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We present a model to describe the concentration-dependent growth of protein filaments. Our model contains two states, a low-entropy/high-affinity ordered state and a high-entropy/low-affinity disordered state. Consistent with experiments, our model shows a diffusion-limited linear growth regime at low concentration, followed by a concentration-independent plateau at intermediate concentrations, and rapid disordered precipitation at the highest concentrations. We show that growth in the linear and plateau regions is the result of two processes that compete amid the rapid binding and unbinding of nonspecific states. The first process is the addition of ordered molecules during periods in which the end of the filament is free of incorrectly bound molecules. The second process is the capture of defects, which occurs when consecutive ordered additions occur on top of incorrectly bound molecules. We show that a key molecular property is the probability that a diffusive collision results in a correctly bound state. Small values of this probability suppress the defect capture growth mode, resulting in a plateau in the growth rate when incorrectly bound molecules become common enough to poison ordered growth. We show that conditions that nonspecifically suppress or enhance intermolecular interactions, such as the addition of depletants or osmolytes, have opposite effects on the growth rate in the linear and plateau regimes. In the linear regime, stronger interactions promote growth by reducing dissolution events, but in the plateau regime stronger interactions inhibit growth by stabilizing incorrectly bound molecules., Competing Interests: Declaration of interests The authors declare no competing interests., (Copyright © 2025 Biophysical Society. All rights reserved.)

In the limit of small trial moves the Metropolis Monte Carlo algorithm is equivalent to gradient descent on the energy function in the presence of Gaussian white noise. This observation was originally used to demonstrate a correspondence between Metropolis Monte Carlo moves of model molecules and overdamped Langevin dynamics, but it also applies in the context of training a neural network: making small random changes to the weights of a neural network, accepted with the Metropolis probability, with the loss function playing the role of energy, has the same effect as training by explicit gradient descent in the presence of Gaussian white noise. We explore this correspondence in the context of a simple recurrent neural network. We also explore regimes in which this correspondence breaks down, where the gradient of the loss function becomes very large or small. In these regimes the Metropolis algorithm can still effect training, and so can be used as a probe of the loss function of a neural network in regimes in which gradient descent struggles. We also show that training can be accelerated by making purposely-designed Monte Carlo trial moves of neural-network weights., (© 2024 Author(s). Published under an exclusive license by AIP Publishing.)

We show that the time-resolved dynamics of an underdamped harmonic oscillator can be used to do multifunctional computation, performing distinct computations at distinct times within a single dynamical trajectory. We consider the amplitude of an oscillator whose inputs influence its frequency. The activity of the oscillator at fixed times is a nonmonotonic function of its inputs, so it can solve problems such as XOR that are not linearly separable. The activity of the oscillator at fixed input is a nonmonotonic function of time, so it is multifunctional in a temporal sense, and able to carry out distinct nonlinear computations at distinct times within the same dynamical trajectory. We show that a single oscillator, observed at different times, can act as all of the elementary logic gates and perform binary addition, the latter usually implemented in hardware using five logic gates. We show that a set of n oscillators, observed at different times, can perform an arbitrary number of analog-to-n-bit digital conversions. We also show that oscillators can be trained by gradient descent to perform distinct classification tasks at distinct times. Computing with time-dependent functionality can be done in or out of equilibrium, and suggests a way of reducing the number of parameters or devices required to do nonlinear computations.

Exact analytic calculation shows that optimal control protocols for passive molecular systems often involve rapid variations and discontinuities. However, similar analytic baselines are not generally available for active-matter systems, because it is more difficult to treat active systems exactly. Here we use machine learning to derive efficient control protocols for active-matter systems, and find that they are characterized by sharp features similar to those seen in passive systems. We show that it is possible to learn protocols that effect fast and efficient state-to-state transformations in simulation models of active particles by encoding the protocol in the form of a neural network. We use evolutionary methods to identify protocols that take active particles from one steady state to another, as quickly as possible or with as little energy expended as possible. Our results show that protocols identified by a flexible neural-network ansatz, which allows the optimization of multiple control parameters and the emergence of sharp features, are more efficient than protocols derived recently by constrained analytical methods. Our learning scheme is straightforward to use in experiment, suggesting a way of designing protocols for the efficient manipulation of active matter in the laboratory., (© 2024. The Author(s).)

The Jarzynski equality allows the calculation of free-energy differences using values of work measured from nonequilibrium trajectories. The number of trajectories required to accurately estimate free-energy differences in this way grows sharply with the size of work fluctuations, motivating the search for protocols that perform desired transformations with minimum work. However, protocols of this nature can involve varying temperature, to which the Jarzynski equality does not apply. We derive a variant of the Jarzynski equality that applies to varying-temperature protocols, and show that it can have better convergence properties than the standard version of the equality. We derive this modified equality and the associated fluctuation relation within the framework of Markovian stochastic dynamics, complementing related derivations done within the framework of Hamiltonian dynamics.

