In many viruses, hundreds of proteins assemble an outer shell (capsid) around the viral nucleic acid to form an infectious virion. How the assembly process selects the viral genome amidst a vast excess of diverse cellular nucleic acids is poorly understood. It has recently been discovered that many viruses perform assembly and genome packaging within liquid-liquid phase separated biomolecular condensates inside the host cell. However, the role of condensates in genome packaging is poorly understood. Here, we construct equilibrium and dynamical rate equation models for condensate-coupled assembly and genome packaging. We show that when the viral genome and capsid proteins favorably partition into the condensate, assembly rates, yields, and packaging efficiencies can increase by orders of magnitude. Selectivity is further enhanced by the condensate when capsid proteins are translated during assembly and packaging. Our results suggest that viral condensates provide a mechanism to ensure robust and highly selective assembly of virions around viral genomes. More broadly, our results may apply to other types of selective co-assembly processes that occur within biomolecular condensates, and suggest that liquid-liquid phase-separated condensates could be exploited for selective encapsulation of microscopic cargo in human-engineered systems.

The spontaneous formation of contractile asters is ubiquitous in reconstituted active materials composed of biopolymers and molecular motors. Asters are radially oriented biopolymers or biopolymer bundles with a dense motor-rich core. The microscopic origins of their material properties and their stability are unknown. Recent efforts highlighted how motor-filament and filament-filament interactions control the formation of asters composed of microtubules and kinesin motors. However, the impact of motor-motor interactions is less understood, despite growing evidence that molecular motors often spontaneously aggregate, both in vitro and in vivo. In this article, we combine experiments and simulations to reveal the origin of the arrested coarsening, aging, and stability of contractile asters composed of microtubules, clusters of adenosine triphosphate (ATP)-powered kinesin-1 motors, and a depletant. Asters coalesce into larger asters upon collision. We show that the spontaneous aggregation of motor clusters drives the solidification of aster cores, arresting their coalescence. We detect aggregation of motor clusters at the single microtubule level, where the uncaging of additional ATP drives the delayed but sudden detachment of large motor aggregates from isolated microtubules. Computer simulations of cytoskeletal assemblies demonstrate that decreasing the motors' unbinding rate slows down the aster's coalescence. Changing the motors' binding rate did not impact the aster's coalescence dynamics. Finally, we show that the aggregation of motor clusters and aster aging result from the combined effects of depletion forces and nonspecific binding of the clusters to themselves. We propose alternative formulations that mitigate these effects, and prevent aster aging. The resulting self-organized structures have a finite lifetime, which reveals that motor aggregation is crucial for maintaining aster's stability. Overall, these experiments and simulations enhance our understanding of how to rationally design long-lived and stable contractile materials from cytoskeletal proteins.

Competition for resources is a fundamental constraint that guides the self-organization of natural, biological, and human systems, ranging from urban planning and ecosystem development to intracellular pattern formation. Here, we reveal that competition for resources is at the origin of the collective dynamics that emerge in a population of colloids propelled by actin treadmilling, an out-of-equilibrium process where filaments grow from one end while shrinking from the other. Using a combination of experiments and theory, we show that symmetry-breaking, self-propulsion, and flocking emerge from the local competition for actin monomers. We demonstrate that beads propelled by actin treadmilling are anti-chemotactic and spontaneously generate asymmetric actin gradients that trigger and sustain directed motility. Flocking emerges from the combined effects of anti-chemotaxis and local competition for monomers. The flocking transition depends on the actin polymerization rate, actin monomer diffusivity, and the bead's motility, whose interplay controls the emergence of short-range attractive interactions between the colloids. Our findings demonstrate that active stress generation coupled to reaction-diffusion is a generic mechanism that can lead to a multiscale cascade of behaviors when active agents remodel their environment. Actin treadmilling offers a platform to study how motile agents that interact through a field self-organize in novel dynamical phases, with potential applications in non-reciprocal and trainable active matter.

