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Lactate is not only an endpoint of glycolysis but is gradually being discovered to play the role of a universal metabolic fuel for energy via the ‘lactate shuttle’ moving between cells and transmitting signals. The glycolytic‐dependent metabolism found in tumours and fast‐growing cells has made lactate a pivotal player in energy metabolism reprogramming, which enables cells to obtain abundant energy in a short time. Moreover, lactate can provide favourable conditions for tumorigenesis by shaping the acidic tumour microenvironment, recruiting immune cells, etc. and the recently discovered lactate‐induced lactylation moves even further on pro‐tumorigenesis mechanisms of lactate production, circulation and utilization. As with other epigenetic modifications, lactylation can modify histone proteins to alter the spatial configuration of chromatin, affect DNA accessibility and regulate the expression of corresponding genes. What's more, the degree of lactylation is inseparable from the spatialized lactate concentration, which builds a bridge between epigenetics and metabolic reprogramming. Here, we review the important role of lactate in energy reprogramming, summarize the latest finding of lactylation in tumorigenesis and try to explore therapeutic strategies in oncotherapy that can kill two birds with one stone.

Pancreatic ductal adenocarcinoma (PDAC) is one of the most lethal cancers, with a minimal difference between its incidence rate and mortality rate. Advances in oncology over the past several decades have dramatically improved the overall survival of patients with multiple cancers due to the implementation of new techniques in early diagnosis, therapeutic drugs, and personalized therapy. However, pancreatic cancers remain recalcitrant, with a 5-year relative survival rate of <9%. The lack of measures for early diagnosis, strong resistance to chemotherapy, ineffective adjuvant chemotherapy and the unavailability of molecularly targeted therapy are responsible for the high mortality rate of this notorious disease. Genetically, PDAC progresses as a complex result of the activation of oncogenes and inactivation of tumor suppressors. Although next-generation sequencing has identified numerous new genetic alterations, their clinical implications remain unknown. Classically, oncogenic mutations in genes such as KRAS and loss-of-function mutations in tumor suppressors, such as TP53 , CDNK2A , DPC4 / SMAD4, and BRCA2 , are frequently observed in PDAC. Currently, research on these key driver genes is still the main focus. Therefore, studies assessing the functions of these genes and their potential clinical implications are of paramount importance. In this review, we summarize the biological function of key driver genes and pharmaceutical targets in PDAC. In addition, we conclude the results of molecularly targeted therapies in clinical trials and discuss how to utilize these genetic alterations in further clinical practice.

Background The insect gut microbiota has been shown to contribute to the host’s digestion, detoxification, development, pathogen resistance, and physiology. However, there is poor information about the ranking of these roles. Most of these results were obtained with cultivable bacteria, whereas the bacterial physiology may be different between free-living and midgut-colonizing bacteria. In this study, we provided both proteomic and genomic evidence on the ranking of the roles of gut bacteria by investigating the anal droplets from a weevil, Cryptorhynchus lapathi . Results The gut lumen and the anal droplets showed qualitatively and quantitatively different subsets of bacterial communities. The results of 16S rRNA sequencing showed that the gut lumen is dominated by Proteobacteria and Bacteroidetes, whereas the anal droplets are dominated by Proteobacteria. From the anal droplets, enzymes involved in 31 basic roles that belong to 7 super roles were identified by Q-TOF MS. The cooperation between the weevil and its gut bacteria was determined by reconstructing community pathway maps, which are defined in this study. A score was used to rank the gut bacterial roles. The results from the proteomic data indicate that the most dominant role of gut bacteria is amino acid biosynthesis, followed by protein digestion, energy metabolism, vitamin biosynthesis, lipid digestion, plant secondary metabolite (PSM) degradation, and carbohydrate digestion, while the order from the genomic data is amino acid biosynthesis, vitamin biosynthesis, lipid digestion, energy metabolism, protein digestion, PSM degradation, and carbohydrate digestion. The PCA results showed that the gut bacteria form functional groups from the point of view of either the basic role or super role, and the MFA results showed that there are functional variations among gut bacteria. In addition, the variations between the proteomic and genomic data, analyzed with the HMFA method from the point of view of either the bacterial community or individual bacterial species, are presented. Conclusion The most dominant role of gut bacteria is essential nutrient provisioning, followed by digestion and detoxification. The weevil plays a pioneering role in diet digestion and mainly digests macromolecules into smaller molecules which are then mainly digested by gut bacteria.

In this paper, by means of the Faà di Bruno formula, with the help of explicit formulas for partial Bell polynomials at specific arguments of two specific sequences generated by derivatives at the origin of the inverse sine and inverse cosine functions, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes power series expansions for real powers of the inverse cosine (sine) functions and the inverse hyperbolic cosine (sine) functions. By comparing different series expansions for the square of the inverse cosine function and for the positive integer power of the inverse sine function, the author not only finds infinite series representations of the circular constant π and its real powers, but also derives several combinatorial identities involving central binomial coefficients and the Stirling numbers of the first kind.