In this work we investigate the behaviour of molecules at the nanoscale using scanning tunnelling microscopy in order to explore the origin of the cooperativity in the formation of self-assembled molecular networks (SAMNs) at the liquid/solid interface. By studying concentration dependence of alkoxylated dimethylbenzene, a molecular analogue to 5-alkoxylated isophthalic derivatives, but without hydrogen bonding moieties, we show that the cooperativity effect can be experimentally evaluated even for low-interacting systems and that the cooperativity in SAMN formation is its fundamental trait. We conclude that cooperativity must be a local effect and use the nearest-neighbor Ising model to reproduce the coverage vs. concentration curves. The Ising model offers a direct link between statistical thermodynamics and experimental parameters, making it a valuable tool for assessing the thermodynamics of SAMN formation., Competing Interests: There are no conflicts to declare., (This journal is © The Royal Society of Chemistry.)

We show that a neural network originally designed for language processing can learn the dynamical rules of a stochastic system by observation of a single dynamical trajectory of the system, and can accurately predict its emergent behavior under conditions not observed during training. We consider a lattice model of active matter undergoing continuous-time Monte Carlo dynamics, simulated at a density at which its steady state comprises small, dispersed clusters. We train a neural network called a transformer on a single trajectory of the model. The transformer, which we show has the capacity to represent dynamical rules that are numerous and nonlocal, learns that the dynamics of this model consists of a small number of processes. Forward-propagated trajectories of the trained transformer, at densities not encountered during training, exhibit motility-induced phase separation and so predict the existence of a nonequilibrium phase transition. Transformers have the flexibility to learn dynamical rules from observation without explicit enumeration of rates or coarse-graining of configuration space, and so the procedure used here can be applied to a wide range of physical systems, including those with large and complex dynamical generators., (© 2024. The Author(s).)

Time-dependent protocols that perform irreversible logical operations, such as memory erasure, cost work and produce heat, placing bounds on the efficiency of computers. Here we use a prototypical computer model of a physical memory to show that it is possible to learn feedback-control protocols to do fast memory erasure without input of work or production of heat. These protocols, which are enacted by a neural-network "demon," do not violate the second law of thermodynamics because the demon generates more heat than the memory absorbs. The result is a form of nonlocal heat exchange in which one computation is rendered energetically favorable while a compensating one produces heat elsewhere, a tactic that could be used to rationally design the flow of energy within a computer.

Buča et al. [Phys. Rev. E 100, 020103(R) (2019)2470-004510.1103/PhysRevE.100.020103] study the dynamical large deviations of a boundary-driven cellular automaton. They take a double limit in which first time and then space is made infinite, and interpret the resulting large-deviation singularity as evidence of a first-order phase transition and the accompanying coexistence of two distinct dynamical phases. This view is characteristic of an approach to dynamical large deviations in which time is interpreted as if it were a spatial coordinate of a thermodynamic system [Jack, Eur. Phys. J. B 93, 74 (2020)1434-602810.1140/epjb/e2020-100605-3]. Here, I argue that the large-deviation function produced in this double limit is not consistent with the basic features of the model of Buča et al. I show that a modified limiting procedure results in a nonsingular large-deviation function consistent with those features, and that neither supports the idea of coexisting dynamical phases.

We show that cellular automata can classify data by inducing a form of dynamical phase coexistence. We use Monte Carlo methods to search for general two-dimensional deterministic automata that classify images on the basis of activity, the number of state changes that occur in a trajectory initiated from the image. When the number of time steps of the automaton is a trainable parameter, the search scheme identifies automata that generate a population of dynamical trajectories displaying high or low activity, depending on initial conditions. Automata of this nature behave as nonlinear activation functions with an output that is effectively binary, resembling an emergent version of a spiking neuron.

Phillip L. Geissler made important contributions to the statistical mechanics of biological polymers, heterogeneous materials, and chemical dynamics in aqueous environments. He devised analytical and computational methods that revealed the underlying organization of complex systems at the frontiers of biology, chemistry, and materials science. In this retrospective we celebrate his work at these frontiers.

We use neuroevolutionary learning to identify time-dependent protocols for low-dissipation self-assembly in a model of generic active particles with interactions. When the time allotted for assembly is sufficiently long, low-dissipation protocols use only interparticle attractions, producing an amount of entropy that scales as the number of particles. When time is too short to allow assembly to proceed via diffusive motion, low-dissipation assembly protocols instead require particle self-propulsion, producing an amount of entropy that scales with the number of particles and the swim length required to cause assembly. Self-propulsion therefore provides an expensive but necessary mechanism for inducing assembly when time is of the essence.

Nanocapsules are hollow nanoscale shells that have applications in drug delivery, batteries, self-healing materials, and as model systems for naturally occurring shell geometries. In many applications, nanocapsules are designed to release their cargo as they buckle and collapse, but the details of this transient buckling process have not been directly observed. Here, we use in situ liquid-phase transmission electron microscopy to record the electron-irradiation-induced buckling in spherical 60-187 nm polymer capsules with ∼3.5 nm walls. We observe in real time the release of aqueous cargo from these nanocapsules and their buckling into morphologies with single or multiple indentations. The in situ buckling of nanoscale capsules is compared to ex situ measurements of collapsed and micrometer-sized capsules and to Monte Carlo (MC) simulations. The shape and dynamics of the collapsing nanocapsules are consistent with MC simulations, which reveal that the excessive wrinkling of nanocapsules with ultrathin walls results from their large Föppl-von Kármán numbers around 10 5 . Our experiments suggest design rules for nanocapsules with the desired buckling response based on parameters such as capsule radius, wall thickness, and collapse rate.