Biology is full of intricate molecular structures whose geometries are inextricably linked to their function. Many of these structures exhibit varying curvature, such as the helical structure of the bacterial flagellum, which is critical for their motility. Because synthetic analogues of these shapes could be valuable platforms for nanotechnologies, including drug delivery and plasmonics, controllable synthesis of variable-curvature structures of various material systems, from fullerenes to supramolecular assemblies, has been a long-standing goal. Like two-dimensional crystals, these structures have a two-periodic symmetry, but unlike standard two-dimensional crystals, they are embedded in three dimensions with complex, spatially-varying curvatures that cause the structures to close upon themselves in one or more dimensions. Here, we develop and implement a design strategy to program the self-assembly of a complex spectrum of two-periodic curved crystals with variable periodicity, spatial dimension, and topology, spanning from toroids to achiral serpentine tubules to both left- and right-handed helical tubules. Our design strategy uses a kirigami-based mapping of 2D planar tilings to 3D curved crystals that preserves the periodicity, two-fold rotational symmetries, and subunit dimensions via the arrangement of disclination defects. We survey the modular geometry of these curved crystals and infer the addressable subunit interactions required to assemble them from triangular subunits. To demonstrate this design strategy, we program the self-assembly of toroids, helical- and serpentine-tubules from DNA origami subunits. A simulation model of the assembly pathways reveals physical considerations for programming the geometric specificity of angular folds in the curved crystal required to avoid defect-mediated misassembly.

Biomolecular condensates are liquid- or gel-like droplets of proteins and nucleic acids formed at least in part through liquid-liquid phase separation. Condensates enable diverse functions of cells and the pathogens that infect them, including self-assembly reactions. For example, it has been shown that many viruses form condensates within their host cells to compartmentalize capsid assembly and packaging of the viral genome. Yet, the physical principles controlling condensate-mediated self-assembly remain incompletely understood. In this article we use coarse-grained molecular dynamics simulations to study the effect of a condensate on the assembly of icosahedral capsids. The capsid subunits are represented by simple shape-based models to enable simulating a wide range of length and time scales, while the condensate is modeled implicitly to study the effects of phase separation independent of the molecular details of biomolecular condensates. Our results show that condensates can significantly enhance assembly rates, yields, and robustness to parameter variations, consistent with previous theoretical predictions. However, extending beyond those predictions, the computational models also show that excluded volume enables control over the number of capsids that assemble within condensates. Moreover, long-lived aberrant off-pathway assembly intermediates can suppress yields within condensates. In addition to elucidating condensate-mediated assembly of viruses and other biological structures, these results may guide the use of condensates as a generic route to enhance and control self-assembly in human-engineered systems.

Recent advances in synthetic methods enable designing subunits that self-assemble into structures with precise, finite sizes and well-defined architectures, but yields are frequently suppressed by the formation of off-target metastable structures. Increasing the complexity (the number of distinct subunit types) can inhibit off-target structures, but leads to slower kinetics and higher synthesis costs. Here, we study icosahedral shells formed of programmable triangular subunits as a model system, and identify design principles that produce the highest target yield at the lowest complexity. We use a symmetry-based construction to create a range of design complexities, starting from the maximal symmetry Caspar-Klug assembly up to the fully addressable, zero-symmetry assembly. Kinetic Monte Carlo simulations reveal that the most prominent defects leading to off-target assemblies are disclinations at sites of rotational symmetry. We derive symmetry-based rules for identifying the optimal (lowest-complexity, highest-symmetry) design that inhibits these disclinations, leading to robust, high-fidelity assembly of targets with arbitrarily large, yet precise, finite sizes. The optimal complexity varies non-monotonically with target size, with `magic' sizes appearing for high-symmetry designs in which symmetry axes do not intersect vertices of the triangular net. The optimal designs at magic sizes require 12 times fewer inequivalent interaction-types than the (minimal symmetry) fully addressable construction, which greatly reduces the timescale and experimental cost required to achieve high fidelity assembly of large targets. This symmetry-based principle for pruning off-target assembly generalizes to diverse architectures with different topologies.

