.346 kT and 2.039 kT for A and T, respectively. Note that there is a significant and surprising increase in binding energy between ssDNA and SWCNTs compared to flat graphite. Unless otherwise stated, we have used the measured value for binding energy for ssDNA strands against surface deposited SWCNTs. Note that the ssDNA base-SWCNT interaction is sensitive to chirality in that a given DNA conformation will have different energies depending on the chirality of the SWCNT (e.g., (9,1) vs (1,9)) because it ‘sees’ a different arrangement of atoms in the two cases. However, this interaction has the following symmetry. If we take a certain ssDNA-SWCNT hybrid, say right-handed (TAT)4 on (9,1), then it will have the same adhesion energy as its mirror image, i.e., a left-handed (TAT)4 conformation on (1,9).

2.2.2. Bending energy of backbone We assign a resistance to bending that is quadratic in the bending angle, and allow the possibility of spontaneous bending, i.e., that the ssDNA is naturally bent. 1

𝐸𝑏𝑒𝑛𝑑𝑖𝑛𝑔 = 2 𝑘𝑏 (𝜃𝑏𝑒𝑛𝑑 − 𝜃0 )2

(2)

The bending stiffness includes intrinsic bending resistance and electrostatic self-repulsion. To establish the order of magnitude value of kb we rely on experimental measurements that show persistence length of ssDNA in solution to be about 8 Angstroms, although it can be quite a bit larger at lower salt concentration.62 For this reason we can treat this value as a lower limit. We estimate kb ~ 4kT (see SI). In eq. (2), bend is the angle formed by three consecutive backbone beads. So, for an ssDNA strand with ‘n’ bases, there will be (n-2) bending angles, which will all be identical for a given helix pitch. Most ssDNA exists in nature as part of the double helical structure, usually as B-DNA. In this structure, each of the two single strands has a curvature and a torsion, giving rise to the idea 7 ACS Paragon Plus Environment


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that each strand DNA ought to have a spontaneous bending angle and torsional angle. That is, we assume that the double stranded B-DNA conformation is the unstressed, relaxed state. Assuming that the spontaneous bending angle comes from the double stranded B-DNA found in nature, we find a value for θ0 to be approximately 30 degrees. (See supporting information S2.) Figure 2 shows how bending angle bend (in blue) changes as a function of the helix pitch angle. Note that it is symmetric with respect to handedness (negative pitch corresponds to left-handed helices; positive to right-handed ones.) Figure 3 shows the corresponding bending energy as described by equation (2) as a function of the helix pitch. Note that the spontaneous curvature makes the minimum in free energy at some bent state, but this is symmetric with respect to handedness. That is, a right handed and a corresponding left handed helix have the same bending energy.

Figure 2 Bending angle bend and torsional angle ang as a function of helical pitch angle (all the angles are shown in degrees but used in radians in equations (2) and (4)). 8 ACS Paragon Plus Environment

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Figure 3 Bending and torsional energy as a function of pitch angle for θ0 = 30 degrees, and φo = 15 degrees, kt/kb = 20 for (AT)10 on (9,1) SWCNT

2.2.3. Torsional energy Similarly, we assign to the single-stranded DNA backbone a spontaneous torsion corresponding to its native double-stranded B-DNA structure. Some studies on single stranded DNA have reported signals in circular dichroism experiments, consistent with spontaneous torsion.63–65 The torsional angle ϕang is formed by four consecutive backbone beads. So, for an ssDNA with ‘n’ bases, there will be (n-3) torsional angles, which will all be identical for a given helix pitch. We assign a resistance to torsion that is periodic in the torsional angle ϕang and allow the possibility of spontaneous torsion, i.e., that the ssDNA is naturally twisted. 9 ACS Paragon Plus Environment


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1

𝐸𝑡𝑜𝑟𝑠𝑖𝑜𝑛 = 2 𝑘𝑡 sin2(𝜙𝑎𝑛𝑔 − 𝜙0 )

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(3)

For simplicity, the value of the spontaneous torsional angle ϕ0 is taken to be independent of sequence with a value of 15 degrees. (See supporting information S2.)