Using a model heat engine, we show that neural-network-based reinforcement learning can identify thermodynamic trajectories of maximal efficiency. We consider both gradient and gradient-free reinforcement learning. We use an evolutionary learning algorithm to evolve a population of neural networks, subject to a directive to maximize the efficiency of a trajectory composed of a set of elementary thermodynamic processes; the resulting networks learn to carry out the maximally efficient Carnot, Stirling, or Otto cycles. When given an additional irreversible process, this evolutionary scheme learns a previously unknown thermodynamic cycle. Gradient-based reinforcement learning is able to learn the Stirling cycle, whereas an evolutionary approach achieves the optimal Carnot cycle. Our results show how the reinforcement learning strategies developed for game playing can be applied to solve physical problems conditioned upon path-extensive order parameters.

We show analytically that training a neural network by conditioned stochastic mutation or neuroevolution of its weights is equivalent, in the limit of small mutations, to gradient descent on the loss function in the presence of Gaussian white noise. Averaged over independent realizations of the learning process, neuroevolution is equivalent to gradient descent on the loss function. We use numerical simulation to show that this correspondence can be observed for finite mutations, for shallow and deep neural networks. Our results provide a connection between two families of neural-network training methods that are usually considered to be fundamentally different., (© 2021. The Author(s).)

We introduce a minimal model of solid-forming anisotropic molecules that displays, in thermal equilibrium, surface orientational order without bulk orientational order. The model reproduces the nonequilibrium behavior of recent experiments in which a bulk nonequilibrium structure grown by deposition contains regions of orientational order characteristic of the surface equilibrium. This order is deposited, in general, in a nonuniform way because of the emergence of a growth-poisoning mechanism that causes equilibrated surfaces to grow slower than non-equilibrated surfaces. We use evolutionary methods to design oscillatory protocols able to grow nonequilibrium structures with uniform order, demonstrating the potential of protocol design for the fabrication of this class of materials.

We use a neural-network ansatz originally designed for the variational optimization of quantum systems to study dynamical large deviations in classical ones. We use recurrent neural networks to describe the large deviations of the dynamical activity of model glasses, kinetically constrained models in two dimensions. We present the first finite size-scaling analysis of the large-deviation functions of the two-dimensional Fredrickson-Andersen model, and explore the spatial structure of the high-activity sector of the South-or-East model. These results provide a new route to the study of dynamical large-deviation functions, and highlight the broad applicability of the neural-network state ansatz across domains in physics.

Natural and artificial proteins with designer properties and functionalities offer unparalleled opportunity for functional nanoarchitectures formed through self-assembly. However, to exploit this potential we need to design the system such that assembly results in desired architecture forms while avoiding denaturation and therefore retaining protein functionality. Here we address this challenge with a model system of fluorescent proteins. By manipulating self-assembly using techniques inspired by soft matter where interactions between the components are controlled to yield the desired structure, we have developed a methodology to assemble networks of proteins of one species which we can decorate with another, whose coverage we can tune. Consequently, the interfaces between domains of each component can also be tuned, with potential applications for example in energy - or electron - transfer. Our model system of eGFP and mCherry with tuneable interactions reveals control over domain sizes in the resulting networks.

Within simulations of molecules deposited on a surface we show that neuroevolutionary learning can design particles and time-dependent protocols to promote self-assembly, without input from physical concepts such as thermal equilibrium or mechanical stability and without prior knowledge of candidate or competing structures. The learning algorithm is capable of both directed and exploratory design: it can assemble a material with a user-defined property, or search for novelty in the space of specified order parameters. In the latter mode it explores the space of what can be made, rather than the space of structures that are low in energy but not necessarily kinetically accessible.

Diamine-appended metal-organic frameworks (MOFs) of the form Mg 2 (dobpdc)(diamine) 2 adsorb CO 2 in a cooperative fashion, exhibiting an abrupt change in CO 2 occupancy with pressure or temperature. This change is accompanied by hysteresis. While hysteresis is suggestive of a first-order phase transition, we show that hysteretic temperature-occupancy curves associated with this material are qualitatively unlike the curves seen in the presence of a phase transition; they are instead consistent with CO 2 chain polymerization, within one-dimensional channels in the MOF, in the absence of a phase transition. Our simulations of a microscopic model reproduce this dynamics, providing a physical understanding of cooperative adsorption in this industrially important class of materials.

Singularities of dynamical large-deviation functions are often interpreted as the signal of a dynamical phase transition and the coexistence of distinct dynamical phases, by analogy with the correspondence between singularities of free energies and equilibrium phase behavior. Here we study models of driven random walkers on a lattice. These models display large-deviation singularities in the limit of large lattice size, but the extent to which each model's phenomenology resembles a phase transition depends on the details of the driving. We also compare the behavior of ergodic and nonergodic models that present large-deviation singularities. We argue that dynamical large-deviation singularities indicate the divergence of a model timescale, but not necessarily one associated with cooperative behavior or the existence of distinct phases.