Programmable self-assembly has recently enabled the creation of complex structures through precise control of the interparticle interactions and the particle geometries. Targeting ever more structurally complex, dynamic, and functional assemblies necessitates going beyond the design of the structure itself, to the measurement and control of the local flexibility of the intersubunit connections and its impact on the collective mechanics of the entire assembly. In this study, we demonstrate a method to infer the mechanical properties of multisubunit assemblies using cryogenic electron microscopy (cryo-EM) and RELION's multi-body refinement. Specifically, we analyze the fluctuations of pairs of DNA-origami subunits that self-assemble into tubules. By measuring the fluctuations of dimers using cryo-EM, we extract mechanical properties such as the bending modulus and interparticle spring constant. These properties are then applied to elastic models to predict assembly outcomes, which align well with experimental observations. This approach not only provides a deeper understanding of nanoparticle mechanics but also opens pathways to refining subunit designs to achieve precise assembly behavior. This methodology could have broader applications in the study of nanomaterials, including protein assemblies, where understanding the interplay of mechanical properties and subunit geometry is essential for controlling complex self-assembled structures.

The spontaneous creation of disclinations is a defining characteristic of active nematics, which is rarely observed in equilibrium systems or other active matter systems. Thus, understanding the mechanics of disclinations is crucial for developing reliable continuum theories and practical applications. In this work, we explore this intrinsic mechanics by performing large-scale 3D simulations of a particle-based model of active semiflexible filaments. We investigate the effects of filament stiffness and activity on the collective behavior of active nematics. Analysis of the steady state and the topological properties of initial disclination loops reveals that the system is governed by a single parameter, an activity-dependent effective stiffness. Then, we develop a method to visualize director field orientations in a physically transparent manner during the formation of disclination loops. Based on this, we establish a unified theory for the mechanics of disclination emergence, across the range of bend and twist. This disclination analysis framework can also be applied to diverse other 3D liquid crystal systems.

Being intrinsically nonequilibrium, active materials can potentially perform functions that would be thermodynamically forbidden in passive materials. However, active systems have diverse local attractors that correspond to distinct dynamical states, many of which exhibit chaotic turbulent-like dynamics and thus cannot perform work or useful functions. Designing such a system to choose a specific dynamical state is a formidable challenge. Motivated by recent advances enabling optogenetic control of experimental active materials, we describe an optimal control theory framework that identifies a spatiotemporal sequence of light-generated activity that drives an active nematic system toward a prescribed dynamical steady state. Active nematics are unstable to spontaneous defect proliferation and chaotic streaming dynamics in the absence of control. We demonstrate that optimal control theory can compute activity fields that redirect the dynamics into a variety of alternative dynamical programs and functions. This includes dynamically reconfiguring between states, selecting and stabilizing emergent behaviors that do not correspond to attractors, and are hence unstable in the uncontrolled system. Our results provide a roadmap to leverage optical control methods to rationally design structure, dynamics, and function in a wide variety of active materials.

In many simple viruses and virus-like particles, the protein capsid self-assembles around a nucleic-acid genome. Although the assembly process has been studied in detail, relatively little is known about how the capsid disassembles, a potentially important step for infection (in viruses) or cargo delivery (in virus-like particles). We investigate capsid disassembly using a coarse-grained molecular dynamics model of a T = 1 dodecahedral capsid and an RNA-like polymer. We alter the interactions between the subunits of the capsid as well as the ionic strength of the solution to induce partial or complete disassembly of self-assembled particles. We find that disassembly follows nucleation-and-growth kinetics, where the nucleation barrier is related to the interaction strengths as well as to the conformation of the polymer. In particular, we find that polymer segments that interact with adjacent subunits reinforce the subunit-subunit contacts. These results have implications for the design of virus-like particles for applications such as drug delivery. A cargo designed with reinforcement in mind might be used to control the stability of such particles and mediate disassembly.

