Discover the world’s best science and medicine | Nature.com

Discover the world’s best science and medicine | Nature.com

Discover the world’s best science and medicine | Nature.com

Discover the world’s best science and medicine | Nature.com

The local structure of the highly “overdoped” 95 K superconductor Sr2CuO3.3 determined by Cu K X-ray absorption fine structure (XAFS) at 62 K in magnetically oriented samples shows that 1) the magnetization is perpendicular to the c axis; 2) at these levels of precision the Cu sublattice is tetragonal in agreement with the crystal structure; the O sublattice has 3) continuous -Cu-O- chains that orient perpendicular to an applied magnetic field; 4) approximately half-filled -Cu-O- chains that orient parallel to this field; 5) a substantial number of apical O vacancies; 6) O ions at some apical positions with expanded Cu-O distances; and 7) interstitial positions that imply highly displaced Sr ions. These results contradict the universally accepted features of cuprates that require intact CuO2 planes, magnetization along the c axis, and a termination of the superconductivity when the excess charge on the CuO2 Cu ions exceeds 0.27. These radical differences in charge and structure demonstrate that this compound constitutes a separate class of Cu-O–based superconductors in which the superconductivity originates in a different, more complicated structural unit than CuO2 planes while retaining exceptionally high transition temperatures.

In the primate brain, a set of areas in the ventrolateral frontal (VLF) cortex and the dorsomedial frontal (DMF) cortex appear to control vocalizations. The basic role of this network in the human brain and how it may have evolved to enable complex speech remain unknown. In the present functional neuroimaging study of the human brain, a multidomain protocol was utilized to investigate the roles of the various areas that comprise the VLF–DMF network in learning rule-based cognitive selections between different types of motor actions: manual, orofacial, nonspeech vocal, and speech vocal actions. Ventrolateral area 44 (a key component of the Broca’s language production region in the human brain) is involved in the cognitive selection of orofacial, as well as, speech and nonspeech vocal responses; and the midcingulate cortex is involved in the analysis of speech and nonspeech vocal feedback driving adaptation of these responses. By contrast, the cognitive selection of speech vocal information requires this former network and the additional recruitment of area 45 and the presupplementary motor area. We propose that the basic function expressed by the VLF–DMF network is to exert cognitive control of orofacial and vocal acts and, in the language dominant hemisphere of the human brain, has been adapted to serve higher speech function. These results pave the way to understand the potential changes that could have occurred in this network across primate evolution to enable speech production.

The United Nations (UN) launched the 2030 Agenda for Sustainable Development to address an ongoing crisis: human pressure leading to unprecedented environmental degradation, climatic change, social inequality, and other negative planet-wide consequences. This crisis stems from a dramatic increase in human appropriation of natural resources to keep pace with rapid population growth, dietary shifts toward higher consumption of animal products, and higher demand for energy (1, 2). There is an increased recognition that Sustainable Development Goals (SDGs) are linked to one another (3, 4), and priorities such as food production, biodiversity conservation, and climate change mitigation cannot be considered in isolation (5⇓⇓–8). Hence, understanding those dynamics is central to achieving the vision of the UN 2030 Agenda.

The LIN28:pre-let-7:TUTase ternary complex regulates pluripotency and oncogenesis by controlling processing of the let-7 family of microRNAs. The complex oligouridylates the 3′ ends of pre-let-7 molecules, leading to their degradation via the DIS3L2 exonuclease. Previous studies suggest that components of this complex are potential therapeutic targets in malignancies that aberrantly express LIN28. In this study we developed a functional epitope selection approach to identify nanobody inhibitors of the LIN28:pre-let-7:TUT4 complex. We demonstrate that one of the identified nanobodies, Nb-S2A4, targets the 106-residue LIN28:let-7 interaction (LLI) fragment of TUT4. Nb-S2A4 can effectively inhibit oligouridylation and monouridylation of pre-let-7g in vitro. Expressing Nb-S2A4 allows maturation of the let-7 species in cells expressing LIN28, highlighting the therapeutic potential of targeting the LLI fragment.

