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The self-assembly of complex structures from a set of non-identical building blocks is a hallmark of soft matter and biological systems, including protein complexes, colloidal clusters, and DNA-based assemblies. Predicting the dependence of the equilibrium assembly yield on the concentrations and interaction energies of building blocks is highly challenging, owing to the difficulty of computing the entropic contributions to the free energy of the many structures that compete with the ground state configuration. While these calculations yield well known results for spherically symmetric building blocks, we discovered that they do not hold when the building blocks have internal rotational degrees of freedom. Here we present an approach for solving this problem that works with arbitrary building blocks, including proteins with known structure and complex colloidal building blocks. Our algorithm combines classical statistical mechanics with recently developed computational tools for automatic differentiation. Automatic differentiation allows efficient evaluation of equilibrium averages over configurations that would otherwise be intractable. We demonstrate the validity of our framework by comparison to molecular dynamics simulations of simple examples, and apply it to calculate the yield curves for known protein complexes and for the assembly of colloidal shells.

Significance Self-assembly is one of the central themes in biologically controlled synthesis, and it also plays a pivotal role in fabricating a variety of advanced engineering materials. In particular, evaporation-induced self-assembly of colloidal particles enables versatile fabrication of highly ordered two- or three-dimensional nanostructures for optical, sensing, catalytic, and other applications. While it is well known that this process results in the formation of the face-centered cubic (fcc) lattice with the close-packed {111} plane parallel to the substrate, the crystallographic texture development of colloidal crystals is less understood. In this study, we show that the preferred growth in the fcc colloidal crystals synthesized through evaporation-induced assembly is achieved through a gradual crystallographic rotation facilitated by mechanical stress-induced geometrically necessary dislocations., Unlike crystalline atomic and ionic solids, texture development due to crystallographically preferred growth in colloidal crystals is less studied. Here we investigate the underlying mechanisms of the texture evolution in an evaporation-induced colloidal assembly process through experiments, modeling, and theoretical analysis. In this widely used approach to obtain large-area colloidal crystals, the colloidal particles are driven to the meniscus via the evaporation of a solvent or matrix precursor solution where they close-pack to form a face-centered cubic colloidal assembly. Via two-dimensional large-area crystallographic mapping, we show that the initial crystal orientation is dominated by the interaction of particles with the meniscus, resulting in the expected coalignment of the close-packed direction with the local meniscus geometry. By combining with crystal structure analysis at a single-particle level, we further reveal that, at the later stage of self-assembly, however, the colloidal crystal undergoes a gradual rotation facilitated by geometrically necessary dislocations (GNDs) and achieves a large-area uniform crystallographic orientation with the close-packed direction perpendicular to the meniscus and parallel to the growth direction. Classical slip analysis, finite element-based mechanical simulation, computational colloidal assembly modeling, and continuum theory unequivocally show that these GNDs result from the tensile stress field along the meniscus direction due to the constrained shrinkage of the colloidal crystal during drying. The generation of GNDs with specific slip systems within individual grains leads to crystallographic rotation to accommodate the mechanical stress. The mechanistic understanding reported here can be utilized to control crystallographic features of colloidal assemblies, and may provide further insights into crystallographically preferred growth in synthetic, biological, and geological crystals.

Significance Engineering at the nanoscale is rich and complex: researchers have designed small-scale structures ranging from smiley faces to intricate sensors. However, designing specific dynamical features within these structures has proven to be significantly harder than designing the structures themselves. Biology, on the other hand, demonstrates fine-tuned kinetic control at nearly all scales: viruses that form too quickly are rarely infectious, and proper embryonic development depends on the relative rate of tissue growth. Clearly, kinetic features are designable and critical for biological function. We demonstrate a method to control kinetic features of complex systems and apply it to two classic self-assembly systems. Studying and optimizing for kinetic features, rather than static structures, opens the door to a different approach to materials design., The inverse problem of designing component interactions to target emergent structure is fundamental to numerous applications in biotechnology, materials science, and statistical physics. Equally important is the inverse problem of designing emergent kinetics, but this has received considerably less attention. Using recent advances in automatic differentiation, we show how kinetic pathways can be precisely designed by directly differentiating through statistical physics models, namely free energy calculations and molecular dynamics simulations. We consider two systems that are crucial to our understanding of structural self-assembly: bulk crystallization and small nanoclusters. In each case, we are able to assemble precise dynamical features. Using gradient information, we manipulate interactions among constituent particles to tune the rate at which these systems yield specific structures of interest. Moreover, we use this approach to learn nontrivial features about the high-dimensional design space, allowing us to accurately predict when multiple kinetic features can be simultaneously and independently controlled. These results provide a concrete and generalizable foundation for studying nonstructural self-assembly, including kinetic properties as well as other complex emergent properties, in a vast array of systems.

