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Mass spectrometry imaging of mice brain lipid profile changes over time under high fat diet

Abstract

Overweight and obesity have been shown to significantly affect brain structures and size. Obesity has been associated with cerebral atrophy, alteration of brain functions, including cognitive impairement, and psychiatric diseases such as depression. Given the importance of lipids in the structure of the brain, here, by using 47 mice fed a high fat diet (HFD) with 60% calories from fat (40% saturated fatty acids) and 20% calories from carbohydrates and age-matched control animals on a normal chow diet, we examined the effects of HFD and diet-induced obesity on the brain lipidome. Using a targeted liquid chromatography mass spectrometry analysis and a non-targeted mass spectrometry MALDI imaging approach, we show that the relative concentration of most lipids, in particular brain phospholipids, is modified by diet-induced obesity (+ 40%of body weight). Use of a non-targeted MALDI-MS imaging approach further allowed define cerebral regions of interest (ROI) involved in eating behavior and changes in their lipid profile. Principal component analysis (PCA) of the obese/chow lipidome revealed persistence of some of the changes in the brain lipidome of obese animals even after their switch to chow feeding and associated weight loss. Altogether, these data reveal that HFD feeding rapidly modifies the murine brain lipidome. Some of these HFD-induced changes persist even after weight loss, implying that some brain sequelae caused by diet-induced obesity are irreversible.

Introduction

Diet-induced obesity is at present one of the most prevalent and costly diseases worldwide1, as this condition is often associated with type 2 diabetes, cardiovascular and neurodegenerative pathologies and several forms of cancer, resulting in a decline in both quality of life and life expectancy2. Obesity is substantially a life-style, environmentally-driven pathology, which results from the ingestion and storage of more calories than what the organism actually needs3. In mammals, energy homeostasis is finely regulated by the central nervous system (CNS). What we eat has a clear impact on brain function and an unbalanced diet with hypercaloric food rich in saturated fats and sugars is a risk factor for the development of brain diseases, including obesity2.

Lipid content represents more than half of the brain’s dry weight and plays a pivotal role in brain integrity. In particular, the brain is highly rich in complex lipids, such as glycerophospholipids, sphingolipids [sphingomyelin (SM), cerebrosides, ceramides (Cer) sulfatudes (SF) and gangliosides] and cholesterol4. The glycerophospholipids class includes 18 different categories, among them: phosphatidylethanolamines (PE), phosphatidylserines (PS), phosphatidylinositols (PI) and phosphatidylcholines (PCs)5. Phospholipids and sphingolipids play a critical role in the bilayer structure of cell membranes, since they allow the amphilicity of the membranes thanks to their polar head group and apolar chain. Within the brain, they also have a signaling role since PIs give rise to phosphorylated inophosphoinositides that act as second messenger6.

The interest in investigating the neuromodulatory role of lipids has therefore grown in the last decades and has been associated with pathological conditions.

For instance, Mendis et al. have described changes in the lipid composition of the hippocampus in Alzheimer’s disease7. Gonzalez et al. have demonstrated a depletion of different sulfatid lipid species in both gray and white matter areas of the frontal cortex in Alzheimer’s disease8. Other changes in the brain phospholipids and proteins were observed in animal models of depression9, while exposure to cocaine was associated with changes in phospholipids composition in the hippocampus and cerebellum10.

However, relatively little is known concerning brain lipid composition and its alteration in the context of obesity.

In particular, few studies have investigated the impact of the consumption of specific nutrients, such as sucrose, or of a high-fat diet (HFD) rich in saturated fats on brain lipid composition. These investigations focused on n-6 and n-3 fatty acid levels11, as changes in the balance of these fatty acids can directly affect neurotransmission and favor neuroinflammation12,13. In turn, these modifications have been associated with weight gain and obesity14. In rats fed a lard-enriched chow (high in saturated/low in polyunsaturated fatty acids) for eight weeks, hypothalamic insulin signaling and insulin-dependent regulation of food intake were drastically impaired, independently of the hypothalamic fatty acids compostion15.

A recent study using shotgun lipidomics has revealed an increase of amyloids plaquets under HFD in Alzheimer disease, and the modulation of 24 lipids subspecies with a specific alteration of cardiolipids in obese mice16 Other HFD-induced lipidome perturbations were described in the cortex, hippocampus, hypothalamus, and olfactory bulb of mice by using untargeted LC–MS–MS analysis, which showed an increase of all lipid species in all these brain structures16.

In these studies, brain fatty acid content was generally analyzed after several weeks of exposure to the HFD, when an obese phenotype has already developed, therefore not allowing distinguishing the impact of the diet on brain composition, before the development of obesity.

Other studies have shown correlations between circulating levels of LPCs, SM, nonesterified fatty acids (NEFAs), PCs and obesity17, suggesting that alterations in lipid metabolism play an important role in the disease.

In depth evaluation of brain lipidome could help understand underlying biological mechanisms for increased disease risk and consequently help determine specific therapeutic approaches.

Therefore, in the current study we have evaluated the impact of a HFD on brain lipid content in mice, by first assessing changes in brain composition after only 3 days of HFD consumption. We then evaluated brain lipid profile after 16 weeks of HFD, once that a clear obese phenotype was established, as well as during and after weight loss, by switching animals back from HFD to a control chow diet. In our lipid profile analysis we focused on phospholipids, such as phosphatidyl choline, phosphatidyl ethanolamine, phosphatidyl serine and phasphatidyl inositol. We performed these sets of analyses on the cortex, the striatum, the thalamus, the amygdala and the hypothalamus. These brain regions, and in particular the hypothalamus, the striatum and the cortex, are well known to play different and complementary roles in the regulation of feeding behavior, body weight and whole body energy metabolism18,19,20,21.

To ensure the specific identification and mapping of the brain lipidome, we have developed a unique approach that combines high resolution untargeted mass spectrometry imaging (MSI) and a targeted LC–MS/MS, enabling the analysis of specific structures of the brain in their native state and after obesity induction.

MSI is a powerful tool that allows untargeted investigations into the spatial distribution of molecular species in tissues. It has the capability to image thousands of molecules, such as metabolites, lipids, peptides, proteins, and glycans, in a single experiment without labeling22. The combination of information gained from MSI and LC–MS/MS analyses described here therefore provides a comprehensive database of the brain lipidome and of its spatial organization in relation to different exposure times to HFD, to the development of obesity and to its reversal.

Results

Body weight evolution

Figure 1 illustrates the changes over time in the body weight (BW) of the mice used for the study. After 3 days on HFD, mice did not show any difference in the BW, fat or lean mass as compared to chow-fed controls (Fig. 2a, T0). After 16 weeks on HFD, mice had clearly become obese, with a significant increase in BW and fat mass (Fig. 2b, T16), as expected. After 10 days of diet switch, previously HFD-fed mice switched to chow were still frankly obese, with a much greater BW and fat mass than chow-fed controls (Fig. 2c,T18). Finally, BW and fat mass decreased in previously HFD-fed mice after 7 weeks of chow, with a fat mass that was no longer significantly different from the one of chow-fed controls (Fig. 2d, T23).

LC–MS–MS results

Principal component analysis

Using LC–MS–MS results of three brains from HFD-fed or diet reversed animals and three brains from age-matched chow-fed controls, a PCA was performed on the log ratio HFD-fed (or reversed)/ chow-fed control at the four time points of the study (T0, T16, T18 and T23).

Table1 represents the eigenvalues and the percentage of variance explained by each principal component, as well as the accumulated variance. The first three components are needed to explain 85.5% of the total variability in the phospholipids of the brain section. The first component (F1) explains 38.53% and 25.38% is explained by the second component (F2), together accounting for 63.91% of the total variation (Table1). The percentage of the accumulated variance in the first factor is justified by the nature and quantity of the variables evaluated. So even with the standardization of the data, four components were necessary to reach the adopted selection criteria (≥ 1.0), with 100% of the accumulated variance. Most of the evaluated phospholipids (63.91%) made significant contributions to the total variation of the data (Fig. 3b).

The loading plot of variables (Fig. 3a) shows that PC2 is mostly explained by the variable at T16 while T0, T18 and T23 constituted the PC2. The biplot depicts phospholipids whose variation explain group clustering, distinguishing T16 from the other time points (Fig. 3b). A heat map based on the mean ratio of phospholipids in HFD or reversed mice and chow-fed control mice at different time points (T0, T16, T18 and T23) showed clearly the phospholipids that significantly vary at T16 (Fig. 3c).

Among the 37 phospholipids identified in brain sections using LC–MS–MS, 17 PCs, 4 PI, 6 PS and 10 PE were identified.

Differentially expressed lipids in positive mode

The majority of lipids observed in positive mode in LC–MS–MS are PCs. Among a list of 66 PCs, 17 were identified and quantified.

The analysis did not show a significant difference across weeks, since only 2 of the 17 identified PCs had a difference upwards or downwards (Fig. 4a).

