Characteristics of Conservation Laws for Difference Equations

Each conservation law of a given partial differential equation is determined (up to equivalence) by a function known as the characteristic. This function is used to find conservation laws, to prove equivalence between conservation laws, and to prove the converse of Noether's Theorem. Transferring these results to difference equations is nontrivial, largely because difference operators are not derivations and do not obey the chain rule for derivatives. We show how these problems may be resolved and illustrate various uses of the characteristic. In particular, we establish the converse of Noether's Theorem for difference equations, we show (without taking a continuum limit) that the conservation laws in the infinite family generated by Rasin and Schiff are distinct, and we obtain all five-point conservation laws for the potential Lotka-Volterra equation

Virtual Ontogeny of Cortical Growth Preceding Mental Illness

Background: Morphology of the human cerebral cortex differs across psychiatric disorders, with neurobiology and developmental origins mostly undetermined. Deviations in the tangential growth of the cerebral cortex during pre/perinatal periods may be reflected in individual variations in cortical surface area later in life. Methods: Interregional profiles of group differences in surface area between cases and controls were generated using T1-weighted magnetic resonance imaging from 27,359 individuals including those with attention-deficit/hyperactivity disorder, autism spectrum disorder, bipolar disorder, major depressive disorder, schizophrenia, and high general psychopathology (through the Child Behavior Checklist). Similarity of interregional profiles of group differences in surface area and prenatal cell-specific gene expression was assessed. Results: Across the 11 cortical regions, group differences in cortical area for attention-deficit/hyperactivity disorder, schizophrenia, and Child Behavior Checklist were dominant in multimodal association cortices. The same interregional profiles were also associated with interregional profiles of (prenatal) gene expression specific to proliferative cells, namely radial glia and intermediate progenitor cells (greater expression, larger difference), as well as differentiated cells, namely excitatory neurons and endothelial and mural cells (greater expression, smaller difference). Finally, these cell types were implicated in known pre/perinatal risk factors for psychosis. Genes coexpressed with radial glia were enriched with genes implicated in congenital abnormalities, birth weight, hypoxia, and starvation. Genes coexpressed with endothelial and mural genes were enriched with genes associated with maternal hypertension and preterm birth. Conclusions: Our findings support a neurodevelopmental model of vulnerability to mental illness whereby prenatal risk factors acting through cell-specific processes lead to deviations from typical brain development during pregnancy

The genetic architecture of the human cerebral cortex

INTRODUCTION The cerebral cortex underlies our complex cognitive capabilities. Variations in human cortical surface area and thickness are associated with neurological, psychological, and behavioral traits and can be measured in vivo by magnetic resonance imaging (MRI). Studies in model organisms have identified genes that influence cortical structure, but little is known about common genetic variants that affect human cortical structure. RATIONALE To identify genetic variants associated with human cortical structure at both global and regional levels, we conducted a genome-wide association meta-analysis of brain MRI data from 51,665 individuals across 60 cohorts. We analyzed the surface area and average thickness of the whole cortex and 34 cortical regions with known functional specializations. RESULTS We identified 306 nominally genome-wide significant loci (P < 5 × 10−8) associated with cortical structure in a discovery sample of 33,992 participants of European ancestry. Of the 299 loci for which replication data were available, 241 loci influencing surface area and 14 influencing thickness remained significant after replication, with 199 loci passing multiple testing correction (P < 8.3 × 10−10; 187 influencing surface area and 12 influencing thickness). Common genetic variants explained 34% (SE = 3%) of the variation in total surface area and 26% (SE = 2%) in average thickness; surface area and thickness showed a negative genetic correlation (rG = −0.32, SE = 0.05, P = 6.5 × 10−12), which suggests that genetic influences have opposing effects on surface area and thickness. Bioinformatic analyses showed that total surface area is influenced by genetic variants that alter gene regulatory activity in neural progenitor cells during fetal development. By contrast, average thickness is influenced by active regulatory elements in adult brain samples, which may reflect processes that occur after mid-fetal development, such as myelination, branching, or pruning. When considered together, these results support the radial unit hypothesis that different developmental mechanisms promote surface area expansion and increases in thickness. To identify specific genetic influences on individual cortical regions, we controlled for global measures (total surface area or average thickness) in the regional analyses. After multiple testing correction, we identified 175 loci that influence regional surface area and 10 that influence regional thickness. Loci that affect regional surface area cluster near genes involved in the Wnt signaling pathway, which is known to influence areal identity. We observed significant positive genetic correlations and evidence of bidirectional causation of total surface area with both general cognitive functioning and educational attainment. We found additional positive genetic correlations between total surface area and Parkinson’s disease but did not find evidence of causation. Negative genetic correlations were evident between total surface area and insomnia, attention deficit hyperactivity disorder, depressive symptoms, major depressive disorder, and neuroticism. CONCLUSION This large-scale collaborative work enhances our understanding of the genetic architecture of the human cerebral cortex and its regional patterning. The highly polygenic architecture of the cortex suggests that distinct genes are involved in the development of specific cortical areas. Moreover, we find evidence that brain structure is a key phenotype along the causal pathway that leads from genetic variation to differences in general cognitive function

