Local-To-Global Agreement Expansion via the Variance Method

Agreement expansion is concerned with set systems for which local assignments to the sets with almost perfect pairwise consistency (i.e., most overlapping pairs of sets agree on their intersections) implies the existence of a global assignment to the ground set (from which the sets are defined) that agrees with most of the local assignments. It is currently known that if a set system forms a two-sided or a partite high dimensional expander then agreement expansion is implied. However, it was not known whether agreement expansion can be implied for one-sided high dimensional expanders. In this work we show that agreement expansion can be deduced for one-sided high dimensional expanders assuming that all the vertices\u27 links (i.e., the neighborhoods of the vertices) are agreement expanders. Thus, for one-sided high dimensional expander, an agreement expansion of the large complicated complex can be deduced from agreement expansion of its small simple links. Using our result, we settle the open question whether the well studied Ramanujan complexes are agreement expanders. These complexes are neither partite nor two-sided high dimensional expanders. However, they are one-sided high dimensional expanders for which their links are partite and hence are agreement expanders. Thus, our result implies that Ramanujan complexes are agreement expanders, answering affirmatively the aforementioned open question. The local-to-global agreement expansion that we prove is based on the variance method that we develop. We show that for a high dimensional expander, if we define a function on its top faces and consider its local averages over the links then the variance of these local averages is much smaller than the global variance of the original function. This decreasing in the variance enables us to construct one global agreement function that ties together all local agreement functions

High Dimensional Random Walks and Colorful Expansion

Random walks on bounded degree expander graphs have numerous applications, both in theoretical and practical computational problems. A key property of these walks is that they converge rapidly to their stationary distribution. In this work we {\em define high order random walks}: These are generalizations of random walks on graphs to high dimensional simplicial complexes, which are the high dimensional analogues of graphs. A simplicial complex of dimension $d$ has vertices, edges, triangles, pyramids, up to $d$-dimensional cells. For any $0 \leq i < d$, a high order random walk on dimension $i$ moves between neighboring $i$-faces (e.g., edges) of the complex, where two $i$-faces are considered neighbors if they share a common $(i+1)$-face (e.g., a triangle). The case of $i=0$ recovers the well studied random walk on graphs. We provide a {\em local-to-global criterion} on a complex which implies {\em rapid convergence of all high order random walks} on it. Specifically, we prove that if the $1$-dimensional skeletons of all the links of a complex are spectral expanders, then for {\em all} $0 \le i < d$ the high order random walk on dimension $i$ converges rapidly to its stationary distribution. We derive our result through a new notion of high dimensional combinatorial expansion of complexes which we term {\em colorful expansion}. This notion is a natural generalization of combinatorial expansion of graphs and is strongly related to the convergence rate of the high order random walks. We further show an explicit family of {\em bounded degree} complexes which satisfy this criterion. Specifically, we show that Ramanujan complexes meet this criterion, and thus form an explicit family of bounded degree high dimensional simplicial complexes in which all of the high order random walks converge rapidly to their stationary distribution.Comment: 27 page

Unique-Neighbor-Like Expansion and Group-Independent Cosystolic Expansion

In recent years, high dimensional expanders have been found to have a variety of applications in theoretical computer science, such as efficient CSPs approximations, improved sampling and list-decoding algorithms, and more. Within that, an important high dimensional expansion notion is cosystolic expansion, which has found applications in the construction of efficiently decodable quantum codes and in proving lower bounds for CSPs. Cosystolic expansion is considered with systems of equations over a group where the variables and equations correspond to faces of the complex. Previous works that studied cosystolic expansion were tailored to the specific group ??. In particular, Kaufman, Kazhdan and Lubotzky (FOCS 2014), and Evra and Kaufman (STOC 2016) in their breakthrough works, who solved a famous open question of Gromov, have studied a notion which we term "parity" expansion for small sets. They showed that small sets of k-faces have proportionally many (k+1)-faces that contain an odd number of k-faces from the set. Parity expansion for small sets could then be used to imply cosystolic expansion only over ??. In this work we introduce a stronger unique-neighbor-like expansion for small sets. We show that small sets of k-faces have proportionally many (k+1)-faces that contain exactly one k-face from the set. This notion is fundamentally stronger than parity expansion and cannot be implied by previous works. We then show, utilizing the new unique-neighbor-like expansion notion introduced in this work, that cosystolic expansion can be made group-independent, i.e., unique-neighbor-like expansion for small sets implies cosystolic expansion over any group