The normalized relative intensities of PC 40:5 was found to increase over time and with weight gain. Then it went down at T18, during weight loss and back up at T23, after 7 weeks of switch to chow. Conversely, the normalized relative intensity of PC 38:5 increased continuously, even in response to the chow switch at T18 and remained unchanged 7 weeks after from the diet switch.

Differentially expressed lipids in negative mode

The quantitative LC–MS–MS analysis showed variation in the concentrations of several negatively charged phospholipids (PS, PI and PE). Relative concentration ratios between brains from HFD-fed mice and chow-fed controls over time revealed negatively correlated phosphatidylserines, like PS 36:1, which was found to vary inversely with PS 34:1 at T16, T18 and T23 (Fig. 4b). Whilst, PS 38:1 increased at week 16 and went back to baseline with the chow switch (Fig. 4c). On the other hand, PS 40:6 and PS 38:4 remained unchanged over the course of the study (Fig. 4c).

For the phosphatidylinositols, the proportions of PI 36:1 and PI 34:1 varied inversely, with the PI 34:1 ratio increased at T16 and T18 and decreased at T23, while the PI 36:1 decreased at T16, went back to baseline at T18 and decreased again at T23 (Fig. 4d).

MSI results

Brain segmentation with MSI

Using the MALDI–TOF–TOF at low resolution and a specific software (SCiLS, Bruker), we draw the edges of the cortex, striatum, hypothalamus, thalamus, amygdala and of the brain ventricles guided by the Allen mouse brain atlas (Fig. 5a). Serial sections from the brain of a chow-fed control mouse in positive and negative mode were analyzed and a normalized average mass spectrum was generated for each region (Fig. 5b16). We therefore observed a regional distribution of lipids with higher relative intensity of lysophospholipids (Lyso) (200–400 m/z) in the cortex and hypothalamus (Fig. 5b3 and b5). While, hypocampus and choroid plexus (Fig. 5b2, b4 and b6) had higher relative intensity of phospholipids (PL) (500–900 m/z). In contrast, in the thalamus, the normalized average spectrum had low relative intensity of both Lyso and PL (Fig. 5b1).

The MSI screening method was then used to obtain the anatomical distribution patterns of lipids in coded images and confirmed the regional specificity of the brain lipids distribution. After this initial analysis, we compared the different brain regions using MALDI-IMS high resolution in order to identify brain lipids changes occurring in response to chronic HFD exposure (T16) as well as after long-term diet switch and body weight loss stabilization (T23).

Global lipid profile

Using MALDI-IMS in negative mode on brain samples, a total of 650 lipids were identified but only 430 lipids were kept after Metaspace FDR. Among these, 216 lipids at W16 and 116 at W23 were identified with Lipid Maps. Of note, we did not observe significant differences in positive mode in LC–MS–MS and in MALDI–TOF–TOF (data not shown).

Simultaneous analysis of brain sections from mice fed with HFD (or reversed) and age-matched chow-controls was carried out, with biological triplicates for each. Only lipids with a signal greater than 1% of the internal standard signal were kept for interpretation.

The signal intensities obtained for each identified lipid were systematically normalized to the signal from the internal standard sprayed with the matrix.

After normalization, as done before, we calculated the ratio of the HFD (or reversed) signal over the chow control signal and the average of the ratios was obtained for each individual lipid.

The distribution of the log ratios at T16 and T23 is represented in Fig. 6A. Log ratios at T16 were much higher than at T23, with the number of lipids with a log ratio centered on the value 0.4 at T16, while the distribution of the log ratio at T23 was centered around 0.3, with lipids’ abundance higher at T16 than at T23. This overall means that a major proportion of lipids from brains of mice fed a HFD for 16 weeks had concentrations 2 times higher than in the brain sections from chow-fed controls. While at T23, after diet switch, only very few lipids were still increased as compared to the chow-control group. A PCA was carried out to examine the correlation between the lipids’ concentrations at T16 and T23 in Fig. 6b. By choosing components 1 & 2 we were able to describe 51.72% of the variability of lipids within brain regions and time points. This analysis revealed specific distribution of lipids in different brain regions depending on the two conditions (T16 and T23) analyzed.

In particular, and despite an apparent grouping of lipids around the center of the biplot (green dots), testifying the presence of unchanged lipids between T16 and T23, we were able to distinguish two major groups of lipids. The first group was located on the upper part of the second component (F2), which correlated with T23, and the second group was located in the second lower part of axis 2, and was related to T16. In each group of lipids we noticed a subdivision into two subgroups according to axis 1 of the biplot, thus marking the presence of subgroups of lipids positively and negatively correlated and with the experimental condition.

Considering the low percentage of variability that was described by the PCA (51.72%), we decided to perform an agglomerative hierarchical clustering (AHC), which has the advantage of describing the total variability in the chosen observations (Fig. 6c).

To facilitate the clustering interpretation, we preferred to transpose data before the AHC processing. The results described in the dendrogram confirmed the general trend with different regions of the brain at T16 that showed lower Euclidean distance and were grouped together in one class. This stems from the presence of a large common class of lipids characteristic of the obese state at T16, independently of the brain region studied. The only exception that emerged from this clustering was the frontal cortex at T16, which represented itself an individual class with a very important Euclidean distance from the remaining regions at T16.

Different brain regions at T23 were also grouped in a single class, indicating the presence of the same families of lipids in these brain structures after diet switch and weight loss.

We then illustrated in Fig. 6d three examples of overexpressed lipids at T16 in an obese brain (section at the bottom) as compared to the chow control (section at the top). Specifically, we observed an overexpression of phosphatidyl Inositol (PI 38:4), in the cortex and the hypothalamus, of PA 40:7 in the frontal cortex and of the sulfated hexose ceramide 40:2 in the thalamus of the obese brain relative to the chow controlAll these changes were normalized at T23.

Comparison of lipid profile over time in the cortex

Among the 216 lipids initially identified, only 44 were found to significantly differ in the cortex (p < 0.05) and were used for the PCA (Fig. 7a). 30 lipids (in red) were specific of T16, testifying for high relative concentration of this group of lipids in the obese brain. Differently, 14 lipids were associated with T23 and therefore typical of the diet reversal phase.

As done before, we calculated the log of ratios of obese (or reversed) intensity vs chow-control intensity for each individual identified lipid and used it to classify lipids occurrence in the obese (or reversed) brain, from higher to lower occurring ones (Fig. 7b). Most lipids were upregulated at T16 as compared to T23, with lipids intensities in the obese state varying from 1.5 to 10 times the lipid intensity at T23. Lyso-phospholipids (Lyso-phosphatidyl ethanolamine PE 24:6) were among the most overexpressed lipids, reaching a relative concentration 10 times higher in the obese brain at T16 (Fig. 7b).

Figure 7c shows the MSI of brain sections at T16 illustrating the distribution and expression of three lipids found overexpressed in the cortex (highlighted area). The images clearly show, as for the graphs, an overexpression of these lipids in the cortex (as defined in Fig. 5b) in the obese state (bottom images) as compared to the chow-control condition (top images).

Comparison of lipid profile over time in the striatum

As for the cortex, most of the lipids were overexpressed at T16 in the striatum as well (red dots, Fig. 7d). On the right down quartile of the biplot a few lipids correlated with diet reversion at T23. The log ratio histogram showed specific lipids overexpressed in the striatum in the obese state at T16 (Fig. 7e), with phosphatidyl glycerol PE 36:1 being 8 times higher in the obese than in the chow control.

Figure 7f shows an example of images with over expression of lipids (especially plasmalogens PE P 38:5) in the striatum region of obese animals at T16.

Comparison of the lipid profile over time in the hypothalamus

Similarly to the cortex and striatum, also in the hypothalamus the majority of phospholipids were overexpressed at T16 as compared to chow-fed control. The principal component analysis showed a symmetric dispersion of lipids according to the first axis F1, which represents 60.12% of the total variability. Half of the points were associated with obesity at T16, while the other half correlated positively with the diet switch at T23, revealing the persistence of high abundance of lipids after switch to chow and weight loss (Fig. 7g).

The histogram of log ratios showed 26 phospholipids with ratio higher than 2 at T16. Among those lipids, phospholipids PA 38:4 and PA 38:6, PG 36:1 and PG38:4, and various lyso-phospholipids, plasmalogens and ceramides were found highly expressed in the hypothalamus from obese mice (Fig. 7h). 26 other lipids were highly expressed after the switch to chow and weight loss. These lipids were mainly plasmalogens, also called ether lipids. The following were already increased with weight gain at T16 and their ratios was even higher at T23: PA P-38:6; PA P-40:6; PAP-38:4; PE P-38:5; PI P-38:5; PE P-40:4; PE P-38:4; PE P-40:5; PE P-40:6; PE P-38:2; PE P-36:2. These are ethers of phosphatidyl ethanol amine, phosphatidyl ethanol or phosphatidyl inositol, all with fatty acyl chain length from 36 to 40 and number of unsaturation from 2 to 6.

