Improved bounds for noisy group testing with constant tests per item

The group testing problem is concerned with identifying a small set of infected individuals in a large population. At our disposal is a testing procedure that allows us to test several individuals together. In an idealized setting, a test is positive if and only if at least one infected individual is included and negative otherwise. Significant progress was made in recent years towards understanding the information-theoretic and algorithmic properties in this noiseless setting. In this paper, we consider a noisy variant of group testing where test results are flipped with certain probability, including the realistic scenario where sensitivity and specificity can take arbitrary values. Using a test design where each individual is assigned to a fixed number of tests, we derive explicit algorithmic bounds for two commonly considered inference algorithms and thereby naturally extend the results of Scarlett \& Cevher (2016) and Scarlett \& Johnson (2020). We provide improved performance guarantees for the efficient algorithms in these noisy group testing models -- indeed, for a large set of parameter choices the bounds provided in the paper are the strongest currently proved

Improved bounds for noisy group testing with constant tests per item

The group testing problem is concerned with identifying a small set of infected individuals in a large population. At our disposal is a testing procedure that allows us to test several individuals together. In an idealized setting, a test is positive if and only if at least one infected individual is included and negative otherwise. Significant progress was made in recent years towards understanding the information-theoretic and algorithmic properties in this noiseless setting. In this paper, we consider a noisy variant of group testing where test results are flipped with certain probability, including the realistic scenario where sensitivity and specificity can take arbitrary values. Using a test design where each individual is assigned to a fixed number of tests, we derive explicit algorithmic bounds for two commonly considered inference algorithms and thereby naturally extend the results of Scarlett \& Cevher (2016) and Scarlett \& Johnson (2020). We provide improved performance guarantees for the efficient algorithms in these noisy group testing models -- indeed, for a large set of parameter choices the bounds provided in the paper are the strongest currently proved

The kk-XORSAT threshold revisited

We provide a simplified proof of the random $k$-XORSAT satisfiability threshold theorem. As an extension we also determine the full rank threshold for sparse random matrices over finite fields with precisely $k$ non-zero entries per row. This complements a result from [Ayre, Coja-Oghlan, Gao, M\"uller: Combinatorica 2020]. The proof combines physics-inspired message passing arguments with a surgical moment computation

The Full Rank Condition for Sparse Random Matrices

We derive a sufficient condition for a sparse random matrix with given numbers of non-zero entries in the rows and columns having full row rank. Inspired by low-density parity check codes, the family of random matrices that we investigate is very general and encompasses both matrices over finite fields and {0,1}-matrices over the rationals. The proof combines statistical physics-inspired coupling techniques with local limit arguments

A novel COL1A2 C-propeptide cleavage site mutation causing high bone mass osteogenesis imperfecta with a regional distribution pattern

Osteogenesis imperfecta (OI) is typically characterized by low bone mass and increased bone fragility caused by heterozygous mutations in the type I procollagen genes (COL1A1/COL1A2). We report two cases of a 56-year-old woman and her 80-year-old mother who suffered from multiple vertebral and non-vertebral fractures with onset in early childhood. A full osteologic assessment including dual-energy X-ray absorptiometry (DXA), high-resolution peripheral quantitative computed tomography (HR-pQCT), and serum analyses pointed to a high bone mineral density (BMD) in the hip (DXA Z-score + 3.7 and + 3.9) but low to normal bone mass in the spine and preserved bone microstructure in the distal tibia. Serum markers of bone formation and bone resorption were elevated. Using whole exome sequencing, we identified a novel mutation in the COL1A2 gene causing a p. (Asp1120Gly) substitution at the protein level and affecting the type I procollagen C-propeptide cleavage site. In line with previously reported cases, our data independently prove the existence of an unusual phenotype of high bone mass OI caused by a mutation in the procollagen C-propeptide cleavage with a clinically persistent phenotype through adulthood

