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Mechanical and phononic metamaterials exhibiting negative elastic moduli, gapped vibrational spectra, or topologically protected modes enable precise control of structural and acoustic functionalities. While much progress has been made in their experimental and theoretical characterization, the inverse design of mechanical metamaterials with arbitrarily programmable spectral properties and mode localization remains an unsolved problem. Here, we present a flexible computational inverse-design framework that allows the efficient tuning of one or more gaps at nearly arbitrary positions in the spectrum of discrete phononic metamaterial structures. The underlying algorithm optimizes the linear response of elastic networks directly, is applicable to ordered and disordered structures, scales efficiently in 2D and 3D, and can be combined with a wide range of numerical optimization schemes. We illustrate the broad practical potential of this approach by designing mechanical bandgap switches that open and close pre-programmed spectral gaps in response to an externally applied stimulus such as shear or compression. We further show that the designed structures can host topologically protected edge modes, and validate the numerical predictions through explicit 3D finite element simulations of continuum elastica with experimentally relevant material parameters. Generally, this network-based inverse design paradigm offers a direct pathway towards manufacturing phononic metamaterials, DNA origami structures and topolectric circuits that can realize a wide range of static and dynamic target functionalities. arXiv:1802.07214v2 [cond-mat.mtrl-sci]

Active biological flow networks pervade nature and span a wide range of scales, from arterial blood vessels and bronchial mucus transport in humans to bacterial flow through porous media or plasmodial shuttle streaming in slime molds. Despite their ubiquity, little is known about the self-organization principles that govern flow statistics in such nonequilibrium networks. Here we connect concepts from lattice field theory, graph theory, and transition rate theory to understand how topology controls dynamics in a generic model for actively driven flow on a network. Our combined theoretical and numerical analysis identifies symmetry-based rules that make it possible to classify and predict the selection statistics of complex flow cycles from the network topology. The conceptual framework developed here is applicable to a broad class of biological and nonbiological far-from-equilibrium networks, including actively controlled information flows, and establishes a correspondence between active flow networks and generalized ice-type models.networks | active transport | stochastic dynamics | topology B iological flow networks, such as capillaries (1), leaf veins (2) and slime molds (3), use an evolved topology or active remodeling to achieve near-optimal transport when diffusion is ineffectual or inappropriate (2, 4-7). Even in the absence of explicit matter flux, living systems often involve flow of information currents along physical or virtual links between interacting nodes, as in neural networks (8), biochemical interactions (9), epidemics (10), and traffic flow (11). The ability to vary the flow topology gives networkbased dynamics a rich phenomenology distinct from that of equivalent continuum models (12). Identical local rules can invoke dramatically different global dynamical behaviors when node connectivities change from nearest-neighbor interactions to the broad distributions seen in many networks (13-16). Certain classes of interacting networks are now sufficiently well understood to be able to exploit their topology for the control of input-output relations (17, 18), as exemplified by microfluidic logic gates (19,20). However, when matter or information flow through a noisy network is not merely passive but actively driven by nonequilibrium constituents (3), as in maze-solving slime molds (6), there are no overarching dynamical self-organization principles known. In such an active network, noise and flow may conspire to produce behavior radically different from that of a classical forced network. This raises the general question of how path selection and flow statistics in an active flow network depend on its interaction topology.Flow networks can be viewed as approximations of a complex physical environment, using nodes and links to model intricate geometric constraints (21-23). These constraints can profoundly affect matter transport (24-27), particularly for active systems (28-30) where geometric confinement can enforce highly ordered collective dynamics (20,(31)(32)(33)(34)(35)(36)(37)(38)(39)(40). In s...

Understanding the genetic and epigenetic programs that control differentiation during development is a fundamental challenge, with broad impacts across biology and medicine. Measurement technologies like single-cell RNA-sequencing and CRISPR-based lineage tracing have opened new windows on these processes, through computational trajectory inference and lineage reconstruction. While these two mathematical problems are deeply related, methods for trajectory inference are not typically designed to leverage information from lineage tracing and vice versa. Here, we present LineageOT, a unified framework for lineage tracing and trajectory inference. Specifically, we leverage mathematical tools from graphical models and optimal transport to reconstruct developmental trajectories from time courses with snapshots of both cell states and lineages. We find that lineage data helps disentangle complex state transitions with increased accuracy using fewer measured time points. Moreover, integrating lineage tracing with trajectory inference in this way could enable accurate reconstruction of developmental pathways that are impossible to recover with state-based methods alone.