Figure 7i shows representative MALDI images of lipids with log ratio > 0.5 at T16 and we have indicated with an (*) those lipids that remained high at T23, after chow switch. Thus, even after the diet switch and body weight loss, the hypothalamus, a structure that critically controls food intake and body weight (16, 17, 19), clearly shows permanent changes in lipid composition.

Comparison of lipid profile over time in the thalamus

The PCA for the thalamus showed that all the lipids except 3 were aligned and correlated at T16 (Fig. 7j).

Log-ratio analysis showed that the lipid with the highest ratio at T16 in this brain region was the phosphatidyl serine PS 40:7 (Fig. 7k). The thalamus had high presence of several poly lipids with a fairly long medium chain length (PG 36:1; PA 42:7; PI 38:6; PA 40:7). As with the previous brain regions, we found abundance of ether lipids (PE P-38:3;PE P-38:4; LPG P-34:5; LPE P-16:0; PA P-38:4; PA P-40:6; PE P-38:2; PA P-36:2; PE P-36:2; PE P-38:5) and lyso-phospholipids (LPA 22:5; LPA 22:6; LPA 22:4) at T16. Figure 7l shows representative images of lipids whose log-ratio was the highest at T16.

Comparison of lipid profile over time in the amygdala

The PCA describing the distribution of lipids in the amygdala and their correlation with experimental conditions of T16 and T23 showed results similar to those already described for the thalamus, with an apparent alignment of lipids correlated with the obese state and very few lipids related to the diet switch at T23, which however were highly present (Fig. 7m). By calculating log-ratios (Fig. 7n), we found that the majority of lipids increased in the obese state at T16, were normalized at T23.However, 5 lipids that were high at T16 aremained high at T23, and were called “persistent lipids” (PE P-38:5; PE P-36:2; PE P-36:1; PI 40:6 and PI 38:6). Finally there were also lipids that did not have a high ratio at T16, but that were increased at T23 (PE P-40:7; PE P-38:2; PE P-38:1; PG 36:1; PS 40:7; HexCer 44:6;O5; S-HexCer 42:2;O2; LPE 24:6; CerP 44:6;O6; CerP 42:2;O2; LPC 16:2).

Figure 7o shows representative images by MALDI-IMS at T16 illustrating the spatial distribution of lipids in the amygdala and their more abundant relative intensity in the obese brain as compared to chow-control. We identified with an (*) those lipids that were still highly present at T23, after diet switch and weight loss.

Overall the analyzed brain areas, we classified the identified lipids in three species (phospholipids, ceramides and plasmogens) and compared their variation between T16 and T23 (Fig. 8a). This analysis revealed that plasmogen relative concentrations are the most modified, with a strong increase in the obese state, and remain high after weight loss. We also analyzed the evolution of lipid structure in term of carbon chain length and number of unsaturation. Persistent lipids after diet switch were clearly those with a high number of unsaturation and long carbon chain (Fig. 8b,c).

Discussion

This study demonstrates that feeding mice with a HFD leading to obesity results in lipid profile changes and that these changes are localized in the specific brain region investigated. Importantly, some changes in the brain lipidome were still present 7 weeks after switch back obese animals to chow and observing weight loss (which by week 23 had plateaued, see also Fig. 1). This evidence implies that these persistent changes in lipid composition are not merely a direct effect of the diet composition, but could indicate a definitive alteration in the structure of certain areas of the brain due to a previous episode of obesity. This was particularly true for the cortex, the striatum and the hypothalamus, brain regions critically involved in the regulation of different aspects of feeding behavior, of body weight and of peripheral metabolic responses (Supplementary Fig. S1).

The results obtained in LC–MSMS showed very little variation of the lipid profile in positive mode which mainly concerned PC 38: 5 and PC 40: 5. These lipids had shown a slight variation between obese mice and controls at T16 (Fig. 4a). Whilst, in the LC–MSMS analysis in negative mode, two phospholipids species were found varying with weight gain: the phosphatidylserine mono unsaturated (PS 36:1 and PS 38:1), and the phosphatidyl inositol (PI 34:1 and PI 36:1). The LC–MSMS analysis in negative mode also revealed a recovery of the initial ratios of lipid signals after 7 weeks of switch to chow for the majority of lipids. Nevertheless the presence of certain types of lipids persisted even after weight loss. These observations led us to focus on the IMS analysis of brain sections in negative mode, which revealed several lipids varying significantly in association with the obese state and within specific brain regions.

The general conclusion from the IMS studies we have conducted, is that all lipids are overexpressed in diet-induced obese mice as compared to age-matched chow-fed controls at T16, and in particular the plasmalogens or ether lipids. These phospholipids are a subclass of glycerophospholipids (GP) that have an alkyl chain attached by an ether bond at the sn-1 position of the glycerol backbone23.

They also represent the major part of the persistent modifications after weight loss (Fig. 8a).

Overall, these findings provide evidence for an ether lipid signature in the brain of diet-induced obese mice. This signature is characterized by an increased content of alkenyl forms of phosphatidylethanolamine (GPEtn).

In addition, these lipid species have an increased relative abundance of a longer carbon chain length (> 36C) (Fig. 8b) with a higher number of double bonds (> 4 insaturations) (Fig. 8c).

Of note, Plasmalogens constitute ~ 15–20% of total phospholipids in cell membranes, with ≥ 50% of the GPEtn fraction in brain, heart, neutrophils and eosinophils24. Farooqui et al., have demonstated that plasmalogen content of tissues generally decreases in aged mammals25. It was also showen a low lipoxidative damage in long-lived animal species, with an accumulation rate that inversely correlates with longevity26.

Plasmalogens derivatives from phosphatidylethanolamine species with long carbon chains and with a high number of double bonds were also found reduced in the plasma of centenarians compared with young people27.

Given the above findings, we may hypotehsize that positive correlation between PE Plasmogens derivatives and obesity will likely correlate with the reduced life expectancy observed among obese subjects28,29.

The second species of lipids which was overexpressed in obese mice at T16 were lipids of the ceramides family, which remained expressed even at T23, after the switch to chow and weight loss, with in addition the appearance of new ceramides not previously modified in the brain of obese animals. This is not at all surprising given the role that ceramides play as metabolic intermediaries in nutrition, weight gain and obesity30. In particular, ceramides accumulation in the brain due to pharmacology or genetic manipulation directly alters food intake, energy expenditure and glucose homeostasis30. Here (Fig. 8d), we were able to distinguish two subclasses of ceramides which were found to be overexpressed in the brain of obese animals. These were the ether ceramides and the hexose ceramides, but only the hexose ceramides were still high after the switch to chow and the weight loss at T23.

A recent lipid-gene regulatory network study has also revealed coregulations of triacylglycerol with phosphatidylinositol/lysophosphatidylinositol and with hexosyl-ceramide31, which may explain the high level of hexosyl-ceramide that we have observed after weight loss.

Another interesting finding was that the expression of Bis(monoacylglycero)phosphate (BMP, and specifically BMP 18:0) was increased at T16 and, although decreased at T23 after weight loss, still remained high as compared to age-matched chow-fed controls (Fig. 8d).

BMP is a negatively charged glycerophospholipid with an unusual sn-1;sn-1’ structural configuration. BMP is thought to play a role in glycosphingolipid degradation and cholesterol transport32. Elevated levels of BMP have been found in diseases associated with lysosomal storage material. Interestingly, BMP content in brain of humans and animals with either GM1 or GM2 ganglioside storage diseases is significantly greater as compared to brains of normal subjects33. Another peculiarity of BMP is its specific biophysical and functional properties confered by the incorporation of docosahexaenoic acid (DHA) compared to other polyunsaturated fatty acids34.

This can explain its presence only in the HFD condition at T16, leading to the assumption that its role is directly linked to food metabolism35.

We have investigated the impact of HFD, obesity development and its reversal by dietary intervention on the brain lipidome, clearly showing that the type of diet eaten impacts the brain lipids profiles. The novel aspect of this work is that it demonstrated, using IMS, the variation in lipids distribution and concentration within different regions of the brain, such as the cortex, the striatum and the hypothalamus, which are known to play critical roles in the regulation of feeding behavior and whole-body energy metabolism18,19,20,21. It was concluded that numerous lipids vary subsequently to chronic exposure to HFD, especially plasmalogens, ceramides and lyso-phospholipids, which were found to increase significantly in the obese state. The most surprising finding was the presence of a large number of lipids which, despite the switch back to chow and the consequent body weight loss, did not recover their initial relative intensity as compared to age-matched chow-fed controls, indicating the production of irreversible changes caused by chronic exposure to HFD and weight gain.

Materials and methods

Animal experimental procedure

Experiments were performed following the European Union recommendations (2010/63/EU) after approval from the French Ministry of Agriculture and Fisheries (authorization no. 3309004) and the local ethical committee of the University of Bordeaux and in accordance with the The ARRIVE guidelines 2.0.