We need to make a reasonable assumption for the value of kt in equation (3). It is known that torsional persistence length is roughly half of bending persistence length for double stranded DNA.66 To obtain estimates of reasonable bounds on kt, we assume single stranded DNA to be represented by a solid body with various shapes. The bending rigidity kb is proportional to the product of Young’s modulus and area moment of inertia about an axis within the cross-section of a rod, while the torsional rigidity kt is proportional to the product of shear modulus and area moment of inertia about its axis.67 If we assume the single stranded DNA to be a circular rod, then kt will be equal to 2kb/3 (see supporting information S3). If single stranded DNA is assumed to be a thin rectangular block, kt can be as large as 60 times the value of kb (see supporting information S3). Figure 2 shows how torsional angle ang changes as a function of the helix pitch. Figure 3 shows the torsional energy as described in equation (3) as a function of the helix pitch. Note that this torsional energy with spontaneous torsion introduces a dependence of energy on handedness. Right handed helices have much lower torsional energy than their corresponding left handed mirror images.

2.2.4. Hydrogen bonding between bases Molecular dynamics simulation studies have shown that non-Watson-Crick base pairing can be quite significant in DNA structures adsorbed on nanotube and graphite surfaces. 44,45,57,58,68 Hence, we assume that hydrogen bonding can occur between any two bases (including non-Watson-Crick pairs) if they are separated by at least two bases, i.e., the base n can have hydrogen bonding with all others on the same DNA strand except bases (n ± 1) and (n ± 2). Hydrogen bonding between 10 ACS Paragon Plus Environment

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base pairs separated by one or two bases are excluded to prevent the formation of three-membered rings which can only be formed by unrealistic backbone chain twisting and interactions through the phosphate backbone.69 We represent the hydrogen bonding energy between the base beads by a 12-6 Lennard Jones potential, multiplied by a factor that is a Gaussian with argument αHB 2 . This allows us to capture the strong directionality of the hydrogen bonding interaction which decays rapidly if the angle αHB differs from 00. 12

𝜎

𝐸 = ∑ 4 ∈𝐻𝐵 [( 𝑟𝐻𝐵 ) 𝑏𝑏

6

𝜎

2

α

− ( 𝑟𝐻𝐵 ) ] ∗ exp (− 2 HB ) 𝑠2 𝑏𝑏

(4)

𝐻𝐵

Where rbb = distance between the base beads, αHB is the angle formed by backbone bead, base and another base, sHB is the deviation allowed for the angle which we assume to be 30 degrees. We fitted the Lennard Jones 12-6 potentials to the Potential of Mean Force (PMF), which is free energy as a function of distance between the two bodies, obtained from Molecular Dynamics simulations for various DNA bases on graphite previously68. Thus, we can obtain the values for εHB and σHB. εHB for A-T = 3.70, εHB for A-A = 2.70, εHB for T-T = 1.37 (kT) σHB for A-T = 0.53 nm, σHB for A-A = 0.59 nm, σHB for T-T = 0.77 nm In case we have Gua and Cyt also in the DNA sequence, we will have to account for the remaining possible 7 hydrogen bonding base pairs as well.

2.2.5. Repulsion between base and backbone of adjacent strand: In order to preclude the possibility of a DNA base crossing over an adjacent strand to interact with a non-adjacent base, we introduced an energy penalty: if the base gets closer than a distance equal to the distance between consecutive backbone beads (dpp) from a backbone bead on an adjacent strand, a hard wall energy potential is introduced. All the energy data reported henceforth are Boltzmann averaged energies , calculated for a 100 nm DNA-SWCNT hybrid with error being the fluctuations in the energy which is equal to - 2. 11 ACS Paragon Plus Environment


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Table 2 Parameters for energy calculations (force field)

Parameters

Values

Source

Adhesion energy between σadhe = 0.34 nm Calibrated using experimental value bases and graphite for adhesion energy51 εadhe (Ade) = 0.636 kT εadhe (Thy) = 0.726 kT Adhesion energy between σadhe = 0.34 nm Calibrated using experimental value bases and SWCNT for adhesion energy52 εadhe (Ade) = 3.346 kT εadhe (Thy) = 2.039 kT Bending energy

kb = 4kT

Supporting information S2 and S3

θ0 = 30 degrees Torsional energy

kt = 2/3 to 60 times kb Supporting information S2 and S3 ϕ0 = 15 degrees

Hydrogen bonding energy

εHB = 2.7 kT, (A-A)