Mathematical models for two experimentally realizable posttranslational protein oscillators are constructed., Recent advances in synthetic posttranslational protein circuits are substantially impacting the landscape of cellular engineering and offer several advantages compared to traditional gene circuits. However, engineering dynamic phenomena such as oscillations in protein-level circuits remains an outstanding challenge. Few examples of biological posttranslational oscillators are known, necessitating theoretical progress to determine realizable oscillators. We construct mathematical models for two posttranslational oscillators, using few components that interact only through reversible binding and phosphorylation/dephosphorylation reactions. Our designed oscillators rely on the self-assembly of two protein species into multimeric functional enzymes that respectively inhibit and enhance this self-assembly. We limit our analysis to within experimental constraints, finding (i) significant portions of the restricted parameter space yielding oscillations and (ii) that oscillation periods can be tuned by several orders of magnitude using recent advances in computational protein design. Our work paves the way for the rational design and realization of protein-based dynamic systems.

Recent advances in designing meta-materials have demonstrated that global mechanical properties of disordered spring networks can be tuned by selectively modifying only a small subset of bonds. Here, using a computationally-efficient approach, we extend this idea in order to tune more general properties of networks. With nearly complete success, we are able to produce a strain between any pair of target nodes in a network in response to an applied source strain on any other pair of nodes by removing only ~1% of the bonds. We are also able to control multiple pairs of target nodes, each with a different individual response, from a single source, and to tune multiple independent source/target responses simultaneously into a network. We have fabricated physical networks in macroscopic two- and three-dimensional systems that exhibit these responses. This targeted behavior is reminiscent of the long-range coupled conformational changes that often occur during allostery in proteins. The ease with which we create these responses may give insight into why allostery is a common means for the regulation of activity in biological molecules., Comment: 28 pages (single column), 5 figures, 2 videos

Controlling motion at the microscopic scale is a fundamental goal in the development of biologically inspired systems. We show that the motion of active, self-propelled colloids can be sufficiently controlled for use as a tool to assemble complex structures such as braids and weaves out of microscopic filaments. Unlike typical self-assembly paradigms, these structures are held together by geometric constraints rather than adhesive bonds. The out-of-equilibrium assembly that we propose involves precisely controlling the 2D motion of active colloids so that their path has a nontrivial topology. We demonstrate with proof-of-principle Brownian dynamics simulations that, when the colloids are attached to long semiflexible filaments, this motion causes the filaments to braid. The ability of the active particles to provide sufficient force necessary to bend the filaments into a braid depends on a number of factors, including the self-propulsion mechanism, the properties of the filament, and the maximum curvature in the braid. Our work demonstrates that nonequilibrium assembly pathways can be designed using active particles.

An orthogonal wavelet basis is characterized by its approximation order, which relates to the ability of the basis to represent general smooth functions on a given scale. It is known, though perhaps not widely known, that there are ways of exceeding the approximation order, i.e., achieving higher-order error in the discretized wavelet transform and its inverse. The focus here is on the development of a practical formulation to accomplish this first for 1D smooth functions, then for 1D functions with discontinuities and then for multidimensional (here 2D) functions with discontinuities. It is shown how to transcend both the wavelet approximation order and the 2D Gibbs phenomenon in representing electromagnetic fields at discontinuous dielectric interfaces that do not simply follow the wavelet-basis grid.

