Polymer Conjugation as a Strategy for Long-Range Order in

Mar 3, 2016 - ... which association constants are independent of assembly size, i.e., ..... ordered, paralleling the order of note sequences in classi...
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Polymer Conjugation as a Strategy for Long-Range Order in Supramolecular Polymers Ari Benjamin and Sinan Keten* Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3109, United States S Supporting Information *

ABSTRACT: Supramolecular polymers are polymers in which the individual subunits self-assemble via noncovalent and reversible bonds. An important axis of control for systems of mixed subunit composition is the order in which the subunit types assemble. Existing ordering techniques, which rely on pairwise interactions through the inclusion of highly specific chemistry, have the downside that patterns of length n require n specific chemistries, making long-range order complicated to attain. Here we present a simple alternative method: we attach varying numbers of polymers to self-assembling subunits, in our case ring shaped macrocycles, and the polymers’ aversion to confinement imposes system order. We evaluate the feasibility of the strategy using coarse-grained molecular dynamics simulations of polymer-conjugated rings designed to model cyclic peptide nanotubes. We discuss the effects of polymer conjugation on the energetics of association and predict the equilibrium orderings for various ratios of ring types. The emergent patterns are associated with a certain stochastic disorder, which we quantify by deriving and employing a formula for the expected statistical weight of any pattern within the ensemble of all possible orderings.

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INTRODUCTION Supramolecular chemistry aims to study and engineer selfassembling molecules that noncovalently associate to form complex structures.1 When each molecule assembles only linearly, they form one-dimensional chains termed supramolecular polymers.2,3 This wide class of materials boasts an equally wide range of applications, from biocompatible scaffolds to nanoelectronics to self-healing interfaces.4−6 While many supramolecular polymers are composed of only a single monomer type, the potential for supramolecular polymers only has room to broaden with the incorporation of two or more distinct chemical species. Such heterogeneous polymers may either be ordered or random with respect to monomer composition, though ordered systems quite clearly offer more capability. Linear order might serve as a strategy for entwining disparate compounds in a regular fashion, for example. Such a technique could assist applications like organic photovoltaics made from supramolecular discotic liquid crystals,11,12 nanorodbased transmembrane DNA shuttles for gene therapy,7,10 or the multi-step catalytic centers both seen in nature and deployed synthetically on layered, multisegmented nanorods.7−9 Utilizing polymer confinement to dictate linear order could also open interesting possibilities for cyclic peptide nanotubes (CPNs). CPNs are rigid supramolecular nanotubes assembled from stacked peptide rings that form a stable core through hydrogen bonding.14,15 CPNs are notoriously functionalizable, both inside and out, and have already been developed for numerous applications, such as nanoscale transport, biosensing, and antimicrobials.15−18 Among many potential benefits, © 2016 American Chemical Society

ordering CP subunits with comparable binding energies but different interior chemistries would assist the development of CPNs that more closely mimic the internal energy landscape and chemical diversity of transmembrane protein channels.19,20 There are a few current methods for encouraging patterning in a linear assembly, though they suffer from certain limitations. Most rely on highly specific interactions between monomer moieties that only allow assembly in the prescribed order. Examples of such systems include chains formed from DNA or peptides, which serve as the ordered system itself21 or as templates for ordered growth.22,23 All such systems depend on the pairwise interaction of distinct subunits with their immediate neighbors. While this strategy allows near-perfect order, the complexity of chemistry required may make certain types of patterns infeasible. Because of the locality of specific pairwise interactions, it is readily apparent that a pattern of length n would require n separate species. An ordering such as ···ABCABC··· is possible under such constraints, for example, but ···AABAAB··· is not, as it is not the unique pattern formed from mixing two pairwise-interacting species (consider the energetically equivalent AAABAB). This requirement is not ideal when long patterns are desired. To form repeating patterns with a pattern length longer than the number of distinct species, it is thus necessary to turn to forces that act over distances longer than one subunit. This is Received: December 22, 2015 Revised: March 3, 2016 Published: March 3, 2016 3425