68

σHB = 0.59 nm (A-A) εHB = 3.7 kT, (A-T) σHB = 0.53 nm (A-T) εHB = 1.37 kT, (T-T) σHB = 0.77 nm (T-T)

3. Results and discussion: Figure 4 shows a typical plot of average energy per unit length as a function of helical pitch angle for the DNA wrap, for one strand of (AT)10 on (9,1) SWCNT. The plot shows two energy


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minima for one strand of DNA on the SWCNT surface, where one minimum (AR) corresponds to the right handed DNA helix (positive pitch angle) and the other (AL) corresponds to the left handed DNA helix (negative pitch angle). Whichever of the two minima is lower corresponds to the more stable handedness of the hybrid for that particular case.

Figure 4: Average total energy points corresponding to one DNA strand around nanotube, corresponding to left handed and right handed configurations. 3.1. Recognition of handedness requires spontaneous torsion In order to explore systematically which contributions to the energy matter most for recognition of handedness, we calculated the total energy per unit length for a single strand of (AT)10 on various achiral and chiral SWCNTs while considering the different contributions to the energy as shown in Table 3. All the values are shown with respect to a reference energy value



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indicated in the table for each case, as the adhesion energy is about two orders of magnitude larger than other energy contributions and we wish to focus on the relatively small differences in energies that account for recognition. First we consider the DNA sequence (AT)10 wrapped on an achiral nanotube (10,0) in row 1 of Table 3. (10,0) is an achiral SWCNT which is close to (9,1) and (1,9) in diameter. We consider the energy contributions from adhesion, hydrogen bonding and exclusion energy. From the values AL and AR in row 1, we see that the total energy as a function of helix pitch is symmetric with respect to pitch angle; there is equal preference for both handedness. Next we considered (AT)10 wrapped on chiral SWCNT enantiomers (9,1) and (1,9). We take into account the energy contributions from adhesion, hydrogen bonding and exclusion energy. As seen in row 2, column AL and AR, the left handed helix (AL) is slightly more stable on (9,1) compared to the right handed helix (AR). The situation is reversed in case of (1,9), as seen in row 3, where the right handed helix is slightly more stable. We can also see in rows 2 and 3 that a lefthanded helix on (9,1) mirrors the right-handed helix on (1,9). It means that there is no basis for separation of (9,1) and (1,9) if we only consider the energy contributions from adhesion, hydrogen bonding and exclusion energy.


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Table 3 Handedness preference for one DNA strand on various SWCNTs SWCNT: Carbon nanotube chirality DNA: DNA sequence Adhesion: Adhesion between nanotube and DNA bases Actual energy values = reference energy + actual number in the table cell

SW-

DNA

Adhesion

HB

Bending

Torsion

Excl.

Ref. Energy

AL

AR

(kT/100nm)

(kT/100nm)

(kT/100nm)

-16839.94

-0.03

0.063

± 0.604

± 0.712

-1.13

0.69

± 1.222

± 1.465

0.53

-0.10

± 1.128

± 1.125

-1.47

1.67

± 1.222

± 1.559

0.19

-0.38

± 1.128

± 1.124

1814.78

-1810.65

± 0.802

± 0.365

1703.11

-1707.24

± 0.270

± 1.192

CNT

1

2

(10,0)

(9,1)

(AT)10

(AT)10

✓

✓

✓

✓









✓

✓ -16623.67

3

4

(1,9)

(9,1)

(AT)10

(AT)10

✓

✓

✓

✓



✓





✓

✓ -16615.68

5

6

(1,9)

(9,1)

(AT)10

(AT)10

✓

✓

✓

✓

✓

✓



✓

✓

✓ -12773.31

7

(1,9)

(AT)10

✓

✓

✓

✓

✓

As seen in row 4, if we include bending energy, the left handed helix on (9,1) is still slightly more stable as compared to the right handed helix. Row 5 shows that the reverse holds for (1,9) and the