A conceptually simple way to classify images is to directly compare test-set data and training-set data. The accuracy of this approach is limited by the method of comparison used, and by the extent to which the training-set data cover configuration space. Here we show that this coverage can be substantially increased using coarse-graining (replacing groups of images by their centroids) and stochastic sampling (using distinct sets of centroids in combination). We use the MNIST and Fashion-MNIST data sets to show that a principled coarse-graining algorithm can convert training images into fewer image centroids without loss of accuracy of classification of test-set images by nearest-neighbor classification. Distinct batches of centroids can be used in combination as a means of stochastically sampling configuration space, and can classify test-set data more accurately than can the unaltered training set. On the MNIST and Fashion-MNIST data sets this approach converts nearest-neighbor classification from a mid-ranking- to an upper-ranking member of the set of classical machine-learning techniques.

Since the pioneering work of Ned Seeman in the early 1980s, the use of the DNA molecule as a construction material experienced a rapid growth and led to the establishment of a new field of science, nowadays called structural DNA nanotechnology. Here, the self-recognition properties of DNA are employed to build micrometer-large molecular objects with nanometer-sized features, thus bridging the nano- to the microscopic world in a programmable fashion. Distinct design strategies and experimental procedures have been developed over the years, enabling the realization of extremely sophisticated structures with a level of control that approaches that of natural macromolecular assemblies. Nevertheless, our understanding of the building process, i.e., what defines the route that goes from the initial mixture of DNA strands to the final intertwined superstructure, is, in some cases, still limited. In this review, we describe the main structural and energetic features of DNA nanoconstructs, from the simple Holliday junction to more complicated DNA architectures, and present the theoretical frameworks that have been formulated until now to explain their self-assembly. Deeper insights into the underlying principles of DNA self-assembly may certainly help us to overcome current experimental challenges and foster the development of original strategies inspired to dissipative and evolutive assembly processes occurring in nature.

We show how to bound and calculate the likelihood of dynamical large deviations using evolutionary reinforcement learning. An agent, a stochastic model, propagates a continuous-time Monte Carlo trajectory and receives a reward conditioned upon the values of certain path-extensive quantities. Evolution produces progressively fitter agents, potentially allowing the calculation of a piece of a large-deviation rate function for a particular model and path-extensive quantity. For models with small state spaces, the evolutionary process acts directly on rates, and for models with large state spaces, the process acts on the weights of a neural network that parameterizes the model's rates. This approach shows how path-extensive physics problems can be considered within a framework widely used in machine learning.

Förster resonant energy transfer (FRET)-mediated exciton diffusion through artificial nanoscale building block assemblies could be used as an optoelectronic design element to transport energy. However, so far, nanocrystal (NC) systems supported only diffusion lengths of 30 nm, which are too small to be useful in devices. Here, we demonstrate a FRET-mediated exciton diffusion length of 200 nm with 0.5 cm 2 /s diffusivity through an ordered, two-dimensional assembly of cesium lead bromide perovskite nanocrystals (CsPbBr 3 PNCs). Exciton diffusion was directly measured via steady-state and time-resolved photoluminescence (PL) microscopy, with physical modeling providing deeper insight into the transport process. This exceptionally efficient exciton transport is facilitated by PNCs' high PL quantum yield, large absorption cross section, and high polarizability, together with minimal energetic and geometric disorder of the assembly. This FRET-mediated exciton diffusion length matches perovskites' optical absorption depth, thus enabling the design of device architectures with improved performances and providing insight into the high conversion efficiencies of PNC-based optoelectronic devices.

We show that neural networks trained by evolutionary reinforcement learning can enact efficient molecular self-assembly protocols. Presented with molecular simulation trajectories, networks learn to change temperature and chemical potential in order to promote the assembly of desired structures or choose between competing polymorphs. In the first case, networks reproduce in a qualitative sense the results of previously known protocols, but faster and with higher fidelity; in the second case they identify strategies previously unknown, from which we can extract physical insight. Networks that take as input the elapsed time of the simulation or microscopic information from the system are both effective, the latter more so. The evolutionary scheme we have used is simple to implement and can be applied to a broad range of examples of experimental self-assembly, whether or not one can monitor the experiment as it proceeds. Our results have been achieved with no human input beyond the specification of which order parameter to promote, pointing the way to the design of synthesis protocols by artificial intelligence.

We describe a simple form of importance sampling designed to bound and compute large-deviation rate functions for time-extensive dynamical observables in continuous-time Markov chains. We start with a model, defined by a set of rates, and a time-extensive dynamical observable. We construct a reference model, a variational ansatz for the behavior of the original model conditioned on atypical values of the observable. Direct simulation of the reference model provides an upper bound on the large-deviation rate function associated with the original model, an estimate of the tightness of the bound, and, if the ansatz is chosen well, the exact rate function. The exact rare behavior of the original model does not need to be known in advance. We use this method to calculate rate functions for currents and counting observables in a set of network- and lattice models taken from the literature. Straightforward ansätze yield bounds that are tighter than bounds obtained from Level 2.5 of large deviations via approximations that involve uniform scalings of rates. We show how to correct these bounds in order to recover the rate functions exactly. Our approach is complementary to more specialized methods and offers a physically transparent framework for approximating and calculating the likelihood of dynamical large deviations.