Adult 7-week-old male C57bl6/J mice (Janvier Labs, Le Genest-Saint-Isles, France) were individually housed at 22 °C with a 12 h light–dark cycle (light off at 1 pm). Animals had free access to water and chow (13.5% of calories from fat, 61% of calories from carbohydrates, Standard Rodent Diet A03, SAFE, France) unless stated otherwise. Mice were acclimated for one week before the start of the study. At the start of the study, mice were randomly distributed in two groups and either maintained on chow for the length of the study or fed a HFD with 60% calories from fat (40% saturated fatty acids) and 20% calories from carbohydrates (D12492 Research Diets, New Brunswick, NJ) for 3 days (T0), or 16 weeks (T16). In order to study the possible reversal of brain lipid changes induced by HFD or obesity, 16 weeks HFD-fed mice were then placed back on chow and animals were analyzed after 10 days on chow (T18), or after 7 weeks on chow (T23) and compared to age-matched chow-fed control animals never exposed to HFD (See also Fig. 1 for the evolution of body weight). Changes in body weight were recorded weekly for the length of the study. At T0, T16, T18 and T23, 6 mice per group were killed by cervical dislocation and brains rapidly removed and fresh frozen for further analysis. In total 47 mice were used to generate data on body weight and body composition at the different time points; while brain analysis was conducted on three mice per group/time point chosen randomly and with technical repetition for each brain section.

Body composition analysis

A nuclear echo magnetic resonance imaging whole-body composition analyzer (Echo MRI 900; EchoMedical Systems, Houston, TX, USA) was used35 to repeatedly assess body fat and lean mass changes in conscious mice during the study at T0, T16, T18 and T23 (Fig. 2).

Chemicals

α-Cyano-4-hydroxycinnamic acid (HCCA), 1,5-Diaminonaphthalene (DAN) were supplied by Sigma-Aldrich. Trifluoroacetic acid (TFA), HPLC grade ethanol, Acetone, acetonitrile and isopropanol were supplied by Fischer Scientific UK.

Extraction protocols

A two-step process was used for the extraction of PLs36. Briefly, sections adjacent to those used for imaging were removed and introduced into an eppendorf tube to perform lipid extraction. Brains sections were homogenized with 200 µL of 180 mM ammonium acetate, pH 8. The lysate was extracted with 990 µL of chloroform/methanol 17/1 (v/v) to eliminate apolar lipids. Then, the aqueous phase was re-extracted with 990 µL of chloroform/methanol 2/1 (v/v) to isolate polar lipids in the organic phase; this PLs extract was dried under vacuum (Speedvac, Thermo).

LC–MS and LC–MS–MS methods

A Shimadzu gradient LC system (CBM-20A) interfaced to a QTrap mass spectrometer equipped with a TurboV electrospray source (SCIEX API 5500) was used for LC–ESI–MS/MS analysis of PLs. The separation of lipids was carried out at 40 °C on a C8 Synergi Fusion-RP 150 × 1 mm column, with 80 Å pore size, 4 µm particles (Phenomenex). Eluent A was isopropanol/methanol/H2O + 0.2% formic acid + 0.028% NH4OH; Eluent B was isopropanol + 0.2% formic acid + 0.028% NH4OH. The gradient elution program was: 0 min, 30% B; 5 min, 50% B; 30 min, 80% B; 31–41 min, 95% B; 42–47 min, 5% B; 52–62 min, 30% B. The flow rate was 40 µL/min36.

PLs extracts were dissolved in 400 µL of solvent B used for the reversed phase liquid chromatography analysis; 5 µL sample volumes were injected. The mass spectrometer was operated over a mass range of m/z 250–1100 at a 1000 Da/s scan speed. The ion spray voltage was set at 5500 V in the positive ion mode and 4500 V in the negative ion mode. The trap fill-time was set at 3 ms in the positive ion mode, and 5 ms in the negative ion mode.

A simple mass analysis was performed in both positive and negative mode to predict phospholipids identification following their retention time and m/z values.

Afterwards, we completed a parent ion scan. The combination of retention time and fragments identification based on the loss of the phospholipids head followed by subsequent loss of Sn2 and Sn1 fatty acid chains, allowed the final confirmation of the phospholipids. Finally the confirmation step consists in matching two fragments with a common precursor ion at the corresponding retention time.

The Multiple Reactions Monitoring (MRM) mode was used for the quantification step, where each phospholipid was distinguished by selecting the m/z of the precursor ion and major fragment ion. Before the quantitative analysis, 6 standards phospholipids were added to sample solution (PA 34:0; PS 34:0; PG 34:0; PC 26:0; PE 34:0; PI 31:1), and used respectively as internal standard to quantify different detected phospholipids.

After normalization with the internal standard signal, the relative abundance of each phospholipid species was calculated, for instance, a phosphatidyl choline relative abundance was calculated considering the ratio: concentration of the given PC to the sum of all PCs that are present in the sample.

$${\rm iPC X:y}\,\%\,=\,{\rm (iPC X:y/PC sum)} \times100.$$

The calculation of concentration, and the relative abundance of each lipid was performed in three different biological samples. The results were used to confirm if there was quantitative changes in brain lipids with weight gain.

Tissue preparation for MSI

Mouse brains were frozen in a dry ice chilled isopentane and stored at − 80 °C. Frozen organs were transported from the − 80 °C freezer to a cryostat (Lyca, CM1950) on dry ice. For mounting, tissues were held atop a drop of HPLC grade water on the cutting stage and placed into the cryostat chamber, held at − 19 °C to equilibrate. Frozen brains were sliced into 12 µm sections. A mouse brain atlas was used to guide the experimenter in the collection of the samples.

The brain sections were directly deposited on indium tin oxide (ITO) coated conductive glass slides for MALDI imaging and dried and preserved in vacuum dissicator to remove moisture before matrix coating.

Matrix application on brain sections

ITO-coated slides bearing mouse brain sections were scanned (HP Scanjet G4050) prior to matrix deposition. α-Cyano-4-hydroxycinnamic acid (HCCA) was used at concentration of 7 mg/ml in acetone/water, 0.2% TFA 1:1 (v:v) as a MALDI matrix for the analysis in positive ion mode and DAN was used for the analysis at 5 mg/ml in Acetone/water 7:3, supplemented with 16:0-d30-18:1 PA at 10 μg/ml as internal standard for normalization and Palmitate-d30 at 10 μg/ml for LockMass. Matrix was spray-coated using a homemade based on a glass Type A Meinhard nebulizer (Golden, CO)37. Matrix was applied in multiple cycles (spray generation, matrix incubation on the tissue section, and drying the sample using a soft nitrogen flow) to reach an adequate thickness without lateral delocalization of analytes (approximatively 1 ml matrix solution per slide).

Brain tissue imaging

An Ultraflex III MALDI-TOF-TOF tandem mass spectrometer (Bruker Daltonics Inc., Bremen, Germany) was used for explorative imaging at low resolution. Briefly, mass spectra acquired for a m/z 140–1300 range were averaged from 300 consecutive laser shots. The raster spatial resolution was 90 μm. Depending on the image size, acquisition took 3 h. MALDI-MSI images was produced from the corresponding spectra to generate a relative intensity ion map at a given m/z value, using FlexImaging 4.1 software (Bruker Daltonics Inc.).

High resolution complementary analysis was performed using a high performance atmospheric pressure imaging ion source named AP-SMALDI5 AF (TransMIT GmbH, Giessen, Germany) connected to an orbital trapping mass spectrometer (Q-Exactive Orbitrap, Thermo Fisher Scientific, Bremen, Germany)38. This latter was operated in negative ion mode at a mass resolution of 70,000 at m/z 100 over a mass range of m/z 190–2000. The ion source is equipped with a diode laser (Flare NX343, λ = 343 nm), operating at a repetition rate of 2 kHz with an energy of 200 µJ/pulse. The imaging data were acquired in high speed continuous mode with a pixel size of 30 µm and a speed rate of 3.7 pixels/s.

Data analysis

For LC–MS–MS and MSI, lipids identification was realized on LipidMaps39 and METASPACE software40,41 was used for high resolution spectra, based on peak exact mass and isotope profiles. A list of 216 identified lipids with a FDR of at least 20% was built. Identified lipids were classified according to their class (phospholipids, ceramides or plasmogens). Additionally, we also characterized the number of unsaturation and carbon chain length for each lipid.

MALDI images were generated using MSi Reader software42.

Normalized negative ion images of lipids were generated by dividing the given lipid signal by 16:0-d30-18:1 PA (m/z 703.670 ± 0.003) signal.

Normalized Positive ion images of lipids were generated by dividing the given lipid signal by the 16:0-d30-18:1 PC ([M + H] + m/z 790.772 ± 0.003 or [M + K] + m/z 828.728 ± 0.003, depending of the adduct of given positive ion) signal (Supplementary Figs. S2S7).

MSI and LC–MS–MS validation

To validate obtained results, three brains/group were analysed in MSI and in LC/MS/MS with technical replicate.

