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Given a set of sources and a set of sinks as points in the Euclidean plane, a directed network is a directed graph drawn in the plane with a directed path from each source to each sink. Such a network may contain nodes other than the given sources and sinks, called Steiner points. We characterize the local structure of the Steiner points in all shortest-length directed networks in the Euclidean plane. This characterization implies that these networks are constructible by straightedge and compass. Our results build on unpublished work of Alfaro, Campbell, Sher, and Soto from 1989 and 1990. Part of the proof is based on a new method that uses other norms in the plane. This approach gives more conceptual proofs of some of their results, and as a consequence, we also obtain results on shortest directed networks for these norms., 21 pages, 14 figures. To appear in Graphs and Combinatorics

Huntington’s disease (HD) is an inherited neurodegenerative disorder caused by an expanded CAG repeat in the HTT gene. It is diagnosed following a standardized exam of motor control and often presents with cognitive decline and psychiatric symptoms. Recent studies have detected genetic loci modifying the age at onset of motor symptoms in HD, but genetic factors influencing cognitive and psychiatric presentations are unknown. We tested the hypothesis that psychiatric and cognitive symptoms in HD are influenced by the same common genetic variation as in the general population by constructing polygenic risk scores from large genome-wide association studies of psychiatric and neurodegenerative disorders, and of intelligence, and testing for correlation with the presence of psychiatric and cognitive symptoms in a large sample (n=5160) of HD patients. Polygenic risk score for major depression was associated specifically with increased risk of depression in HD, as was schizophrenia risk score with psychosis and irritability. Cognitive impairment and apathy were associated with reduced polygenic risk score for intelligence. In general, polygenic risk scores for psychiatric disorders, particularly depression and schizophrenia, are associated with increased risk of the corresponding psychiatric symptoms in HD, suggesting a common genetic liability. However, the genetic liability to cognitive impairment and apathy appears to be distinct from other psychiatric symptoms in HD. No associations were observed between HD symptoms and risk scores for other neurodegenerative disorders. These data provide a rationale for treatments effective in depression and schizophrenia to be used to treat depression and psychotic symptoms in HD.

Huntington disease (HD)is a dominantly inherited neurodegenerative disorder caused by the expansion of a polyglutamine encoding CAG repeat in the huntingtin gene. Recently, it has been established that disease severity in HD is best predicted by the number of pure CAG repeats rather than total glutamines encoded. Along with uncovering DNA repair gene variants as

SummaryThe age at onset of motor symptoms in Huntington’s disease (HD) is driven by HTT CAG repeat length but modified by other genes. We used exome sequencing of 683 HD patients with extremes of onset or phenotype relative to CAG length to identify rare variants associated with clinical effect. We identified damaging coding variants in candidate modifier genes from prior genome-wide association studies associated with altered HD onset or severity. Variants in FAN1 clustered in its DNA-binding and nuclease domains and were associated predominantly with earlier onset HD. Nuclease activities of these variants correlated with residual age at motor onset of HD. Mutating endogenous FAN1 to a nuclease-inactive form in an induced pluripotent stem cell model of HD led to rates of CAG expansion comparable to those observed with complete FAN1 knock out. Together, these data implicate FAN1 nuclease activity in slowing somatic repeat expansion and hence onset of HD.

Although Huntington’s disease (HD) is a well studied Mendelian genetic disorder, less is known about its associated epigenetic changes. Here, we characterize DNA methylation levels in six different tissues from 3 species: a mouse huntingtin (Htt) gene knock-in model, a transgenic HTT sheep model, and humans. Our epigenome-wide association study (EWAS) of human blood reveals that HD mutation status is significantly (p, Although Huntington’s disease (HD) is a well-studied genetic disorder, less is known about the epigenetic changes underlying it. Here, the authors characterize DNA methylation levels in tissues from patients, a mouse huntingtin (Htt) gene knock-in model, and a transgenic HTT sheep model, and provide evidence that HD is accompanied by DNA methylation changes in these three species.

SUMMARYThe effects of variable, glutamine-encoding, CAA interruptions indicate that a property of the uninterrupted HTT CAG repeat sequence, distinct from huntingtin’s polyglutamine segment, dictates the rate at which HD develops. The timing of onset shows no significant association with HTT cis-eQTLs but is influenced, sometimes in a sex-specific manner, by polymorphic variation at multiple DNA maintenance genes, suggesting that the special onset-determining property of the uninterrupted CAG repeat is a propensity for length instability that leads to its somatic expansion. Additional naturally-occurring genetic modifier loci, defined by GWAS, may influence HD pathogenesis through other mechanisms. These findings have profound implications for the pathogenesis of HD and other repeat diseases and question a fundamental premise of the “polyglutamine disorders”.

Background: Huntington disease (HD) is a dominantly inherited neurodegenerative disorder, presenting with a movement disorder, cognitive decline and psychiatric complications. HD is caused by the expansion of a CAG repeat tract in exon 1 of the huntingtin gene (HTT). In other repeat expansion disorders, variants within the associated repeat tracts may affect stability, which in turn modifies disease severity. Although sequence variation has been reported within the sequence encoding the HTT polyglutamine and polyproline tract, little is known about its effect on disease. Aim: To determine the sequence diversity precisely within the HTT exon 1 trinucleotide repeat in black South African individuals and assess the effect of sequence variation in modifying disease. Methods/techniques: A sample of black South African individuals (59 patients and 153 controls) was analysed using a specialised high-throughput DNA sequencing assay and data analysis pipeline, specific for the HTT exon 1 repeat. Results/outcomes: Significant diversity was observed in comparison to a European sample of patients, with novel African alleles defined by the description of allele sub-structures. The most common disease allele structure in our patients was associated with the absence of the CCGCCA cassette that precedes the polyproline CCG repeat in typical HD-causing alleles. Regression analysis determined that age at diagnosis was approximately 6 to 13 years earlier in patients carrying HD-causing alleles lacking the CCGCCA cassette. Conclusion: This is the first study to describe allele sequence diversity within the HTT exon 1 repeat in an African population. The absence of the CCGCCA cassette, which appears to be African-specific and associated with an earlier age at diagnosis in African HD patients, is a potential cis-acting modifier of HD onset.