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opposite handed helices on the two SWCNT enantiomers still are nearly equally stable within the error margin. This shows that bending energy also cannot explain the possibility of separation of two SWCNT enantiomers wrapped by the same DNA sequence. Figure 3 shows that bending energy is actually symmetric around zero pitch. Therefore, if one starts with a mixture of enantiomers, there is again no basis for separation of chiral nanotube enantiomers. In rows 6 and 7, we include additionally the contribution of torsional energy with a spontaneous torsional angle of 15 degrees, using a ratio of constants for torsional and bending energies (kt/kb) = 20. Now we see that the right handed helix (AR) has lower energy for both the SWCNT enantiomers (9,1) and (1,9), and the right handed helix is more stable on (9,1) as compared to (1,9). This energy difference (which is expressed here in units of kT /100 nm) for typically long SWCNTs (average length of SWCNTs used in separation experiments vary between 200 to 700 nm) is substantial. The crucial factor that allows for difference in energy minimum is torsional energy, when one invokes a spontaneous torsion angle, coupled with the intrinsically chiral arrangement of atoms on an SWCNT surface.

3.2. Recognition of handedness depends on DNA sequence, geometry and SWCNT chirality The model used here relies on various parameters which are known with various levels of confidence. In particular, the parameters that characterize the torsional energy are poorly known. Here, we study the effect of some of the model parameters to check on the robustness of our main conclusion. We also study how the results depend on the DNA sequence and the SWCNT chirality. In Table 4, we consider a DNA strand wrapped around the SWCNT for several different cases. From the previous section, we know that torsional energy is critical in enabling separation of DNA wrapped SWCNT enantiomers. We consider the effects of altering the value of torsional rigidity kt, eq. (3). Specifically, we consider wrapping of the (9,1) SWCNT by (AT)10 for kt = 1, 10 and 20 times the values of kb (rows 1,2 and 4 of Table 4). For the lowest value of kt, the righthanded configuration is lower in energy by ~ 180 kT/100 nm. On increasing kt to 10 times kb, the preference for the right handed configuration increases to ~1800 kT/100 nm, i.e., not surprisingly, the quantitative energy difference depends strongly on the strength of backbone torsional energy.


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In all the data presented previously, we used the values of distance from backbone to base beads from a three bead model by Knotts et al.61 Since we have a two bead model here, the distance is somewhat underestimated. To understand the effect of base-backbone distance, we increase these values from 0.68 to 0.8 nm in case of Ade and 0.49 to 0.6 nm in case of Thy (see row 3 of table 4). The right handed configuration is again preferred over the left-handed one by a similar energy difference (compare rows 2 and 3). For the data shown in table 3 and table 4 rows 1 to 4, we have used adhesion energy data from the peeling energy data of single stranded DNA from carbon nanotube surfaces by Iliafar et al.52 which is considerably higher than on graphite substrate51. If we use the adhesion energy data from graphite instead, the adhesion energy becomes lower but the same preference for the right handed configuration is still exhibited (see row 5 of table 4). Thus, our results are quite robust and do not depend on whether graphite or SWCNT substrate is used to calibrate the adhesion energy parameters.



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Table 4 Energy differences for varying parameters [Actual energy value = Reference (Ref.) Energy value + actual number value in the table cell] CNT DNA

kt /

Substrate

Base -

Ref.

AL

AR

kb

used

Backbone

Energy

(kT/100

(kT/100

for adhesion

Distance

(kT/100nm) nm)

nm)

calibration

(Ade/Thy)

