A Mathematical Formulation and Solution of the CoPhMoRe Inverse

J.A. is grateful for the support of the Samsung scholarship, S.K. is grateful for a postdoctoral fellowship from the Deutsche Forschungsgemeinschaft (...
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A Mathematical Formulation and Solution of the CoPhMoRe Inverse Problem for Helically Wrapping Polymer Corona Phases on Cylindrical Substrates Gili Bisker,‡ Jiyoung Ahn,‡ Sebastian Kruss, Zachary W. Ulissi, Daniel P. Salem, and Michael S. Strano* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States S Supporting Information *

ABSTRACT: Corona phase molecular recognition (CoPhMoRe) is a new technique that generates a nanoparticle-coupled polymer phase, capable of recognizing a specific molecule with high affinity and selectivity. CoPhMoRe has been successfully demonstrated using polymer wrapped single walled carbon nanotubes, resulting in molecular recognition complexes, to date, for dopamine, estradiol, riboflavin, and L-thyroxine, utilizing combinatorial library screening. A rational alternative design to this empirical library screening is to solve the mathematical formulation that we introduce as the CoPhMoRe inverse problem. This inverse problem seeks a linear function representing the position of monomers or functional groups along a polymer backbone that results in a 3-dimensional structure capable of recognizing a specific molecule when mapped to a nanoparticle surface. The potential solution space for such an inverse problem is infinite in general, but for the specific constraint of a helically wrapping polymer, mapped to a cylindrical nanoparticle, we show in this work that two types of inverse problems are exactly solvable. In one case, the polymer pitch and composition can be designed to allow for the specific binding of a small molecule analyte in the occluded space on the nanotube surface. In the other, a larger macromolecule can interact with a deformed helix, which partially conforms to it. A simplified, coarse-grained molecular model of a helically wrapping polymer demonstrates the inhomogeneous binding potential formed by a wrapping with a given pitch. Calculating the potential maps for various pitch values illustrates that there is an optimal pitch that enables the selective and specific binding of the target analyte. An additional coarse-grained model of a helical wrapping by a polymer consisting of alternating hydrophobic−hydrophilic segments demonstrates the resulting deformed helix corona around the nanotube, which forms accessible binding pockets between the hydrophilic loops. While these are the idealized forms of actual CoPhMoRe phases, the formation and solution of such inverse problems may serve to reduce the dimensionality of library screening for CoPhMoRe discoveries, as well as provide a theoretical basis for understanding certain types of CoPhMoRe recognition.

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INTRODUCTION The ability to selectively recognize molecules with high specificity is crucial for innumerable biological and chemical processes and has widespread applications in diagnostics and therapeutics. Antibodies have been used for these purposes for over 50 years,1 however, their production involves a living organism, which can pose a limitation in discovery research.2 Powerful combinatorial approaches have allowed for synthetic antibody development, without the need for a living host, by diversifying the genes encoding for the antigen binding sites within known antibodies, displaying them on the surfaces of phages, and screening against a library of analytes.3 Moreover, aptamers, which are protein binding oligonucleotide sequences, were discovered and developed by an in vitro screening method of systematic evolution of ligands by exponential enrichment (SELEX).4 Additionally, dynamic combinatorial chemistry (DCC) approaches have led to the discovery of new receptors by shifting thermodynamic equilibria of interconverting building blocks in order to find stable complexation with a target molecule.5,6 More recently, our laboratory has introduced © 2015 American Chemical Society

the concept of corona phase molecular recognition, or CoPhMoRe, whereby a nanoparticle, such as a nanotube, imposes a structure onto a selected polymer that then generates a recognition site for a specific analyte. CoPhMoRe has been successfully demonstrated using polymer wrapped single walled carbon nanotubes, resulting in molecular recognition complexes to date for dopamine,7 estradiol, riboflavin, L-thyroxine,8 and the protein fibrinogen, utilizing combinatorial library screening. CoPhMoRe, like the other techniques described, relies on combinatorial selection, much like the combinatorial processes in antibody development over billions of years of evolution. Synthesizing novel synthetic molecular recognition sites using judiciously rational design and engineering has been a central goal for researchers, who have been using numerous strategies to address this task.9 Received: February 19, 2015 Revised: April 24, 2015 Published: May 13, 2015 13876

