This paper presents two remarkable phenomena associated with the heat equation with a time delay: namely, the propagation of singularities and periodicity. These are manifested through a distinctive mode of propagation of singularities in the solutions. Precisely, the singularities of the solutions propagate periodically in a bidirectional fashion along the time axis. Furthermore, this propagation occurs in a stepwise manner. More specifically, when propagating in the positive time direction, the order of the joint derivatives of the solution increases by 2 for each period; conversely, when propagating in the reverse time direction, the order of the joint derivatives decreases by 2 per period. Additionally, we elucidate the way in which the initial data and historical values impact such a propagation of singularities. The phenomena we have discerned not only corroborate the pronounced differences between heat equations with and without time delay but also vividly illustrate the substantial divergence between the heat equation with a time delay and the wave equation, especially when viewed from the point of view of singularity propagation.

We develop a structure-preserving computational framework for optimal mixing control in incompressible flows. Our approach exactly conserves the continuous system's key invariants (mass and L^2-energy), while also maintaining discrete state-adjoint duality at every time step. These properties are achieved by integrating a centered finite-volume discretization in space with a time-symmetric Crank-Nicolson integrator for both the forward advection and its adjoint, all inside a gradient-based optimization loop. The result is a numerical solver that is faithful to the continuous optimality conditions and efficiently computes mixing-enhancing controls. In our numerical tests, the optimized time-dependent stirring produces a nearly exponential decay of a chosen mix-norm, achieving orders-of-magnitude faster mixing than any single steady flow. To our knowledge, this work provides the first evidence that enforcing physical structure at the discrete level can lead to both exact conservation and highly effective mixing outcomes in optimal flow design.

This paper studies the sampling observability for the heat equations with memory in the lower-order term, where the observation is conducted at a finite number of time instants and on a small open subset at each time instant. We present a two-sided sampling observability inequality and give a sharp sufficient condition to ensure the aforementioned inequality. We also provide a method to select the time instants and then to design the observation regions, based on a given memory kernel, such that the above-mentioned inequality holds for these time instants and observation regions. Additionally, we demonstrate that the positions of these time instants depend significantly on the memory kernel.

The tumor microenvironment (TME) imposes profound metabolic and functional constraints on immune cells, with mitochondrial dysfunction emerging as a pivotal driver of immunosuppression. While mitochondrial metabolism is well recognized for its role in energy production and cellular homeostasis, its dynamic regulation of immune cell activation, differentiation, and exhaustion within the TME remains underexplored. In this review we summarize insights into how TME stressors such as hypoxia, nutrient competition, and metabolic byproducts subvert mitochondrial dynamics, redox balance, and mitochondrial DNA (mtDNA) signaling in T cells, natural killer (NK) cells, and macrophages, thereby directly impairing their antitumor efficacy. We emphasize that the restoration of mitochondrial fitness in immune cells, achieved by targeting metabolites in the TME and mitochondrial quality control, represents a pivotal axis for adoptive cell therapies (ACTs) and TME reprogramming.

Dysregulated mitophagy is essential for mitochondrial quality control within human cancers. However, identifying hub genes regulating mitophagy and developing mitophagy-based treatments to combat drug resistance remains challenging. Herein, BayeDEM (Bayesian-optimized Deep learning for identifying Essential genes of Mitophagy) was proposed for such a task. After Bayesian optimization, BayeDEM demonstrated its excellent performance in identifying critical genes regulating mitophagy of osteosarcoma (area under curve [AUC] of ROC: 98.96%; AUC of PR curve: 100%). CERS1 was identified as the most essential gene regulating mitophagy (mean (|SHAP value|): 4.14). Inhibition of CERS1 sensitized cisplatin-resistant osteosarcoma cells to cisplatin, restricting their growth, proliferation, invasion, migration and colony formation and inducing apoptosis. Mechanistically, inhibition of CERS1 restricted mitophagy to destroy the mitochondrial quality control in cisplatin-resistant osteosarcoma cells, including mitochondrial membrane potential loss and unfavourable mitochondrial dynamics, rendering them susceptible to cisplatin-induced apoptosis. More importantly, mitophagy facilitated the immunosuppressive microenvironment formation by significantly modulating T-cell differentiation, adhesion and antigen presentation, and mitophagy mainly affects malignant osteoblasts in the early-mid developmental stage. Immunologically, mitophagy potentially modulated the MIF signalling transmission between malignant osteoblasts and B cells, DCs, CD8+ T cells, NK cells and monocytes through the MIF-(CD74 + CXCR4) receptor-ligand interaction, thereby modulating the biological functions of these immune cells. Collectively, BayeDEM emerged as a promising tool for oncologists to identify pivotal genes governing mitophagy, thereby enabling mitophagy-centric therapeutic strategies to counteract drug resistance.

