On sums of two Fibonacci numbers that are powers of numbers with limited Hamming weight

In 2018, Luca and Patel conjectured that the largest perfect power representable as the sum of two Fibonacci numbers is $3864^2 = F_{36} + F_{12}$. In other words, they conjectured that the equation \begin{equation}\tag{$\ast$}\label{eq:abstract} y^a = F_n + F_m \end{equation} has no solutions with $a\geq 2$ and $y^a > 3864^2$. While this is still an open problem, there exist several partial results. For example, recently Kebli, Kihel, Larone and Luca proved an explicit upper bound for $y^a$, which depends on the size of $y$. In this paper, we find an explicit upper bound for $y^a$, which only depends on the Hamming weight of $y$ with respect to the Zeckendorf representation. More specifically, we prove the following: If $y = F_{n_1}+ \dots + F_{n_k}$ and equation \eqref{eq:abstract} is satisfied by $y$ and some non-negative integers $n,m$ and $a\geq 2$, then $$y^a \leq \exp\left(C{(\varepsilon)} \cdot k^{(3+\varepsilon)k^2} \right).$$ Here, $\varepsilon >0$ can be chosen arbitrarily and $C(\varepsilon)$ is an effectively computable constant.Comment: 14 page

Influence of sublexical frequencies on the speech production in aphasia and apraxia of speech

The aim of this study was to evaluate the influence of sublexical frequencies on the speech production of patients with apraxia of speech and patients with a phonological disorder. Recent studies have discussed syllable frequency as an important parameter which influences phonetically and / or phonologically disturbed speech. In the current investigation frequencies for the units syllable, phoneme, biphoneme, onset and rhyme were evaluated. The focus of our analysis was how the frequencies of the target units relate to those of the units realised by the patients. The results are discussed with regard to the pathomechanisms underlying of apraxia of speech and phonological impairment

In Time with the Beat: Entrainment in Patients with Phonological Impairment, Apraxia of Speech, and Parkinson’s Disease

In the present study, we investigated if individuals with neurogenic speech sound impairments of three types, Parkinson’s dysarthria, apraxia of speech, and aphasic phonological impairment, accommodate their speech to the natural speech rhythm of an auditory model, and if so, whether the effect is more significant after hearing metrically regular sentences as compared to those with an irregular pattern. This question builds on theories of rhythmic entrainment, assuming that sensorimotor predictions of upcoming events allow humans to synchronize their actions with an external rhythm. To investigate entrainment effects, we conducted a sentence completion task relating participants’ response latencies to the spoken rhythm of the prime heard immediately before. A further research question was if the perceived rhythm interacts with the rhythm of the participants’ own productions, i.e., the trochaic or iambic stress pattern of disyllabic target words. For a control group of healthy speakers, our study revealed evidence for entrainment when trochaic target words were preceded by regularly stressed prime sentences. Persons with Parkinson’s dysarthria showed a pattern similar to that of the healthy individuals. For the patient groups with apraxia of speech and with phonological impairment, considerably longer response latencies with differing patterns were observed. Trochaic target words were initiated with significantly shorter latencies, whereas the metrical regularity of prime sentences had no consistent impact on response latencies and did not interact with the stress pattern of the target words to be produced. The absence of an entrainment in these patients may be explained by the more severe difficulties in initiating speech at all. We discuss the results in terms of clinical implications for diagnostics and therapy in neurogenic speech disorders. View Full-Tex

CopR, a Global Regulator of Transcription to Maintain Copper Homeostasis in Pyrococcus furiosus