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Figure 1. (a) An analyte projected onto a surface with functional groups at p1 and p2. (b) Linear polymer with functional groups at g1 and g2, such that they are colocalized with the functional groups of the analyte when wrapped around a carbon nanotube. (c) Polymer and analyte presented in the (η, z) plane. (d) Parametric functions of the edges of the polymer and the matching of the functional group locations. (e) Illustration of available functional groups on a potential target analyte, human serum albumin.39

In this work, we propose the first steps in the rational design of CoPhMoRe phases by introducing the formulation and solution to this inverse problem. This inverse problem, in this case, is posed as seeking a linear function of the polymer sequence or composition that results in a 3-dimensional structure capable of specific molecular recognition when mapped to the nanoparticle surface. Polymer molecular imprinting is an example of a molecular recognition approach with a deterministic connection between analyte and receptor synthesis, since polymerization in the presence of a target molecule is followed by its removal, resulting in a molecularly imprinted cavity template for specific recognition of the chosen target.10−12 Analogously, molecularly imprinted organic polymer nanoparticles were successfully used for capturing a target peptide in vivo.13 In contrast, the recent demonstrations of CoPhMoRe used an empirically determined corona phase formed from a synthetic polymer adsorbed onto the surface of semiconducting single walled carbon nanotubes (SWCNT), rendering them near-infrared (nIR) fluorescent biosensors.14−16 SWCNTs are cylindrical tubes, consisting of a monolayer carbon honeycomb lattice rolled into a cylinder, whose diameter is in the nanometer scale. Owing to the 1-dimensional confinement, SWCNTs have unique optical and electronic properties,17,18 which can be dramatically affected by small perturbations such as the adsorption of a single molecule.19 Semiconducting SWCNTs fluoresce in the nIR range,20 overlapping with the tissue transparency window and allowing for deep tissue penetration.21 Other advantages of SWCNTs for biomedical applications are nondetectable blinking, lack of photobleaching, and a large Stokes shift.22 SWCNTs, which have hydrophobic surfaces, must be modified with hydrophilic groups to enable solubility in an aqueous solution and prevent aggregation. Typically, amphiphilic polymers or surfactants are used for noncovalent modification, since they can bind or wrap around the

SWCNT, yielding a stable colloidal suspension by mediating hydrophobic interaction.23−25 Interestingly, the nIR fluorescence of such SWCNT−polymer hybrids was shown to be affected by certain molecules.15 In this scheme, the organic (polymer) phase around a carbon nanotube determines how and if a molecule changes the nIR fluorescence. The SWCNT itself serves as a scaffold on which the polymer is folded and forms a unique 3-dimensional organic phase that interacts and responds to other molecules. Therefore, the prototype of a fluorescent SWCNT sensor is a SWCNT−polymer hybrid structure. The CoPhMoRe concept8 was successfully demonstrated utilizing polymers that have no known inherent affinity for an analyte but can enable detection if the polymers are confined onto the SWCNT sensor surface.8 In turn, this concept implies that heteropolymers form a specific 3-dimensional structure, either rigidly or dynamically, on the SWCNT, which is able to recognize the analyte (i.e., a “synthetic antibody”). Using this concept, fluorescent SWCNT sensors for different substances such as reactive oxygen species, neurotransmitters, or sugars have been reported.19,26,27 In contrast, the corona in other approaches serves only as a scaffold for immobilization of known recognition units such as antibodies, as in the case of protein detection and glycosylation pattern characterization.22,28 Until now, polymer−SWCNT complexes capable of selectively recognizing specific target analytes were discovered by extensive screening of libraries of functionalized SWCNT against libraries of molecules of interest. Interesting events, in which the interaction of the polymer−SWCNT complex with the target molecule resulted in the modulation of the intrinsic fluorescent emission of the SWCNTs,20 were recorded. Detectable modulation of the emission intensity and a shift in the emission peak wavelength16 are considered as detection events if the response is selective and specific. A model for a molecular recognition mechanism previously proposed by our 13877