This paper presents a complete analysis of the observability property of heat equations with time-dependent real analytic memory kernels. More precisely, we characterize the geometry of the space-time measurable observation sets ensuring sharp observability inequalities, which are relevant both for control and inverse problems purposes. Despite the abundant literature on the observation of heat-like equations, existing methods do not apply to models involving memory terms. We present a new methodology and observation strategy, relying on the decomposition of the flow, the time-analyticity of solutions and the propagation of singularities. This allows us to obtain a sufficient and necessary geometric condition on the measurable observation sets for sharp two-sided observability inequalities. In addition, some applications to control and relevant open problems are presented.

Several researchers have focused on understanding the pathogenesis and treatment strategies for osteoarthritis (OA). Gastrodin (GAS) is a potential anti-inflammatory agent. In this study, we constructed an in vitro OA chondrocyte model by treating chondrocytes with IL-1β. Next, we determined the expression of aging-related markers and mitochondrial functions in chondrocytes treated with GAS. Further, we constructed a "drug-component-target-pathway-disease" interactive network and determined the effect of GAS on the functions and pathways related to OA. Finally, we constructed the OA rat model by removing the medial meniscus of the right knee and transection of the anterior cruciate ligament. The results revealed that GAS reduced senescence and improved mitochondrial functions in OA chondrocytes. We used network pharmacology and bioinformatics to screen for key molecules Sirt3 and the PI3K-AKT pathway involved in regulating the effect of GAS on OA. Further studies showed an increase in SIRT3 expression and reduced chondrocyte aging, mitochondrial damage, and the phosphorylation of the PI3K-AKT pathway. The results showed that GAS ameliorates pathological changes related to aging, increases SIRT3 expression, and protects the ECM in the OA rat model. These results were consistent with our bioinformatics results and previous studies. In summary, GAS slows down the aging of chondrocytes and mitochondrial damage in OA by regulating the phosphorylation of the PI3K-AKT pathway via SIRT3.

Objective:To explore the molecular mechanism of traditional Chinese medicine monomer curcumin in alleviating osteoarthritis in rats.Methods:Totally, 30 SD rats were randomly divided into control group, arthritis group and curcumin group, with 10 rats in each group by random number table method. Rats in arthritis group and curcumin group were injected with papain to replicate the animal model of osteoarthritis. The animals in curcumin group were injected with 50 mg/kg curcumin through the articular cavity, and those in control group and arthritis group were injected with equal volume of normal saline through the articular cavity. The follow-up study was carried out after 3 weeks of continuous treatment. The maximum range of motion and synovial thickness of the knee joint in the three groups were measured. The content of glycosaminoglycan in the synovial supernatant in the three groups was determined by spectrophotometry. The degree of arthritis was analyzed by Mankin histological score. The levels of inflammatory factors in synovial fluid were analyzed by enzyme linked immunosorbent assay (ELISA). Masson staining was used to analyze the content of collagen fibers in knee joint. One way ANOVA was used for comparison between groups.Results:The maximum knee flexion and extension angle [(136.87±6.42) °] in curcumin group were significantly lower than those in arthritis group [(156.81±13.90) °, t=4.714, P<0.05]. The Mankin score of knee joint in curcumin group (2.36±0.24) was significantly higher than that in arthritis group (1.50±0.29, t=18.051, P<0.05). The glycosaminoglycan level in curcumin group [(501.90±26.92) ng/mg] was significantly higher than that in arthritis group [(445.80±49.22) ng/mg, t=3.162, P<0.05]. Tumor necrosis factor-α (TNF-α) and interleukin (IL)-1 β in knee joint of rats in curcumin group [(143.33±20.58), (41.20±5.98) ng/mg] were significantly lower than those in arthritis group[(255.28±51.48), (101.03±14.21) ng/mg, t=6.386, 12.272, P<0.05]. The expression level of p-p65 protein in curcumin group (1.49±0.13) was significantly lower than that in arthritis group (2.36±0.37, t=7.063, P<0.05). Conclusion:Curcumin can significantly reduce the level of arthritis, promote the synthesis of collagen fibers in damaged joints and alleviate the symptoms of osteoarthritis.