CNT

0.64 / 0.49 nm

-16522.39

89.45

-89.44

± 1.222

± 1.465

902.39

-902.40

± 1.351

± 1.465

932.32

-932.32

± 0.779

± 0.365

1812.72

-1812.72

± 0.802

± 0.365

89.59

-89.59

± 1.713

± 0.7534

1 (9,1) (AT)10 1

2 (9,1) (AT)10 10

3 (9,1) (AT)10 10

4 (9,1) (AT)10 20

5 (9,1) (AT)10 1

CNT

0.64 / 0.49 nm

CNT

0.8 / 0.6 nm

CNT

0.8 / 0.6 nm

Graphite

0.64/0.49 nm

-15677.91

-13662.03

-12771.25

-4087.31

In Table 5, we consider the effect of changing the DNA sequence or the SWCNT chirality. In rows 1 and 2, the same set of parameters were used for (AT)10 on (9,1) and (1,9) enantiomers. Both showed preference for the right handed helix, but there is an energy difference of ~103 kT/100nm between the right handed configuration on (9,1) and (1,9). For the same set of parameters, we changed the DNA sequence from (AT)10 on (9,1) and (1,9) to A20 on (9,1) and (1,9) (see rows 3 and 4). In case of A20, which is also a 20mer like (AT)10, the most preferred configuration is still the right handed single helix just as in case of (AT)10, but the energy difference between the left and right handed configurations is now reduced to ~68 kT/100nm which in turn will reduce their separability. Similarly, when we changed the SWCNT chiralities from (AT)10 on (9,1) and (1,9) to from (AT)10 on (6,5) and (5,6) (see rows 5 and 6), the energy difference between the left and right handed configurations is further reduced to ~23 kT/100nm which in turn will further reduce their separability. Thus, for a given set of parameters, 18 ACS Paragon Plus Environment

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the preference for a particular configuration depends on the DNA sequence and SWCNT chirality, as has been established experimentally by Geyou et al.40 where different DNA sequences are used to separate different SWCNT chiralities and enantiomers. Table 5 Energy differences for different DNA-SWCNT combinations

CNT DNA

kt / kb

Substrate

Base –

Ref.

AL

AR

used for

Backbone

Energy

(kT/100

(kT/100nm)

adhesion

Distance

(kT/100

nm)

calibration (Ade/Thy) 1 (9,1) (AT)10 20

CNT

0.8 / 0.6 nm

2 (1,9) (AT)10 20

3 (9,1) (A)20

20

CNT

CNT

nm) 1814.78 -12773.31 ± 0.802

20

5 (6,5) (AT)10 20

CNT

CNT

1703.11

-1707.98

nm

± 0.270

± 1.192

0.8 nm

1789.17

-1787.98

0.8 nm

0.8 / 0.6 nm

6 (5,6) (AT)10 20

CNT

± 0.365

0.8 / 0.6

-16055.14 ± 1.804 4 (1,9) (A)20

-1810.65

± 0.284

1718.40

-1719.58

± 0.837

± 1.545

1771.75

-1771.87

-12640.88 ± 1.341

± 0.554

0.8 / 0.6

1748.57

-1748.45

nm

± 1.007

± 1.018

Our results have shown that both a chiral arrangement of SWCNT atoms and spontaneous torsion are required for the DNA/SWCNT hybrid to depend on SWCNT handedness. If the SWCNT is achiral, then of course there aren’t two enantiomers to separate in the first place. If the SWCNT is chiral, but there is no torsional energy, then again there is no basis for separation (see again rows 4,5 in Table 3). Therefore, one can think of SWCNT chirality as providing an energy difference between left and right handed wraps and torsion additionally providing the difference that enables



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Page 20 of 32

separability. The torsional stiffness kt characterizes the role of torsion; we now define another parameter, kc to quantify the role of SWCNT chirality. Consider the total energy contribution per nm of the SWCNT-DNA hybrid from adhesion, bending, hydrogen bonding and exclusion energy, i.e., all terms excluding torsion. We can obtain two minima for this energy as a function of the pitch angle, one for the right handed helix (AR) and one for the left handed helix (AL). We define kc as the energy difference between the AR and AL. The reported kc values are Boltzmann averaged energies , calculated for a 100 nm DNA-SWCNT hybrid with error being the fluctuations in the energy which is equal to - 2. Figure 5 and Table 6 show kc for a few different SWCNT chiralities. We see that there are significant differences between the different chiralities but no clear trend, except for that achiral SWCNTs such as (6,6), (6,0) and (10,0), kc is zero, as it must be. The large difference between (AT)10 on (6,5) versus (9,1) seen in Table 5 can be attributed to the large difference in kc, as shown in Figure 5 and Table 6.