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The Journal of Physical Chemistry C group8 treated only the hydrophobic polymer anchors and did not include the hydrophilic chains in the equation of state.29 Here, we present a simple mathematical model that addresses the inverse problem of engineering a corona phase for molecular recognition. It applies only to the case of a helical wrapping of a polymer around a cylindrical nanotube, and we hypothesize that binding pockets for the analyte of interest can be formed by matching its physical dimension to the helix pitch. DNA is an example of a linear polymer known to wrap SWCNT helically.24 We list several helically wrapping DNA sequences and their corresponding pitch values, as reported in the literature, and suggest additional functional groups that can enhance selectivity and specificity. Moreover, we formulate an illustrative, coarse-grained model of a SWCNT helically wrapped by a functionalized linear polymer, and we calculate the total enthalpy of the system as a function of the position of a model target analyte on the SWCNT surface. A rigorous molecular dynamic simulation or structure determination is beyond the scope of the current work. However, the approach allows us to generate a map of the binding potential for a given pitch value, which demonstrates the existence of an optimal pitch for a selective and specific recognition. A different case, which we describe mathematically, is of a polymer with hydrophilic and hydrophobic segments, whose hydrophilic portions create binding pockets on the SWCNT surface when the polymer is helically wrapped around the nanotube. The polymer chain can be tailored such that the resulting corona configuration would fit a cross section of the target analyte, promoting a selective binding based on the shape and size of the target. This system is also illustrated by a simplified coarsegrained model, which demonstrates the final conformation of such a polymer around the cylindrical nanoparticle. Our work provides design and engineering tools for future development of novel SWCNT-based sensors. Such sensors have huge potential to impact biomedical research and applications in both diagnostics and treatment fields.30−33

describe how to program the recognition site in the 1dimensional space of the polymer. In the reference frame of the 3D cylinder, (x, y, z), with the z axis orientated along the axis of the cylinder, we can generate a 2D mapping of the 3D cylinder surface as follows: y = R sin(η /R ) x = R cos(η/R )

(1)

In the 2D system of (η, z), we can define

η = R arctan(y/x)

(2)

We can develop a parametric set of equations that can describe the centerline of the helically wrapping polymer, with a parameter t. Let γ be the slope of parametric representation of the wrapping polymer:

η = γt z=t

(3)

The centerline of the wrapping polymer then traces the following periodic function in (η, z) space (Figure 1c): η = arctan(tan(γz))

(4)

The pitch of the wrapping, or distance between adsorbed segments along the axis of the cylinder, can be expressed in terms of the cylinder radius and slope in the 2D projection as follows: p=

2πR γ

(5)

This can be used to solve for a critical design consideration. The width of the analyte, h, which resides orthogonal to the polymer path, and the width of the polymer, w, can be solved in terms of the pitch,

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h=

THEORETICAL BASIS. CASE 1: MOLECULAR RECOGNITION INTERSTITIALLY WITHIN A REGULAR HELIX The specific CoPhMoRe inverse problem of interest in this paper is described using a simplified polymer wrapping around SWCNT. In this work, we constrain the solution to a polymer that forms a helical wrapping motif around the cylinder. Analytes of interest are projected onto a surface for adsorption approximated as a box of width h and length La, displaying functionalities located at p1 and p2 along the longitudinal direction of the molecule (Figure 1a). The inverse problem seeks to identify a polymer, correspondingly functionalized at g1 and g2, such that, when adsorbed onto a surface of a cylinder of radius R, it creates a specific recognition site for the analytes, by near proximity of g1 and p1 and simultaneously g2 and p2 (Figure 1b). For simplicity, we considered DNA as the wrapping polymer, and assumed that the functional groups form H-bonds with the DNA backbone. The list of possible candidates for p1, p2, g1, and g2 includes hydroxy, carbonyl, aldehyde, amine, etc. It was assumed that a fluorescent response of DNA wrapped SWCNT only depended on H-bonds between the analytes and the DNA, and that each H-bond contributes equally to the response of the SWCNT, regardless of its location. Such wrapping has been robustly observed for DNA, as shown in the literature.8,21,34−38 We ultimately seek to

2πR sin(arctan γ ) − w γ

(6)

or ⎛ ⎛ 2πR ⎞⎞ h = p sin⎜⎜arctan⎜ ⎟⎟⎟ − w ⎝ p ⎠⎠ ⎝

(7)

This equation implies that, for the given analyte and polymer, a requisite pitch, p, is required to match the former with the latter. This minimal design criterion simply ensures that the available spacing on the cylinder surface allows for analyte adsorption. When satisfied, the equation produces an adsorbed analyte with projected edges roughly parallel to the polymer wrapping path. The upper and lower boundaries of the wrapping polymer on the SWCNT are defined as f1 and f 2, which can be expressed with respect to the centerline of the wrapping polymer cos(arctan γ ) =

w 2(f1 − γt )

cos(arctan γ ) = −

w 2(f2 − γt )

(8)

The parametric functions f1 and f 2 can then be expressed as follows: 13878

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Figure 2. Schematic illustration of a polymer of which repeating units of hydrophobic and hydrophilic segments form unique binding pockets for a specific analyte when adsorbed onto the surface of a single walled carbon nanotube.