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RESULTS Our simulation system, as described fully in methods consists of nanotubes formed from self-assembling cyclic peptide rings conjugated with zero (CP-0), two (CP-2), or four (CP-4) short polymer chains. The polymer chains have purely repulsive, excluded volume interactions representing a Θ solvent scenario, and have a contour length corresponding to 10 stacked rings (∼5 nm). The system is coarse-grained, for generality. A schematic of the system setup can be found in Figure 1a. All system-specific parameters such as polymer

equivalent to adding forces that, in the language of supramolecular polymers, shift an isodemic system (in which association constants are independent of assembly size, i.e., with only pairwise interactions) to a cooperative or anticooperative system (in which association constants increase or decrease with assembly length, respectively).3 Attractive long-range forces induce cooperativity, responsible for effects such as nucleation-limited growth, while long-range repulsive forces induce anticooperativity and have been employed as a length-control strategy.24,25 As many monoblock supramolecular polymer systems display cooperativity or anticooperativity, there is considerable choice of nonlocal force mechanisms.26−29 Here we present analyses revealing the potential of creating long-range forces through the conjugation of (covalent) polymer arms to individual self-assembling monomers. The attached polymers carry conformational entropy that decreases when in close proximity to neighboring polymers, thus generating neighborhood-dependent forces. This force is repulsive and acts to adjust the intermolecular attractive force that drives self-assembly. Polymer conjugation is a facile chemical procedure and has already been employed in several contexts to modify the properties of the final assembly and drive self-assembly in certain media.29−34 In CPN systems, for example, it has been used as a tool for membrane inclusion and to modify the aggregation process and assembly properties.18,31,32,35−37 To our knowledge, however, the entropic repulsion due to polymer conjugation has never been employed as a method of encouraging order in a heterogeneous supramolecular polymer, and in general the properties of assemblies of two or more subunits with different conjugation levels remain to be understood. CPNs serve as a realistic model system for exploring such features. We hypothesize that, in mixtures of monomers with differing numbers of polymer arms, assembly will favor patterns that maximize the conformational entropy. Similar entropic effects have already been shown to induce order in other systems, such as mixed self-assembled monolayers (SAMs) adsorbed onto nanorod surfaces.38 With this system in mind, then, we will test our ordering strategy on nanotubes formed from self-assembling rings. We emphasize that in other systems the subunits need not be rings, rather, a broad range of colloidal or supramolecular units with overall significantly favorable binding energies should be amenable to the same ordering strategies. The ultimate goal of this work is to predict the emergent nanotube order from a mixture of rings of varying degrees of conjugation (“ring types”). Since the lengthy time scales required would prohibit simulations of the full self-assembly process,39 we turn instead to methods which directly calculate the free energy. We developed a method to estimate the relative free energy of a nanotube of any ring type order using coarsegrained, collective variable molecular dynamics simulations of nanotubes formed from polymer-conjugated rings. Knowledge of the free energy of all possible orderings allows the prediction of the most probable, equilibrium ordering and its likelihood relative to other orderings. Using this approach, we also provide answers to several questions regarding the energetics of polymer conjugation, including the spatial limits of the nonlocal repulsion effect and the quantitative size of free energy penalties incurred. We conclude by discussing the scope of the cases in which polymer conjugation may serve as a strategy for long-range ordering of supramolecular systems.

Figure 1. Sample system setup and energetic effect of polymer conjugation. (a) Coarse-graining procedure and sample nanotube. The collective variable is defined as the center-of-mass distance between the two central green rings; the tube will be separated at its center. (b) Potential curve of separating an unconjugated nanotube, dashed line, and that of a conjugated nanotube, dot-dashed line. Their difference is the contribution of polymers alone, blue, and the difference at the equilibrium position is the free energy penalty ΔF. Also noted is the activation energy to assocation ΔF†, referenced in the discussion of rate.

length, excluded volume, and interring binding energy represent only a single representative system; as we discuss later, other systems will likely form similar patterns with quantitatively different (though thematically related) statistical weights. Our first step was to extract free energies relative to the unconjugated (CP-0) binding energies and determine the free energy penalty due to polymer conjugation. This we achieved with adaptive biasing force (ABF) simulations of the separation 3426