A bstract In the geometric approach to defining complexity, operator complexity is defined as the distance in the operator space. In this paper, based on the analogy with the circuit complexity, the operator size is adopted as the metric of the operator space where the path length is the complexity. The typical sectional curvatures of this complexity geometry are positive. It is further proved that the typical sectional curvatures are always positive if the metric is an arbitrary function of operator size, while complexity geometry is usually expected to be defined on negatively curved manifolds. By analyzing the sectional curvatures distribution for the N -qubit system, it is shown that surfaces generated by Hamiltonians of size smaller than the typical size can have negative curvatures. In the large N limit, the form of complexity metric is uniquely constrained up to constant corrections if we require sectional curvatures are of order 1 /N 2 . With the knowledge of states, the operator size should be modified due to the redundant action of operators, and thus is generalized to be state-dependent. Then we use this state-dependent operator size as the metric of the Hilbert space to define state complexity. It can also be shown that in the Hilbert space, 2-surfaces generated by operators of size much smaller than the typical size acting on typical states also have negative curvatures.

Macrophage polarization is central to immune homeostasis and disease pathogenesis. It is achieved through complex regulatory networks mediated by post-translational modifications (PTMs). This review synthesizes the roles of phosphorylation, ubiquitination, methylation, acetylation, and lactylation in shaping polarization outcomes through signal-responsive and metabolism-sensitive molecular networks. We integrate evidence that PTMs form interconnected circuits between signaling, epigenetic, and metabolic pathways, thereby enabling sophisticated immune interpretation. Therapeutically, we demonstrate that targeting PTM hubs rather than isolated pathways has transformative potential for reprogramming macrophages in cancer, inflammatory disorders, and tissue repair. However, applying these insights to clinical practice will require overcoming key challenges related to targeting specific pathogenic PTMs with precision, delivering them to specific cells, and validating their mechanisms in vivo .

In this work, the author reviews the origination of normalized tails of the Maclaurin power series expansions of infinitely differentiable functions, presents that the ratio between two normalized tails of the Maclaurin power series expansion of the exponential function is decreasing on the positive axis, and proves that the normalized tail of the Maclaurin power series expansion of the exponential function is absolutely monotonic on the whole real axis.

Glucose is the major precursor of lactose, which is synthesized in Golgi vesicles of mammary secretory alveolar epithelial cells during lactation. Glucose is taken up by mammary epithelial cells through a passive, facilitative process, which is driven by the downward glucose concentration gradient across the plasma membrane. This process is mediated by facilitative glucose transporters (GLUTs), of which there are 14 known isoforms. Mammary glands mainly express GLUT1 and GLUT8, and GLUT1 is the predominant isoform with a K m of ~10 mM and transport activity for mannose and galactose in addition to glucose. Mammary glucose transport activity increases dramatically from the virgin state to the lactation state, with a concomitant increase in GLUT expression. The increased GLUT expression during lactogenesis is not stimulated by the accepted lactogenic hormones. New evidence indicates that a possible low oxygen tension resulting from increased metabolic rate and oxygen consumption may play a major role in stimulating glucose uptake and GLUT1 expression in mammary epithelial cells during lactogenesis. In addition to its primary presence on the plasma membrane, GLUT1 is also expressed on the Golgi membrane of mammary epithelial cells and is likely involved in facilitating the uptake of glucose and galactose to the site of lactose synthesis. Because lactose synthesis dictates milk volume, regulation of GLUT expression and trafficking represents potentially fruitful areas for further research in dairy production. In addition, this research will have pathological implications for the treatment of breast cancer because glucose uptake and GLUT expression are up-regulated in breast cancer cells to accommodate the increased glucose need.

This paper presents an extensive investigation into several interrelated topics in mathematical analysis and number theory. The author revisits and builds upon known results regarding the Maclaurin power series expansions for a variety of functions and their normalized remainders, explores connections among central factorial numbers, the Stirling numbers, and specific matrix inverses, and derives several closed-form formulas and inequalities. Additionally, this paper reveals new insights into the properties of these mathematical objects, including logarithmic convexity, explicit expressions for certain quantities, and identities involving the Bell polynomials of the second kind.

Epstein-Barr virus (EBV)-associated epithelial cancers, including nasopharyngeal carcinoma (NPC) and approximately 10% of gastric cancers, termed EBVaGC, represent 80% of all EBV-related malignancies. However, the exact role of EBV in epithelial cancers remains elusive. Here, we report that EBV functions in vasculogenic mimicry (VM). Epithelial cancer cells infected with EBV develop tumor vascular networks that correlate with tumor growth, which is different from endothelial-derived angiogenic vessels and is VEGF-independent. Mechanistically, activation of the PI3K/AKT/mTOR/HIF-1α signaling cascade, which is partly mediated by LMP2A, is responsible for EBV-induced VM formation. Both xenografts and clinical samples of NPC and EBVaGC exhibit VM histologically, which are correlated with AKT and HIF-1α activation. Furthermore, although anti-VEGF monotherapy shows limited effects, potent synergistic antitumor activities are achieved by combination therapy with VEGF and HIF-1α-targeted agents. Our findings suggest that EBV creates plasticity in epithelial cells to express endothelial phenotype and provides a novel EBV-targeted antitumor strategy.