Figure 3. Plot of the 3D structure of the engineered polymer when helically adsorbed onto the SWCNT surface. Blue and red segments represent 2

hydrophilic and hydrophobic segments, respectively. (a) f(z) = const, (b) f(z) = 1 + sin(z), (c) f(z) = z2 e−(z/2) . (d, e, f) The corresponding linear mapping of the hydrophilic and hydrophobic segments to a 1-dimensional space, respectively. (g) An example of f(z) chosen to fit a 2-dimensional cross section of the protein albumin, and (h) the corresponding distorted helix.

13879

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protein circumferentially. A segment of this curve was then incorporated into the deformation f(z) of the distorted helix function (Figure 3g), creating a binding pocket for the albumin (Figure 3h). Similarly, such a function can be chosen for any analyte based on its unique conformation, given that the physical dimensions of the hydrophilic domains of the wrapping polymer are comparable to those of the target analyte.

f1 = γt +

(9)

In this frame, p1, p2, g1, and g2 lie along the same axis. Matching them is simply a matter of having them share the same origin of axis (Figure 1d). Hence, eqs 6 and 9 exactly solve the inverse problem of analyte adsorption for this helical constraint.

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RESULTS AND DISCUSSION 1. Observed Pitch Values of DNA Wrapping around SWCNT. Many attempts have been made to measure the pitch of DNA wrapping on the surface of SWCNT. The pitch values of DNA with various sequences are summarized as reported in the literature in Table 1.

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THEORETICAL BASIS. CASE 2: MOLECULAR RECOGNITION ON A DISTORTED HELIX Another type of solution to the CoPhMoRe inverse problem, subject to a different set of constraints, can be formalized by considering a distorted helix. Suppose that now the polymer contains alternating polymer segments of nanotube adhesive and repulsive chains. As the polymer adsorbs, it does so necessarily such that the adhesive domains lie along the contours of the nanotube, and the repulsive segments desorb into the solution, requiring the system to otherwise adopt a helical configuration. When mapped to the cylindrical nanotube, this time the 3D structure can be shown to possess the property that any arbitrary analyte cross section can be described by the distortions of the helix, as shown in Figure 2. The mathematical description of such a system is a helix as before, but with a nonconstant radius, R′. If f(z) is the function along the axial dimension of the nanotube that describes the grooves or binding sites for the analyte (see Figure 2), then R′ can be found such that the distorted loops map the f(z) function:

Table 1. Pitch of DNA Wrapping around SWCNT in the Literature Pitch (nm)

Based on

Reference

18 13 ± 5 10 ± 4 2−8 2.2 14−20 3.3 3.2

AFM AFM AFM MD HRTEM AFM STM MD

38 34 34 21 35 36 37 37

The pitch of helical DNA wrapping varies from 2 nm up to 20 nm. Although the measured pitch varies widely even for the same sequence, these literature values can still pose the upper and lower boundaries of the possible pitches for DNA. Based on these values, many molecules can be targeted, since a diverse library of analytes exists in this window. Averaged size globular proteins41 are one example. Particularly, human serum albumin (HSA), whose dimensions are 8−13 nm,39,42,43 can be a potential target analyte as shown in Figure 1e. We note that other synthetic polymers have the capability to wrap helically as well, and can serve as candidates for the direct design of CoPhMoRe phases given a target analyte. 2. Ratio of Hydrophilic and Hydrophobic Moieties and the Resulting Recognition. Helical wrapping of cylindrical substrates has been reported not only with DNA8,21,34−38 but also with other polymers.44−46 Therefore, once we create the polymers with the functional groups and hydrophilic/hydrophobic moieties, it is possible to construct the resulting corona phases with various configurations as depicted in Figure 2 and Figure 3. If the functional groups for recognition (g1, g2) exist in the hydrophobic moiety, the adsorption of analytes must precede the recognition. This means that only the analytes, which have the right size to match the pitch of the hydrophobic wrapping, can be detected. In the other case where the functional groups lie on the hydrophilic moiety, the size of potential analytes is not restricted by the pitch of the hydrophobic moiety (Figure 3a), but rather by the large gaps between the hydrophilic loops (Figure 3c). If the analyte is detected in the gap depicted in Figure 3c, recognition can take place in the media without direct adsorption onto the SWCNT surface. This would eventually enhance the probability of the recognition and enlarge the library of potential analytes using CoPhMoRe. It is worth pointing out that the pitch of a polymer wrapping depends on the radius of the cylindrical substrate, as shown in eq 5. Therefore, the radius