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The Journal of Physical Chemistry B of assembled nanotubes.40 These calculations yield the free energy as a function of the separation of the two halves of a reference tube, which is affected by the presence of the many chains in the neighborhood of the separation interface. An example free energy curve, shown in Figure 1b, displays the clear effect of polymer conjugation upon the energetics of attraction. In the absence of polymers, the potential curve of two rings closely resembles the Lennard-Jones potential. This potential is plotted as the dashed curve in Figure 1b. The deviation of the conjugated potential from the dashed curve is due entirely to the free energy penalty of the polymer arms in mutual confinement. The blue line in Figure 1b represents the difference of these potentials. It can be seen that the free energy increase due to the polymers decreases monotonically with respect to ring distance except for a small turnaround near the minimum caused by the slight change in equilibrium distance. The long-range effect of the polymers’ steric repulsion is clear, extending many times the equilibrium interring spacing of about 5 Å. Throughout this paper, we refer to the free energy penalty ΔF due to polymer conjugation, which is defined as the difference between the well depth of a given separation and the well depth of separating an unconjugated tube, noted on Figure 1b. Further evidence of the long-range effect can be found by comparing ΔF for a set of reference situations. Figure 2a displays ΔF for tubes with an increasing number of conjugated rings on either side of the separation point. For simplicity, in this figure we examine only nanotubes comprised of only one CP type (the homomeric case). It is clear that the free energy of separating an interface indeed depends on the degree of conjugation of several rings around that interface. For each homomeric case, we can characterize the conjugation dependence of ΔF on the number of adjacent conjugated rings n by fitting the series to an exponential function, ΔF(n) = ΔF∞ − a exp(−γn), where ΔF∞ is the extrapolated value of ΔF for an infinite homomeric nanotube, γ is the rate of convergence toward this value, and a is an empirical parameter. High values of γ indicate that ΔF depends on the conjugation level of only a small number of neighboring rings, while low values indicate greater nonlocality of forces. We examine the dependence of ΔF∞ and γ on the ring type comprising each tube in Figure 2, parts b and c, respectively. The value of ΔF∞ can be seen to increase with conjugation degree, as expected. Additionally, γ decreases with conjugation degree, indicating that the entropic force exerted by the polymers becomes yet more nonlocal in regions of high grafting density. In the case of CP-4, γ is low enough that ΔF is affected by the conjugation of rings more distant than the contour length of any individual polymer (the force thus being transferred through intermediate polymers). In summary, we can deduce that ΔF depends roughly exponentially on the number of neighboring conjugated rings. The rate of this decay (as indicated by γ) also depends the grafting density, decaying more quickly for low densities. Though it prevents simple estimations of ΔF for arbitrary sequences, this nonlinearity and nonlocality is an important cause of the emergent lowest-energy ring type orderings we present below. Using knowledge of the ΔF’s for many individual short sequences, we next developed a method to rapidly estimate the total free energy penalty of any ring type sequence of arbitrary length (see Methods). This information allows us to predict in what order the rings will naturally assemble from a certain stoichiometric mixture of ring types, as the equilibrium ring

Figure 2. Free energy penalty of separation ΔF for homomeric conjugated nanotubes of varying length. (a) ΔF for the series ··· 00x:x00···, ...00xx:xx00···, etc., i.e., sequences of length 16 with an increasing number of conjugated rings on either side of the central interface, for ring types x = CP-1, CP-2, CP-3, and CP-4. Error bars are calculated from the standard deviation of the ΔF from several ABF runs. The parameters of the exponential fits (red lines) are displayed for each series in parts b and c. Error bars of γ reflect the 95% confidence bounds of the exponential fit; low confidence reflects small ΔF values relative to the largely constant error of ABF.

order is simply the sequence of rings with the lowest total free energy. In Figure 3, we present the equilibrium tube ordering for select ratios of CP-0’s and CP-4’s. In each ratio studied, a single most probable motif emerges. The most probable sequences indicate a chief advantage of this nonspecific ordering strategy: it is possible increase the number of spacer rings between rings of a certain type simply by modifying the ratio of unconjugated rings to conjugated rings. In a large ensemble of assembled nanotubes, the lowest energy ring ordering will be in constant equilibrium with other, higher energy orderings. It is possible to enumerate the exact likelihood of the most probable pattern within the ensemble of all possible ring orderings (see Methods for the formula for statistical weight). We express this likelihood by examining the distance between every CP-4 ring and its nearest CP-4, which we then assemble into a histogram. The histograms are displayed in Figure 3 along with the percentages of the most probable distances. It is clear that the order method is less robust for highly asymmetric ratios. For this system, for example, increasing the ratio CP-0:CP-4 from 1:1 to 2:1 decreases the probability of the alternating, equilibrium pattern from 48% to 41%. This trend places an upper limit on the spacing possible with this set of system parameters. 3427

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Figure 3. Most likely patterns for various ratios of unconjugated and fully conjugated rings (CP-0’s and CP-4’s). Only a certain portion of sequences would display the mostly likely pattern; at right, we present the statistical weight. The relative statistical weight of the preferred pattern tends to decrease as the space between CP-4’s increases. The bars display a histogram of the distances between CP-4’s in the resultant ensemble. In the ensemble with a 2:1 ratio, for example, CP4’s are 3 rings away from one another 41% of the time, and very rarely 1 or more than 4 rings away. The top row, in blue, measures the distance of CP-0’s rather than CP-4’s.

