Spatiotemporal control and superselectivity in supramolecular polymers using multivalency.

Multivalency has an important but poorly understood role in molecular self-organization. We present the noncovalent synthesis of a multicomponent supramolecular polymer in which chemically distinct monomers spontaneously coassemble into a dynamic, functional structure. We show that a multivalent recruiter is able to bind selectively to one subset of monomers (receptors) and trigger their clustering along the self-assembled polymer, behavior that mimics raft formation in cell membranes. This phenomenon is reversible and affords spatiotemporal control over the monomer distribution inside the supramolecular polymer by superselective binding of single-strand DNA to positively charged receptors. Our findings reveal the pivotal role of multivalency in enabling structural order and nonlinear recognition in water-soluble supramolecular polymers, and it offers a design principle for functional, structurally defined supramolecular architectures.

Influence of ligand distribution on uptake efficiency.

Cellular uptake is a crucial process in nanomedicine and drug-delivery; however, the factors that affect its efficiency/speed are not well understood. We report computer simulations on passive uptake via receptor-mediated endocytosis of nanoparticle coated with ligands. In particular, we study how the distribution of ligands on the nanoparticle surface influences the uptake rate. The speed of membrane wrapping and uptake was found to be the fastest for nanoparticles with homogeneous ligand distributions, where ligands are spread evenly on the surface. We show that the diffusion of the ligands on the nanoparticle can hinder its uptake, since upon the interaction with the membrane the ligand distribution becomes extremely inhomogeneous, with a large ligand-free patch. When the ligand-free-area was more than 20% of the surface, we did not observe uptake within the scale of our simulations.

Nanoparticle organization in sandwiched polymer brushes.

The organization of nanoparticles inside grafted polymer layers is governed by the interplay of polymer-induced entropic interactions and the action of externally applied fields. Earlier work had shown that strong external forces can drive the formation of colloidal structures in polymer brushes. Here we show that external fields are not essential to obtain such colloidal patterns: we report Monte Carlo and molecular dynamics simulations that demonstrate that ordered structures can be achieved by compressing a "sandwich" of two grafted polymer layers, or by squeezing a coated nanotube, with nanoparticles in between. We show that the pattern formation can be efficiently controlled by the applied pressure, while the characteristic length-scale, that is, the typical width of the patterns, is sensitive to the length of the polymers. Based on the results of the simulations, we derive an approximate equation of state for nanosandwiches.

Designing stimulus-sensitive colloidal walkers.

Colloidal particles with DNA "legs" that can bind reversibly to receptors on a surface can be made to 'walk' if there is a gradient in receptor concentration. We use a combination of theory and Monte Carlo simulations to explore how controllable parameters, e.g. coating density and binding strength, affect the dynamics of such colloids. We find that competition between thermodynamic and kinetic trends imply that there is an optimal value for both the binding strength and the number of "legs" for which transport is the fastest. Using available thermodynamic data on DNA binding, we indicate how directionally reversible, temperature-controlled transport of colloidal walkers can be achieved. In particular, the present results should make it possible to design a chromatographic technique that can be used to separate colloids with different DNA functionalizations.

Designing super selectivity in multivalent nano-particle binding.

A key challenge in nano-science is to design ligand-coated nano-particles that can bind selectively to surfaces that display the cognate receptors above a threshold (surface) concentration. Nano-particles that bind monovalently to a target surface do not discriminate sharply between surfaces with high and low receptor coverage. In contrast, "multivalent" nano-particles that can bind to a larger number of ligands simultaneously, display regimes of "super selectivity" where the fraction of bound particles varies sharply with the receptor concentration. We present numerical simulations that show that multivalent nano-particles can be designed such that they approach the "on-off" binding behavior ideal for receptor-concentration selective targeting. We propose a simple analytical model that accounts for the super selective behavior of multivalent nano-particles. The model shows that the super selectivity is due to the fact that the number of distinct ligand-receptor binding arrangements increases in a highly nonlinear way with receptor coverage. Somewhat counterintuitively, our study shows that selectivity can be improved by making the individual ligand-receptor bonds weaker. We propose a simple rule of thumb to predict the conditions under which super selectivity can be achieved. We validate our model predictions against the Monte Carlo simulations.