Light limitation represents a significant threat to plant survival. Shade-intolerant species have, therefore, evolved mechanisms to detect and avoid shading by neighbors. Plants detect the proximity and density of neighboring vegetation by monitoring alterations in light quality (1). Phytochrome photoreceptors detect changes in the ratio of red (R) to far-red light (FR), with phytochrome B performing a dominant role. R is absorbed by living vegetation and used for photosynthesis, whereas the majority of FR is transmitted through and reflected within canopies. R:FR is, therefore, reduced proportionally with increasing depth of canopy (1). Early perception of encroaching shade enables plants to rapidly elongate stems and elevate leaves to overtop competitors and avoid light limitation. Such responses are termed shade avoidance and can promote survival in mixed stands (1). Shade avoidance is regulated by a group of transcription factors named PHYTOCHROME INTERACTING FACTORS (PIFs), with PIF4, PIF5, and PIF7 performing dominant roles (2⇓–4). These PIFs collectively promote synthesis of the growth-promoting hormone auxin (4, 5). In sunlight (high R:FR), phytochrome B becomes …

Population expansion in space, or range expansion, is widespread in nature and in clinical settings. Space competition among heterogeneous subpopulations during range expansion is essential to population ecology, and it may involve the interplay of multiple factors, primarily growth and motility of individuals. Structured microbial communities provide model systems to study space competition during range expansion. Here we use bacterial swarms to investigate how single-cell motility contributes to space competition among heterogeneous bacterial populations during range expansion. Our results revealed that motility heterogeneity can promote the spatial segregation of subpopulations via a dynamic motility selection process. The dynamic motility selection is enabled by speed-dependent persistence time bias of single-cell motion, which presumably arises from physical interaction between cells in a densely packed swarm. We further showed that the dynamic motility selection may contribute to collective drug tolerance of swarming colonies by segregating subpopulations with transient drug tolerance to the colony edge. Our results illustrate that motility heterogeneity, or “motility fitness,” can play a greater role than growth rate fitness in determining the short-term spatial structure of expanding populations.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

This article has been retracted; see accompanying Retraction Note, which can be accessed via a link at the top of the paper.

Multicomponent reactions have become a mainstay in both academic and industrial synthetic organic chemistry owing to their step- and atom-economy advantages over traditional synthetic sequences1. Recently, bicyclo[1.1.1]pentane (BCP) motifs have come to the fore as valuable pharmaceutical bioisosteres of benzene rings, and, in particular, 1,3-disubstituted BCP moieties have become widely adopted in medicinal chemistry as para-phenyl ring replacements2. Often these structures are generated from [1.1.1]propellane via opening of the internal C–C bond, through the addition of either radicals or metal-based nucleophiles3–13. The resulting propellane-addition adducts are subsequently transformed to the requisite polysubstituted BCP compounds via a range of synthetic sequences that traditionally involve multiple chemical steps. While this approach has been effective so far, it is clear that a multicomponent reaction that enables single-step access to complex and diverse polysubstituted BCP products would be synthetically advantageous over the current stepwise approaches. Here we report a one-step three-component radical coupling of [1.1.1]propellane to afford diverse functionalized bicycles using various radical precursors and heteroatom nucleophiles via a metallaphotoredox catalysis protocol. The reaction operates on short timescales (five minutes to one hour) across multiple (>10) nucleophile classes and can accommodate a diverse array of radical precursors, including those which generate alkyl, α-acyl, trifluoromethyl, and sulfonyl radicals. This method has been used to rapidly prepare BCP analogues of known pharmaceuticals, one of which has pharmacokinetic properties substantially different to those of its commercial progenitor.

Building and expanding on principles of statistics, machine learning, and scientific inquiry, we propose the predictability, computability, and stability (PCS) framework for veridical data science. Our framework, composed of both a workflow and documentation, aims to provide responsible, reliable, reproducible, and transparent results across the data science life cycle. The PCS workflow uses predictability as a reality check and considers the importance of computation in data collection/storage and algorithm design. It augments predictability and computability with an overarching stability principle. Stability expands on statistical uncertainty considerations to assess how human judgment calls impact data results through data and model/algorithm perturbations. As part of the PCS workflow, we develop PCS inference procedures, namely PCS perturbation intervals and PCS hypothesis testing, to investigate the stability of data results relative to problem formulation, data cleaning, modeling decisions, and interpretations. We illustrate PCS inference through neuroscience and genomics projects of our own and others. Moreover, we demonstrate its favorable performance over existing methods in terms of receiver operating characteristic (ROC) curves in high-dimensional, sparse linear model simulations, including a wide range of misspecified models. Finally, we propose PCS documentation based on R Markdown or Jupyter Notebook, with publicly available, reproducible codes and narratives to back up human choices made throughout an analysis. The PCS workflow and documentation are demonstrated in a genomics case study available on Zenodo.