Double Balanced Sets in High Dimensional Expanders

Recent works have shown that expansion of pseudorandom sets is of great importance. However, all current works on pseudorandom sets are limited only to product (or approximate product) spaces, where Fourier Analysis methods could be applied. In this work we ask the natural question whether pseudorandom sets are relevant in domains where Fourier Analysis methods cannot be applied, e.g., one-sided local spectral expanders. We take the first step in the path of answering this question. We put forward a new definition for pseudorandom sets, which we call "double balanced sets". We demonstrate the strength of our new definition by showing that small double balanced sets in one-sided local spectral expanders have very strong expansion properties, such as unique-neighbor-like expansion. We further show that cohomologies in cosystolic expanders are double balanced, and use the newly derived strong expansion properties of double balanced sets in order to obtain an exponential improvement over the current state of the art lower bound on their minimal distance

Transcriptome Analysis Provides a Blueprint of Coral Egg and Sperm Functions

Background Reproductive biology and the evolutionary constraints acting on dispersal stages are poorly understood in many stony coral species. A key piece of missing information is egg and sperm gene expression. This is critical for broadcast spawning corals, such as our model, the Hawaiian species Montipora capitata, because eggs and sperm are exposed to environmental stressors during dispersal. Furthermore, parental effects such as transcriptome investment may provide a means for cross- or trans-generational plasticity and be apparent in egg and sperm transcriptome data. Methods Here, we analyzed M. capitata egg and sperm transcriptomic data to address three questions: (1) Which pathways and functions are actively transcribed in these gametes? (2) How does sperm and egg gene expression differ from adult tissues? (3) Does gene expression differ between these gametes? Results We show that egg and sperm display surprisingly similar levels of gene expression and overlapping functional enrichment patterns. These results may reflect similar environmental constraints faced by these motile gametes. We find significant differences in differential expression of egg vs. adult and sperm vs. adult RNA-seq data, in contrast to very few examples of differential expression when comparing egg vs. sperm transcriptomes. Lastly, using gene ontology and KEGG orthology data we show that both egg and sperm have markedly repressed transcription and translation machinery compared to the adult, suggesting a dependence on parental transcripts. We speculate that cell motility and calcium ion binding genes may be involved in gamete to gamete recognition in the water column and thus, fertilization

Evolution of Protein-Mediated Biomineralization in Scleractinian Corals

While recent strides have been made in understanding the biological process by which stony corals calcify, much remains to be revealed, including the ubiquity across taxa of specific biomolecules involved. Several proteins associated with this process have been identified through proteomic profiling of the skeletal organic matrix (SOM) extracted from three scleractinian species. However, the evolutionary history of this putative “biomineralization toolkit,” including the appearance of these proteins’ throughout metazoan evolution, remains to be resolved. Here we used a phylogenetic approach to examine the evolution of the known scleractinians’ SOM proteins across the Metazoa. Our analysis reveals an evolutionary process dominated by the co-option of genes that originated before the cnidarian diversification. Each one of the three species appears to express a unique set of the more ancient genes, representing the independent co-option of SOM proteins, as well as a substantial proportion of proteins that evolved independently. In addition, in some instances, the different species expressed multiple orthologous proteins sharing the same evolutionary history. Furthermore, the non-random clustering of multiple SOM proteins within scleractinian-specific branches suggests the conservation of protein function between distinct species for what we posit is part of the scleractinian “core biomineralization toolkit.” This “core set” contains proteins that are likely fundamental to the scleractinian biomineralization mechanism. From this analysis, we infer that the scleractinians’ ability to calcify was achieved primarily through multiple lineage-specific protein expansions, which resulted in a new functional role that was not present in the parent gene