R′ = R 0(1 + f (b·t )sin 2(πt )) x = R′ cos(2πt ) y = R′ sin(2πt ) z = b·t

DNA Sequence (GT)30 T120 T30 (GT)30 (GT)30 (GT)15 (GA)2(AG)3CAGAAG(GA)2 (GA)2(AG)3CAGAAG(GA)2

(10)

In Figure 3, we illustrate this mapping for three different functions of f(z), namely, f(z) = c, where c is an arbitrary constant (Figure 3a), f(z) = 1 + sin(z) (Figure 3b), and f(z) = 2 z2 e−(z/2) (Figure 3c). The functions f(z) can be mapped into a 1-dimensional linear space (Figure 3, panels d, e, and f, respectively) to depict the required design of the interchanging hydrophobic (red)−hydrophilic (blue) segments. Comparing the corona phases, we can conclude that longer hydrophobic segments (Figure 3e, relative to Figure 3d) give rise to larger gaps between the hydrophilic segments, allowing for larger analytes to adsorb into the binding pockets between the hydrophilic loops. The more complicated scheme presented in Figure 3f, which is constructed of nonperiodic hydrophobic− hydrophilic chains, can create a single binding pocket upon adsorption and can be fine-tuned according to a specific analyte size and shape. This was achieved using a function which has a single minimum at x = 0, and decays to zero as |x| → ∞ (Figure 3c). In this case, the function f(z) can be chosen explicitly to promote binding to the analyte. In other words, one can choose f(z) based on the size and shape of the analyte. In this way, it provides a solution to this type of inverse problem. As an example, we chose the protein albumin40 as a model analyte. A plane passing through the centroid of the protein was arbitrarily selected, creating a continuous function outlining the 13880

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The Journal of Physical Chemistry C of the SWCNT is an essential design parameter that confines the recognition ability of the polymer to a selected group of nanotube chiralities. The radius of a SWCNT can be calculated from its chirality, (n, m), according to the following equation:15 R=

0.123 nm 2 n + nm + m2 π

(11)

Given that many commercially available SWCNT products contain a mixture of chiralities, we expect that a fraction of the polymer−SWCNT complexes synthesized from these products would be unable to selectively recognize the analyte of interest. This problem can be mitigated by using commercially available SWCNT products that are enriched in a particular chirality,47 such as (6,5) or (7,6), and by using the enriched chirality as the basis for polymer design. Additionally, several laboratory techniques have been developed to separate SWCNTs according to their chirality.48−52 The use of single chirality SWCNTs as a raw material preserves the recognition capability of all nanotubes within the system and provides a more controlled environment for polymer design and optimization. 3. Functionalization To Enhance Selectivity. When a polymer adsorbs onto the SWCNT surface, it reduces its active surface area. However, a helically wrapped polymer, such as DNA, can provide many accessible side groups that can be further functionalized. When a molecule diffuses to the SWCNT surface, it can interact with those functional groups if the geometric requirements are fulfilled (e.g., when it fits into the pitch). A particular example would be boronic acid grafted polymers that would enhance the affinity of the SWCNT complexes to diol-presenting analytes, such as sugars.53,54 In the case of DNA, we hypothesize that the selectivity of DNA−SWCNT binding for certain molecules can be attributed to the presentation and spatial distribution of those functional groups. DNA contains three different functional groups: (1) the nucleotides adenine (A), guanine (G), cytosine (C) or thymine (T); (2) phosphate groups; (3) deoxyribose. All of these display oxygen and nitrogen containing functional groups that can form noncovalent bonds with other DNA strands or molecules in the proximity of the SWCNT surface. Wrapping of SWCNTs by single-stranded DNA is attributed to π-stacking of the heteroaromatic nucleotides.38 If we assume that the nucleotide backbone adsorbs onto the hydrophobic SWCNT surface, we can view the chemical structure as a backbone with different functional groups looking out into the space perpendicular to it. The stable adsorption of an analyte can take place only if both the pitch of the DNA backbone matches the size of the analyte (Figure 4a) and the functional groups presented by the specific DNA sequence are colocalized with the functional groups of the analyte (Figure 4b). These two conditions limit the total number of potential stable adsorbents dramatically and could explain how DNA sequence affects the selectivity for certain analytes. 4. Binding Potential Maps. Molecular Simulation for Case 1. As an illustrative example of how a helical wrapping can present surface regions tailored to a particular analyte, we numerically calculate the spatially inhomogeneous surface adsorption potential of a simplified model analyte composed of a central hydrophobic portion and various pendant functional groups onto a SWCNT surface. As an example, we consider a model of trinitrotoluene detection (TNT, Figure 5a). Given the functional groups of the TNT molecule, a tailored design of a DNA helical wrapping with an optimal