The continued emergence of Middle East Respiratory Syndrome (MERS) cases with a high case fatality rate stresses the need for the availability of effective antiviral treatments. Remdesivir (GS-5734) effectively inhibited MERS coronavirus (MERS-CoV) replication in vitro, and showed efficacy against Severe Acute Respiratory Syndrome (SARS)-CoV in a mouse model. Here, we tested the efficacy of prophylactic and therapeutic remdesivir treatment in a nonhuman primate model of MERS-CoV infection, the rhesus macaque. Prophylactic remdesivir treatment initiated 24 h prior to inoculation completely prevented MERS-CoV−induced clinical disease, strongly inhibited MERS-CoV replication in respiratory tissues, and prevented the formation of lung lesions. Therapeutic remdesivir treatment initiated 12 h postinoculation also provided a clear clinical benefit, with a reduction in clinical signs, reduced virus replication in the lungs, and decreased presence and severity of lung lesions. The data presented here support testing of the efficacy of remdesivir treatment in the context of a MERS clinical trial. It may also be considered for a wider range of coronaviruses, including the currently emerging novel coronavirus 2019-nCoV.

Kirigami, the creative art of paper cutting, is a promising paradigm for mechanical metamaterials. However, to make kirigami-inspired structures a reality requires controlling the topology of kirigami to achieve connectivity and rigidity. We address this question by deriving the maximum number of cuts (minimum number of links) that still allow us to preserve global rigidity and connectivity of the kirigami. A deterministic hierarchical construction method yields an efficient topological way to control both the number of connected pieces and the total degrees of freedom. A statistical approach to the control of rigidity and connectivity in kirigami with random cuts complements the deterministic pathway, and shows that both the number of connected pieces and the degrees of freedom show percolation transitions as a function of the density of cuts (links). Together, this provides a general framework for the control of rigidity and connectivity in planar kirigami.

Forecasting the spatiotemporal spread of infectious diseases during an outbreak is an important component of epidemic response. However, it remains challenging both methodologically and with respect to data requirements, as disease spread is influenced by numerous factors, including the pathogen’s underlying transmission parameters and epidemiological dynamics, social networks and population connectivity, and environmental conditions. Here, using data from Sierra Leone, we analyze the spatiotemporal dynamics of recent cholera and Ebola outbreaks and compare and contrast the spread of these two pathogens in the same population. We develop a simulation model of the spatial spread of an epidemic in order to examine the impact of a pathogen’s incubation period on the dynamics of spread and the predictability of outbreaks. We find that differences in the incubation period alone can determine the limits of predictability for diseases with different natural history, both empirically and in our simulations. Our results show that diseases with longer incubation periods, such as Ebola, where infected individuals can travel farther before becoming infectious, result in more long-distance sparking events and less predictable disease trajectories, as compared to the more predictable wave-like spread of diseases with shorter incubation periods, such as cholera.

Voiced sound production is the primary form of acoustic communication in terrestrial vertebrates, particularly birds and mammals, including humans. Developing a causal physics-based model that ultimately links descending vocal motor control to tissue vibration and sound requires embodied approaches that include realistic representations of voice physiology. Here, we first implement and then experimentally test a high-fidelity three-dimensional (3D) continuum model for voiced sound production in birds. Driven by individual-based physiologically quantifiable inputs, combined with noninvasive inverse methods for tissue material parameterization, our model accurately predicts observed key vibratory and acoustic performance traits. These results demonstrate that realistic models lead to accurate predictions and support the continuum model approach as a critical tool toward a causal model of voiced sound production.

Bacillus anthracis, the etiological agent of anthrax, is a well-established model organism. For B. anthracis and most other infectious diseases, knowledge regarding transmission and infection parameters in natural systems, in large part, comprises data gathered from closely controlled laboratory experiments. Fatal, natural anthrax infections transmit the bacterium through new host−pathogen contacts at carcass sites, which can occur years after death of the previous host. For the period between contact and death, all of our knowledge is based upon experimental data from domestic livestock and laboratory animals. Here we use a noninvasive method to explore the dynamics of anthrax infections, by evaluating the terminal diversity of B. anthracis in anthrax carcasses. We present an application of population genetics theory, specifically, coalescence modeling, to intrainfection populations of B. anthracis to derive estimates for the duration of the acute phase of the infection and effective population size converted to the number of colony-forming units establishing infection in wild plains zebra (Equus quagga). Founding populations are small, a few colony-forming units, and infections are rapid, lasting roughly between 1 d and 3 d in the wild. Our results closely reflect experimental data, showing that small founding populations progress acutely, killing the host within days. We believe this method is amendable to other bacterial diseases from wild, domestic, and human systems.