States of self stress, organizations of internal forces in many-body systems that are in equilibrium with an absence of external forces, can be thought of as the constitutive building blocks of the elastic response of a material. In overconstrained disordered packings they have a natural mathematical correspondence with the zero-energy vibrational modes in underconstrained systems. While substantial attention in the literature has been paid to diverging length scales associated with zero- and finite-energy vibrational modes in jammed systems, less is known about the spatial structure of the states of self stress. In this work we define a natural way in which a unique state of self stress can be associated with each bond in a disordered spring network derived from a jammed packing, and then investigate the spatial structure of these bond-localized states of self stress. This allows for an understanding of how the elastic properties of a system would change upon changing the strength or even existence of any bond in the system., Comment: 21 pages (single column), 10 figures

We study the vibrational properties near a free surface of disordered spring networks derived from jammed sphere packings. In bulk systems, without surfaces, it is well understood that such systems have a plateau in the density of vibrational modes extending down to a frequency scale $\omega^*$. This frequency is controlled by $\Delta Z = \langle Z \rangle - 2d$, the difference between the average coordination of the spheres and twice the spatial dimension, $d$, of the system, which vanishes at the jamming transition. In the presence of a free surface we find that there is a density of disordered vibrational modes associated with the surface that extends far below $\omega^*$. The total number of these low-frequency surface modes is controlled by $\Delta Z$, and the profile of their decay into the bulk has two characteristic length scales, which diverge as $\Delta Z^{-1/2}$ and $\Delta Z^{-1}$ as the jamming transition is approached., Comment: 7 pages, 5 figures

Jammed systems are typically thought of as being amorphous. Simulations of packings with varying disorder reveal a crossover from crystalline behaviour, which suggests the physics of jamming also applies to highly ordered systems—providing a new framework for understanding amorphous solids.

We propose a Widom-like scaling ansatz for the critical jamming transition. Our ansatz for the elastic energy shows that the scaling of the energy, compressive strain, shear strain, system size, pressure, shear stress, bulk modulus, and shear modulus are all related to each other via scaling relations, with only three independent scaling exponents. We extract the values of these exponents from already known numerical or theoretical results, and we numerically verify the resulting predictions of the scaling theory for the energy and residual shear stress. We also derive a scaling relation between pressure and residual shear stress that yields insight into why the shear and bulk moduli scale differently. Our theory shows that the jamming transition exhibits an emergent scale invariance, setting the stage for the potential development of a renormalization group theory for jamming.

Characterizing structural inhomogeneity is an essential step in understanding the mechanical response of amorphous materials. We introduce a threshold-free measure based on the field of vectors pointing from the center of each particle to the centroid of the Voronoi cell in which the particle resides. These vectors tend to point in toward regions of high free volume and away from regions of low free volume, reminiscent of sinks and sources in a vector field. We compute the local divergence of these vectors, for which positive values correspond to overpacked regions and negative values identify underpacked regions within the material. Distributions of this divergence are nearly Gaussian with zero mean, allowing for structural characterization using only the moments of the distribution. We explore how the standard deviation and skewness vary with packing fraction for simulations of bidisperse systems and find a kink in these moments that coincides with the jamming transition.

Lerner's theoretical analysis neglects a normalizing factor which distinguishes states of self stress from the stress response to a unit dipolar force along a bond. This factor leads to different spatial profiles upon ensemble averaging.

We introduce a principle unique to disordered solids wherein the contribution of any bond to one global perturbation is uncorrelated with its contribution to another. Coupled with sufficient variability in the contributions of different bonds, this "independent bond-level response" paves the way for the design of real materials with unusual and exquisitely tuned properties. To illustrate this, we choose two global perturbations: compression and shear. By applying a bond removal procedure that is both simple and experimentally relevant to remove a very small fraction of bonds, we can drive disordered spring networks to both the incompressible and completely auxetic limits of mechanical behavior.