For MSI, brain sections from obese mice and chow-fed controls were acquired side by side in the same image (control section in the top and obese section in the bottom). A serial section of these brains were reacquired to obtain duplicate for each section. Then the brain sections from three obese and three chow-fed mice and their replicate were scanned. Ultimately, both the obese group and the control group contained six acquisitions. For each lipid, we performed a normalized relative quantification in the different selected region of interest (ROI) (based on brain atlas) to obtain an average quantification from each ROI. Then, Log-ratios of obese vs control were calculated for each lipid, and a paired t-test was performed on n = 6 (3 biological riplicate × 2 technical riplicate). Therefore, only lipids showing a p-value < 0.05 where kept for data analysis and average of significantly different lipids log-ratios was calculated and used for interpretation.

Principal component analysis (PCA) and statistical analysis

A PCA using XLSTAT software (Addinsoft 2020. XLSTAT statistical and data analysis solution. NEW York, USA)43 was conducted on MALDI and LC–MS–MS data from brains of chow-fed controls and HFD-fed obese mice, before and after weight loss. PCA was used here to identify hidden variables between spectra taken from the brain regions of interest (ROI). It is a statistical tool used to reduce multidimensional data sets to lower dimensions and to identify new, meaningful, hidden differences between clusters. The first step of analysis consisted in a series of pre-processing data on the loaded spectra: deconvolution on the mean spectrum to reduce the background noise and baseline subtraction, then signal intensity normalization on internal standard signal. The software selects the vectors (principal component (PC)) that account for the greatest variation between the groups. A "best fitting" vector can be defined as one that minimizes the average squared distance from a point to the line44.

Body weight and body composition data were analysed using Prism (version 8). Unpaired t-test were used for statistical analysis. P < 0.05 indicate significance (Suppl. Information).

References

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Fotuhi, M., & Lubinski, B. The effects of obesity on brain structure and size. Practical neurology. https://practicalneurology.com/articles/2013-july-aug/the-effects-of-obesity-on-brain-structure-and-size (2013).

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Liu, J. J., Green, P., John Mann, J., Rapoport, S. I. & Sublette, M. E. Pathways of polyunsaturated fatty acid utilization: Implications for brain function in neuropsychiatric health and disease. Brain Res. 1597, 220–246 (2015).

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Lindquist, D. M., Asch, R. H., Schurdak, J. D. & McNamara, R. K. Effects of dietary-induced alterations in rat brain docosahexaenoic acid accrual on phospholipid metabolism and mitochondrial bioenergetics: An in vivo 31P MRS study. J. Psychiatr. Res. 95, 143–146 (2017).

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Dornellas, A. P. S. et al. Deleterious effects of lard-enriched diet on tissues fatty acids composition and hypothalamic insulin actions. Prostaglandins Leukot. Essent. Fatty Acids. 102–103, 21–29 (2015).

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Lee, J. C. et al. High-fat diet-induced lipidome perturbations in the cortex, hippocampus, hypothalamus, and olfactory bulb of mice. Biochim. Biophys. Acta BBA-Mol. Cell Biol. Lipids. 1863(9), 980–990 (2018).

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Ellis, S. R., Brown, S. H., Panhuis, M., Blanksby, S. J. & Mitchell, T. W. Surface analysis of lipids by mass spectrometry: More than just imaging. Prog. Lipid Res. 52(4), 329–353 (2013).

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Dean, J. M. & Lodhi, I. J. Structural and functional roles of ether lipids. Protein Cell 9(2), 196–206 (2018).

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Pradas, I. et al. Exceptional human longevity is associated with a specific plasma phenotype of ether lipids. Redox Biol. 21, 101127 (2019).

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Cruciani-Guglielmacci, C., López, M., Campana, M. & le Stunff, H. Brain ceramide metabolism in the control of energy balance. Front. Physiol. 8, 787 (2017).

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Wang, W. et al. Lipid-gene regulatory network reveals coregulations of triacylglycerol with phosphatidylinositol/lysophosphatidylinositol and with hexosyl-ceramide. Biochim. Biophys. Acta BBA-Mol. Cell Biol. Lipids. 1864(2), 168–180 (2019).

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Hullin-Matsuda, F., Luquain-Costaz, C., Bouvier, J. & Delton-Vandenbroucke, I. Bis(monoacylglycero)phosphate, a peculiar phospholipid to control the fate of cholesterol: Implications in pathology. Prostaglandins Leukot. Essent. Fatty Acids. 81(5–6), 313–324 (2009).

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Akgoc, Z. et al. Bis(monoacylglycero)phosphate: A secondary storage lipid in the gangliosidoses. J. Lipid Res. 56(5), 1006–1013 (2015).

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Bascoul-Colombo, C. et al. Dietary DHA supplementation causes selective changes in phospholipids from different brain regions in both wild type mice and the Tg2576 mouse model of Alzheimer’s disease. Biochim. Biophys. Acta BBA Mol. Cell Biol. Lipids 1861(6), 524–537 (2016).

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Buré, C., Ayciriex, S., Testet, E. & Schmitter, J.-M. A single run LC-MS/MS method for phospholipidomics. Anal. Bioanal. Chem. 405(1), 203–213 (2013).

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Tsai, Y.-H., Garrett, T. J., Carter, C. S. & Yost, R. A. Metabolomic analysis of oxidative and glycolytic skeletal muscles by matrix-assisted laser desorption/ionizationmass spectrometric imaging (MALDI MSI). J. Am. Soc. Mass Spectrom. 26(6), 915–923 (2015).

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Robichaud, G., Garrard, K. P., Barry, J. A. & Muddiman, D. C. MSiReader: An open-source interface to view and analyze high resolving power MS imaging files on matlab platform. J. Am. Soc. Mass. Spectrom. 24(5), 718–721 (2013).

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Halade, G. V. et al. Comprehensive targeted and non-targeted lipidomics analyses in failing and non-failing heart. Anal. Bioanal. Chem. 410(7), 1965–1976 (2018).

Acknowledgements

We acknowledge the support from International master program “Analytical Chemistry for Drugs and Natural Products” and Idex program from University of Bordeaux, France. We thank the animal and histological platforms of the INSERM U1215 Neurocentre Magendie, funded by INSERM and Labex Brain ANR-10-LABX-43, for animal care and use of cryostat for brain sections preparation, respectively. DC is supported by INSERM, Nouvelle Aquitaine Region, and ANR (ANR-13-BSV4-0006, ANR-17-CE14-0007, ANR-18-CE14-0029, ANR-10-EQX-008-1 OPTOPATH). We would like to acknowledge our colleague C. Buré and B. Mortel for their advice and help in this project. The salary of L. Blanc was founded by the FRM (Fondation Recherche Medicale,).

Author information

Authors

Contributions

The authors of this paper designed the study (B.R. & D.C.); collected and oversaw collection of the data (G.S., S.J., L.B., B.R., D.C.); performed the statistical analysis (B.R., D.C., L.B.); wrote the original draft (G.S., B.R.); wrote and edited the final version of the manuscript (B.R., D.C.).

Corresponding author

Correspondence to Boutayna Rhourri-Frih.

Ethics declarations

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

Sighinolfi, G., Clark, S., Blanc, L. et al. Mass spectrometry imaging of mice brain lipid profile changes over time under high fat diet. Sci Rep 11, 19664 (2021). https://doi.org/10.1038/s41598-021-97201-x


text:
This is the repository for Hector, an open source, object-oriented, simple global climate carbon-cycle model. It runs essentially instantaneously while still representing the most critical global scale earth system processes, and is one of a class of models heavily used for for emulating complex climate models and uncertainty analyses. For example, Hector’s global temperature rise for the RCP 8.5 scenario, compared to observations and other model results, looks like this:

The primary link to Hector model documentation is the online manual, which is included in the repository in the vignettes/manual directory. The code is also documented with Doxygen-style comments. A formal model description paper (Hartin et al. 2015) documents its science internals and performance relative to observed data, the CMIP5 archive, and the reduced-complexity MAGICC model (as of version 1.0). In addition, we have developed two package vignettes demonstrating the basics of the Hector R interface and an example application of solving for an emissions pathway.

This research was supported by the U.S. Department of Energy, Office of Science, as part of research in Multi-Sector Dynamics, Earth and Environmental System Modeling Program. The Pacific Northwest National Laboratory is operated for DOE by Battelle Memorial Institute under contract DE-AC05-76RL01830.

Tools and Software That Work with Hector

• GCAM: Hector can be used as the climate component in GCAM.
• pyhector: a Python interface to Hector.
• HectorUI: run Hector in a web interface!