Quantitative prediction of the phase diagram of DNA-functionalized nanosized colloids.

We present a coarse-grained model of DNA-functionalized colloids that is computationally tractable. Importantly, the model parameters are solely based on experimental data. Using this highly simplified model, we can predict the phase behavior of DNA-functionalized nanocolloids without assuming pairwise additivity of the intercolloidal interactions. Our simulations show that, for nanocolloids, the assumption of pairwise additivity leads to substantial errors in the estimate of the free energy of the crystal phase. We compare our results with available experimental data and find that the simulations predict the correct structure of the solid phase and yield a very good estimate of the melting temperature. Current experimental estimates for the contour length and persistence length of single-stranded (ss) DNA sequences are subject to relatively large uncertainties. Using the best available estimates, we obtain predictions for the crystal lattice constants that are off by a few percent: this indicates that more accurate experimental data on ssDNA are needed to exploit the full power of our coarse-grained approach.

A theoretical and simulation study of the self-assembly of a binary blend of diblock copolymers.

Pure diblock copolymer melts exhibit a narrow range of conditions at which bicontinuous and cocontinuous phases are stable; such conditions and the morphology of such phases can be tuned by the use of additives. In this work, we have studied a bidisperse system of diblock copolymers using theory and simulation. In particular, we elucidated how a short, lamellar-forming diblock copolymer modifies the phase behavior of a longer, cylinder-forming diblock copolymer. In a narrow range of intermediate compositions, self-consistent field theory predicts the formation of a gyroid phase although particle-based simulations show that three phases compete: the gyroid phase, a disordered cocontinuous phase, and the cylinder phase, all having free energies within error bars of each other. Former experimental studies of a similar system have yielded an unidentified, partially irregular bicontinuous phase, and our simulations suggest that at such conditions the formation of a partially transformed network phase is indeed plausible. Close examination of the spatial distribution of chains reveals that packing frustration (manifested by chain stretching and low density spots) occurs in the majority-block domains of the three competing phases simulated. In all cases, a double interface around the minority-block domains is also detected with the outer one formed by the short chains, and the inner one formed by the longer chains.

Receptor-mediated endocytosis of nanoparticles of various shapes.

Cellular uptake through endocytosis is crucial for drug delivery and nanomedicine. However, the conditions under which passive endocytosis (i.e., not ATP driven) takes place are not well understood. We report MD simulations of the passive uptake of ligand-coated nanoparticles with varying size, shape, coverage, and membrane-binding strength. We find that the efficiency of passive endocytosis is higher for spherocylindrical particles than for spheres and that endocytosis is suppressed for particles with sharp edges.

Design rule for colloidal crystals of DNA-functionalized particles.

We report a Monte Carlo simulation study of the phase behavior of colloids coated with long, flexible DNA chains. We find that an important change occurs in the phase diagram when the number of DNAs per colloid is decreased below a critical value. In this case, the triple point disappears and the condensed phase that coexists with the vapor is always liquid. Our simulations thus explain why, in the dilute solutions typically used in experiments, colloids coated with a small number of DNA strands cannot crystallize. We understand this behavior in terms of the discrete nature of DNA binding.

Variance minimization of free energy estimates from optimized expanded ensembles.

We explored the possibility of improving the accuracy and precision of free-energy differences estimated via expanded ensembles by manipulation of the biasing weights. Three different weighing approaches were compared: the flat histogram (FH) method, the optimized ensemble (OE) method, and a method introduced in this work, denoted MinVar, which aims to explicitly minimize the expected variance. The performance of these three methods was tested for the simulation of chemical potentials in systems of symmetric diblock copolymers with chain lengths of either 10 or 4 beads, and a system of one large hard sphere of diameter 10 d immersed in a fluid of hard spheres of diameter d. In addition, the effect of the weighing method on the observed accuracy was investigated for different choices of macrostate staging and for both optimized and nonoptimized acceptance ratio methods for calculating free-energy differences. In the diblock copolymer systems, we found that the maximum attainable accuracy can be limited by correlations between the samples, causing the "real" observed variances to be much larger than the expected "ideal" ones. Hence, if the formal minimization of the variance, as aimed by the MinVar method, occurs at the expense of increasing the correlations in the data, the accuracy may actually decrease. Although maximizing the number of round trips between initial and final macrostates (as aimed by the OE method) was found to be directly related to data decorrelation, this only translates into increased accuracy if the correlations are the major source of errors in the free energy estimates. Finally, for the hard sphere system, we found that the MinVar method performs better than both the OE and FH methods even though the MinVar method in this case never completes a round trip, illustrating that maximizing the number of round trips for fixed computational cost does not necessarily lead to increased precision.