Molecular self-assembly is usually done at low supersaturation, leading to low rates of growth, in order to allow time for binding mistakes to anneal. However, such conditions can lead to prohibitively long assembly times where growth proceeds by the slow nucleation of successive layers. Here we use a lattice model of molecular self-assembly to show that growth in this regime can be sped up by impurities, which lower the free-energy cost of layer nucleation. Under certain conditions impurities behave almost as a catalyst in that they are present at high concentration at the surface of the assembling structure, but at low concentration in the bulk of the assembled structure. Extrapolation of our numerics using simple analytic arguments suggests that this mechanism can reduce growth times by orders of magnitude in parameter regimes applicable to molecular systems.

The regular packing of atoms, molecules and nanoparticles provides the basis for the understanding of structural order within condensed phases of matter. Typically the constituent particles are considered to be rigid with a fixed shape. Here we show, through a combined experimental and numerical study of the adsorption of cyclic porphyrin polymers, nanorings, on a graphite surface, that flexible molecules can exhibit a rich and complex packing behaviour. Depending on the number of porphyrin sub-units within the nanoring we observe either a highly ordered hexagonal phase or frustrated packing driven by directional interactions which for some arrangements is combined with the internal deformation of the cyclic polymer. Frustration and deformation occur in arrays of polymers with ten sub-units since close packing and co-alignment of neighbouring groups cannot be simultaneously realised for nanorings with this internal symmetry.

Conformational control in peptoids, N-substituted glycines, is crucial for the design and synthesis of biologically-active compounds and atomically-defined nanomaterials. While there are a growing number of structural studies in solution, most have been performed with conformationally-constrained short sequences (e.g., sterically-hindered sidechains or macrocyclization). Thus, the inherent degree of heterogeneity of unconstrained peptoids in solution remains largely unstudied. Here, we explored the folding landscape of a series of simple peptoid tetramers in aqueous solution by NMR spectroscopy. By incorporating specific 13 C-probes into the backbone using bromoacetic acid-2- 13 C as a submonomer, we developed a new technique for sequential backbone assignment of peptoids based on the 1,n-Adequate pulse sequence. Unexpectedly, two of the tetramers, containing an N-(2-aminoethyl)glycine residue (Nae), had preferred conformations. NMR and molecular dynamics studies on one of the tetramers showed that the preferred conformer (52%) had a trans-cis-trans configuration about the three amide bonds. Moreover, >80% of the ensemble contained a cis amide bond at the central amide. The backbone dihedral angles observed fall directly within the expected minima in the peptoid Ramachandran plot. Analysis of this compound against similar peptoid analogs suggests that the commonly used Nae monomer plays a key role in the stabilization of peptoid structure via a side-chain-to-main-chain interaction. This discovery may offer a simple, synthetically high-yielding approach to control peptoid structure, and suggests that peptoids have strong intrinsic conformational preferences in solution. These findings should facilitate the predictive design of folded peptoid structures, and accelerate application in areas ranging from drug discovery to biomimetic nanoscience., (© 2019 Wiley Periodicals, Inc.)

Like polypeptides, peptoids, or N-substituted glycine oligomers, have intrinsic conformational preferences due to their amide backbones and close spacing of side chain substituents. However, the conformations that peptoids adopt are distinct from polypeptides due to several structural differences: the peptoid backbone is composed of tertiary amide bonds that have trans and cis conformers similar in energy, they lack a backbone hydrogen bond donor, and have an N-substituent. To better understand how these differences manifest in actual peptoid structures, we analyzed 46 high quality, experimentally determined peptoid structures reported in the literature to extract their backbone conformational preferences. One hundred thirty-two monomer dihedral angle pairs were compared to the calculated energy landscape for the peptoid Ramachandran plot, and were found to fall within the expected minima. Interestingly, only two regions of the backbone dihedral angles ϕ and ψ were found to be populated that are mirror images of each other. Furthermore, these two conformers are present in both cis and trans forms. Thus, there are four primary conformers that are sufficient to describe almost all known backbone conformations for peptoid oligomers, despite conformational constraints imposed by a variety of side chains, macrocyclization, or crystal packing forces. Because these conformers are predominant in peptoid structure, and are distinct from those found in protein secondary structures, we propose a simple naming system to aid in the description and classification of peptoid structure., (© 2019 Wiley Periodicals, Inc.)