## Contributing to Hector

The Hector team welcomes and values community contributions but please see our Contribution Guide.

format: plain


text:
1. ## solve

0=n^2-n

can you show me a step by step process

2. Originally Posted by elfen_lied
0=n^2-n

can you show me a step by step process
Factor,
n(n-1)=0
Whenver the product of two number is zero one of them must be zero (if you want to be cool then say the reals have no zero divisors).
Thus,
n=0
n-1=0
Are the only possible solutions,
Thus,
n=0,1
But you need to check, and indeed they are solutions.


text:
ML4Jets2021

Jul 6 – 8, 2021
Europe/Zurich timezone

SPANet: Generalized Permutationless Set Assignment for Particle Physics using Symmetry Preserving Attention

Jul 6, 2021, 9:40 AM
20m

Speaker

Michael James Fenton (University of California Irvine (US))

Description

One of the most ubiquitous challenges in analyses at the LHC is event reconstruction, whereby heavy resonance particles (such as top quarks, Higgs bosons, or vector bosons) must be reconstructed from the detector signatures left behind by their decay products. This is particularly challenging when all decay products have similar or identical signatures, such as all-jet events. Existing methods typically require the evaluation of every possible permutation of these events to find the "best" assignment. In this work, we present a novel neural network architecture SPANet'', or Symmetry Preserving Attention networks. By casting this problem as a set assignment problem on a variable size set, and embedding our knowledge of the symmetries in the problem into the neural network architecture, we demonstrate that these problems can be solved efficiently even in cases for which existing methods are intractable. We demonstrate the approach using a suite of progressively more complex benchmarks, going from all-hadronic ttbar, via ttH, to 4top final states, and provide an easy to use and flexible software package to design and train networks for arbitrary final states.

Affiliation UCI Postdoc

Primary authors

Alex Shmakov (UCI) Michael James Fenton (University of California Irvine (US)) Pierre Baldi (UCI) Ta-Wei Ho (National Tsing Hua University (TW))

Presentation materials

 Fenton_ML4Jets2021.pdf Michael_Fenton.mp4


text:
lgrind(1)                                                         lgrind(1)

NAME
lgrind - grind nice program listings using LaTeX

SYNOPSIS
lgrind [ -e ] [ -i ] [ - ] [ -n ] [ -h <header> ] [ -d <description
file> ] [ -l<language> ] <name> ...

DESCRIPTION
Lgrind formats program sources in a nice style using (La)TeX(1).
Comments are placed in roman, keywords in bold face, variables in
italics, and strings in typewriter font. Source file line numbers
appear in the right margin (every 10 lines).

Lgrind processes its input file(s) and writes the result to standard
output.  This output can be saved for later editting, inclusion in a
larger document, etc.

The options are:

-e   process a LaTeX file for embedded text.

-i   process for inclusion in a LaTeX document.

-    take input from standard input.

-n   don't boldface keywords.

-h   specifies text to go on the left side of every output page
(default is none).

-d   specifies the language definitions file (default is
/cs/lib/lgrindefs).

-l   specifies the language to use.  Currently known are C (-lc or the
default), C++ (-lc++ or -lCC), PASCAL (-lp), Modula-2 (-lm2),
Fortran (-lf), RATFOR (-lr), Yacc (-ly), PostScript (-lps),
Prolog (-lprolog), MLisp (-lmlisp), Icon (-lI), TeX (-ltex),
Metafont (-lmf), Perl (-lperl), CSH (-lcsh), Bourne Shell (-lsh)
assembler (-lasm), 68000 assembler (-la68), asm68 (-lasm68), VMS
assembler (-lvmsasm), ISP (-lisp), LDL (-lldl), Linda (-llinda),
MODEL (-lm), and Russell (-lrussell).

If neither -e nor -i are specified, a complete LaTeX file is produced.

USAGE
For example, to include a C file named foo.c into your LaTeX document,
first give the command:

lgrind -i -lc foo.c > foo.tex

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lgrind(1)                                                         lgrind(1)

This will generate foo.tex, which will have the pretty-printed version
of foo.c with a lot of LaTeX commands.

Then include lgrind.sty as you include any other style, namely with
the \documentstyle[...,lgrind,...]{...} line at the beginning of your
LaTeX document. Having done this, within the document you can include
foo.tex using one of the following commands:

\lgrindfile{foo.tex}
which will simply include the file at that point of text, and
will draw horizontal lines before and after the listing.

\lagrind[htbp]{foo.tex}{caption}{label}
which will put the listing also within a figure environment,
using the float options, caption and label you gave.

To produce a standalone LaTeX file from, say, a Yacc file:

lgrind bary.y > bary.tex
This uses Piet van Oostrum's fancyheadings.sty to make the

For a more detailed explanation of these commands, refer to
/cs/lib/tex/inputs/lgrind.doc.

EMBEDDED PROGRAMS WITHIN A LaTeX FILE
(From Jerry Leichter's notes.)

Within the text of your LaTeX file, you mark groups of lines as either
text- or display-style program code:

Text style: l l.       The expression      %(      a + 3      %)
produces 10.  prints something like:  "The expression a + 3
produces 10." (with "a + 3" set as a program.)

The same effect can be achieved with inline @'s.  l l.       The
expression @a + 3@ produces 10.

Display style: l l.       The statement      %[      a += 3;      %]
is an example of an incrementing operator.  prints something
like: l l.       The statement            a += 3;      is an example
of an incrementing operator.

Important rules:

% and the following character must be the first two characters on
the line to be recognized.

Put nothing on the line after the % and the key character.  If
you do that, lgrind will provide a default environment that will
produce an \hbox for %( )%, and a \vbox for %[ - %].  If you put

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lgrind(1)                                                         lgrind(1)

stuff on the line, lgrind assumes you want to control the format
completely.  Doing this requires understanding exactly what the
code lgrind produces is doing. (Sometimes I'm not sure I do!)

%) and %] are, if I remember right, simply ignored outside of a
code group, but any extra %( or %[ produces a warning, so a
missing %) or %] is usually caught.

You can insert your own code by using a line starting with %= in the
program text.  Whatever you enter after that is left in the output,
exactly as you typed it.  It will be executed in a strange
environment, so doing anything fancy is very tricky.  A macro, \Line,
is provided to help you do simple things.  For example, tab (/); l.
%[ %=\Line{________\vdots}
a = 1; %] produces: tab (/); l.
.
.
.
a = 1;

(Within the program text, _ is active and expands to a fixed-width
space.  A whole bunch of macros are also defined.  If you understand
how lgrind sets lines up, you can replace the 8 _'s with a call to
\Tab - but I'll let you hang yourself on that one.)

The output of lgrind always contains exactly one output line for each
input line.  Hence, you can look up line numbers in TeX error messages
in your original file, rather than in the lgrind'ed (lground?) file.
(Of course, if the problem is in the lgrind output....)

Many things are controllable by re-defining various macros.  You can
change what fonts lgrind will use for various kinds of things, how
much it indents the output, whether it adds line numbers, and if so at
what interval it prints them and whether it sticks them on the left or
right, and so on.  This stuff is all described in lgrind.doc, though
probably not very well. The default settings produce output that looks
reasonable to me, though I can't say I'm ecstatic about it.  Doing a
really good job would require defining some special fonts.

FILES
/cs/bin/lgrind
Executable

/cs/lib/tex/inputs/lgrind.doc
Documented LaTeX style file

/cs/lib/tex/inputs/lgrind.sty
LaTeX style file

/cs/lib/lgrindefs
Language descriptions

- 3 -          Formatted:  June 30, 2022

lgrind(1)                                                         lgrind(1)

AUTHOR
Van Jacobson, Lawrence Berkeley Laboratory (based on "vgrind" by Dave
Presotto & William Joy of UC Berkeley), wrote it for TeX.

Jerry Leichter of Yale University modified it for LaTeX.

George V. Reilly of Brown University changed the name to lgrind, fixed
within-LaTeX features.

latex(1), tex(1), vgrind(1), lgrindefs(5)

- 4 -          Formatted:  June 30, 2022


text:
# Question #81fe9

Dec 30, 2016

(see below)

#### Explanation:

Consider the relationship between decreasing powers of any number.

If we let $n$ be such a number then we see that (for example):
$\textcolor{w h i t e}{\text{XXX}} {n}^{4} = {n}^{5} \div n$

$\textcolor{w h i t e}{\text{XXX}} {n}^{3} = {n}^{4} \div n$

$\textcolor{w h i t e}{\text{XXX}} {n}^{2} = {n}^{3} \div n$

$\textcolor{w h i t e}{\text{XXX}} {n}^{1} = {n}^{2} \div n$

if we continue...
$\textcolor{w h i t e}{\text{XXX}} {n}^{0} = {n}^{1} \div n$ (n.b. this is just $n \div n = 1$)
and
$\textcolor{w h i t e}{\text{XXX}} {n}^{- 1} = {n}^{0} \div n$ (that is $1 \div n = \frac{1}{n}$; the reciprocal of $n$)


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# Taylor–Green vortex

In fluid dynamics, the Taylor–Green vortex is an unsteady flow of a decaying vortex, which has an exact closed form solution of the incompressible Navier–Stokes equations in Cartesian coordinates. It is named after the British physicist and mathematician Geoffrey Ingram Taylor and his collaborator A. E. Green.[1]

Vector plot of the Taylor-Green Vortex

## Original work

In the original work of Taylor and Green,[1] a particular flow is analyzed in three spatial dimensions, with the three velocity components ${\displaystyle \mathbf {v} =(u,v,w)}$ at time ${\displaystyle t=0}$ specified by

${\displaystyle u=A\cos ax\sin by\sin cz,}$
${\displaystyle v=B\sin ax\cos by\sin cz,}$
${\displaystyle w=C\sin ax\sin by\cos cz.}$

The continuity equation ${\displaystyle \nabla \cdot \mathbf {v} =0}$ determines that ${\displaystyle Aa+Bb+Cc=0}$. The small time behavior of the flow is then found through simplification of the incompressible Navier–Stokes equations using the initial flow to give a step-by-step solution as time progresses.