Crop to Weed Gene Flow in Sorghum: Implications for Transgenic Release in Africa

Potential problems with the release of transgenic crops are varied. Among these problems is the threat of gene flow via pollen transfer from the genetically altered crop to wild or weedy plant relatives growing nearby. Because some engineered genes may confer an adaptive advantage, troublesome weeds may become more problematic if they bear such transgenes. Although the rate of crop to weed gene flow has been measured for a few systems, large gaps in our understanding of this phenomenon remain. The likelihood of crop to weed gene flow in a crop/weed system involving the crop, Sorghum bicolor, and a related weed, S. halepense, in North America is described. Measurements of the rate and incidence of hybridization were made using progeny analysis. Spontaneous hybridization under field conditions was detected at distances of 100 metres. Measurements of several fitness correlates were taken on hybrid weeds and compared to non-hybrid weeds in order to address the potential for persistence of transgenes in the environment. No significant differences were recorded between hybrid and non-hybrid weeds, suggesting that beneficial genes introduced into weed populations may persist. The implications for the release of transgenic sorghums in Africa are discussed.Les problemes potenciels lies a la diffusion des cultures transgeniques sont varies. Parmi ces problemes il y a la menace de la dispersion de genes par le transfert de pollen de la culture genetiquement alteree aux parents sauvages ou herbaces qui poussent dans les environs. Comme certains genes peuvent conrefer une adaptation plus avantageuse, les mauvaises herbes qui les acquierent peuvent devenir plus problematiques. Bien que le taux de dispersion de genes de culture vers les mauvaises herbes a ete mesure dans le cas de quelques systemes, la comprehension de ce phenomene n'est pas encore totale. Le papier decrit le passage de gene d'une culture telle que le sorgho vers une mauvaise herbe, S. halepense, au nord de l'Amerique. En examinant les progenitures, il etait possible de mesurer le taux et l'incidence de l'hybridation. L'hybridation spontanee en conditions de champ etait detectee a 100 metres de distance. Les mesures de plusieurs correlations etaient faites sur les mauvaises herbes hybrides et comparees aux non-hybrides en vue de determiner la persistance de genes transferes dans cet environnement. Il n'y avait pas de differences significatives entre hybrides et non-hybrides de mauvaises herbes; ceci indique que les genes benefiques introduits dans les populations de mauvaises herbes pourraient persister. Le papier discute aussi les implications dues a la dissemination des sorgho transgeniques en Afrique

The Potential for Crop-to-Wild Gene Flow in Sorghum in Ethiopia and Niger: A Geographic Survey