Although copper is in many cases an essential micronutrient for cellular life, higher concentrations are toxic. Therefore, all living cells have developed strategies to maintain copper homeostasis. In this manuscript, we have analyzed the transcriptome-wide response of Pyrococcus furiosus to increased copper concentrations and described the essential role of the putative copper-sensing metalloregulator CopR in the detoxification process. To this end, we employed biochemical and biophysical methods to characterize the role of CopR. Additionally, a copR knockout strain revealed an amplified sensitivity in comparison to the parental strain towards increased copper levels, which designates an essential role of CopR for copper homeostasis. To learn more about the CopR-regulated gene network, we performed differential gene expression and ChIP-seq analysis under normal and 20 μM copper-shock conditions. By integrating the transcriptome and genome-wide binding data, we found that CopR binds to the upstream regions of many copper-induced genes. Negative-stain transmission electron microscopy and 2D class averaging revealed an octameric assembly formed from a tetramer of dimers for CopR, similar to published crystal structures from the Lrp family. In conclusion, we propose a model for CopR-regulated transcription and highlight the regulatory network that enables Pyrococcus to respond to increased copper concentrations

Delineating the genotypic and phenotypic spectrum of HECW2-related neurodevelopmental disorders

Background Variants in HECW2 have recently been reported to cause a neurodevelopmental disorder with hypotonia, seizures and impaired language; however, only six variants have been reported and the clinical characteristics have only broadly been defined. Methods Molecular and clinical data were collected from clinical and research cohorts. Massive parallel sequencing was performed and identified individuals with a HECW2-related neurodevelopmental disorder. Results We identified 13 novel missense variants in HECW2 in 22 unpublished cases, of which 18 were confirmed to have a de novo variant. In addition, we reviewed the genotypes and phenotypes of previously reported and new cases with HECW2 variants (n=35 cases). All variants identified are missense, and the majority of likely pathogenic and pathogenic variants are located in or near the C-terminal HECT domain (88.2%). We identified several clustered variants and four recurrent variants (p.(Arg1191Gln);p.(Asn1199Lys);p.(Phe1327Ser);p.(Arg1330Trp)). Two variants, (p.(Arg1191Gln);p.(Arg1330Trp)), accounted for 22.9% and 20% of cases, respectively. Clinical characterisation suggests complete penetrance for hypotonia with or without spasticity (100%), developmental delay/intellectual disability (100%) and developmental language disorder (100%). Other common features are behavioural problems (88.9%), vision problems (83.9%), motor coordination/movement (75%) and gastrointestinal issues (70%). Seizures were present in 61.3% of individuals. Genotype-phenotype analysis shows that HECT domain variants are more frequently associated with cortical visual impairment and gastrointestinal issues. Seizures were only observed in individuals with variants in or near the HECT domain. Conclusion We provide a comprehensive review and expansion of the genotypic and phenotypic spectrum of HECW2 disorders, aiding future molecular and clinical diagnosis and management.Peer reviewe

Lifelong Reduction of LDL-Cholesterol Related to a Common Variant in the LDL-Receptor Gene Decreases the Risk of Coronary Artery Disease—A Mendelian Randomisation Study

Rare mutations of the low-density lipoprotein receptor gene (LDLR) cause familial hypercholesterolemia, which increases the risk for coronary artery disease (CAD). Less is known about the implications of common genetic variation in the LDLR gene regarding the variability of cholesterol levels and risk of CAD.Imputed genotype data at the LDLR locus on 1 644 individuals of a population-based sample were explored for association with LDL-C level. Replication of association with LDL-C level was sought for the most significant single nucleotide polymorphism (SNP) within the LDLR gene in three European samples comprising 6 642 adults and 533 children. Association of this SNP with CAD was examined in six case-control studies involving more than 15 000 individuals.Each copy of the minor T allele of SNP rs2228671 within LDLR (frequency 11%) was related to a decrease of LDL-C levels by 0.19 mmol/L (95% confidence interval (CI) [0.13-0.24] mmol/L, p = 1.5x10(-10)). This association with LDL-C was uniformly found in children, men, and women of all samples studied. In parallel, the T allele of rs2228671 was associated with a significantly lower risk of CAD (Odds Ratio per copy of the T allele: 0.82, 95% CI [0.76-0.89], p = 2.1x10(-7)). Adjustment for LDL-C levels by logistic regression or Mendelian Randomisation models abolished the significant association between rs2228671 with CAD completely, indicating a functional link between the genetic variant at the LDLR gene locus, change in LDL-C and risk of CAD.A common variant at the LDLR gene locus affects LDL-C levels and, thereby, the risk for CAD