Optimization of expanded ensemble methods.

Expanded ensemble methods, designed to sample a range of an order parameter lambda of interest, can be optimized to overcome the difficulties associated with traversing large free-energy barriers or rugged landscapes. The optimization strategy of Trebst et al. [Phys. Rev. E 70, 046701 (2004)] is based on finding suitable biasing weights for inter-lambda transitions that maximize the number of round trips that the system performs between the lower and upper lambda bounds. In this work, this optimized-ensemble methodology is extended by finding weights that minimize the mean round-trip time tau (between the lambda end states) for a Markovian walk. Applications are presented for an atomistically detailed model and for systems where one needs to sample a wide range of concentrations or compositions. A less rigorous method that implements a dual tau minimization (for both upward and downward trajectories) is found to be harder to converge but produce more round trips than a method based on a single tau minimization for all trajectories. While the proposed methods do not always minimize the true tau, they have performances that are either similar or better than those of the original optimized-ensemble method and provide useful information to characterize deviations from Markovian dynamics in the sampling of the lambda space.

Design of Multivalent Inhibitors for Preventing Cellular Uptake.

Cellular entry, the first crucial step of viral infection, can be inhibited by molecules adsorbed on the virus surface. However, apart from using stronger affinity, little is known about the properties of such inhibitors that could increase their effectiveness. Our simulations showed that multivalent inhibitors can be designed to be much more efficient than their monovalent counterparts. For example, for our particular simulation model, a single multivalent inhibitor spanning 5 to 6 binding sites is enough to prevent the uptake compared to the required 1/3 of all the receptor binding sites needed to be blocked by monovalent inhibitors. Interestingly, multivalent inhibitors are more efficient in inhibiting the uptake not only due to their increased affinity but mainly due to the co-localization of the inhibited receptor binding sites at the virion's surface. Furthermore, we show that Janus-like inhibitors do not induce virus aggregation. Our findings may be generalized to other uptake processes including bacteria and drug-delivery.

Intracellular release of endocytosed nanoparticles upon a change of ligand-receptor interaction.

During passive endocytosis, nanosized particles are initially encapsulated by a membrane separating it from the cytosol. Yet, in many applications the nanoparticles need to be in direct contact with the cytosol in order to be active. We report a simulation study that elucidates the physical mechanisms by which such nanoparticles can shed their bilayer coating. We find that nanoparticle release can be readily achieved by a pH-induced lowering of the attraction between nanoparticle and membrane only if the nanoparticle is either very small or nonspherical. Interestingly, we find that in the case of large spherical nanoparticles, the reduction of attraction needs to be accompanied by exerting an additional tension on the membrane (e.g., via nanoparticle expansion) to achieve release. We expect these findings will contribute to the rational design of drug delivery strategies via nanoparticles.

Layering, freezing, and re-entrant melting of hard spheres in soft confinement.

Confinement can have a dramatic effect on the behavior of all sorts of particulate systems, and it therefore is an important phenomenon in many different areas of physics and technology. Here, we investigate the role played by the softness of the confining potential. Using grand canonical Monte Carlo simulations, we determine the phase diagram of three-dimensional hard spheres that in one dimension are constrained to a plane by a harmonic potential. The phase behavior depends strongly on the density and on the stiffness of the harmonic confinement. While we find the familiar sequence of confined hexagonal and square-symmetric packings, we do not observe any of the usual intervening ordered phases. Instead, the system phase separates under strong confinement, or forms a layered re-entrant liquid phase under weaker confinement. It is plausible that this behavior is due to the larger positional freedom in a soft confining potential and to the contribution that the confinement energy makes to the total free energy. The fact that specific structures can be induced or suppressed by simply changing the confinement conditions (e.g., in a dielectrophoretic trap) is important for applications that involve self-assembled structures of colloidal particles.