We demonstrate the guiding principles behind simple two dimensional self-assembly of MOF nanoparticles (NPs) and oleic acid capped iron oxide (Fe 3 O 4 ) NCs into a uniform two-dimensional bi-layered superstructure. This self-assembly process can be controlled by the energy of ligand-ligand interactions between surface ligands on Fe 3 O 4 NCs and Zr 6 O 4 (OH) 4 (fumarate) 6 MOF NPs. Scanning transmission electron microscopy (TEM)/energy-dispersive X-ray spectroscopy and TEM tomography confirm the hierarchical co-assembly of Fe 3 O 4 NCs with MOF NPs as ligand energies are manipulated to promote facile diffusion of the smaller NCs. First-principles calculations and event-driven molecular dynamics simulations indicate that the observed patterns are dictated by combination of ligand-surface and ligand-ligand interactions. This study opens a new avenue for design and self-assembly of MOFs and NCs into high surface area assemblies, mimicking the structure of supported catalyst architectures, and provides a thorough fundamental understanding of the self-assembly process, which could be a guide for designing functional materials with desired structure., (© 2018 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.)

We use computer simulation to study the layer-by-layer growth of particle structures in a lattice gas, taking the number of incorporated vacancies as a measure of the quality of the grown structure. A dynamic scaling relation describes the quality of structures in and out of equilibrium and reveals that the highest-quality structures are obtained, for fixed observation time, using strong interactions and far-from-equilibrium growth conditions. This result contrasts with the usual assumption that weak interactions and mild nonequilibrium conditions are the best way to minimize errors during assembly.

Cooperative adsorption of gases by porous frameworks, which permits more efficient uptake and removal than the more usual noncooperative (Langmuir-type) adsorption, usually results from a phase transition of the framework. Here we show how cooperativity emerges in the class of metal-organic frameworks mmen-M_{2}(dobpdc) in the absence of a phase transition. Our study provides a microscopic understanding of the emergent features of cooperative binding, including the position, slope, and height of the isotherm step, and indicates how to optimize gas storage and separation in these materials.

We study simple models of intermittency, involving switching between two states, within the dynamical large-deviation formalism. Singularities appear in the formalism when switching is cooperative or when its basic time scale diverges. In the first case the unbiased trajectory distribution undergoes a symmetry breaking, leading to a change in shape of the large-deviation rate function for a particular dynamical observable. In the second case the symmetry of the unbiased trajectory distribution remains unbroken. Comparison of these models suggests that singularities of the dynamical large-deviation formalism can signal the dynamical equivalent of an equilibrium phase transition but do not necessarily do so.

The conformations adopted by the molecular constituents of a supramolecular assembly influence its large-scale order. At the same time, the interactions made in assemblies by molecules can influence their conformations. Here we study this interplay in extended flat nanosheets made from nonnatural sequence-specific peptoid polymers. Nanosheets exist because individual polymers can be linear and untwisted, by virtue of polymer backbone elements adopting alternating rotational states whose twists oppose and cancel. Using molecular dynamics and quantum mechanical simulations, together with experimental data, we explore the design space of flat nanostructures built from peptoids. We show that several sets of peptoid backbone conformations are consistent with their being linear, but the specific combination observed in experiment is determined by a combination of backbone energetics and the interactions made within the nanosheet. Our results provide a molecular model of the peptoid nanosheet consistent with all available experimental data and show that its structure results from a combination of intra- and intermolecular interactions., Competing Interests: The authors declare no conflict of interest.

Peptoid nanosheets are supramolecular protein-mimetic materials that form from amphiphilic polypeptoids with aromatic and ionic side chains. Nanosheets have been studied at the nanometer scale, but the molecular structure has been difficult to probe. We report the use of 13 C- 13 C dipolar recoupling solid-state NMR measurements to reveal the configuration of backbone amide bonds selected by 13 C isotopic labeling of adjacent α-carbons. Measurements on the same molecules in the amorphous state and in nanosheets revealed that amide bonds in the center of the amino block of peptoid (NaeNpe) 7 -(NceNpe) 7 (B28) favor the trans configuration in the amorphous state and the cis configuration in the nanosheet. This unexpected result contrasts with previous NMR and theoretical studies of short solvated peptoids. Furthermore, examination of the amide bond at the junction of the two charged blocks within B28 revealed a mixture of both cis and trans configurational states, consistent with the previously predicted brickwork-like intermolecular organization.

Off-lattice active Brownian particles form clusters and undergo phase separation even in the absence of attractions or velocity-alignment mechanisms. Arguments that explain this phenomenon appeal only to the ability of particles to move persistently in a direction that fluctuates, but existing lattice models of hard particles that account for this behavior do not exhibit phase separation. Here we present a lattice model of active matter that exhibits motility-induced phase separation in the absence of velocity alignment. Using direct and rare-event sampling of dynamical trajectories, we show that clustering and phase separation are accompanied by pronounced fluctuations of static and dynamic order parameters. This model provides a complement to off-lattice models for the study of motility-induced phase separation.

We describe a simple method that can be used to sample the rare fluctuations of discrete-time Markov chains. We focus on the case of Markov chains with well-defined steady-state measures, and derive expressions for the large-deviation rate functions (and upper bounds on such functions) for dynamical quantities extensive in the length of the Markov chain. We illustrate the method using a series of simple examples, and use it to study the fluctuations of a lattice-based model of active matter that can undergo motility-induced phase separation.