Relevant system parameters strongly affect the listed pattern probabilities. Identical simulations with different polymer bead radii indicate that increasing the polymer excluded volume increases the free energy penalties by a consistent factor, thus increasing the likelihood of the most probable sequence (Figure S1). Longer polymer arms tend to decrease the likelihood for ratios around 1:1, a fact that makes intuitive sense when one considers that extending the polymer length to infinity would eliminate any energetic advantage to switching ring locations and thus eliminate the propensity for patterning. However, longer polymer arms do allow the creation of longer patterns. The maximum designable spacing of CP-4 is expected to scale roughly with N3/5, the Flory radius of a tethered chain, where N is the number of monomers in each polymer chain.41 The potential range of patterns becomes more interesting upon the addition of a third ring type. We chose to introduce a third type with the intermediate conjugation level of two polymer arms (CP-2). The resulting patterns are somewhat surprising, though certain trends remain similar to binary mixtures. In Figure 4, we examine in detail the ensemble created by mixing all three types in an equal ratio, which serves to represent the general features of other ratios. Figure 4a lists the three most probable emergent patterns. All three patterns exhibit a sort of phase separation, each with at least one phase composed of alternating CP-4’s and CP-0’s and the single most probable pattern showing a complete segregation of CP-2’s. These motifs, which we will write as e.g. (4040)n(22)m in the case of the most probable pattern, indicate the potential of this method to yield hierarchical two-phase patterns. We investigate the origin of this curious feature in Figure 4b by displaying the energy penalty ΔF/monomer as a function of the percentage of ring−ring interfaces that are CP-0/CP-4 interfaces and CP-2/CP-2 interfaces. Only the physically accessible regions are colored. As long as the equal ratio of rings types is maintained, it is not possible to construct a nanotube with more than one-third CP-2/CP-2 interfaces, for example, or two-thirds CP-0/CP-4 interfaces. Importantly, the maximum percentage of CP-0/CP-4 interfaces depends on the

Figure 4. Analysis of the patterns created by an equal ratio of ring types CP-0, CP-2, and CP-4. In part a, we show only the three most probable pattern styles. The prevalence of the phase separation of CP2’s (seen in the first row) is due to the enormous preference of CP-4’s for CP-0’s; conglomeration of CP-2’s allows for more CP-0/CP-4 interfaces. We examine this effect in part b, which shows the free energy penalty per monomer as a function of the percentage of CP-2/ CP-2 and CP-0/CP-4 interfaces. Uncolored sections are physically inaccessible for this ratio. The largest predictor of energy is the percentage of CP-0/CP-4 interfaces (horizontal axis). Increasing the percentage of CP-2/CP-2 interfaces alone (moving vertically) serves to increase the free energy penalty, but doing so also creates room for more CP-0/CP-4 interfaces. This allows the system to find the lowest energy location in the upper right, which corresponds to the first pattern shown in part a.

percentage of CP-2/CP-2 interfaces. It is clear that the percentage of CP-0/CP-4 interfaces is the larger predictor of free energy. This comes as no surprise, as the total conformational entropy of the nanotube is most reduced when the environment around each highly conjugated CP-4 is cleared of competing polymers. Solely increasing the number of CP-2/CP-2 interfaces always serves to increase the nanotube’s free energy, which might indicate that phase separation of CP2’s is actually unfavored. As the CP-2’s conglomerate, however, they create additional space for the creation of CP-0/CP-4 interfaces, whose energetic benefit outweighs the cost of CP-2/ 3428

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The Journal of Physical Chemistry B CP-2 interfaces. The favorability of CP-2 phase separation thus depends on the relationship between the energy balance of the CP-2/CP-2 and CP-0/CP-4 interfaces and the cutoff for physically possible nanotube configurations; i.e. on the slope of the contour lines and the slope of the rightward cutoff. Similar effects are responsible for the other two hierarchically organized patterns in Figure 4a (that is, (0242)n(04)m and (0422)n(04)m). While intriguing, it is crucial to note that the most likely patterns of Figure 4a have very similar free energies. Since the probability of each sequence is determined by its free energy relative to all other sequences, similar free energies indicate that the statistical weight of any one motif is likely low. This may prohibit effective implementation of the two-phase, multiscale patterns (like (4040)n(22)m) for this set of system parameters, though it might be possible to design a system with more favorable probabilities. We can characterize the exact disorder of our system by employing the tool of Shannon block entropy. Given an alphabet of symbols and knowledge of the statistical weight of all possible sequences of length n, the block entropy for sequences of length n is defined as42 Hn = −∑ pi(n) log n pi(n) i