Stabilizing and shaping autonomous flows of active fluids is a fundamental challenge and a prerequisite for applications. We embed a light-responsive microtubule-based nematic in a proportional-integral control loop that adjusts the applied light intensity in response to real-time measurements of the spatially averaged flow speed. The self-regulating hardware-software-wetware system maintains a target flow speed against external or internal perturbations, including protein aging and aggregation, sample-to-sample variability, and temperature variation. Varying the controller’s gains reveals antagonistic roles between feedback and intrinsic processes, leading to nontrivial dynamics observed in fluctuation spectra. In particular, oscillations emerge from the interplay between the controller, motor binding kinetics, and active hydrodynamic relaxation. Accounting for the underlying binding timescale, our coarse-grained model and nematohydrodynamics simulations corroborate these observations. This work provides insight into the coupled dynamics of controlled active matter, laying the foundation for spatiotemporal patterning of active stress to generate and stabilize new dynamical configurations.

Deep learning-based optical flow (DLOF) extracts features in adjacent video frames with deep convolutional neural networks. It uses those features to estimate the inter-frame motions of objects. We evaluate the ability of optical flow to quantify the spontaneous flows of microtubule (MT)-based active nematics under different labeling conditions, and compare its performance to particle image velocimetry (PIV). We obtain flow velocity ground truths either by performing semi-automated particle tracking on samples with sparsely labeled filaments, or from passive tracer beads. DLOF produces more accurate velocity fields than PIV for densely labeled samples. PIV cannot reliably distinguish contrast variations at high densities, particularly along the nematic director. DLOF overcomes this limitation. For sparsely labeled samples, DLOF and PIV produce comparable results, but DLOF gives higher-resolution fields. Our work establishes DLOF as a versatile tool for measuring fluid flows in a broad class of active, soft, and biophysical systems. Deep learning-based optical flow (DLOF) extracts features in video frames with deep convolutional neural networks to estimate the inter-frame motions of objects. DLOF computes velocity fields more accurately than PIV for densely labeled systems.

We propose and investigate an extension of the Caspar-Klug symmetry principles for viral capsid assembly to the programmable assembly of size-controlled triply-periodic polyhedra, discrete variants of the Primitive, Diamond, and Gyroid cubic minimal surfaces. Inspired by a recent class of programmable DNA origami colloids, we demonstrate that the economy of design in these crystalline assemblies -- in terms of the growth of the number of distinct particle species required with the increased size-scale (e.g. periodicity) -- is comparable to viral shells. We further test the role of geometric specificity in these assemblies via dynamical assembly simulations, which show that conditions for simultaneously efficient and high-fidelity assembly require an intermediate degree of flexibility of local angles and lengths in programmed assembly. Off-target misassembly occurs via incorporation of a variant of disclination defects, generalized to the case of hyperbolic crystals. The possibility of these topological defects is a direct consequence of the very same symmetry principles that underlie the economical design, exposing a basic tradeoff between design economy and fidelity of programmable, size controlled assembly.

Computational modeling of assembly is challenging for many systems, because their timescales can vastly exceed those accessible to simulations. This article describes the multiMSM, which is a general framework that uses Markov state models (MSMs) to enable simulating self-assembly and self-organization of finite-sized structures on timescales that are orders of magnitude longer than those accessible to brute-force dynamics simulations. As with traditional MSM approaches, the method efficiently overcomes free energy barriers and other dynamical bottlenecks. In contrast to previous MSM approaches to simulating assembly, the framework describes simultaneous assembly of many clusters and the consequent depletion of free subunits or other small oligomers. The algorithm accounts for changes in transition rates as concentrations of monomers and intermediates evolve over the course of the reaction. Using two model systems, we show that the multiMSM accurately predicts the concentrations of the full ensemble of intermediates on timescales required to reach equilibrium. Importantly, after constructing a multiMSM for one system concentration, yields at other concentrations can be approximately calculated without any further sampling. This capability allows for orders of magnitude additional speedup. In addition, the method enables highly efficient calculation of quantities such as free energy profiles, nucleation timescales, flux along the ensemble of assembly pathways, and entropy production rates. Identifying contributions of individual transitions to entropy production rates reveals sources of kinetic traps. The method is broadly applicable to systems with equilibrium or nonequilibrium dynamics and is trivially parallelizable and, thus, highly scalable.