An exact solution in two spatial dimensions is known, and is presented below.

## Incompressible Navier–Stokes equations

The incompressible Navier–Stokes equations in the absence of body force, and in two spatial dimensions, are given by

${\displaystyle {\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}=0,}$
${\displaystyle {\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}=-{\frac {1}{\rho }}{\frac {\partial p}{\partial x}}+\nu \left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}\right),}$
${\displaystyle {\frac {\partial v}{\partial t}}+u{\frac {\partial v}{\partial x}}+v{\frac {\partial v}{\partial y}}=-{\frac {1}{\rho }}{\frac {\partial p}{\partial y}}+\nu \left({\frac {\partial ^{2}v}{\partial x^{2}}}+{\frac {\partial ^{2}v}{\partial y^{2}}}\right).}$

The first of the above equation represents the continuity equation and the other two represent the momentum equations.

## Taylor–Green vortex solution

In the domain ${\displaystyle 0\leq x,y\leq 2\pi }$, the solution is given by

${\displaystyle u=\cos x\sin y\,F(t),\qquad \qquad v=-\sin x\cos y\,F(t),}$

where ${\displaystyle F(t)=e^{-2\nu t}}$, ${\displaystyle \nu }$ being the kinematic viscosity of the fluid. Following the analysis of Taylor and Green[1] for the two-dimensional situation, and for ${\displaystyle A=a=b=1}$, gives agreement with this exact solution, if the exponential is expanded as a Taylor series, i.e. ${\displaystyle F(t)=1-2\nu t+O(t^{2})}$.

The pressure field ${\displaystyle p}$ can be obtained by substituting the velocity solution in the momentum equations and is given by

${\displaystyle p=-{\frac {\rho }{4}}\left(\cos 2x+\cos 2y\right)F^{2}(t).}$

The stream function of the Taylor–Green vortex solution, i.e. which satisfies ${\displaystyle \mathbf {v} =\nabla \times {\boldsymbol {\psi }}}$ for flow velocity ${\displaystyle \mathbf {v} }$, is

${\displaystyle {\boldsymbol {\psi }}=-\cos x\cos yF(t)\,{\hat {\mathbf {z} }}.}$

Similarly, the vorticity, which satisfies ${\displaystyle {\boldsymbol {\mathbf {\omega } }}=\nabla \times \mathbf {v} }$, is given by

${\displaystyle {\boldsymbol {\mathbf {\omega } }}=-2\cos x\cos y\,F(t){\hat {\mathbf {z} }}.}$

The Taylor–Green vortex solution may be used for testing and validation of temporal accuracy of Navier–Stokes algorithms.[2][3]

## References

1. ^ a b c Taylor, G. I. and Green, A. E., Mechanism of the Production of Small Eddies from Large Ones, Proc. R. Soc. Lond. A, 158, 499–521 (1937).
2. ^ Chorin, A. J., Numerical solution of the Navier–Stokes equations, Math. Comp., 22, 745–762 (1968).
3. ^ Kim, J. and Moin, P., Application of a fractional-step method to incompressible Navier–Stokes equations, J. Comput. Phys., 59, 308–323 (1985).


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# Sliding wear of cadmium against alumina

Singh, Arvind R and Biswas, SK and Kailas, Satish V (1999) Sliding wear of cadmium against alumina. In: Wear, 225-22 (2). pp. 770-776.

 PDF sliding_wear.pdf Restricted to Registered users only Download (3178Kb) | Request a copy

## Abstract

As cast cadmium rods were slid against alumina at a load of 50 N and sliding speeds from $0.1 \hspace{2mm}ms^{-1}$ to $4 \hspace{2mm} ms^{-1}$. The wear characteristics of cadmium were related to its strain rate response under uniaxial compression between strain rates of $0.001 \hspace{2mm} s^{-1}$ to $100 \hspace{2mm} s^{-1}$ and temperatures in the range of 323 K to 573 K. In uniaxial compression, cadmium undergoes adiabatic shear banding at strain rates $>10 \hspace{2mm} s^{-1}$ and temperature up to 423 K. The intensity of this adiabatic shear banding reduces as the temperature is increased or the strain rate is reduced. This adiabatic shear banding is the only major microstructural response, in the present test region, that promotes cracking. At temperatures greater than 473 K the flow becomes homogeneous and no adiabatic shear banding is observed. The strain rates and temperatures existing during sliding are estimated using a simple non-iterative procedure and superimposed on the strain rate response map. At sliding speeds $<1.0 \hspace{2mm} ms^{-1}$ the near surface regions are likely to undergo adiabatic shear banding induced cracking. As the velocity of sliding increases the subsurface temperature increases making the regions close to the surface move towards regions of lower adiabatic shear banding intensity and finally to regions of homogeneous flow. It is seen that wear rates when the near surface strain rate response is expected to be adiabatic shear banding, which promotes cracking, is higher than when the near surface regions are in the homogeneous flow region. This indicates that when the strain rate response of a material promotes cracking at near surface regions the wear rates are higher. Thus, it can be said that the strain rate response of a material plays an important role in deciding the wear mechanism and wear rate of a material.

Item Type: Journal Article Copyright of this article belongs to Elsevier. Division of Mechanical Sciences > Mechanical Engineering 28 May 2007 19 Sep 2010 04:36 http://eprints.iisc.ernet.in/id/eprint/10277


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# Function from boundary of the unit disc to the complex plane is holomorphic

Let us consider the unit disc $\mathcal{D}=\{z: |z|\leq 1\}$ and an arbitrary continuous function $f:\partial \mathcal{D} \rightarrow \mathbb{C}$.

I want to find an extension of $f$ that is continuous on $\mathcal{D}$ and holomorphic on $int(\mathcal{D})$, by applying the Cauchy-Riemann integral formula: $f(z_0)=\int_{\partial \mathcal{D}}\frac{f(z)}{z-z_0}dz$.

Moreover, by Morera's theorem, a continuous, complex-valued function ƒ defined on an open set D in the complex plane that satisfies $\int_{\gamma}f(z)dz=0$ for every closed piecewise $\mathcal{C^1}$ curve $\gamma$ in $\mathcal{D}$ must be holomorphic on $\mathcal{D}$.

How can I apply the Morera's theorem to find out whether there is an extension of $f$ that is holomorphic on $int(\mathcal{D})$?

• You have given a formula for $f$, so what then is the question? – Lord Shark the Unknown Oct 13 '17 at 21:33
• I would like to understand Morera's theorem. Does it suffice to show that the integral on the unit disc is zero? – FunnyBuzer Oct 13 '17 at 21:39
• You want to show that the integral of $f$ over any nice curve inside $D$ is zero, that is $$\int_\gamma\int_{\partial D}\frac{f(w)}{w-z}\,dw\,dz.$$ Faced with this integral, I'd try changing the order of integration. – Lord Shark the Unknown Oct 13 '17 at 21:42
• If there is a holomorphic extension of $f$ to the unit disk, then by Cauchy's integral theorem, one must have $$\int_{\partial \mathcal{D}} z^k f(z)\,dz = 0\tag{1}$$ for every integer $k \geqslant 0$. This condition is also sufficient, if $f\in C(\partial \mathcal{D})$ satisfies $(1)$ (for all $k \geqslant 0$), then there is an $F \in C(\mathcal{D})$, holomorphic in the interior, with $F\lvert_{\partial \mathcal{D}} = f$. – Daniel Fischer Oct 13 '17 at 21:57

See what happens if $f(z)=\frac1z$ when $|z|=1$. Then, if $0<|z_0|<1$,\begin{align}\int_{\partial D}\frac{f(z)}{z-z_0}\mathrm dz&=\int_{\partial D}\frac1{z_0}\left(\frac1{z-z_0}-\frac1z\right)\,\mathrm dz\\&=\frac{2\pi i}{z_0}\bigl(\operatorname{Ind}_{\partial D}(z_0)-\operatorname{Ind}_{\partial D}(0)\bigr)\\&=0,\end{align}and it is easy to see that $\int_{\partial D}\frac{f(z)}z\mathrm dz=0$ too. So, you get the null function in $\operatorname{Int}D$.
• In your example, you chose a function that allows an extension that is continuous on $\mathcal{D}$ and holomorphic on $int(\mathcal{D})$. Suppose $\mathcal{Im}(z)=0$. Then, computing the integral on the unit disc, one gets that the integral is $\pi i \neq 0$. Isn't it a contradiction with the fact that $f$ is holomorphic on $\mathbb{C}$? – FunnyBuzer Oct 14 '17 at 17:26
• @FunnyBuzer Are you saying that there is a continuous function $F\colon D\longrightarrow\mathbb C$ which is continuous on $\mathring{D}$ and such that $|z|=1\implies f(z)=\frac1z$? – José Carlos Santos Oct 14 '17 at 17:37
• Not exactly. Rather consider a function on the disc of radius $1-\varepsilon$ (which is still a compact set) and $f(z)=\mathcal{Re}(z)$. – FunnyBuzer Oct 14 '17 at 17:55


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Convert Pmids To Nihmsid And Pmcid Using Eutils
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Entering edit mode
9.2 years ago
Smandape ▴ 110

I would like to know if I can convert PMID to NIHMSID, if there is no PMCID associated with that PMID and retrieve both of them if PMCID exists. I want to do this programmatically may be using eutils if possible. I know about this link http://www.ncbi.nlm.nih.gov/pmc/pmctopmid/ that will do the work but its not programmatically. I also tried the services by PMC http://www.ncbi.nlm.nih.gov/pmc/tools/oai/#examples but not much of success.

