Entanglement vs. gap for one-dimensional spin systems

We study the relationship between entanglement and spectral gap for local Hamiltonians in one dimension. The area law for a one-dimensional system states that for the ground state, the entanglement of any interval is upper-bounded by a constant independent of the size of the interval. However, the possible dependence of the upper bound on the spectral gap Delta is not known, as the best known general upper bound is asymptotically much larger than the largest possible entropy of any model system previously constructed for small Delta. To help resolve this asymptotic behavior, we construct a family of one-dimensional local systems for which some intervals have entanglement entropy which is polynomial in 1/Delta, whereas previously studied systems, such as free fermion systems or systems described by conformal field theory, had the entropy of all intervals bounded by a constant times log(1/Delta).Comment: 16 pages. v2 is final published version with slight clarification

Extractors for Adversarial Sources via Extremal Hypergraphs

Randomness extraction is a fundamental problem that has been studied for over three decades. A well-studied setting assumes that one has access to multiple independent weak random sources, each with some entropy. However, this assumption is often unrealistic in practice. In real life, natural sources of randomness can produce samples with no entropy at all or with unwanted dependence. Motivated by this and applications from cryptography, we initiate a systematic study of randomness extraction for the class of adversarial sources defined as follows. A weak source $\mathbf{X}$ of the form $\mathbf{X}_1,...,\mathbf{X}_N$, where each $\mathbf{X}_i$ is on $n$ bits, is an $(N,K,n,k)$-source of locality $d$ if the following hold: (1) Somewhere good sources: at least $K$ of the $\mathbf{X}_i$\u27s are independent, and each contains min-entropy at least $k$. We call these $\mathbf{X}_i$\u27s good sources, and their locations are unknown. (2) Bounded dependence: each remaining (bad) source can depend arbitrarily on at most $d$ good sources. We focus on constructing extractors with negligible error, in the regime where most of the entropy is contained within a few sources instead of across many (i.e., $k$ is at least polynomial in $K$). In this setting, even for the case of $0$-locality, very little is known prior to our work. For $d \geq 1$, essentially no previous results are known. We present various new extractors for adversarial sources in a wide range of parameters, and some of our constructions work for locality $d = K^{\Omega(1)}$. As an application, we also give improved extractors for small-space sources. The class of adversarial sources generalizes several previously studied classes of sources, and our explicit extractor constructions exploit tools from recent advances in extractor machinery, such as two-source non-malleable extractors and low-error condensers. Thus, our constructions can be viewed as a new application of non-malleable extractors. In addition, our constructions combine the tools from extractor theory in a novel way through various sorts of explicit extremal hypergraphs. These connections leverage recent progress in combinatorics, such as improved bounds on cap sets and explicit constructions of Ramsey graphs, and may be of independent interest

Wear Minimization for Cuckoo Hashing: How Not to Throw a Lot of Eggs into One Basket

We study wear-leveling techniques for cuckoo hashing, showing that it is possible to achieve a memory wear bound of $\log\log n+O(1)$ after the insertion of $n$ items into a table of size $Cn$ for a suitable constant $C$ using cuckoo hashing. Moreover, we study our cuckoo hashing method empirically, showing that it significantly improves on the memory wear performance for classic cuckoo hashing and linear probing in practice.Comment: 13 pages, 1 table, 7 figures; to appear at the 13th Symposium on Experimental Algorithms (SEA 2014

The Complete Spectrum of Yeast Chromosome Instability Genes Identifies Candidate CIN Cancer Genes and Functional Roles for ASTRA Complex Components

