La regola dei segni dall'enunciato di R. Descartes (1637) alla dimostrazione di C. F. Gauss (1828)

Conclusioni Le dimostrazioni algebriche della regola, sia che riguardino il caso a radici reali o quello generale si fondano tutte su una proposizione che descrive il comportamento dei segni di un'equazione quando la si moltiplica per x – a oppure per x + a.Le prime dimostrazioni algebriche relative ad equazioni a radici reali, cioé quelle diSegner (1728),Campbell, Stübner eDe Gua, benchè trovate indipendentemente, presentano notevoli somiglianze. Per esempio tutti gli autori menzionati dimostrano laproposizione di Leibniz ricorrendo al lemma che afferma che, in un'equazione completa a radici reali, il quadrato di ogni coefficiente è maggiore del prodotto dei due coefficienti adiacenti, anche se di esso forniscono dimostrazioni diverse.Nella dimostrazione algebrica relativa al caso generale laproposizione di Leibniz è invece dimostrata direttamente. Già nella dimostrazione diSegner (1756) vengono introdotti, anche se in forma oscura e prolissa, quegli elementi che verranno poi semplificati e chiariti nelle dimostrazioni diWaring, Fourier eGauss.Il punto essenziale delle dimostrazioni analitiche, sia che riguardino il caso a radici reali o quello generale, è una proposizione che trasporta la validità del teorema dall'equazione derivata di un'equazione, all'equazione stessa. La regola viene poi dimostrata per induzione. Nelle prime dimostrazioni analitiche, cioè quelle diDe Gua, Kaestner edAepinus, gli autori fanno diretto riferimento al grafico della curva che rappresenta l'equazione. Successivamente con il consolidarsi delle conoscenze dell'analisi infinitesimale,Euler, Milner, Lagrange eRuffini omettono ogni riferimento al grafico dell'equazione.Osserviamo infine che è proprio l'applicazione dell'analisi infinitesimale alla teoria delle equazioni a suggerire aFourier una generalizzazione della regola dei segni, che condurrà successivamente al più generaleteorema di Sturm.     

ZNRF3 promotes Wnt receptor turnover in an R-spondin-sensitive manner

R-spondin proteins strongly potentiate Wnt signalling and function as stem-cell growth factors. Despite the biological and therapeutic significance, the molecular mechanism of R-spondin action remains unclear. Here we show that the cell-surface transmembrane E3 ubiquitin ligase zinc and ring finger 3 (ZNRF3) and its homologue ring finger 43 (RNF43) are negative feedback regulators of Wnt signalling. ZNRF3 is associated with the Wnt receptor complex, and inhibits Wnt signalling by promoting the turnover of frizzled and LRP6. Inhibition of ZNRF3 enhances Wnt/β-catenin signalling and disrupts Wnt/planar cell polarity signalling in vivo. Notably, R-spondin mimics ZNRF3 inhibition by increasing the membrane level of Wnt receptors. Mechanistically, R-spondin interacts with the extracellular domain of ZNRF3 and induces the association between ZNRF3 and LGR4, which results in membrane clearance of ZNRF3. These data suggest that R-spondin enhances Wnt signalling by inhibiting ZNRF3. Our study provides new mechanistic insights into the regulation of Wnt receptor turnover, and reveals ZNRF3 as a tractable target for therapeutic exploration.     

Oncogene-induced senescence is a DNA damage response triggered by DNA hyper-replication

Early tumorigenesis is associated with the engagement of the DNA-damage checkpoint response (DDR). Cell proliferation and transformation induced by oncogene activation are restrained by cellular senescence. It is unclear whether DDR activation and oncogene-induced senescence (OIS) are causally linked. Here we show that senescence, triggered by the expression of an activated oncogene (H-RasV12) in normal human cells, is a consequence of the activation of a robust DDR. Experimental inactivation of DDR abrogates OIS and promotes cell transformation. DDR and OIS are established after a hyper-replicative phase occurring immediately after oncogene expression. Senescent cells arrest with partly replicated DNA and with DNA replication origins having fired multiple times. In vivo DNA labelling and molecular DNA combing reveal that oncogene activation leads to augmented numbers of active replicons and to alterations in DNA replication fork progression. We also show that oncogene expression does not trigger a DDR in the absence of DNA replication. Last, we show that oncogene activation is associated with DDR activation in a mouse model in vivo. We propose that OIS results from the enforcement of a DDR triggered by oncogene-induced DNA hyper-replication.     