So my questions, again, is I have PMIDs and want to find it there exists a NIHMSID and PMCID for it and retrieve them. If PMCID has not yet been assigned then there should be a NIHMSID, retrieve it. For example I have the following two ids:

21707345
23482678

Has anyone tried anything similar to this before?

eutils webservice • 4.2k views
1
Entering edit mode
9.2 years ago
Neilfws 49k

If no web service exists for this purpose, the next best solution is to interact programmatically with the PMCID/PMID/NIHMSID Converter web page. Of course, the downside is that your code breaks when the web page HTML is altered.

Several languages (Perl, Python, Ruby) provide a library named Mechanize for this purpose.

Here is some example code using the Ruby Mechanize library.

require 'mechanize'

agent = Mechanize.new
page  = agent.get("http://www.ncbi.nlm.nih.gov/pmc/pmctopmid/")
form  = page.forms[1]  # the second form on the page

# now fill in the form
form.radiobutton_with(:value => "from_pmid").check  # from PMID
form["PAFAppLayout.AppController.Page.PMCToPmidC.MainPortlet.Ids"] = "21707345\n23482678"  # the PMID(s)
form.checkbox_with(:name => "PAFAppLayout.AppController.Page.PMCToPmidC.MainPortlet.ToFile").check  # send to CSV file

# submit
f = agent.submit(form, form.buttons[1])  # second button on the form
f.save "ids.txt"

Result: a file named ids.txt in the directory from where you ran the code.

PMID,PMCID,NIHMSID
21707345,-,NIHMSID331689
23482678,PMC3592971,NIHMSID392341

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Entering edit mode

@Neilfws: Thank you so much for this direction. I didn't know how to move forward with my question. I also tried to put this question in Stackoverflow with possibility of webscraping but, no one replied and I was clueless how to do it with scrappy and other libraries in python. I was trying to get the query link to convert these ids, but not much of success.

Atleast in this direction I can start working. Thanks, again, a ton for giving me this direction.

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Entering edit mode

Happy to help. Someone from NCBI commented on my blog post to say that in fact there is an undocumented web service for this task. I think you will find their comment helpful.

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Entering edit mode

I think perhaps you should look at the Public Access Compliance Monitor for what you are trying to do. You can get more information about it on the NIH Public Access page.

As I commented on Neilfws' blog, we are coming out with a new web service for ID conversions - it should be deployed in a couple of weeks, and when it is released, you'll be able to find a link to it from the pmctopmid page.


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# Solution - Area Under Simple Curves

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Using integration find the area of the region {(x, y) : x2+y2 2ax, y2 ax, x, y  0}.

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# Difference between 'infinite' and 'not finite' [closed]

In a chat on Mathematics.SE about countability of infinite sets, someone responded that regarding the 'counting', infinite in mathematics does not mean uncountable, but means 'not finite'.

Can someone explain the difference between these concepts?

 Do Not Respond to The Clarification Below (It will elicit a discussion which I do not want)

Started of chat on math.se with this:

Infinity by definition means uncountable, but Cantor jumps & says he's counted it and the answer is aleph-something.

then I was told:

infinity means not finite countable means can be indexed (so, counted) by integers

and:

Well, disregarding the (philosophical?) question of whether something infinite can be counted or not, countable is a term with a specific meaning.

I then later responded with:

I'll try to be clearer: the context in which something is applied, fully defines that something for that context, and is valid only in that specific context. (it's a coffee-mug if you drink coffee from it; it's a paperweight if it prevents papers from being blown away; a paperweight is not a mug, they cannot be equated) ==> it's integers if it is used with respect to the set of integers, it's counters if it is used to count something, a counter is not an integer, integers just happen to be used (like the mug just happen to be used as a paperweight). My problem is that I cannot see how the integers-as-counters can be given all the attributes of the integers-as-a-set, they are fully different contexts. All this stem from having the cardinality of the set of integers is the same as that of the set of rational numbers: the sameness deriving from using counters and saying that the counters and the set of integers are the same thing, which it is not.

which got the following reponse:

the integers are the prototypical and canonical example of an index set for counting things. calling elements of other index sets "counters" is misleading (e.g. you can index a collection of objects with an uncountable set). you seem to want the word "count" to be more general than it actually is. the thing in set theory that allows us to compare sizes of sets is the idea of bijections.

after which I gave up on math.se...

"An infinite set is defined as countable if it is in one to one correspodence with the natural numbers, N={1,2,3,...,n,...}. An infinite set X is uncountable if there exists no bijective map between X and the natural numbers N."

• -1 Isn't that a question about definitions of mathematical terms? Could you expand on why you are posting this on Philo.SE instead of Math.SE? – DBK Mar 28 '12 at 8:41
• However, a quick off-topic answer to your question goes as follows: There is no difference between "infinite" and "not finite". However, there are infinite sets with different cardinality. The "smallest" infinite set is the set of natural numbers , which is countable. All infinite sets with greater cardinality are uncountable. See en.wikipedia.org/wiki/Infinite_set. – DBK Mar 28 '12 at 8:53
• Questions are expected to generally relate to philosophy, within the scope defined in the faq. – stoicfury Mar 28 '12 at 17:10

So in mathematics we measure the cardinality of a set with bijective functions/maps or one to one correspondences.

For example suppose you know that there are 100 seats in some movie theatre. When the movie starts, suppose it is a hit movie and fills up. In other words, there is a person for every seat in the theatre. Without counting the number of people, we can deduce that there are 100 people in the theatre. This is an example of a one to one correspondence (also known as a bijective function or map) between people and seats in the theatre, i.e. the cardinality of the people is the same as the cardinality of seats because for every seat there is one person sitting in it, and for every person there is one seat that they are sitting on.

There are two types of sets, countable and uncountable sets. Countable sets can either be finite or infinite, but uncountable sets are always infinite just a 'larger' infinite.

More precisely, A set X is finite if there is a bijection between the set X and the finite whole numbers, N_n={1,2,3,...,n}. If X is not finite, then X is infinite (they mean the same thing). Now concerning infinite sets, there are two types, countable and uncountable (here is the difference you seek). An infinite set is defined as countable if it is in one to one correspodence with the natural numbers, N={1,2,3,...,n,...}. An infinite set X is uncountable if there exists no bijective map between X and the natural numbers N. Note: Finite sets are also countable.

I think examples will be helpful here:

The set A={1,2,3,4,5} is finite and countable.

The set of integers is considered *infinite *and countable.

The set of real numbers (rational numbers and irrational numbers) is infinite and uncountable.

You can, informally, think of a countable set as a set where you are able to potentially list all of the elements of the set, and think of an uncountable set as saying there is no list that contains all the elements of the set. Naively, we can see that the real numbers are uncountable, because between any two real numbers there is another real number. Whereas there is no integer between the numbers 1 and 2.

Hope this helps!

http://gowers.wordpress.com/2011/11/28/a-short-post-on-countability-and-uncountability/

http://en.wikipedia.org/wiki/Cardinality

• If interested, there is a wealth of resources on these topics. I can recommend books if you like, but most books on set theory or analysis will work. – Alborz Yarahmadi Mar 28 '12 at 9:46
• This part helped: "An infinite set is defined as countable if it is in one to one correspodence with the natural numbers, N={1,2,3,...,n,...}. An infinite set X is uncountable if there exists no bijective map between X and the natural numbers N." Thanks – slashmais Mar 28 '12 at 10:25
• Giving the definition of a finite set as a set from which every injective function to itself is also surjective is also an option, especially if we want to avoid axiom of choice. – Konformist Liberal Jan 2 '19 at 11:47

`infinite` and `not finite` are per se, regarding the word, the very same thing.

Regarding mathematics: There are several possible kinds of infinite sets, such as countable and uncountable sets, how ever both are infinite (and not finite, which is the same).

You'd better restate your question on Math.SE as they can help you much better. Also have a look in an encyclopedia for “countable” and “uncountable” (or “countable set” and “uncountable set”).

• I added a clarification to my question – slashmais Mar 28 '12 at 10:22


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