Chromosome instability (CIN) is observed in most solid tumors and is linked to somatic mutations in genome integrity maintenance genes. The spectrum of mutations that cause CIN is only partly known and it is not possible to predict a priori all pathways whose disruption might lead to CIN. To address this issue, we generated a catalogue of CIN genes and pathways by screening ∼2,000 reduction-of-function alleles for 90% of essential genes in Saccharomyces cerevisiae. Integrating this with published CIN phenotypes for other yeast genes generated a systematic CIN gene dataset comprised of 692 genes. Enriched gene ontology terms defined cellular CIN pathways that, together with sequence orthologs, created a list of human CIN candidate genes, which we cross-referenced to published somatic mutation databases revealing hundreds of mutated CIN candidate genes. Characterization of some poorly characterized CIN genes revealed short telomeres in mutants of the ASTRA/TTT components TTI1 and ASA1. High-throughput phenotypic profiling links ASA1 to TTT (Tel2-Tti1-Tti2) complex function and to TORC1 signaling via Tor1p stability, consistent with the role of TTT in PI3-kinase related kinase biogenesis. The comprehensive CIN gene list presented here in principle comprises all conserved eukaryotic genome integrity pathways. Deriving human CIN candidate genes from the list allows direct cross-referencing with tumor mutational data and thus candidate mutations potentially driving CIN in tumors. Overall, the CIN gene spectrum reveals new chromosome biology and will help us to understand CIN phenotypes in human disease

The Elg1 Clamp Loader Plays a Role in Sister Chromatid Cohesion

Mutations in the ELG1 gene of yeast lead to genomic instability, manifested in high levels of genetic recombination, chromosome loss, and gross chromosomal rearrangements. Elg1 shows similarity to the large subunit of the Replication Factor C clamp loader, and forms a RFC-like (RLC) complex in conjunction with the 4 small RFC subunits. Two additional RLCs exist in yeast: in one of them the large subunit is Ctf18, and in the other, Rad24. Ctf18 has been characterized as the RLC that functions in sister chromatid cohesion. Here we present evidence that the Elg1 RLC (but not Rad24) also plays an important role in this process. A genetic screen identified the cohesin subunit Mcd1/Scc1 and its loader Scc2 as suppressors of the synthetic lethality between elg1 and ctf4. We describe genetic interactions between ELG1 and genes encoding cohesin subunits and their accessory proteins. We also show that defects in Elg1 lead to higher precocious sister chromatid separation, and that Ctf18 and Elg1 affect cohesion via a joint pathway. Finally, we localize both Ctf18 and Elg1 to chromatin and show that Elg1 plays a role in the recruitment of Ctf18. Our results suggest that Elg1, Ctf4, and Ctf18 may coordinate the relative movement of the replication fork with respect to the cohesin ring

DNA replication stress restricts ribosomal DNA copy number

Ribosomal RNAs (rRNAs) in budding yeast are encoded by ~100–200 repeats of a 9.1kb sequence arranged in tandem on chromosome XII, the ribosomal DNA (rDNA) locus. Copy number of rDNA repeat units in eukaryotic cells is maintained far in excess of the requirement for ribosome biogenesis. Despite the importance of the repeats for both ribosomal and non-ribosomal functions, it is currently not known how “normal” copy number is determined or maintained. To identify essential genes involved in the maintenance of rDNA copy number, we developed a droplet digital PCR based assay to measure rDNA copy number in yeast and used it to screen a yeast conditional temperature-sensitive mutant collection of essential genes. Our screen revealed that low rDNA copy number is associated with compromised DNA replication. Further, subculturing yeast under two separate conditions of DNA replication stress selected for a contraction of the rDNA array independent of the replication fork blocking protein, Fob1. Interestingly, cells with a contracted array grew better than their counterparts with normal copy number under conditions of DNA replication stress. Our data indicate that DNA replication stresses select for a smaller rDNA array. We speculate that this liberates scarce replication factors for use by the rest of the genome, which in turn helps cells complete DNA replication and continue to propagate. Interestingly, tumors from mini chromosome maintenance 2 (MCM2)-deficient mice also show a loss of rDNA repeats. Our data suggest that a reduction in rDNA copy number may indicate a history of DNA replication stress, and that rDNA array size could serve as a diagnostic marker for replication stress. Taken together, these data begin to suggest the selective pressures that combine to yield a “normal” rDNA copy number

PhenoM: a database of morphological phenotypes caused by mutation of essential genes in Saccharomyces cerevisiae