Aldehyde suppression of copepod recruitment in blooms of a ubiquitous planktonic diatom

The growth cycle in nutrient-rich, aquatic environments starts with a diatom bloom that ends in mass sinking of ungrazed cells and phytodetritus. The low grazing pressure on these blooms has been attributed to the inability of overwintering copepod populations to track them temporally. We tested an alternative explanation: that dominant diatom species impair the reproductive success of their grazers. We compared larval development of a common overwintering copepod fed on a ubiquitous, early-blooming diatom species with its development when fed on a typical post-bloom dinoflagellate. Development was arrested in all larvae in which both mothers and their larvae were fed the diatom diet. Mortality remained high even if larvae were switched to the dinoflagellate diet. Aldehydes, cleaved from a fatty acid precursor by enzymes activated within seconds after crushing of the cell, elicit the teratogenic effect. This insidious mechanism, which does not deter the herbivore from feeding but impairs its recruitment, will restrain the cohort size of the next generation of early-rising overwinterers. Such a transgenerational plant-herbivore interaction could explain the recurringly inefficient use of a predictable, potentially valuable food resource--the spring diatom bloom--by marine zooplankton.     

X-linked protocadherin 19 mutations cause female-limited epilepsy and cognitive impairment

Epilepsy and mental retardation limited to females (EFMR) is a disorder with an X-linked mode of inheritance and an unusual expression pattern. Disorders arising from mutations on the X chromosome are typically characterized by affected males and unaffected carrier females. In contrast, EFMR spares transmitting males and affects only carrier females. Aided by systematic resequencing of 737 X chromosome genes, we identified different protocadherin 19 (PCDH19) gene mutations in seven families with EFMR. Five mutations resulted in the introduction of a premature termination codon. Study of two of these demonstrated nonsense-mediated decay of PCDH19 mRNA. The two missense mutations were predicted to affect adhesiveness of PCDH19 through impaired calcium binding. PCDH19 is expressed in developing brains of human and mouse and is the first member of the cadherin superfamily to be directly implicated in epilepsy or mental retardation.     

Complement factor H in host defense and immune evasion

Complement is the major humoral component of the innate immune system. It recognizes pathogen- and damage-associated molecular patterns, and initiates the immune response in coordination with innate and adaptive immunity. When activated, the complement system unleashes powerful cytotoxic and inflammatory mechanisms, and thus its tight control is crucial to prevent damage to host tissues and allow restoration of immune homeostasis. Factor H is the major soluble inhibitor of complement, where its binding to self markers (i.e., particular glycan structures) prevents complement activation and amplification on host surfaces. Not surprisingly, mutations and polymorphisms that affect recognition of self by factor H are associated with diseases of complement dysregulation, such as age-related macular degeneration and atypical haemolytic uremic syndrome. In addition, pathogens (i.e., non-self) and cancer cells (i.e., altered-self) can hijack factor H to evade the immune response. Here we review recent (and not so recent) literature on the structure and function of factor H, including the emerging roles of this protein in the pathophysiology of infectious diseases and cancer.     

Dysfunction of lipid sensor GPR120 leads to obesity in both mouse and human

Free fatty acids provide an important energy source as nutrients, and act as signalling molecules in various cellular processes. Several G-protein-coupled receptors have been identified as free-fatty-acid receptors important in physiology as well as in several diseases. GPR120 (also known as O3FAR1) functions as a receptor for unsaturated long-chain free fatty acids and has a critical role in various physiological homeostasis mechanisms such as adipogenesis, regulation of appetite and food preference. Here we show that GPR120-deficient mice fed a high-fat diet develop obesity, glucose intolerance and fatty liver with decreased adipocyte differentiation and lipogenesis and enhanced hepatic lipogenesis. Insulin resistance in such mice is associated with reduced insulin signalling and enhanced inflammation in adipose tissue. In human, we show that GPR120 expression in adipose tissue is significantly higher in obese individuals than in lean controls. GPR120 exon sequencing in obese subjects reveals a deleterious non-synonymous mutation (p.R270H) that inhibits GPR120 signalling activity. Furthermore, the p.R270H variant increases the risk of obesity in European populations. Overall, this study demonstrates that the lipid sensor GPR120 has a key role in sensing dietary fat and, therefore, in the control of energy balance in both humans and rodents.