Figure 5. Block entropy of sequence ensembles compared to other ordered sequence ensembles. The energetic similarity of viable alternative sequences decreases the overall order of the resultant ensemble. We characterize this disorder by plotting the block entropy of two example ratios along with a perfectly disordered system, the English language, DNA, and music. Ratios are indicated as (CP-0:CP2:CP-4). We see that the ensemble of sequences created from a mixture of CP-4 and CP-0 in equal ratios (1:0:1) is quite ordered, paralleling the order of note sequences in classical music. The (1:1:1) ensemble, however, is quite disordered, despite the preferences of CP4’s and CP-2’s. DNA, music, and English data taken from Schmitt and Herzel, 1997 (specifically representing the block entropies of the Epstein−Barr virus, Beethoven Sonata No. 32, and the corpus of Alice in Wonderland).

(1)

p(n) i

Here is the probability of sequence i, which is n letters long. The block entropy evaluates to zero for an ensemble in which only one sequence is found, and to n in the completely disordered ensemble in which all sequences are equiprobable. Increasing the order of the system, then, is akin to lowering the block entropy. Block entropy is frequently employed as a measure of information efficiency for a language, as well as for DNA characterization,43 though in this application of maximizing the relative likelihood of a specific sequence or order, we seek to minimize, not maximize, the Shannon entropy. We plot the block entropy of this ensemble as a function of block length n in Figure 5, along with several reference data sets obtained from ref.43 and the block entropy of the binary ensemble of CP-0s and CP-4 in equal ratios. The sequence ensemble of the trinary mixture is more ordered than DNA, though somewhat less ordered than natural language (e.g., the collection of words in this paper within the ensemble of all possible letter orderings). We can infer that there is therefore a very large number of orderings with similar free energy penalties and statistical weights. A sample nanotube taken at random from the assembled ensemble will thus appear to be disordered, apart from certain trends. An analysis of these ordering trends can be found in Figure S2. We found that adjusting the ratio of rings can be a powerful technique to modify the probable trends of ring type association in mixtures of three ring types, just as it was for two. In Figure S3 we characterize the sequence ensemble that arises from a mixture of CP-0, CP-2, and CP-4 in an example ratio of 4:2:1, respectively, by examining the spacing between ring types. This analysis confirms that CP-2’s often come together in pairs to cede space to the CP-4’s, that CP-4’s are almost never adjacent, and that CP-0’s rarely form long chains. Similar analysis of other ratios reveals similar trends. The percentage changes from the unconjugated ensemble do change slightly between ratios, however, and in somewhat surprising ways, indicating that the ratio must still be considered when designing the properties of the ensemble. A comparison of the percentage changes among different ratios can be found in Figure S2. Taken together, these results indicate that incorporating multiple levels of polymer conjugation in various

ratios could be useful in any application in which it is desirable to modify the trends of mixing but unnecessary to guarantee an exact and specific order.

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DISCUSSION In summary, here we have shown that grafting polymers to selfassembling subunits is an effective method to introduce longrange repulsive forces that depend on the local environment. We have developed a method to enumerate the exact contribution of polymers to the energetics of association, finding that the entropic penalty depends nonlinearly on the grafting density of many neighbors around the association interface. When a mixture of rings with different levels of conjugation is allowed to self-assemble, the entropic penalty has the effect of encouraging a certain level of order within the nanotube. In binary mixtures of rings with high and low conjugation, we observe the emergence of long-range twospecies patterns (e.g., “...aaabaaab···”). This class of pattern would have required many molecular species or complicated templates in systems reliant on pairwise interactions. In mixtures of three ring types, the lowest-energy orderings display a more interesting range of patterns, often with some degree of phase-like separation between pattern modes. Alternating sequences within a block-like pattern can be seen, e.g., (4040)n(22)m, which are examples of hierarchical material patterns at both the monomer and nanotube scales. It is essential to consider the lowest-energy orderings along with their statistical weight relative to all other possible orderings. Binary mixtures offered lowest-energy orderings with generally high statistical weights, indicating that such orderings could be reliably programmed into a physical system. Mixtures of three types, however, present very large ensembles of 3429