About one-fifth of the genes in the budding yeast are essential for haploid viability and cannot be functionally assessed using standard genetic approaches such as gene deletion. To facilitate genetic analysis of essential genes, we and others have assembled collections of yeast strains expressing temperature-sensitive (ts) alleles of essential genes. To explore the phenotypes caused by essential gene mutation we used a panel of genetically engineered fluorescent markers to explore the morphology of cells in the ts strain collection using high-throughput microscopy. Here, we describe the design and implementation of an online database, PhenoM (Phenomics of yeast Mutants), for storing, retrieving, visualizing and data mining the quantitative single-cell measurements extracted from micrographs of the ts mutant cells. PhenoM allows users to rapidly search and retrieve raw images and their quantified morphological data for genes of interest. The database also provides several data-mining tools, including a PhenoBlast module for phenotypic comparison between mutant strains and a Gene Ontology module for functional enrichment analysis of gene sets showing similar morphological alterations. The current PhenoM version 1.0 contains 78 194 morphological images and 1 909 914 cells covering six subcellular compartments or structures for 775 ts alleles spanning 491 essential genes. PhenoM is freely available at http://phenom.ccbr.utoronto.ca/

Exponential Decay of Correlations Implies Area Law

We prove that a finite correlation length, i.e. exponential decay of correlations, implies an area law for the entanglement entropy of quantum states defined on a line. The entropy bound is exponential in the correlation length of the state, thus reproducing as a particular case Hastings proof of an area law for groundstates of 1D gapped Hamiltonians. As a consequence, we show that 1D quantum states with exponential decay of correlations have an efficient classical approximate description as a matrix product state of polynomial bond dimension, thus giving an equivalence between injective matrix product states and states with a finite correlation length. The result can be seen as a rigorous justification, in one dimension, of the intuition that states with exponential decay of correlations, usually associated with non-critical phases of matter, are simple to describe. It also has implications for quantum computing: It shows that unless a pure state quantum computation involves states with long-range correlations, decaying at most algebraically with the distance, it can be efficiently simulated classically. The proof relies on several previous tools from quantum information theory - including entanglement distillation protocols achieving the hashing bound, properties of single-shot smooth entropies, and the quantum substate theorem - and also on some newly developed ones. In particular we derive a new bound on correlations established by local random measurements, and we give a generalization to the max-entropy of a result of Hastings concerning the saturation of mutual information in multiparticle systems. The proof can also be interpreted as providing a limitation on the phenomenon of data hiding in quantum states.Comment: 35 pages, 6 figures; v2 minor corrections; v3 published versio

Mutability and mutational spectrum of chromosome transmission fidelity genes

It has been more than two decades since the original chromosome transmission fidelity (Ctf) screen of Saccharomyces cerevisiae was published. Since that time the spectrum of mutations known to cause Ctf and, more generally, chromosome instability (CIN) has expanded dramatically as a result of systematic screens across yeast mutant arrays. Here we describe a comprehensive summary of the original Ctf genetic screen and the cloning of the remaining complementation groups as efforts to expand our knowledge of the CIN gene repertoire and its mutability in a model eukaryote. At the time of the original screen, it was impossible to predict either the genes and processes that would be overrepresented in a pool of random mutants displaying a Ctf phenotype or what the entire set of genes potentially mutable to Ctf would be. We show that in a collection of 136 randomly selected Ctf mutants, >65% of mutants map to 13 genes, 12 of which are involved in sister chromatid cohesion and/or kinetochore function. Extensive screening of systematic mutant collections has shown that ~350 genes with functions as diverse as RNA processing and proteasomal activity mutate to cause a Ctf phenotype and at least 692 genes are required for faithful chromosome segregation. The enrichment of random Ctf alleles in only 13 of ~350 possible Ctf genes suggests that these genes are more easily mutable to cause genome instability than the others. These observations inform our understanding of recurring CIN mutations in human cancers where presumably random mutations are responsible for initiating the frequently observed CIN phenotype of tumors