On 13 November 2015, a group of Islamic jihadists launched a series of coordinated terror attacks across the city of Paris, France. Witness statements and police reports were almost unbearable to hear, but for people directly affected by the attacks, their traumatic experiences are unforgettable. How do people cope with the memories of such terrible experiences when reminders of the event are omnipresent? Selectively blocking memories of the event is a common coping strategy, but is it a good one? Clinicians would probably be skeptical about recommending this strategy because it is counterproductive for many patients who have experienced a traumatic event. On page 756 of this issue, Mary et al. (1) report the neural differences that control the retrieval of traumatic memories in 102 individuals who were affected by the Paris terror attacks but who dealt with these memories in different ways: 55 developed posttraumatic stress disorder (PTSD), and 47 did not.

The formation of junctions between p- and n-type semiconductors is the elementary building block of solid-state electronics. The unidirectional transportation of electrons across the junction interface, known as rectification, is the functional basis of electronic diodes, transistors, and integrated logic circuits. By contrast, biological systems use ions as signal carriers for sensing, signal transduction, and information processing. For example, ion-selective proteins embedded in the neuronal membrane transport sodium and potassium ions asymmetrically to propagate nerve impulses (1). Although the pursuit of dimensional shrinkage in modern electronics is reaching its physical limitation, the development of an ionic analogy to p-n junctions is expected to bring about unconventional circuits that simulate the nervous system (2) and has the potential to deliver intrinsically deformable processing units. On page 773 of this issue, Kim et al. (3) report the fabrication of ionic diodes and transistors using solvent-free ionoelastomers, thereby establishing a basis for stretchable ionotronic devices.

Surface defects of nanomaterials can serve as active sites for adsorption and chemical transformations in heterogeneous catalysis (1, 2). However, the defects in catalyst supports can also induce carbon deposition to deactivate the catalysts. This issue is particularly relevant for supported metal catalysts, a major category of heterogeneous catalysts, which deactivate because of the formation of carbon-based materials on catalyst surfaces after prolonged use, through a process called coking (3). Developing supported metal catalysts with coking and sintering resistance with high catalytic activity in high-temperature applications remains a great challenge (4). On page 777 of this issue, Song et al. (5) address this critical issue by choosing a defect-free single-crystalline magnesium oxide (MgO) as a support and then blocking the active step edges with nickel-molybdenum (Ni–Mo) nanocatalysts, achieving coke- and sintering-resistant activity in quantitative production of synthesis gas from dry reforming of methane (CH4).

Oceans play a critical role in Earth's carbon cycle. Quantifying essential processes in carbon cycling and extending these to future predictions remain great scientific challenges. Nearly 30% of anthropogenic carbon is absorbed from the atmosphere into the ocean, where sempiternal, ubiquitous populations of microscopic particles transport carbon into the isolated deep sea (1). This complex pathway is driven by various biophysical and chemical interactions, including phytoplankton productivity, zooplankton grazing, oceanic mixing and turbulence, advection, and the sinking of particles and aggregates (2) (see the figure). On page 791 of this issue, Briggs et al. (3) quantitatively describe the key role of particle fragmentation in carbon storage by the ocean, potentially accounting for half of the particle flux that fails to sink into the deep ocean.

As the Earth system moves through continuous changes, scientists have attempted to predict pathways the planet will follow by unraveling trajectories of individual ecosystems and their interactions and by identifying the thresholds beyond which irreversible changes might occur (1). For example, increases in global aridity are known to affect terrestrial ecosystems, but it remains unknown whether modifications in global aridity will cause gradual or abrupt systemic or idiosyncratic transitions. Now, on page 787 of this issue, Berdugo et al. (2) analyze large datasets of observational and empirical evidence from studies of drylands. The authors show that changes occur in a sequential series of nonlinear thresholds beyond which dryland vegetation can vanish, leaving bare soil to prevail.