We compute probability distributions of trajectory observables for reversible and irreversible growth processes. These results reveal a correspondence between reversible and irreversible processes, at particular points in parameter space, in terms of their typical and atypical trajectories. Thus key features of growth processes can be insensitive to the precise form of the rate constants used to generate them, recalling the insensitivity to microscopic details of certain equilibrium behavior. We obtained these results using a sampling method, inspired by the "s-ensemble" large-deviation formalism, that amounts to umbrella sampling in trajectory space. The method is a simple variant of existing approaches, and applies to ensembles of trajectories controlled by the total number of events. It can be used to determine large-deviation rate functions for trajectory observables in or out of equilibrium.

DNA nanoparticles with three-fold coordination have been observed to self-assemble in experiment into a network equivalent to the hexagonal (6.6.6) tiling, and a network equivalent to the 4.8.8 Archimedean tiling. Both networks are built from a single type of vertex. Here we use analytic theory and equilibrium and dynamic simulation to show that a model particle, whose rotational properties lie between those of the vertices of the 6.6.6 and 4.8.8 networks, can self-assemble into a network built from three types of vertex. Important in forming this network is the ability of the particle to rotate when bound, thereby allowing the formation of more than one type of binding motif. The network in question is equivalent to a false tiling, a periodic structure built from irregular polygons, and possesses 40 particles in its unit cell. The emergence of this complex structure, whose symmetry properties are not obviously related to those of its constituent particles, highlights the potential for creating new structures from simple variants of existing nanoparticles.

Models of bacterial growth tend to be "irreversible," allowing for the number of bacteria in a colony to increase but not to decrease. By contrast, models of molecular self-assembly are usually "reversible," allowing for the addition and removal of particles to a structure. Such processes differ in a fundamental way because only reversible processes possess an equilibrium. Here we show at the mean-field level that dynamic trajectories of reversible and irreversible growth processes are similar in that both feel the influence of attractors, at which growth proceeds without limit but the intensive properties of the system are invariant. Attractors of both processes undergo nonequilibrium phase transitions as model parameters are varied, suggesting a unified way of describing typical properties of reversible and irreversible growth. We also establish a connection at the mean-field level between an irreversible model of growth (the magnetic Eden model) and the equilibrium Ising model, supporting the findings made by other authors using numerical simulations.

A challenge of molecular self-assembly is to understand how to design particles that self-assemble into a desired structure and not any of a potentially large number of undesired structures. Here we use simulation to show that a strategy of minimal positive design allows the self-assembly of networks equivalent to the 8 semiregular Archimedean tilings of the plane, structures not previously realized in simulation. This strategy consists of identifying the fewest distinct types of interparticle interaction that appear in the desired structure, and does not require enumeration of the many possible undesired structures. The resulting particles, which self-assemble into the desired networks, possess DNA-like selectivity of their interactions. Assembly of certain molecular networks may therefore require such selectivity.

Conventional approaches to the capture of CO2 by metal-organic frameworks focus on equilibrium conditions, and frameworks that contain little CO2 in equilibrium are often rejected as carbon-capture materials. Here we use a statistical mechanical model, parameterized by quantum mechanical data, to suggest that metal-organic frameworks can be used to separate CO2 from a typical flue gas mixture when used under nonequilibrium conditions. The origin of this selectivity is an emergent gas-separation mechanism that results from the acquisition by different gas types of different mobilities within a crowded framework. The resulting distribution of gas types within the framework is in general spatially and dynamically heterogeneous. Our results suggest that relaxing the requirement of equilibrium can substantially increase the parameter space of conditions and materials for which selective gas capture can be effected.

Three-dimensional protein structures usually contain regions of local order, called secondary structure, such as α-helices and β-sheets. Secondary structure is characterized by the local rotational state of the protein backbone, quantified by two dihedral angles called ϕ and ψ. Particular types of secondary structure can generally be described by a single (diffuse) location on a two-dimensional plot drawn in the space of the angles ϕ and ψ, called a Ramachandran plot. By contrast, a recently-discovered nanomaterial made from peptoids, structural isomers of peptides, displays a secondary-structure motif corresponding to two regions on the Ramachandran plot [Mannige et al., Nature 526, 415 (2015)]. In order to describe such 'higher-order' secondary structure in a compact way we introduce here a means of describing regions on the Ramachandran plot in terms of a single Ramachandran number, [Formula: see text], which is a structurally meaningful combination of ϕ and ψ. We show that the potential applications of [Formula: see text] are numerous: it can be used to describe the geometric content of protein structures, and can be used to draw diagrams that reveal, at a glance, the frequency of occurrence of regular secondary structures and disordered regions in large protein datasets. We propose that [Formula: see text] might be used as an order parameter for protein geometry for a wide range of applications.