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sequences in which no one ordering is of significantly lower free energy than the rest, indicating that such patterns will require further engineering to be feasible via this method. Analysis of histograms of distances between ring types, though, does indicate that polymer conjugation has measurable and distinct effects upon observed ordering even with this set of system parameters. Rings of high conjugation avoid conglomeration, for example, by interspersing with rings of low conjugation, a feature that could be used to encourage selective ring type mixing. The polymer parameters tested here represent just one choice among many. Suitable choices include flexible polymers with an alkyl backbone, such as poly(ethylene glycol), with a suitable Θ-like solvent, though this leaves some room for modifications. Polymers with significant bending rigidity are not suitable, nor are polymers with lengths so long that entanglement becomes a significant factor (i.e., above the reptation length of the polymer). Lengthening the polymers increases the range of the nonlocal forces, which allows the creation of longer patterns, but decreases the specificity of ordering. This implies the existence of an ideal polymer length for a certain pattern length, an interesting question outside the scope of this work. The excluded volume correlates with the incurred free energy penalty, indicating that larger polymer monomers would help to encourage system order. However, very long or bulky polymers may induce a free energy penalty so large as to completely prohibit assembly if introduced in high concentrations. We recommend using such polymers only in low concentrations and for long patterns. The availability of near-Θ solvent conditions is also important, as it precludes polymer aggregation or overdispersal and further emphasizes the role of conformational entropy. An additional important parameter is the subunit attraction strength. In general, the enthalpic attraction between subunits must be large enough to overcome the free energy penalties of the polymer arms.39 Overly strong interactions, however, may kinetically arrest rearrangement and energy minimization at low temperatures. It is vital that the supramolecular polymer be able to dissociate for rearrangement and equilibration. It should be noted that the limiting rate is not governed by the entropic barrier to assembly ΔF† (noted in Figure 2) but by the well depth ΔF0, which exceeds ΔF† as long as assembly is favorable. It is possible to decrease ΔF0 to speed equilibration, making sure that ΔF still exceeds the largest ΔF present in the desired ordering; any smaller and the nanotube would not assemble. Changing ΔF0 has no effect on the values of ΔF used earlier, indicating that ΔF0 has no effect on the equilibrium order as long as assembly is favorable. When modifications of the association energy are not possible, it is also possible to anneal the system at a higher temperature for equilibration. Temperature annealing is complicated slightly by the fact that the entropic polymer repulsion grows with temperature. In that case, it is important to make sure that the ΔF in the highest conjugated case does not grow so fast as to preclude assembly before the unconjugated nanotubes begin to dissociate at a reasonable rate. We have performed simulations at higher temperatures to establish the rate of growth of ΔF, the results of which are shown in Figure S4. For the current set of system parameters and in the limiting case of maximum conjugation, ΔF grows nearly linearly with temperature at the rate of 0.05 kcal mol−1 K−1. Experimentally, the exact rate will depend on a number of factors, though we anticipate that the linearity should hold given the entropic nature of the repulsion.

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CONCLUSION

We have shown that polymer conjugation of self-assembling rings incurs an anticooperative, repulsive force between rings that increases in a nonlinear fashion with the local degree of conjugation. By conjugating only select rings within a heterogeneous mixture, this effect leads to the formation of interesting patterns within the assembled nanotube. This work may also lend insight into the assembly process of systems in which polymer conjugation has already been introduced. The lens of ordering and patterned mixing might expand the capability of polymer−peptide conjugates, for example, which are already well-studied for a variety of reasons.44,45 Another relevant system is that of discotic liquid crystals, which are rod-like supramolecular polymers typically composed of disk-like aromatic cores conjugated with flexible side chains that show promise as organic light emitting diodes (OLED), molecular wires, and organic photovoltaics.2,12 Certain studies have already demonstrated the advantages of supramolecular order in mixtures of discotic compounds, which include the entwining of electron donors and acceptors and increasing hole mobility.11,46 Though polymers are already necessary for liquid crystalline behavior and linear order of subunit types already proven to be beneficial, it does not appear as if the polymers themselves have yet been employed as a principle ordering method. This work can thus inform the feasibility of such a technique. More generally, our results concern the assembly characteristics of any self-assembling subunit with entropically derived long-range repulsion and provide a new example of the potential uses of anticooperativity in multiblock supramolecular polymers.