OBJECTIVE:Osteoarthritis (OA) is characterized by cartilage degeneration and inflammation. Procyanidin B2 (PCB2), a natural flavonoid compound, exhibits potential anti-inflammatory and anti-oxidative effects against several diseases. However, its curative effects on OA remain unclear. PURPOSE:Herein, we explored the anti-arthritic effects of PCB2 on OA onset and progress and its potential mechanism. METHODS:CCK-8 assays and EdU staining were used to assess the cytotoxic effects and cell proliferation activity of PCB2. Flow cytometry was used to detect apoptosis in chondrocytes. ELISA, qPCR, and western blotting, were applied to explore the expression of apoptosis and senescence-associated secretion phenotype (SASP) factors. The Nrf2/NF-κB signaling cascade was explored using immunofluorescence and western blotting. Additionally, we silenced the Nrf2 gene using siRNAs to verify its function in PCB2 regulation of senescence and apoptosis phenotypes. Safranin O-Fast Green (SO) and immunohistochemical staining were used to explore the effects of PCB2 on OA model rats. RESULTS:PCB2 dampened interleukin (IL)-1β-triggered expression of SASP factors in vitro. Additionally, PCB2 diminished IL-1β-triggered destruction of the extracellular matrix (ECM) via downregulating the expression of MMPs, while upregulating the expression of collagen II and aggrecan. In addition, PCB2 treatment reduced IL-1β-induced apoptosis of chondrocytes. Mechanistically, PCB2 could attenuated chondrocyte senescence in vitro via the Nrf2/NF-κB pathway. Moreover, PCB2 exhibited anti-apoptotic properties via the Nrf2/BAX/Bcl-2 pathway. PCB2 alleviated knee cartilage degeneration in an OA rat model. CONCLUSIONS:Our results suggest that PCB2 may be used as a therapeutic agent for OA.

We build up a decomposition for the flow generated by the heat equation with a real analytic memory kernel. It consists of three components: The first one is of parabolic nature; the second one gathers the hyperbolic component of the dynamics, with null velocity of propagation; the last one exhibits a finite smoothing effect. This decomposition reveals the hybrid parabolic-hyperbolic nature of the flow and clearly illustrates the significant impact of the memory term on the parabolic behavior of the system in the absence of memory terms.

Background We evaluated skeletal muscle vascular permeability in diabetic rabbits with critical limb ischaemia using quantitative dynamic contrast agent-enhanced (DCE) magnetic resonance imaging (MRI) and explored the feasibility of using DCE-MRI Ktrans-based texture analysis for assessing early slight ischaemia-related skeletal muscle structural changes. Method Twenty-four male New Zealand white rabbits (2.7 ± 0.3 kg; n = 12 each in sham-operated and experimental groups) underwent axial MRI of the vastus lateralis muscle at 1, 2, and 3 weeks after alloxan injection. Between-group and intra-group postoperative permeability and texture parameters were compared. Texture features of experimental groups in the third week were modelled by receiver operating characteristic (ROC) curve analysis. Correlations of permeability and of statistical texture parameters with peripheral blood endothelial progenitor cells (EPCs) and microvascular density (MVD) were analysed. Results In the experimental group, the transfer constant (Ktrans) was statistically significant at all time-points (F = 5.800, P = 0.009). Their vastus lateralis muscle Ktrans was significantly lower in the third than in the first week (P = 0.018) and correlated positively with peripheral blood EPCs in the experimental group [r = 0.598, (95% CI: 0.256, 0.807)]. The rate constant was negatively associated with vastus lateralis muscle MVD [r = -0.410, (95% CI: -0.698, -0.008)]. The area under the ROC curve of texture parameters based on Ktrans in ischaemic limbs was 0.882. Conclusions Quantitative DCE-MRI parameters could evaluate microvascular permeability of ischaemic limb skeletal muscle, and texture analysis based on DCE-MRI Ktrans allowed evaluation of early slight skeletal muscle structural changes.

