IntroductionLate-onset Alzheimer's disease (LOAD, onset age > 60 years) is the most prevalent dementia in the elderly 1 , and risk is partially driven by genetics 2 . Many of the loci responsible for this genetic risk were identified by genome-wide association studies (GWAS) [3][4][5][6][7][8] . To identify additional LOAD risk loci, the we performed the largest GWAS to date (89,769 individuals), analyzing both common and rare variants. We confirm 20 previous LOAD risk loci and identify four new genome-wide loci (IQCK, ACE, ADAM10, and ADAMTS1). Pathway analysis of these data implicates the immune system and lipid metabolism, and for the first time tau binding proteins and APP metabolism. These findings show that genetic variants affecting APP and Aβ processing are not only associated with early-onset autosomal dominant AD but also with LOAD. Analysis of AD risk genes and pathways show enrichment for rare variants (P = 1.32 x 10 -7 ) indicating that additional rare variants remain to be identified. Main TextOur previous work identified 19 genome-wide significant common variant signals in addition to APOE 9 , that influence risk for LOAD. These signals, combined with 'subthreshold' common variant associations, account for ~31% of the genetic variance of LOAD 2 , leaving the majority of genetic risk uncharacterized 10 . To search for additional signals, we conducted a GWAS metaanalysis of non-Hispanic Whites (NHW) using a larger sample (17 new, 46 total datasets) from our group, the International Genomics of Alzheimer's Project (IGAP) (composed of four AD consortia: ADGC, CHARGE, EADI, and GERAD). This sample increases our previous discovery sample (Stage 1) by 29% for cases and 13% for controls (N=21,982 cases; 41,944 controls) ( Supplementary Table 1 and 2, and Supplementary Note). To sample both common and rare variants (minor allele frequency MAF ≥ 0.01, and MAF < 0.01, respectively), we imputed the discovery datasets using a 1000 Genomes reference panel consisting of . CC-BY-NC-ND 4.0 International license peer-reviewed) is the author/funder. It is made available under a 11 36,648,992 single-nucleotide variants, 1,380,736 insertions/deletions, and 13,805 structural variants. After quality control, 9,456,058 common variants and 2,024,574 rare variants were selected for analysis (a 63% increase from our previous common variant analysis in 2013).Genotype dosages were analyzed within each dataset, and then combined with meta-analysis ( Supplementary Figures 1 and 2 and Supplementary Table 3). The Stage 1 discovery metaanalysis was first followed by Stage 2 using the I-select chip we previously developed in Lambert et al (including 11,632 variants, N=18,845) and finally stage 3A (N=6,998). The final sample was 33,692 clinical AD cases and 56,077 controls.Meta-analysis of Stages 1 and 2 produced 21 associations with P ≤ 5x10 -8 (Table 1 and Figure 1). Of these, 18 were previously reported as genome-wide significant and three of them are signals not initially described in Lambert et al: the rare R47H TREM2 coding va...

We study the computational complexity o f t wo popular problems in multiple sequence alignment: multiple alignment with SP-score and multiple tree alignment. It is shown that the rst problem is NP-complete and the second is MAX SNP-hard. The complexity of tree alignment with a given phylogeny is also considered.

This paper addresses the following relay sensor placement problem: given the set of duty sensors in the plane and the upper bound of the transmission range, compute the minimum number of relay sensors such that the induced topology by all sensors is globally connected. This problem is motivated by practically considering the tradeoff among performance, lifetime, and cost when designing sensor networks. In our study, this problem is modelled by a NP-hard network optimization problem named Steiner Minimum Tree with Minimum number of Steiner Points and bounded edge length (SMT-MSP).We propose two approximate algorithms, together with their detailed performance analysis. The first one has performance ratio 3 and the second one has performance ratio 2.5.

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The problem of finding a center string that is "close" to every given string arises in computational molecular biology and coding theory. This problem has two versions: the Closest String problem and the Closest Substring problem. Given a set of strings S = {s 1 , s 2 , . . . , s n }, each of length m, the Closest String problem is to find the smallest d and a string s of length m which is within Hamming distance d to each s i ∈ S. This problem comes from coding theory when we are looking for a code not too far away from a given set of codes. Closest Substring problem, with an additional input integer L, asks for the smallest d and a string s, of length L, which is within Hamming distance d away from a substring, of length L, of each s i . This problem is much more elusive than the Closest String problem. The Closest Substring problem is formulated from applications in finding conserved regions, identifying genetic drug targets and generating genetic probes in molecular biology. WhetherThe results in Section 2 and some preliminary results in Sections 3 and 4 have been presented in M. LI ET AL.there are efficient approximation algorithms for both problems are major open questions in this area. We present two polynomial-time approximation algorithms with approximation ratio 1 + for any small to settle both questions.