For several decades, researchers have studied the molecular mechanisms underlying circadian rhythms, the daily oscillations ubiquitous in biology. This basic clockwork is well understood in animal cells: Conserved clock proteins form a transcription-translation feedback loop that drives circadian oscillations of gene expression and downstream processes. These cellular clocks in peripheral tissues are hierarchically synchronized by a “master clock” in the brain [the suprachiasmatic nucleus (SCN) in mammals] responding to daylight, and also by other physiological signals such as feeding. On page 800 of this issue, Ray et al. (1) demonstrate that many circadian oscillations—in transcription, translation, and protein phosphorylation—can continue in mouse cells in the absence of an essential circadian clock gene, Bmal1 (brain and muscle ARNT-like 1). Thus, there might be other unknown clocks that also control circadian gene expression.

Sidney Holt, who reshaped fisheries science and helped reverse the decline of marine mammal populations, died on 22 December 2019. He was 93 years old. Sidney was a leader of numerous nongovernmental marine conservation organizations and a former senior staff member at the Food and Agriculture Organization (FAO) and other United Nations (UN) organizations. On the Dynamics of Exploited Fish Populations, the book he wrote in 1957 in collaboration with fellow biologist Raymond Beverton, remains integral to the work of fisheries scientists today.

At the dawn of a new decade and in a pivotal election year, we face un­precedented challenges that threaten the environment, public health, and security. Meanwhile, dark money is being funneled through powerful lobbyists, plaguing the process of enacting informed, evidence-based policies. David Michaels's new book, The Triumph of Doubt, is a tour de force that examines how frequently, and easily, sci­ence has been manipulated to discredit expertise and account­ability on issues ranging from obesity and concussions to opi­oids and climate change.

In Einstein in Bohemia, Michael Gordin seeks to illuminate the elusive sig­nificance of Einstein's brief tenure in Prague, both for the biography of the famous physi­cist and for the cultural history of Bohemia. An expert in the his­tory of modern physical sciences and of Russian, European, and American history, Gordin pulls together a wealth of infor­mation about the wider context of Einstein's stay in Prague and of the cultural, scientific, and political history of Bohemia.

Clonal animals do not sequester a germ line during embryogenesis. Instead, they have adult stem cells that contribute to somatic tissues or gametes. How germ fate is induced in these animals, and whether this process is related to bilaterian embryonic germline induction, is unknown. We show that transcription factor AP2 (Tfap2), a regulator of mammalian germ lines, acts to commit adult stem cells, known as i-cells, to the germ cell fate in the clonal cnidarian Hydractinia symbiolongicarpus. Tfap2 mutants lacked germ cells and gonads. Transplanted wild-type cells rescued gonad development but not germ cell induction in Tfap2 mutants. Forced expression of Tfap2 in i-cells converted them to germ cells. Therefore, Tfap2 is a regulator of germ cell commitment across germ line–sequestering and germ line–nonsequestering animals.

Effector-triggered immunity (ETI), induced by host immune receptors in response to microbial effectors, protects plants against virulent pathogens. However, a systematic study of ETI prevalence against species-wide pathogen diversity is lacking. We constructed the Pseudomonas syringae Type III Effector Compendium (PsyTEC) to reduce the pan-genome complexity of 5127 unique effector proteins, distributed among 70 families from 494 strains, to 529 representative alleles. We screened PsyTEC on the model plant Arabidopsis thaliana and identified 59 ETI-eliciting alleles (11.2%) from 19 families (27.1%), with orthologs distributed among 96.8% of P. syringae strains. We also identified two previously undescribed host immune receptors, including CAR1, which recognizes the conserved effectors AvrE and HopAA1, and found that 94.7% of strains harbor alleles predicted to be recognized by either CAR1 or ZAR1.

One-dimensional electronic systems can support exotic collective phases because of the enhanced role of electron correlations. We describe the experimental observation of a series of quantized conductance steps within strongly interacting electron waveguides formed at the lanthanum aluminate–strontium titanate (LaAlO3/SrTiO3) interface. The waveguide conductance follows a characteristic sequence within Pascal’s triangle: (1, 3, 6, 10, 15, …) ⋅ e2/h, where e is the electron charge and h is the Planck constant. This behavior is consistent with the existence of a family of degenerate quantum liquids formed from bound states of n = 2, 3, 4, … electrons. Our experimental setup could provide a setting for solid-state analogs of a wide range of composite fermionic phases.

Soft ionic conductors have enabled stretchable and transparent devices, but liquids in such devices tend to leak and evaporate. In this study, we demonstrate diodes and transistors using liquid-free ionoelastomers, in which either anions or cations are fixed to an elastomer network and the other ionic species are mobile. The junction of the two ionoelastomers of opposite polarity yields an ionic double layer, which is capable of rectifying and switching ionic currents without electrochemical reactions. The entropically driven depletion of mobile ions creates a junction of tough adhesion, and the stretchability of the junction enables electromechanical transduction.