This paper studies the observability of heat equations with real analytic memory ker-nels. More precisely, we characterize the geometry on observation sets that ensure several observability inequalities, including two–side inequalities. By an observation set, we mean the space–time support of an observer, which is chosen to be measurable. In addition, some applications to control are presented. The main components used to obtain our results are as follows: 1) the decomposition theorem given in [35], 2) the time–analyticity for solutions of our equations, 3) a modified three–step strategy, and 4) an estimate for solutions of the pure heat equation.

Objective:To investigate the effect of hypoxia preconditioning on the cell function of bone marrow mesenchymal stem cells (BMSCs) and the preventive and therapeutic effects of hypoxia preconditioning BMSCs in steroid-induced femoral head necrosis (SONFH) in rats after intravenous transplantation.Methods:BMSCs were isolated from the bone marrow cavity of the femur of rats. Flow cytometry identification of surface antigens (CD34, CD44, CD45 and CD90). Evaluation of cell function of BMSCs in normoxia group (No) and hypoxia group (Hp) by cell counting kit-8 (CCK-8) analysis, scratch test, alizarin red staining and alkaline phosphatase staining. A total of 32 SD rats were randomly divided into four groups ( n=8): (a) a control group, (b) a model group, (c) a normoxic group, and(d)a hypoxic group. Model group used lipopolysaccharide [2 mg/(kg·d)] combined with methylprednisolone [20 mg/(kg·d)] to construct the SONFH model. After SONFH model was constructed, the normoxic-rat bone marrow mesenchymal stem cells (rBMSCs) group and hypoxic-rBMSCs group were injected with 10 7 normoxic or hypoxia preconditioned rBMSCs through the tail vein. All rats were sacrificed and the femoral heads were collected and evaluated by Micro CT. After femoral heads decalcification, the slices were stained with hematoxylin-eosin to observe and count the incidence of osteonecrosis. Results:The results of flow cytometric identification of rBMSCs showed that rBMSCs expressed high expression of CD44 (100%) and CD90 (99.4%), and low expression of CD34 (0.48%) and CD45 (0.39%). The CCK-8 analysis showed that hypoxia preconditioning can enhance the proliferation ability of rBMSCs ( t=7.199, P<0.05). The scratch test analysis showed that compared with No group, Hp group enhanced rBMSCs′ migration ability at 12 h ( t=2.462, P<0.05), 24 h ( t=2.471, P<0.05) and 36 h ( t=3.434, P<0.01). The alizarin red staining showed hypoxia precondition significantly enhanced the mineralization ability of rBMSCs in vitro ( t=6.722, P<0.05). The alkaline phosphatase (ALP) staining showed that hypoxia pretreatment can significantly enhance the osteogenic ability of rBMSCs in vitro ( t=13.37, P<0.05). In animal experiments, the incidence of osteonecrosis in group C, group M, group N, and group H were 0% (0/16), 87.5% (14/16), 25.0% (4/16), 12.5% (2/16). The hypoxia precondition of rBMSCs can reduce the incidence of SONFH. The empty bone lacuna rates of HE staining in group C, M, N, and H were (2.765±0.423)%, (30.817±1.203)%, (14.724±1.312)% and (8.487±1.671)%, indicated the incidence of empty bone lacuna in the rBMSCs treatment groups were significantly reduced ( F=286.3, P<0.05); The Micro CT scan measurement showed that BV/TV, Tb. Th and Tb. N were significantly increased after rBMSCs treatment ( F=197.6, 290.7, 68.7, P<0.05), while Tb. Sp was significantly reduced ( F=123.8, P<0.05). It is proved that intravenous transplantation of rBMSCs can improve the microstructure of the femoral head, and the improvement effect of rBMSCs after hypoxia preconditioning is more significant ( P<0.01). Conclusion:Hypoxic precondition can significantly improve the proliferation, migration and osteogenic ability of rBMSCs in vitro, which is beneficial to enhance the preventive and therapeutic effects of rBMSCs on SONFH in rats.