Figure 4. (a) Illustration of a DNA backbone stacked onto the surface of SWCNT, presenting various functional groups to the surrounding environment. (b) DNA nucleotide with available sites for functionalization.

pitch, may enable a selective interaction between the TNT and appropriate side chains presented on the DNA. Our goal is only the illustration of the design principles of the inverse CoPhMoRe problem as outlined above, and not a rigorous molecular simulation of binding and recognition, which is beyond the scope of the current work.

Figure 5. (a) Trinitrotoluene (TNT) molecule. (b) Representation of TNT molecules in our simulation. The central bead stands for the benzene ring, the three blue beads are the nitro groups, and the cyan bead is the methyl group.

We represent the benzene ring of TNT as a central hydrophobic bead that adsorbs strongly to the SWCNT surface, with four connected beads representing the functional groups (one methyl group and three nitro groups), as illustrated in Figure 5b. The geometry of this coarse-grained molecule is assumed to be planar, with the three nitro groups at 120 degree increments, and the methyl group between two of the nitro groups. Bond distances (center of mass of the benzene ring to center of mass of each functional group) are chosen according to the work of Nash et al.55 To ensure that this structure is not deformed, harmonic forces are used to stabilize the various bonds and angles, with parameters given in the Supporting Information. Further, three improper dihedral terms are added to ensure the planarity of the molecule (e.g., 13881

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Figure 6. Potential maps for the highly simplified model system considered in this work, illustrating the binding energy of TNT to the helically wrapped single walled carbon nanotubes for pitch value (a) 0.19 nm and (b) 0.4 nm. The white rectangles represent the region that is plotted in panels c and d, respectively. (e, f) Snapshots of the TNT molecule on the nanotube surface between the helical chains of the polymer for (e) 0.19 nm pitch and (f) 0.3 nm pitch. The polymer backbone is plotted in pink, and its side groups appear in yellow and gray. (g) Minimum value of the binding potential for various pitch values. The minimum of this function illustrates the existence of an optimal pitch for a given analyte.

dihedrals for the configurations N1−N2−B−N3, N2−N3−B− N1, N3−N1−B−N2, where N1, N2, N3 represent the three nitro beads, and B represents the central benzene bead). The carbon nanotube and DNA structure are modeled with fixed structures and geometry. The carbon nanotube is modeled as a (6,5) SWCNT of approximately 10 nm total length. Each carbon atom is represented as a bead with a radius of 2 Å.56,57 The DNA is a cylindrical helix with a radius of 2 Å58

and a varying pitch between 0.14 and 0.4 Å. This is implemented as a series of overlapping beads every 0.1 radian at the correct radius. The radius of the helical wrapping is the sum of the DNA bead size, the SWCNT atom bead size, and the radius of the (6,5) SWCNT, which is 7.47 Å.15 The SWCNT and DNA beads interact with the various beads of the TNT molecule through Lennard-Jones interactions. 13882

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since it directly affects the electrostatic contribution of pe to the persistence length. For DNA, it is well-known experimentally that the persistence length is inversely proportional to the ionic strength.67,68 Therefore, we anticipate that the corona phase formed by a given sequence of DNA would vary with the salt concentration, forming a smaller pitch size for increasing ionic strength. 6. Distorted Helix Conformation. Molecular Simulation for Case 2. In order to illustrate the distorted helix conformation formed by the adsorption of a hydrophilic− hydrophobic segmented polymer, we performed coarse-grained molecular simulations using HOOMD-Blue59,60 for a polymer with alternating hydrophobic and hydrophilic sections, shown in Figure 7a. We chose a single system with an initial