We study the effect of dilute pinning on the jamming transition. Pinning reduces the average contact number needed to jam unpinned particles and shifts the jamming threshold to lower densities, leading to a pinning susceptibility, $\chi_p$. Our main results are that this susceptibility obeys scaling form and diverges in the thermodynamic limit as $\chi_p \propto |\phi - \phi_c^\infty|^{-\gamma_p}$ where $\phi_c^\infty$ is the jamming threshold in the absence of pins. Finite-size scaling arguments yield these values with associated statistical (systematic) errors $\gamma_p = 1.018 \pm 0.026 (0.291) $ in $d=2$ and $\gamma_p =1.534 \pm 0.120 (0.822)$ in $d=3$. Logarithmic corrections raise the exponent in $d=2$ to close to the $d=3$ value, although the systematic errors are very large., Comment: 5 pages, 3 figures (1a, 1b, 2, 3a, 3b, 3c)

We investigate the vibrational modes of quasi-two-dimensional disordered colloidal packings of hard colloidal spheres with short-range attractions as a function of packing fraction. Certain properties of the vibrational density of states (vDOS) are shown to correlate with the density and structure of the samples (i.e., in sparsely versus densely packed samples). Specifically, a crossover from dense glassy to sparse gel-like states is suggested by an excess of phonon modes at low frequency and by a variation in the slope of the vDOS with frequency at low frequency. This change in phonon mode distribution is demonstrated to arise largely from localized vibrations that involve individual and/or small clusters of particles with few local bonds. Conventional order parameters and void statistics did not exhibit obvious gel-glass signatures as a function of volume fraction. These mode behaviors and accompanying structural insights offer a potentially new set of indicators for identification of glass-gel transitions and for assignment of gel-like versus glass-like character to a disordered solid material.

In their Letter, Schreck, Bertrand, O'Hern and Shattuck [Phys. Rev. Lett. 107, 078301 (2011)] study nonlinearities in jammed particulate systems that arise when contacts are altered. They conclude that there is "no harmonic regime in the large system limit for all compressions" and "at jamming onset for any system size." Their argument rests on the claim that for finite-range repulsive potentials, of the form used in studies of jamming, the breaking or forming of a single contact is sufficient to destroy the linear regime. We dispute these conclusions and argue that linear response is both justified and essential for understanding the nature of the jammed solid.

Athermal packings of soft repulsive spheres exhibit a sharp jamming transition in the thermodynamic limit. Upon further compression, various structural and mechanical properties display clean power-law behavior over many decades in pressure. As with any phase transition, the rounding of such behavior in finite systems close to the transition plays an important role in understanding the nature of the transition itself. The situation for jamming is surprisingly rich: the assumption that jammed packings are isotropic is only strictly true in the large-size limit, and finite-size has a profound effect on the very meaning of jamming. Here, we provide a comprehensive numerical study of finite-size effects in sphere packings above the jamming transition, focusing on stability as well as the scaling of the contact number and the elastic response., Comment: 20 pages, 12 figures

Particle tracking and displacement covariance matrix techniques are employed to investigate the phonon dispersion relations of two-dimensional colloidal glasses composed of soft, thermoresponsive microgel particles whose temperature-sensitive size permits \textit{in situ} variation of particle packing fraction. Bulk, $B$, and shear, $G$, moduli of the colloidal glasses are extracted from the dispersion relations as a function of packing fraction, and variation of the ratio $G/B$ with packing fraction is found to agree quantitatively with predictions for jammed packings of frictional soft particles. In addition, $G$ and $B$ individually agree with numerical predictions for frictional particles. This remarkable level of agreement enabled us to extract an energy scale for the inter-particle interaction from the individual elastic constants and to derive an approximate estimate for the inter-particle friction coefficient.

We used single-channel electrical recordings and Langevin molecular dynamics simulations to explore the electrophoretic translocation of various beta-hairpin peptides across the staphylococcal alpha-hemolysin (alphaHL) protein pore at single-molecule resolution. The beta-hairpin peptides, which varied in their folding properties, corresponded to the C terminal residues of the B1 domain of protein G. The translocation time was strongly dependent on the electric force and was correlated with the folding features of the beta-hairpin peptides. Highly unfolded peptides entered the pore in an extended conformation, resulting in fast single-file translocation events. In contrast, the translocation of the folded beta-hairpin peptides occurred more slowly. In this case, the beta-hairpin peptides traversed the alphaHL pore in a misfolded or fully folded conformation. This study demonstrates that the interaction between a polypeptide and a beta-barrel protein pore is dependent on the folding features of the polypeptide.