Optimized expanded ensembles for simulations involving molecular insertions and deletions. I. Closed systems.

Monte Carlo simulation methods that involve the insertion-deletion of molecules are of wide spread use for the study of thermophysical behavior of complex systems; e.g., for the estimation of chemical potentials in closed-system ensembles. In this work, efficient expanded ensemble methods are described to overcome the lack of ergodicity that often plagues such molecular moves, wherein an arbitrary physical parameter Lambda is used to gradually couple and decouple a partial molecule to and from the system. In particular, we describe the use of (1) acceptance ratio methods for the robust estimation of free-energy changes associated with transitions between Lambda states of the partial molecule, (2) non-Boltzmann sampling of the probability density of Lambda states so that one can achieve either a flat histogram or an optimized histogram based on the maximization of round trips between the Lambda bounds, and (3) an approach to select suitable intermediate stages of the Lambda parameter that maximizes such round trips. The validity of the advocated methods is demonstrated by their application to two model systems, namely, the solvation of large hard spheres into a fluid of small spheres, and the mesophase formation of a block copolymer-homopolymer mixture.

Simulation of the gyroid phase in off-lattice models of pure diblock copolymer melts.

Particle-based molecular simulations of pure diblock copolymer (DBC) systems were performed in continuum space via dissipative particle dynamics and Monte Carlo methods for a bead-spring chain model. This model consisted of chains of soft repulsive particles often used with dissipative particle dynamics. The gyroid phase was successfully simulated in DBC melts at selected conditions provided that the simulation box size was commensurate with the gyroid lattice spacing. Simulations were concentrated at conditions where the gyroid phase is expected to be stable which allowed us to outline approximate phase boundaries. When more than one phase was observed by varying simulation box size, thermodynamic stability was discerned by comparing the Helmholtz free energy of the competing phases. For this purpose, chemical potentials were efficiently simulated via an expanded ensemble that gradually inserts/deletes a target chain to/from the system. These simulations employed a novel combination of Bennett's [J. Comput. Phys. 22, 245 (1976)] acceptance-ratio method to estimate free-energy differences and a recently proposed method to get biasing weights that maximize the number of times that the target chain is regrown. The analysis of the gyroid nodes revealed clear evidence of packing frustration in the form of an (entropically) unfavorably overstretching of chains, a phenomenon that has been suggested to provide the structural basis for the limited region of stability of the gyroid phase in the DBC phase diagram. Finally, the G phase and nodal chain stretching were also found in simulations with a completely different DBC particle-based model.

Protein translocation through a tunnel induces changes in folding kinetics: a lattice model study.

Compaction of a nascent polypeptide chain inside the ribosomal exit tunnel, before it leaves the ribosome, has been proposed to accelerate the folding of newly synthesized proteins following their release from the ribosome. Thus, we used Kinetic Monte Carlo simulations of a minimalist on-lattice model to explore the effect that polypeptide translocation through a variety of channels has on protein folding kinetics. Our results demonstrate that tunnel confinement promotes faster folding of a well-designed protein relative to its folding in free space by displacing the unfolded state towards more compact structures that are closer to the transition state. Since the tunnel only forbids rarely visited, extended configurations, it has little effect on a "poorly designed" protein whose unfolded state is largely composed of low-energy, compact, misfolded configurations. The beneficial effect of the tunnel depends on its width; for example, a too-narrow tunnel enforces unfolded states with limited or no access to the transition state, while a too-wide tunnel has no effect on the unfolded state entropy. We speculate that such effects are likely to play an important role in the folding of some proteins or protein domains in the cellular environment and might dictate whether a protein folds co-translationally or post-translationally.