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Figure 5: kc (kT/100 nm) as a function of the nanotube chirality coordinates ‘n’ and ‘m’



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Table 6 Difference in kc for various nanotube chiralities

Nanotube chirality

kc (kT/100 nm)

(6,5)

23.26 ± 1.77

(9,1)

103.41 ± 1.14

(8,3)

23.37 ± 2.36

(7,5)

7.28 ± 1.70

(7,3)

47.85 ± 1.54

(6,4)

14.48 ± 1.40

(10,2)

32.14 ± 1.02

(6,0)

0.06 ± 1.26

(6,6)

0.00 ± 0.10

(10,0)

0.15 ± 2.02

The discussion so far regarding Tables 3 to 5 have taken into account the chirality contributions from both the SWCNT and the DNA together (i.e. both kc and kt). To illustrate the fact that both kc and kt are needed for separability of the two nanotube enantiomers consider the following scenarios. a) (AT)10 wrapped on (10,0) and (0,10) without torsional energy. (i.e. kc = 0, kt = 0). Clearly this is an achiral case so there are no enantiomers to begin with.

b) Next, instead of an achiral nanotube like (10,0), let us consider (AT)10 wrapped on the chiral SWCNT enantiomers (9,1) and (1,9) but with no torsional energy. (i.e. kc ≠ 0, kt = 0). The


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right handed DNA helix on (9,1) and the left handed DNA helix on (1,9) are equally stable. Thus, there is still no separability.

Chirality Ref. Energy

(9,1)

Right handed helix (AR) Left handed helix (AL)

(kT /100nm) (kT /100nm)

(kT /100nm)

-16623.67

-1.13 ± 1.222

0.69 ± 1.465

0.53 ± 1.128

-0.10 ± 1.125

(1,9)

c) Now consider the case when (AT)10 is wrapped on (9,1) and (1,9) but there is no torsional energy. (i.e. kc ≠ 0, kt = 0) Chirality Ref. Energy

(9,1)


(kT /100nm) (kT /100nm)

(kT /100nm)

-16615.68

-1.47 ± 1.222

1.67 ± 1.559

0.19 ± 1.128

-0.38 ± 1.124

(1,9)

In this case, just like the previous case, there is no separability as the left handed configuration on (9,1) is as energetically stable as the right handed one on (1,9) within the error margin. d) Now for the same scenario, we include the contribution from torsional energy (i.e. kc ≠ 0, kt ≠ 0). The right handed configuration is now more stable for both the nanotube enantiomers as the torsional energy prefers the right handed DNA configuration. But there is still an energetic difference of ~103 kT/100 nm between the right handed configuration on (9,1) and (1,9), which allows separability.



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Chirality Ref. Energy


(kT /100nm) (kT /100nm) (9,1)

-12773.31

(1,9)

Page 24 of 32

(kT /100nm)

-1810.65 ± 0.365

1814.78 ± 0.802

-1707.24 ± 1.192

1703.11 ± 0.270

4. Conclusion Previous surfactant exchange experiments48 have shown that certain recognition DNA sequences/partner SWCNT chiralities have significantly higher activation energy of binding.48,49 The same recognition sequence/SWCNT pairs are also useful for separation of nanotube chiralities using the aqueous two phase technique40, showing that the hydrophobicity of these special pairs is different from others, pointing to differences in the structure of DNA on different SWCNT chiralities and enantiomers. These differences are subtle, but more than sufficient to permit separation of SWCNT enantiomers using DNA-assisted aqueous two phase separation43, which is very sensitive to small differences in solvation energy. The model presented in this work investigates the energetic basis for separability. We show that both torsional energy and SWCNT chirality are necessary for separability. For a given chiral SWCNT, torsional energy causes a difference in the stability of the opposite handed DNA helices on the nanotube enantiomers. We also find strong dependence on DNA sequence/SWCNT pairing of the energy differences that permit separation. Supporting Information Estimating value of bending rigidity, spontaneous bending angle, spontaneous torsional angle and torsional rigidity Acknowledgements This work was supported by the National Science Foundation through grant CMMI-1014960.


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