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METHODS We constructed our test system so as to represent many possible specific systems by using a coarse-graining (CG) protocol similar to the MARTINI method.47 The rings’ enthalpically favorable Lennard-Jones attraction potentials overall match realistic binding energies of cyclic peptides, and the polymers are modeled on PEG. The overall schematic, however, could represent the dynamics of any similar ring-like self-assembling system. Each bead represents one-half of an amino acid or a single monomer of polymer. Only the rings have attractive van-der-Waals forces; the polymers are modeled with a purely repulsive interaction in a Θ solvent. All polymers are 10 CG monomers in length, corresponding to a length of 40 PEG monomers. Free Energy Determination and Order Prediction. We calculate the most probable ordering for a sequence of a given length and ring ratio by evaluating the free energy penalty of all possible permutations of rings. It was thus necessary to develop a rapid method for determining the free energy of the sequence. The free energy of a linear system can be written as the sum of the free energy of each of its interfaces. We therefore calculate the free energy of an assembled tube by summing the free energy required to separate the whole tube at each of its interfaces. It should be noted that it was also possible to determine the free energy of attaching a single new ring to the tube. The free energy relevant to our discussion, however, is the one that relates to the equilibration process. As the dominant equilibration process is the splitting and joining of long tubes, the free energy of choice is the free energy of splitting a whole tube. 3430

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The Journal of Physical Chemistry B

free energy of the sequence. The procedure is identical for the closest-five approximation, which uses a 10-ring environment. As we are interested in general ordering trends and not just the exact order for a repeating sequence of finite length, we devised metrics that converge with tested sequence length n, such as the distance histograms. The overall method is summarized in Figure S6. The closest-three approximation offers the speed required for the equilibrium sequence prediction but likely underestimates the free energy of densely conjugated tubes. We can evaluate the validity of this approximation by, for select sequences, comparing the free energies obtained with this approximation to those obtained without it. For each test sequence, we sum the free energy required to separate the whole tube at each of its interfaces. For a representative set of sequences, the closestthree approximation consistently underestimated the total energy by an average of 0.2 kcal/mol per monomer. Energies obtained via both methods are compared in Figure S7. In order to determine how many rings might be necessary to consider in more accurate approximations, we can plot the free energy of separation for a sequence as a function of the number of conjugated neighbors around the interface. As the closestthree method is likely to be least accurate for densely conjugated sequences, since high grafting densities would increase the range at which rings influence one another, we tested the expected worst case of four polymer arms. The result is displayed in as the CP-4 series in Figure 2a. The closest-five approach (n = 5) for two-species prediction and the closestthree approximation (n = 3) clearly neglect a significant contribution of distant polymers in densely grafted tubes. After estimating the converged value of ΔF in the case of an infinite tube with an exponential fit, indicated on Figure 2a, we find that ΔF attains 70% and 87% of the converged value at n = 3 and n = 5, respectively. This effect is interesting, as the polymers themselves do not span more than three neighboring rings away. To confirm that the free energy penalty converges more quickly in cases of lower grafting density, we also examined the free energy penalty as a function of the number of CP-2’s around the separation interface, also displayed in Figure 2a. In this case, ΔF reaches 80% and 94% of its converged value at n = 3 and n = 5. The closest-three approximation is nevertheless quite effective for the purposes of order prediction. First, for situations of medium to low grafting densities, the method performs well. Such configurations are already selected as most probable by the order prediction algorithm, and increasing the free energy of highly confined sequences would have little effect (and would likely increase the statistical weight of the most probable sequence). Calculation of Statistical Weight. Here we derive an expression to evaluate the exact proportion of sequences in a large ensemble that would exhibit the lowest-energy pattern. This is equivalent to determining the probability that a single sequence is in a particular order. For simplicity, let us consider a test ensemble in which there are only two allowed sequences, α and β; we will generalize later. Both α and β have a different order of rings types, a different (but unknown) number of thermodynamic microstates, and a different free energy. We start by writing the statistical weight of α according to its classical Boltzmann distribution, calculated with the total energies of all possible microstates.