Information about the potential for crop–wild hybridization is needed to understand how crop genes, including transgenes, affect the population genetics and ecology of sexually compatible relatives. Transgenic sorghum is under development for use by traditional farmers in Africa, the center of origin for sorghum [Sorghum bicolor (L.) Moench], but systematic surveys of the current extent of contact with wild and weedy relatives are lacking. We studied wild and weedy sorghums that are interfertile with the crop and constitute a crop–wild–weed complex. The survey was conducted in 2005 in areas of traditional sorghum cultivation in three regions of Ethiopia and two regions of Niger. Within each region, we examined eight representative sorghum fields at each of 10 locations during peak flowering of the crop. In all regions, wild and weedy sorghum occurred intermixed with and adjacent to cultivated sorghum. Wild and weedy sorghums were detected at 56, 44, and 13% of the Ethiopian sites (Amhara, Tigray, and Hararghe regions, respectively), and 74 and 63% of sites in Niger (Maradi-Tahoua and Tillabery- Dosso regions, respectively). Flowering periods of wild and weedy sorghum populations overlapped with those of cultivated sorghum at most sites where the two co-occurred, especially in Ethiopia, and many putative crop–wild hybrids were observed. Therefore, current gene transfer from cultivated sorghum to wild and weedy sorghum populations in Ethiopia and Niger is likely to be widespread

Large-scale analysis of structural brain asymmetries in schizophrenia via the ENIGMA consortium.

Left-right asymmetry is an important organizing feature of the healthy brain that may be altered in schizophrenia, but most studies have used relatively small samples and heterogeneous approaches, resulting in equivocal findings. We carried out the largest case-control study of structural brain asymmetries in schizophrenia, with MRI data from 5,080 affected individuals and 6,015 controls across 46 datasets, using a single image analysis protocol. Asymmetry indexes were calculated for global and regional cortical thickness, surface area, and subcortical volume measures. Differences of asymmetry were calculated between affected individuals and controls per dataset, and effect sizes were meta-analyzed across datasets. Small average case-control differences were observed for thickness asymmetries of the rostral anterior cingulate and the middle temporal gyrus, both driven by thinner left-hemispheric cortices in schizophrenia. Analyses of these asymmetries with respect to the use of antipsychotic medication and other clinical variables did not show any significant associations. Assessment of age- and sex-specific effects revealed a stronger average leftward asymmetry of pallidum volume between older cases and controls. Case-control differences in a multivariate context were assessed in a subset of the data (N = 2,029), which revealed that 7% of the variance across all structural asymmetries was explained by case-control status. Subtle case-control differences of brain macrostructural asymmetry may reflect differences at the molecular, cytoarchitectonic, or circuit levels that have functional relevance for the disorder. Reduced left middle temporal cortical thickness is consistent with altered left-hemisphere language network organization in schizophrenia

Virtual Ontogeny of Cortical Growth Preceding Mental Illness

Background: Morphology of the human cerebral cortex differs across psychiatric disorders, with neurobiology and developmental origins mostly undetermined. Deviations in the tangential growth of the cerebral cortex during pre/perinatal periods may be reflected in individual variations in cortical surface area later in life. Methods: Interregional profiles of group differences in surface area between cases and controls were generated using T1-weighted magnetic resonance imaging from 27,359 individuals including those with attention-deficit/hyperactivity disorder, autism spectrum disorder, bipolar disorder, major depressive disorder, schizophrenia, and high general psychopathology (through the Child Behavior Checklist). Similarity of interregional profiles of group differences in surface area and prenatal cell-specific gene expression was assessed. Results: Across the 11 cortical regions, group differences in cortical area for attention-deficit/hyperactivity disorder, schizophrenia, and Child Behavior Checklist were dominant in multimodal association cortices. The same interregional profiles were also associated with interregional profiles of (prenatal) gene expression specific to proliferative cells, namely radial glia and intermediate progenitor cells (greater expression, larger difference), as well as differentiated cells, namely excitatory neurons and endothelial and mural cells (greater expression, smaller difference). Finally, these cell types were implicated in known pre/perinatal risk factors for psychosis. Genes coexpressed with radial glia were enriched with genes implicated in congenital abnormalities, birth weight, hypoxia, and starvation. Genes coexpressed with endothelial and mural genes were enriched with genes associated with maternal hypertension and preterm birth. Conclusions: Our findings support a neurodevelopmental model of vulnerability to mental illness whereby prenatal risk factors acting through cell-specific processes lead to deviations from typical brain development during pregnancy