Background Huntington disease (HD) is caused by the expansion of a polyglutamine encoding CAG repeat in exon 1 of the HTT gene. Affected individuals inherit ≥40 repeats and longer alleles are associated with earlier onset and higher HD severity. The HTT CAG repeat is genetically unstable in both the germline and soma. Somatic mosaicism is dependent on the number of CAG repeats and age, and is expansion biased and cell-type specific. Recent genome-wide association studies (GWAS) have identified components of the DNA repair system as trans¬-acting modifiers of HD severity, some of which are known to modify somatic instability of the HTT CAG repeat in HD mouse models. We thus hypothesise that the trans¬-acting modifiers identified by GWAS affect HD severity by their direct effect on HTT CAG somatic instability. Aims Determine the exact trinucleotide structure of the HTT exon 1 repeat and quantify its somatic mosaicism to investigate their association with HD severity. Methods/techniques Using genotyping-by-sequencing, we determined the exact genotype of the polyglutamine and polyproline encoding repeats in HTT exon 1 and quantified the somatic mosaicism associated with the CAG repeat in blood DNA from 807 HD expansion carriers. Results/outcome The sequence encoding the HTT polyglutamine and polyproline tract has an atypical structure in ˜8% of the non-HD-causing alleles and ˜3% of the HD-causing alleles, differing from the typical structure by the number of glutamine encoding CAA codons and/or the number of proline encoding CCA and CCT codons. Multiple linear regression analysis revealed that the number of CAA codons is negatively correlated with HD severity and that the number of CAG repeats is a better predictor of HD severity (r2=0.559) than the number of glutamines (r2=0.537). Moreover, somatic mosaicism in blood correlates with HD severity (r2 ≥0.014, p≤2.5 × 10–3) and some of the polymorphisms associated with HD severity (p=1.5 × 10–5 for FAN1 rs3512, p=1.8 × 10–4 for MLH3 rs175080, p=3.6 × 10–3 for MLH1 rs1799977 and p=0.016 for MSH3 rs1382539). Conclusion Our data show that atypical HD-causing alleles have major implications for genetic diagnosis and counselling and confirm the correlation of somatic expansion with HD severity. The latter further supports the therapeutic potential of targeting expansion causing components of the DNA repair system.

Summary Variable, glutamine-encoding, CAA interruptions indicate that a property of the uninterrupted HTT CAG repeat sequence, distinct from the length of huntingtin’s polyglutamine segment, dictates the rate at which Huntington’s disease (HD) develops. The timing of onset shows no significant association with HTT cis-eQTLs but is influenced, sometimes in a sex-specific manner, by polymorphic variation at multiple DNA maintenance genes, suggesting that the special onset-determining property of the uninterrupted CAG repeat is a propensity for length instability that leads to its somatic expansion. Additional naturally occurring genetic modifier loci, defined by GWAS, may influence HD pathogenesis through other mechanisms. These findings have profound implications for the pathogenesis of HD and other repeat diseases and question the fundamental premise that polyglutamine length determines the rate of pathogenesis in the “polyglutamine disorders.”, Graphical Abstract, Highlights • Uninterrupted CAG repeat, not polyglutamine, size drives the timing of HD onset • HD age at onset is influenced by at least six genes involved in DNA maintenance • Genetic modifier loci often show both onset-delaying and onset-hastening haplotypes • The rate-determining mechanism is likely to be somatic expansion of the CAG repeat, The onset of Huntington’s disease is shown to be dependent on the size of the uninterrupted CAG repeat sequence, not polyglutamine.

It is known that in addition to repeat length variation, the exact sequence of the polyglutamine repeat tract and the adjacent polyproline also vary. Likewise, it is known that the expanded CAG is somatically unstable in a process that is age-dependent, tissue-specific and expansion biassed. Notably, very large alleles exceeding greater than 1,000 repeats are observed in a subset of striatal neurons. These data strongly suggest that somatic instability contributes toward the tissue specificity and progressive nature of the symptoms. Indeed, it has been shown that the frequency of large expansions in cortical cells correlates with variation in age at onset not accounted for by inherited repeat length. Most recently, it has been demonstrated that mismatch repair genes lie under some of the association peaks for genome wide analysis of variants contributing to variation in age at onset. Thus, in order to further address these issues, we have developed a high-throughput sequencing pipeline that allows us to determine the precise sequence of the polyglutamine and polyproline tracts. These studies have revealed an unexpectedly high frequency of atypical non-pathogenic alleles. We have also detected novel CCG interruptions within the CAG array in a large ‘premutation’ length allele. The majority of expanded HD alleles retain the expected structure of CAG and CCG repeats, but a subset of atypical pathogenic alleles have been detected. Using this approach we can also estimate the degree of somatic mosaicism present in the blood DNA of each participant. These data confirm that as expected, somatic mosaicism in the blood DNA of HD individuals is age- and allele length-dependent. We are currently attempting to understand how atypical alleles and individual-specific mutational dynamics contribute toward phenotypic variability in HD. We are also investigating how variants in the mismatch repair genes contribute toward variation in somatic mosaicism.