To speed the calculation, we assumed that the free energy of separation depends on the degree of polymer conjugation of only the closest three rings on each side of the interface in question. This approximation, which we dub the closest-three approximation, has the advantage of allowing the calculation of the free energy of any sequence of any length from a finite set of reference calculations. Our task is thus reduced to the separation of all sequences six rings in length, of which there are 378 in the case of three ring types, accounting for symmetry. This index of free energies of separation can then be matched to each interface’s six-ring environment in a longer sequence and summed to yield the total free energy. For the prediction of ordering of the binary mixtures in Figure 3, we employed a closest-five approximation, similar in all other respects though more accurate and only computationally feasible for two species. The Helmholtz free energy of separation includes entropic information and is not trivially obtained. Our strategy is to extract the free energy difference from the potential of mean force (PMF) as supplied by a collective variables method. We employed the adaptive biasing force method (ABF) as implemented in LAMMPS.40,48 In an ABF calculation, the average thermodynamic force acting on a collective variable ζ is recorded and gradually counteracted by an adaptive applied force. The biasing force as a function of ζ is then integrated to yield the free energy as a function of ζ, a curve known as the potential of mean force (PMF). Simulations were performed under an NVT ensemble with a Langevin thermostat, a time step of 5 fs (appropriate due to the large bead masses and weak bonds of the CG scheme), and ran until convergence of the PMF. We found the ABF method to be consistent with similar calculations performed using metadynamics, though considerably more precise for an equivalent running time.49 A comparison of both methods can be found in Figure S5. ABF also has the advantage of having an error that converges to zero as O(T−1/2,), while metadynamics error converges to a finite value.40,49 To guarantee the adequate sampling of all ζ and the reconstruction of the whole free energy curve ΔF(ζ), we constrained the center of mass of each ring to the z-axis, the original axis of the tube. The constraint of the system to one dimension eliminates the contribution of two dimensions of translational and rotational entropy. For short, finite tubes this contribution would have the effect of increasing the free energy of separation in 1D; the entropy increase upon splitting is larger in three dimensions than in one. This entropy could be easily approximated by equations for the entropy of rigid bodies and thus accounted for. However, we are here interested in the free energy of reference tubes that are meant to approximate segments in a very long, and perhaps infinite, sequence. Allowing off-axis rotation of the subsegments would be unphysical. The free energy differences in our constrained system are therefore exactly the free energy of splitting we desire. With the closest-three approximation we can now quickly calculate the free energy of a long sequence composed of rings conjugated with 0, 2, or 4 polymer arms. In this paper we are interested in the long-tube limit, and in the following calculation we impose periodic conditions to eliminate the end effects; the sequence is cyclic, in a sense. Given any sequence of length n, we cycle through all n ring−ring interfaces and match each interface’s six-ring environment to a reference sequence. The sum of the n free energies is the total 3431

DOI: 10.1021/acs.jpcb.5b12547 J. Phys. Chem. B 2016, 120, 3425−3433

The Journal of Physical Chemistry B Pα =

∑α e−Ui / kT ∑α , β e

−Uj / kT

=

■

∑α e−Ui / kT Zαβ

Zα 1 = Zα + Zβ 1+

Zα Zβ

*(S.K.) Telephone: 847-491-5282. E-mail: s-keten@ northwestern.edu. Notes

The authors declare no competing financial interest.

We can write this ratio of partition functions in terms of the free energies. Consider the definition of the total free energy F in terms of the partition function Z: (4)

Zβ Zα

(5)

and Zβ Zα

= e−(Fβ − Fα)/ kT

(6)

We can rewrite the expression for Pα in terms of this ratio.

Pα =

1 (Fα − Fβ)/ kT

1+e

(7)

This process can then be generalized for an ensemble of i distinct states, as long as the relative free energy between the states is known: Pα =

1 ∑i e(Fα − Fi)/ kT

(8)

This expression, the statistical weight of any ordering α, can be interpreted as the proportion of sequences that feature the desired order (for a large system). This calculation is possible only because we are able to calculate the relative free energy of every possible ordering (i.e., of the complete set of substates in the ensemble).

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ACKNOWLEDGMENTS

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REFERENCES

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Having measured the free energy difference between two equithermal states α and β, we write Fβ − Fα = −kT ln Zβ + kT ln Zα = −kT ln

■

The authors acknowledge funding by the Office of Naval Research (Award No. N00014-13-1-0760) and the National Science Foundation (Award No. CBET−1234305). A.B. acknowledges support from a Walter P. Murphy Fellowship by the McCormick School of Engineering and Applied Science at Northwestern University. The authors acknowledge support from the Departments of Civil and Environmental Engineering and Mechanical Engineering at Northwestern University.

(3)

F = −kT ln Z

AUTHOR INFORMATION

Corresponding Author

(2)

Here the sums are over all microstates with the denoted orders, and we have introduced the total partition function Zαβ, which incorporates all microstates with sequence orderings of either α or β. Now, we note that the numerator is simply the partition function of only sequence α, Zα. The total partition function can then be split into a sum Zα + Zβ (and not the typical product ZαZβ as only one ordering can exist at any one moment in time in a given sequence). Pα =

Article

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.5b12547. Graphs and figures providing data on the effect of polymer excluded volume, the distance histograms of other ratios of three ring types, a corroboration of the ABF method with the metadynamics method, additional discussion of the validity of the closest-three method, and a flowchart of the computational approach behind this work (PDF) 3432

DOI: 10.1021/acs.jpcb.5b12547 J. Phys. Chem. B 2016, 120, 3425−3433

Article

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