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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 synthetic post-translational protein circuits are significantly impacting the landscape of biomimicry engineering. However, designing sustained dynamic phenomena in these circuits remains an outstanding challenge. Inspired by the KaiABC system regulating the circadian clock in cyanobacteria, we develop two experimentally realizable post-translational oscillators. The oscillators rely on a small number of components interacting only through reversible binding and phosphorylation/dephosphorylation reactions.

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 study the shear jamming of athermal frictionless soft spheres, and find that in the thermodynamic limit, a shear-jammed state exists with different elastic properties from the isotropically-jammed state. For example, shear-jammed states can have a non-zero residual shear stress in the thermodynamic limit that arises from long-range stress-stress correlations. As a result, the ratio of the shear and bulk moduli, which in isotropically-jammed systems vanishes as the jamming transition is approached from above, instead approaches a constant. Despite these striking differences, we argue that in a deeper sense, the shear jamming and isotropic jamming transitions actually have the same symmetry, and that the differences can be fully understood by rotating the six-dimensional basis of the elastic modulus tensor., 15 pages, 5 figures -- minor improvements on main text and bibliography

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 present a model of soft active particles that leads to a rich array of collective behavior found also in dense biological swarms of bacteria and other unicellular organisms. Our model uses only local interactions, such as Vicsek-type nearest neighbor alignment, short-range repulsion, and a local boundary term. Changing the relative strength of these interactions leads to migrating swarms, rotating swarms and jammed swarms, as well as swarms that exhibit run-and-tumble motion, alternating between migration and either rotating or jammed states. Interestingly, although a migrating swarm moves slower than an individual particle, the diffusion constant can be up to three orders of magnitude larger, suggesting that collective motion can be highly advantageous, for example, when searching for food., 8 pages, 7 figures

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

As a function of packing fraction at zero temperature and applied stress, an amorphous packing of spheres exhibits a jamming transition where the system is sensitive to boundary conditions even in the thermodynamic limit. Upon further compression, the system should become insensitive to boundary conditions provided it is sufficiently large. Here we explore the linear response to a large class of boundary perturbations in 2 and 3 dimensions. We consider each finite packing with periodic-boundary conditions as the basis of an infinite square or cubic lattice and study properties of vibrational modes at arbitrary wave vector. We find that the stability of such modes be understood in terms of a competition between plane waves and the anomalous vibrational modes associated with the jamming transition; infinitesimal boundary perturbations become irrelevant for systems that are larger than a length scale that characterizes the transverse excitations. This previously identified length diverges at the jamming transition., 8 pages, 5 figures

In 2005, Wyart et al. (Europhys. Lett., 72 (2005) 486) showed that the low frequency vibrational properties of jammed amorphous sphere packings can be understood in terms of a length scale, called l*, that diverges as the system becomes marginally unstable. Despite the tremendous success of this theory, it has been difficult to connect the counting argument that defines l* to other length scales that diverge near the jamming transition. We present an alternate derivation of l* based on the onset of rigidity. This phenomenological approach reveals the physical mechanism underlying the length scale and is relevant to a range of systems for which the original argument breaks down. It also allows us to present the first direct numerical measurement of l*., 8 pages, 5 figures