Colloidal particles of two types, driven in opposite directions, can segregate into lanes [Vissers et al., Soft Matter 7, 2352 (2011)1744-683X10.1039/c0sm01343a]. This phenomenon can be reproduced by two-dimensional Brownian dynamics simulations of model particles [Dzubiella et al., Phys. Rev. E 65, 021402 (2002)1063-651X10.1103/PhysRevE.65.021402]. Here we use computer simulation to assess the generality of lane formation with respect to variation of particle type and dynamical protocol. We find that laning results from rectification of diffusion on the scale of a particle diameter: oppositely driven particles must, in the time taken to encounter each other in the direction of the drive, diffuse in the perpendicular direction by about one particle diameter. This geometric constraint implies that the diffusion constant of a particle, in the presence of those of the opposite type, grows approximately linearly with the Péclet number, a prediction confirmed by our numerics over a range of model parameters. Such environment-dependent diffusion is statistically similar to an effective interparticle attraction; consistent with this observation, we find that oppositely driven nonattractive colloids display features characteristic of the simplest model system possessing both interparticle attractions and persistent motion, the driven Ising lattice gas [Katz, Leibowitz, and Spohn, J. Stat. Phys. 34, 497 (1984)JSTPBS0022-471510.1007/BF01018556]. These features include long-ranged correlations in the disordered regime, a critical regime characterized by a change in slope of the particle current with the Péclet number, and fluctuations that grow with system size. By analogy, we suggest that lane formation in the driven colloid system is a phase transition in the macroscopic limit, but that macroscopic phase separation would not occur in finite time upon starting from disordered initial conditions.

The growth of multicomponent structures in simulations and experiments often results in kinetically trapped, nonequilibrium objects. In such cases we have no general theoretical framework for predicting the outcome of the growth process. Here we use computer simulations to study the growth of two-component structures within a simple lattice model. We show that kinetic trapping happens for many choices of growth rate and intercomponent interaction energies, and that qualitatively distinct kinds of kinetic trapping are found in different regions of parameter space. In a region in which the low-energy structure is an "antiferromagnet" or "checkerboard," we show that the grown nonequilibrium structure displays a component-type stoichiometry that is different from the equilibrium one but is insensitive to growth rate and solution conditions. This robust nonequilibrium stoichiometry can be predicted via a mapping to the jammed random tiling of dimers studied by Flory, a finding that suggests a way of making defined nonequilibrium structures in experiment.

Two-dimensional (2D) atomically defined organic nanomaterials are an important material class with broad applications. However, few general synthetic methods exist to produce such materials in high yields and to precisely functionalize them. One strategy to form ordered 2D organic nanomaterials is through the supramolecular assembly of sequence-defined synthetic polymers. Peptoids, one such class of polymer, are designable bioinspired heteropolymers whose main-chain length and monomer sequence can be precisely controlled. We have recently discovered that individual peptoid polymers with a simple sequence of alternating hydrophobic and ionic monomers can self-assemble into highly ordered, free-floating nanosheets. A detailed understanding of their molecular structure and supramolecular assembly dynamics provides a robust platform for the discovery of new classes of nanosheets with tunable properties and novel applications. In this Account, we discuss the discovery, characterization, assembly, molecular modeling, and functionalization of peptoid nanosheets. The fundamental properties of peptoid nanosheets, their mechanism of formation, and their application as robust scaffolds for molecular recognition and as templates for the growth of inorganic minerals have been probed by an arsenal of experimental characterization techniques (e.g., scanning probe, electron, and optical microscopy, X-ray diffraction, surface-selective vibrational spectroscopy, and surface tensiometry) and computational techniques (coarse-grained and atomistic modeling). Peptoid nanosheets are supramolecular assemblies of 16-42-mer chains that form molecular bilayers. They span tens of microns in lateral dimensions and freely float in water. Their component chains are highly ordered, with chains nearly fully extended and packed parallel to one another as a result of hydrophobic and electrostatic interactions. Nanosheets form via a novel interface-catalyzed monolayer collapse mechanism. Peptoid chains first assemble into a monolayer at either an air-water or oil-water interface, on which peptoid chains extend, order, and pack into a brick-like pattern. Upon mechanical compression of the interface, the monolayer buckles into stable bilayer structures. Recent work has focused on the design of nanosheets with tunable properties and functionality. They are readily engineerable, as functional monomers can be readily incorporated onto the nanosheet surface or into the interior. For example, functional hydrophilic "loops" have been displayed on the surfaces of nanosheets. These loops can interact with specific protein targets, serving as a potentially general platform for molecular recognition. Nanosheets can also bind metal ions and serve as 2D templates for mineral growth. Through our understanding of the formation mechanism, along with predicted features ascertained from molecular modeling, we aim to further design and synthesize nanosheets as robust protein mimetics with the potential for unprecedented functionality and stability.

Self-poisoning is a kinetic trap that can impair or prevent crystal growth in a wide variety of physical settings. Here we use dynamic mean-field theory and computer simulation to argue that poisoning is ubiquitous because its emergence requires only the notion that a molecule can bind in two (or more) ways to a crystal; that those ways are not energetically equivalent; and that the associated binding events occur with sufficiently unequal probability. If these conditions are met then the steady-state growth rate is in general a non-monotonic function of the thermodynamic driving force for crystal growth, which is the characteristic of poisoning. Our results also indicate that relatively small changes of system parameters could be used to induce recovery from poisoning.