Adult Osteosclerotic Metaphyseal Dysplasia With Progressive Osteonecrosis of the Jaws and Abnormal Bone Resorption Pattern Due to a LRRK1 Splice Site Mutation

Osteosclerotic metaphyseal dysplasia (OSMD) is a rare autosomal recessive sclerosing skeletal dysplasia. We report on a 34-year-old patient with sandwich vertebrae, platyspondyly, osteosclerosis of the tubular bones, pathologic fractures, and anemia. In the third decade, he developed osteonecrosis of the jaws, which was progressive in spite of repeated surgical treatment over a period of 11 years. An iliac crest bone biopsy revealed the presence of hypermineralized cartilage remnants, large multinucleated osteoclasts with abnormal morphology, and inadequate bone resorption typical for osteoclast-rich osteopetrosis. After exclusion of mutations in TCIRG1 and CLCN7 we performed trio-based exome sequencing. The novel homozygous splice-site mutation c.261G>A in the gene LRRK1 was found and co-segregated with the phenotype in the family. cDNA sequencing showed nearly complete skipping of exon 3 leading to a frameshift (p.Ala34Profs*33). Osteoclasts differentiated from the patient's peripheral blood monocytes were extremely large. Instead of resorption pits these cells were only capable of superficial erosion. Phosphorylation of L-plastin at position Ser5 was strongly reduced in patient-derived osteoclasts showing a loss of function of the mutated LRRK1 kinase protein. Our analysis indicates a strong overlap of LRRK1-related OSMD with other forms of intermediate osteopetrosis, but an exceptional abnormality of osteoclast resorption. Like in other osteoclast pathologies an increased risk for progressive osteonecrosis of the jaws should be considered in OSMD, an intermediate form of osteopetrosis

Enzyme replacement therapy in mice lacking arylsulfatase B targets bone-remodeling cells, but not chondrocytes

Mucopolysaccharidosis type VI (MPS-VI), caused by mutational inactivation of the glycosaminoglycan-degrading enzyme arylsulfatase B (Arsb), is a lysosomal storage disorder primarily affecting the skeleton. We have previously reported that Arsb-deficient mice display high trabecular bone mass and impaired skeletal growth. In the present study, we treated them by weekly injection of recombinant human ARSB (rhARSB) to analyze the impact of enzyme replacement therapy (ERT) on skeletal growth and bone remodeling. We found that all bone-remodeling abnormalities of Arsb-deficient mice were prevented by ERT, whereas chondrocyte defects were not. Likewise, histologic analysis of the surgically removed femoral head from an ERT-treated MPS-VI patient revealed that only chondrocytes were pathologically affected. Remarkably, a side-by-side comparison with other cell types demonstrated that chondrocytes have substantially reduced capacity to endocytose rhARSB, together with low expression of the mannose receptor. We finally took advantage of Arsb-deficient mice to establish quantification of chondroitin sulfation for treatment monitoring. Our data demonstrate that bone-remodeling cell types are accessible to systemically delivered rhARSB, whereas the uptake into chondrocytes is inefficient

Genetic Diagnostics in Routine Osteological Assessment of Adult Low Bone Mass Disorders

Context Many different inherited and acquired conditions can result in premature bone fragility / low bone mass disorders (LBMD). Objective We aimed at elucidating the impact of genetic testing on differential diagnosis of adult LBMD and at defining clinical criteria for predicting monogenic forms. Methods Four clinical centers broadly recruited a cohort of 394 unrelated adult women before menopause and men younger than 55 years with a bone mineral density (BMD) Z-score 2), and a high normal BMI. In contrast, mutation frequencies did not correlate with age, prevalent vertebral fractures, BMD, or biochemical parameters. In individuals without monogenic disease-causing rare variants, common variants predisposing for low BMD, e.g. in LRP5, were overrepresented. Conclusion The overlapping spectra of monogenic adult LBMD can be easily disentangled by genetic testing and the proposed clinical criteria can help to maximize the diagnostic yield

ARSB IS REQUIRED FOR BONE REMODELING

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