Large-scale carbon fixation requires high-volume chemicals production from carbon dioxide. Dry reforming of methane could provide an economically feasible route if coke- and sintering-resistant catalysts were developed. Here, we report a molybdenum-doped nickel nanocatalyst that is stabilized at the edges of a single-crystalline magnesium oxide (MgO) support and show quantitative production of synthesis gas from dry reforming of methane. The catalyst runs more than 850 hours of continuous operation under 60 liters per unit mass of catalyst per hour reactive gas flow with no detectable coking. Synchrotron studies also show no sintering and reveal that during activation, 2.9 nanometers as synthesized crystallites move to combine into stable 17-nanometer grains at the edges of MgO crystals above the Tammann temperature. Our findings enable an industrially and economically viable path for carbon reclamation, and the “Nanocatalysts On Single Crystal Edges” technique could lead to stable catalyst designs for many challenging reactions.

The ground state of charge-neutral graphene under perpendicular magnetic field was predicted to be a quantum Hall topological insulator with a ferromagnetic order and spin-filtered, helical edge channels. In most experiments, however, an insulating state is observed that is accounted for by lattice-scale interactions that promote a broken-symmetry state with gapped bulk and edge excitations. We tuned the ground state of the graphene zeroth Landau level to the topological phase through a suitable screening of the Coulomb interaction with the high dielectric constant of a strontium titanate (SrTiO3) substrate. Robust helical edge transport emerged at magnetic fields as low as 1 tesla and withstanding temperatures up to 110 kelvin over micron-long distances. This versatile graphene platform may find applications in spintronics and topological quantum computation.

Aridity, which is increasing worldwide because of climate change, affects the structure and functioning of dryland ecosystems. Whether aridification leads to gradual (versus abrupt) and systemic (versus specific) ecosystem changes is largely unknown. We investigated how 20 structural and functional ecosystem attributes respond to aridity in global drylands. Aridification led to systemic and abrupt changes in multiple ecosystem attributes. These changes occurred sequentially in three phases characterized by abrupt decays in plant productivity, soil fertility, and plant cover and richness at aridity values of 0.54, 0.7, and 0.8, respectively. More than 20% of the terrestrial surface will cross one or several of these thresholds by 2100, which calls for immediate actions to minimize the negative impacts of aridification on essential ecosystem services for the more than 2 billion people living in drylands.

A critical driver of the ocean carbon cycle is the downward flux of sinking organic particles, which acts to lower the atmospheric carbon dioxide concentration. This downward flux is reduced by more than 70% in the mesopelagic zone (100 to 1000 meters of depth), but this loss cannot be fully accounted for by current measurements. For decades, it has been hypothesized that the missing loss could be explained by the fragmentation of large aggregates into small particles, although data to test this hypothesis have been lacking. In this work, using robotic observations, we quantified total mesopelagic fragmentation during 34 high-flux events across multiple ocean regions and found that fragmentation accounted for 49 ± 22% of the observed flux loss. Therefore, fragmentation may be the primary process controlling the sequestration of sinking organic carbon.

A topological insulator reveals its nontrivial bulk through the presence of gapless edge states: This is called the bulk-boundary correspondence. However, the recent discovery of “fragile” topological states with no gapless edges casts doubt on this concept. We propose a generalization of the bulk-boundary correspondence: a transformation under which the gap between the fragile phase and other bands must close. We derive specific twisted boundary conditions (TBCs) that can detect all the two-dimensional eigenvalue fragile phases. We develop the concept of real-space invariants, local good quantum numbers in real space, which fully characterize these phases and determine the number of gap closings under the TBCs. Realizations of the TBCs in metamaterials are proposed, thereby providing a route to their experimental verification.

Symmetries crucially underlie the classification of topological phases of matter. Most materials, both natural as well as architectured, possess crystalline symmetries. Recent theoretical works unveiled that these crystalline symmetries can stabilize fragile Bloch bands that challenge our very notion of topology: Although answering to the most basic definition of topology, one can trivialize these bands through the addition of trivial Bloch bands. Here, we fully characterize the symmetry properties of the response of an acoustic metamaterial to establish the fragile nature of the low-lying Bloch bands. Additionally, we present a spectral signature in the form of spectral flow under twisted boundary conditions.