The potential surface is determined by scanning for the total enthalpic energy of the system as a function of the position of the central benzene bead. The position of the benzene bead is held at a fixed radius, varied for a range of angles and position along the SWCNT, with the other TNT beads (the four side groups) allowed to relax to an energy minimum. It is important to note that the potential calculated in this manner is not the free energy (including entropic contributions). A full calculation of the adsorption free energy would require a more sophisticated thermodynamic integration technique, such as potential of mean force calculations. All calculations are performed using HOOMD59,60 1.0.0 with the FIRE minimization method to a tolerance of 10−3. The resulting spatial potential maps of the molecular simulations are shown in Figure 6a and Figure 6b for two different pitch values of the helical wrapping (0.19 and 0.3 nm, respectively). The periodic extremum values of the potential correspond to the helical wrapping, and the occluded space between the polymer chains. Zooming into the regions marked by the white rectangles, the available binding region is clearly depicted in Figure 6c and Figure 6d. The corresponding 3dimensional visualizations are presented in Figure 6e and Figure 6f. For the 0.19 nm pitch, the analyte can bind to the surface of the SWCNT only in specific locations and conformations, enabling a selective recognition. In the case of the larger pitch, the analyte can bind to the surface of the nanotube in multiple locations and orientations, which reduces the selectivity of this wrapping to that analyte. The minimal value of the binding potential was calculated for several pitch values, ranging from 0.12 to 0.4 nm, and is plotted in Figure 6g. This reveals the existence of an optimal pitch value for the chosen model analyte, which is 0.19 nm. Smaller pitches would forbid the analyte from binding the surface of the nanotube, due to steric hindrance, while larger pitches would hinder the selectivity and specificity of the binding. 5. Effect of the Persistence Length on Helical Wrapping and Recognition. The molecular simulation results from case 1 (Figure 6) show that the pitch of the adsorbing helix is critical for the detection and should be designed to match the size of the target analyte, which indeed is in agreement with the analytical model. Studies of the helical wrapping motif show that the size of the pitch is mainly determined by the radius of SWCNT and the persistent length of polymer (mechanical rigidity).61 Small radius SWCNTs have large bending angles, which require large end-to-end distance and result in the large pitch. In contrast, for large radius SWCNTs (high curvatures), the relative entropic drives toward a random conformation result in a small pitch. Hence, case 1 CoPhMoRe predicts a strong dependence of the recognition on the diameter of the nanotube. For a fixed SWCNT diameter, the pitch is determined mainly by the persistence length of the polymer chains.61−64 This would eventually control the available spacing for analyte binding on the cylinder surface. With a fixed tube diameter, the wrapping of a polymer with larger persistence length would result in a larger pitch as the bending energy of the wrapping conformation increases.65 To solve the inverse problem practically, one has to design a polymer with a persistence length such that a desired pitch is obtained. Persistence length can be expressed as the sum of pi (the intrinsic rigidity of the chain) and pe (electrostatic contributions).66 Consequently, for ionomers or polyelectrolytes (charged polymers), the salt concentration of the environment can be considered as an additional parameter,

Figure 7. 3-Dimensional configuration of a distorted helix corresponding to case 2 of distorted helix CoPhMoRe. (a) Initial configuration of a helical wrapping of a polymer consisting of hydrophobic (blue)−hydrophilic (red) alternating segments. (b) Final configuration of the distorted helix following structure relaxation.

configuration as presented in Figure 3b. Similar to the numerical simulation of case 1, we considered a (6,5) SWCNT with approximately 10 nm total length. The wrapping polymer is modeled by a freely jointed chain of hydrophilic (red) and hydrophobic (blue) beads with radii of 1.5 Å. The bead bonds were modeled as a stiff harmonic spring with force constant k = 100. Polymer self-avoidance was modeled with Lennard-Jones interactions between beads, regardless of type, with σ = 3 Å and ε = 10−4 (essentially a purely repulsive interaction). The beads interacted with the SWCNT surface with a similar Lennard-Jones interaction, using the van der Waals radius of 1.7 Å for sp2-bonded carbon, resulting in σ = 3.2 Å and ε = 10 for the hydrophobic beads and ε = 0.1 for the hydrophilic ones. The initial configuration was chosen with F(z) = 3(1 + sin[2πz/L + π/2]), such that there were two completely hydrophobic regions with periodic boundary conditions in the z-direction. Beads were placed along the trajectory every 3 Å, so that the initial configuration did not contain strained bonds. Beads with an initial configuration within 3 Å of the SWCNT surface were assigned as hydrophobic beads, with the remainder hydrophilic ones. The system was integrated for 100,000 steps using an NVT 13883