Different skeletal protein toolkits achieve similar structure and performance in the tropical coral Stylophora pistillata and the temperate Oculina patagonica

Stony corals (order: Scleractinia) differ in growth form and structure. While stony corals have gained the ability to form their aragonite skeleton once in their evolution, the suite of proteins involved in skeletogenesis is different for different coral species. This led to the conclusion that the organic portion of their skeleton can undergo rapid evolutionary changes by independently evolving new biomineralization-related proteins. Here, we used liquid chromatography-tandem mass spectrometry to sequence skeletogenic proteins extracted from the encrusting temperate coral Oculina patagonica. We compare it to the previously published skeletal proteome of the branching subtropical corals Stylophora pistillata as both are regarded as highly resilient to environmental changes. We further characterized the skeletal organic matrix (OM) composition of both taxa and tested their effects on the mineral formation using a series of overgrowth experiments on calcite seeds. We found that each species utilizes a different set of proteins containing different amino acid compositions and achieve a different morphology modification capacity on calcite overgrowth. Our results further support the hypothesis that the different coral taxa utilize a species-specific protein set comprised of independent gene co-option to construct their own unique organic matrix framework. While the protein set differs between species, the specific predicted roles of the whole set appear to underline similar functional roles. They include assisting in forming the extracellular matrix, nucleation of the mineral and cell signaling. Nevertheless, the different composition might be the reason for the varying organization of the mineral growth in the presence of a particular skeletal OM, ultimately forming their distinct morphologies

Divergent Evolutionary Histories of DNA Markers in a Hawaiian Population of the Coral Montipora capitata

We investigated intra- and inter-colony sequence variation in a population of the dom- inant Hawaiian coral Montipora capitata by analyzing marker gene and genomic data. Ribosomal ITS1 regions showed evidence of a reticulate history among the colonies, suggesting incomplete rDNA repeat homogenization. Analysis of the mitochondrial genome identified a major (M. capitata) and a minor (M. flabellata) haplotype in single polyp-derived sperm bundle DNA with some colonies containing 2-3 different mtDNA haplotypes. In contrast, Pax-C and newly identified single-copy nuclear genes showed either no sequence differences or minor variations in SNP frequencies segregating among the colonies. Our data suggest past mitochondrial introgression in M. capitata, whereas nuclear single-copy loci show limited variation, highlighting the divergent evolutionary histories of these coral DNA markers

Mineral formation in the primary polyps of pocilloporoid corals

In reef-building corals, larval settlement and its rapid calcification provides a unique opportunity to study the bio-calcium carbonate formation mechanism involving skeleton morphological changes. Here we investigate the mineral formation of primary polyps, just after settlement, in two species of the pocilloporoid corals: Stylophora pistillata (Esper, 1797) and Pocillopora acuta (Lamarck, 1816). We show that the initial mineral phase is nascent Mg-Calcite, with rod-like morphology in P. acuta, and dumbbell morphology in S. pistillata. These structures constitute the first layer of the basal plate which is comparable to Rapid Accretion Deposits (Centers of Calcification, CoC) in adult coral skeleton. We found also that the rod-like/dumbbell Mg-Calcite structures in subsequent growth step will merge into larger aggregates by deposition of aragonite needles. Our results suggest that a biologically controlled mineralization of initial skeletal deposits occurs in three steps: first, vesicles filled with divalent ions are formed intracellularly. These vesicles are then transferred to the calcification site, forming nascent Mg-Calcite rod/pristine dumbbell structures. During the third step, aragonite crystals develop between these structures forming spherulite-like aggregates