We study the controlled heat equations with analytic memory from two perspectives: reachable subspaces and control regions. Due to the hybrid parabolic-hyperbolic phenomenon of the equations, the support of a control needs to move in time to efficiently control the dynamics.We show that under a sharp sufficient geometric condition imposed to the control regions, the difference between reachable subspaces of the controlled heat equations, with and without memory, is precisely given by a Sobolev space. The appearance of this Sobolev space is attributed to the memory which makes the equation having the wave-like nature. The main ingredients in the proofs of our main results are as: first, the decomposition of the flow (generated by the equation with the null control) given in [31], second, an observability inequality built up in this paper.

Objective:To observe the effects of β-elemonic acid (β-EA) on proliferation, migration and apoptosis of rat chondrocytes induced by interleukin-1β (IL-1β).Methods:Chondrocytes of primary rats were extracted and identified by toluidine blue staining, and divided into control group, IL-1β group, low and high dose β-EA group by random number table. Cell counting kit-8 (CCK-8) assay was used to detect the cell viability. The proliferation of Chondrocytes was detected by 5-Ethynyl-2′-deoxyuridine (EdU) assay. Transwell experiment was used to evaluate the cell migration. The apoptosis of chondrocytes was detected by flow cytometry. Western blotting was used to detect the protein levels of apoptosisrelated genes cysteinyl aspartate-specific protease (Caspase)-3, Caspase-9, B cell lymphoma/leukemia-2 associated X protein (bax) and B cell lymphoma/leukemia-2 (bcl-2) in chondrocytes. One-way analysis of variance was used for data processing.Results:After treating normal chondrocytes with different concentrations of β-EA, the cell viability in control group, β-EA low-dose group and β-EA high-dose group was 97%, 97.36% and 95.80% respectively. The viability of chondrocytes in control group, IL-1β group, β-EA low-dose group and β-EA high-dose group was 97.22%, 49.57%, 60.20% and 79.95% respectively. The percentage of proliferating cells was 28.38%, 9.16%, 13.62% and 24.56% respectively. The number of migrating cells was 354.67, 86.67, 136.67 and 192.67 respectively. The apoptosis rates were 12.18%, 36.74%, 24.41% and 16.98% respectively. These results showed that β-EA had no significant toxicity to normal chondrocytes ( F=0.438, P>0.05) and promoted the recovery of inhibited chondrocytes induced by IL-1β ( F=379.200, P<0.05). As compared with the control group, the number of proliferating and migrating cells in IL-1β group was significantly decreased ( F=84.920, 117.100, P< 0.05). As compared with IL-1β group, the proliferation and migration of cells in low-dose and high-dose β-EA groups were significantly increased ( F=84.920, 117.100, P<0.05). IL-1β significantly induced the apoptosis of rat chondrocytes ( F=169.600, P<0.05). After treatment with high and low doses of β-EA, the apoptosis of rat chondrocytes induced by IL-1β was significantly inhibited ( F=169.600, P<0.05). The expression of apoptosis-related proteins including Caspase-3, Caspase-9 and bax was significantly decreased ( F=111.600, 50.830, 132.500, P<0.05), and that of bcl-2 was significantly increased in the β-EA treated group ( F=59.850, P<0.05). Conclusion:β-EA can inhibit IL-1β-mediated chondrocytes apoptosis, promotes chondrocytes proliferation and migration and protects chondrocytes to some extent.