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The Journal of Physical Chemistry C ensemble at a temperature of T = ε/kB, allowing the hydrophilic chains to relax. A representative distorted helix configuration of the polymer, as presented in Figure 7b, clearly shows the hydrophobic regions tightly bound to the SWCNT surface, whereas the hydrophilic segments create loops in the aqueous surroundings, forming docking sites for the analyte in between. The flexible model of the polymer leads to a more disordered structure than suggested from the enthalpic calculations above. Essentially, each hydrophilic loop off the surface behaves as a disordered entropic chain with fixed ends on the SWCNT surface. However, there is still a recognizable spatial variation in the hydrophilic regions presented by the SWCNT. Such a structure, when carefully designed, can discriminate between target analytes based on their size and shape, increasing the selectivity of the adsorption to the accessible regions of the nanotubes. Ensuring that such a polymer does in fact spontaneously self-assemble into these structures is a topic for more detailed future simulations.

groups can be incorporated along the polymer length to enhance selectivity and specificity. The choice of the functional groups and their location can be specified completely by the solutions to the inverse problems presented in this work. These concepts can be experimentally tested in future work and applied to the rational design of novel corona phases for molecular recognition, leading to the discovery of new recognition motifs for separation, catalysis, and label-free detection and sensing.

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ASSOCIATED CONTENT

S Supporting Information *

Molecular simulation parameters. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.5b01705.

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AUTHOR INFORMATION

Corresponding Author

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*E-mail: [email protected]. Phone: 617.324.4323. Mailing address: Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA.

CONCLUSIONS Corona phase molecular recognition (CoPhMoRe) is a new technique that generates a nanoparticle-coupled polymer phase capable of recognizing a specific molecule with high affinity and selectivity. A rational alternative design to high throughput, empirical library screening of such phases is to solve the mathematical formulation of the CoPhMoRe inverse problem. This inverse problem seeks a linear function representing the position of monomers or functional groups along a polymer backbone that results in a 3-dimensional structure capable of recognizing a specific molecule when mapped to a nanoparticle surface. We proposed two distinct cases that can result in unique molecular recognition sites along a cylindrical surface. In the first approach, case 1, we show that a polymer can be designed to helically wrap the surface at a programmed pitch, creating a binding site between the helical segments along the cylinder. According to values reported in the literature for DNA wrapped SWCNT, this pitch can vary over 1 order of magnitude, between 2 and 20 nm, allowing for the adsorption of a wide variety of analytes, ranging from small molecules up to medium-sized globular proteins.41 A simplified numerical simulation of the model representing this case illustrated the existence of an optimal pitch for a given wrapping, side groups, and analyte. The optimal pitch renders the binding selective and specific, as higher pitches would allow other analytes to bind, and smaller pitches would prohibit the binding of the target molecule. Case 2 requires a carefully tailored polymer consisting of alternating hydrophobic−hydrophilic segments of varying chain lengths. Assuming helical wrapping, the hydrophobic segments are binding anchors to the nanotube surface, while the hydrophilic parts, depending on their length and separation, create distortions in the helix that can be designed to conform to a target analyte. A simplified numerical simulation of the corona configuration formed by such a polymer demonstrates the formation of binding pockets on the SWCNT surface between the assembled hydrophilic loops in the aqueous surroundings. Both cases involve challenging polymer chemistries and synthesis techniques,69 since each analyte dictates the specific configuration and length of the adsorbing polymer to form the corona. In addition to the design of the amphiphilic polymer backbone that forms the corona phase, additional functional

Author Contributions ‡

G.B. and J.A. contributed equally.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported in part by the U.S. Army Research Laboratory and the U.S. Army Research Office through the Institute for Soldier Nanotechnologies, under Contract Number W911NF-13-D-0001. The authors gratefully acknowledge support from the National Science Foundation under Grant No. 1213622 (M.S.S). The authors thank Dr. Nicole M. Iverson for valuable discussions. J.A. is grateful for the support of the Samsung scholarship, S.K. is grateful for a postdoctoral fellowship from the Deutsche Forschungsgemeinschaft (DFG), and D.P.S. is grateful for a National Science Foundation Graduate Research Fellowship. The analysis of Case II for macromolecules was funded in part by a grant from the Juvenile Diabetes Research Foundation.

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ABBREVIATIONS SWCNT, single walled carbon nanotubes; CoPhMoRe, corona phase molecular recognition

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REFERENCES

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