Self-assembly of complex and functional materials remains a grand challenge in soft material science. Efficient assembly depends on a delicate balance between thermodynamic and kinetic effects, requiring fine-tuning affinities and concentrations of subunits. By contrast, we introduce an assembly paradigm that allows large error-tolerance in the subunit affinity and helps avoid kinetic traps. Our combined experimental and computational approach uses a model system of triangular subunits programmed to assemble into T = 3 icosahedral capsids comprising 60 units. The experimental platform uses DNA origami to create monodisperse colloids whose three-dimensional geometry is controlled to nanometer precision, with two distinct bonds whose affinities are controlled to kBT precision, quantified in situ by static light scattering. The computational model uses a coarse-grained representation of subunits, short-ranged potentials, and Langevin dynamics. Experimental observations and modeling reveal that when the bond affinities are unequal, two distinct hierarchical assembly pathways occur, in which the subunits first form dimers in one case and pentamers in another. These hierarchical pathways produce complete capsids faster and are more robust against affinity variation than egalitarian pathways, in which all binding sites have equal strengths. This finding suggests that hierarchical assembly may be a general engineering principle for optimizing self-assembly of complex target structures.

Active nematics are dense systems of rodlike particles that consume energy to drive motion at the level of the individual particles. They exist in natural systems like biological tissues and artificial materials such as suspensions of self-propelled colloidal particles or synthetic microswimmers. Active nematics have attracted significant attention in recent years due to their spectacular nonequilibrium collective spatiotemporal dynamics, which may enable applications in fields such as robotics, drug delivery, and materials science. The director field, which measures the direction and degree of alignment of the local nematic orientation, is a crucial characteristic of active nematic and is essential for studying topological defects. However, determining the director field is a significant challenge in many experimental systems. Although director fields can be derived from images of active nematics using traditional imaging processing methods, the accuracy of such methods are highly sensitive to the settings of the algorithms. These settings must be tuned from image-to-image due to experimental noise, intrinsic noise of the imaging technology, and perturbations caused by changes in experimental conditions. This sensitivity currently limits automatic analysis of active nematics. To address this, we developed a machine learning model for extracting reliable director fields from raw experimental images, which enables accurate analysis of topological defects. Application of the algorithm to experimental data demonstrates that the approach is robust and highly generalizable to experimental settings that are different from those in the training data. It could be a promising tool for investigating active nematics and may be generalized to other active matter systems.

Deep learning-based optical flow (DLOF) extracts features in adjacent video frames with deep convolutional neural networks. It uses those features to estimate the inter-frame motions of objects at the pixel level. In this article, we evaluate the ability of optical flow to quantify the spontaneous flows of MT-based active nematics under different labeling conditions. We compare DLOF against the commonly used technique, particle imaging velocimetry (PIV). We obtain flow velocity ground truths either by performing semi-automated particle tracking on samples with sparsely labeled filaments, or from passive tracer beads. We find that DLOF produces significantly more accurate velocity fields than PIV for densely labeled samples. We show that the breakdown of PIV arises because the algorithm cannot reliably distinguish contrast variations at high densities, particularly in directions parallel to the nematic director. DLOF overcomes this limitation. For sparsely labeled samples, DLOF and PIV produce results with similar accuracy, but DLOF gives higher-resolution fields. Our work establishes DLOF as a versatile tool for measuring fluid flows in a broad class of active, soft, and biophysical systems.

This review accompanies the Special Issue on the subject of physical virology, which features work presented at the recent Gordon Research Conference (GRC) on this topic [...]