骨关节炎是骨科常见的关节疾病,随着人口老龄化加重其患病率也在增加.骨关节炎发病受多因素的影响,包括氧化应激和活性氧的过量产生.这些因素调节细胞内信号传导过程,软骨细胞衰老和细胞凋亡,细胞外基质合成和降解以及滑膜炎症和软骨下骨功能障碍.由于治疗骨关节炎的药物相对较少,本文针对复杂的氧化应激信号通路为治疗骨关节炎提供策略.

In this paper, we present several observability and unique continuation inequalities for the free Schrödinger equation in the whole space. The observations in these inequalities are made either at two points in time or one point in time. These inequalities correspond to different kinds of controllability for the free Schrödinger equation. We also find that the observability inequality at two points in time is equivalent to the uncertainty principle built up in [18].

BACKGROUND:The cDNA Library of venomous animals could provide abundant bioactive peptides coding information and is an important resource for screening bioactive peptides that target and regulate disease-related ion channels. To further explore the potential medicinal usage of the transcriptome database of Scorpiops Pocoki's venom gland, this research identified the function of a new potassium channel toxin Ktx-Sp2, whose gene was screened from the database by sequence alignment.RESULTS:The mature peptide of Ktx-Sp2 was obtained by genetic engineering. Whole-cell patch-clamp experiment showed that Ktx-Sp2 peptide could effectively block three types of exogenous voltage-gated potassium channels-Kv1.1, Kv1.2 and Kv1.3, among which, the blocking activity for Kv1.3 was relatively high, showing selectivity to some extent. Taking Jurkat T cells as the cell model, this study found that Ktx-Sp2 peptide could also effectively block endogenous Kv1.3, significantly reduce the free calcium concentration in Jurkat T cells, inhibit the activation of Jurkat T cells and reduce the release of inflammatory cytokines IL-2, showing a strong immunosuppressant effect.CONCLUSIONS:This study further proves that the transcriptome database of the Scorpiops Pocoki venom gland is an important resource for discovery of novel bioactive polypeptide coding genes. The newly screened Kv1.3 channel blocker Ktx-Sp2 expanded the range of leading compounds for the treatment of autoimmune diseases and promoted the development and application of scorpion toxin peptides in the field of biomedicine.

软骨细胞能量代谢方式主要表现为通过合成足量的一氧化氮以及对氧的低摄入量,在细胞内缺氧诱导因子1α稳定表达的同时可以不受体外高氧环境(氧浓度高于21％)的抑制,继而产生“瓦氏效应”.同时,在体外高浓度的葡萄糖环境下培养软骨细胞时会产生与“瓦氏效应”表现相近的Crabtree效应.两者均表现为在有氧条件下,细胞内以糖酵解代谢为主而有氧呼吸被抑制,从而生成乳酸.唯一的区别在于Crabtree效应是由于高浓度的葡萄糖导致的.之前一些学者的研究未考虑高浓度葡萄糖这一因素.因此,在以后的研究中,外源性葡萄糖浓度将是一个需要更精确设计的参数,以尽可能排除Crabtree效应的干扰.同样骨关节炎患者合并糖尿病是否存在Crabtree效应及其相互之间的关系及机制也值得进一步研究.

目的:探讨在肿瘤坏死因子(TNF)-α刺激环境中,天麻素能否保护骨关节炎中滑膜细胞衰老和线粒体膜破坏.方法:分离并鉴定正常人滑膜细胞,对滑膜细胞进行分组:对照组(正常人滑膜细胞组)、TNF-α干预组、TNF-α+天麻素组.采用实时定量PCR检测滑膜细胞中细胞色素C(CytC)的mRNA水平;Western Blot检测滑膜细胞中CytC的蛋白水平;SA-β-半乳糖苷酶染色观察计算衰老细胞的比例;流式细胞仪检测线粒体膜电位(MMP)以及线粒体通透性转换孔(MPTP)的变化.结果:与TNF-α组比较,天麻素作用后CytC的mRNA水平和蛋白表达水平显著降低(P＜0.05),衰老细胞比例下降,MMP水平升高(P＜0.05),MPTP开放降低.结论:天麻素可以保护TNF-α诱导的骨关节炎中滑膜细胞衰老和线粒体膜破坏,其可能成为治疗骨关节炎的新药物.