Xifeng Qiao

Bingying Wang

Zhengqiang Yuan

Feng Yu

Ying Zhang

Yu Wang

Yiting Yang

Jian Tang

Zhihong Jiang

Li Lin

Lanyue Zhang

Professor Zhiyun Du
email: zhiyundu@gdut.edu.cn

Yongmin Zhang
email: yongmin.zhang@upmc.fr

Professor Yongmin Zhang

The Polysaccharides from Yiqi Yangyin Complex Attenuated Mammary Gland Hyperplasia: Integrating Underlying Biological Mechanisms and Network Pharmacology

Keywords: Yiqi Yangyin complex (YYC), Mammary Gland Hyperplasia (MGH), Sex hormones, Apoptosis, Network pharmacology

Yiqi Yangyin complex (YYC), the homology of medicine and food, is comprised of Polygonati Rhizoma, Lycii Fructus, Codonopsis pilosula, and Rehmanniae Radix.

Herein, the YYC polysaccharide treatment effectively attenuated the progression of MGH in a mice MGH model-induced with estrogen and progestogen. YYC significantly relived hormonal disorders by reducing the levels of estrogen receptor α (ERα) and progesterone receptor (PR), and substantially elevated the protein level of BCL2-associated X (Bax) and significantly down-regulated expression of B-cell lymphoma-2 (BCL-2). Finally, the key targets of ERα, PR, Bax and BCL-2 were predicted and significantly enriched on estrogen signaling pathway and apoptosis pathway by network pharmacology. This finding suggests that YYC may influence the sex hormones level through estrogen signaling pathway and then induce apoptosis to balance normal functions of mammary gland. This study thus provided evidences for the potential therapeutic efficacy of YYC on MGH and revealed the correlated regulatory signaling pathways.

Introduction

Mammary gland hyperplasia (MGH) is a common disease characterized by pathological hyperplasia for lobules of mammary gland [START_REF] Jia | The anti-hyperplasia of mammary gland effect of protein extract HSS from Tegillarca granosa[END_REF]. With increase of work stress and competitive pressure in the fast pace of modern life, the incidence of MGH in middle-aged women is increasing rapidly, and its severe cancerous tendencies to threaten human health [START_REF] Chen | Anti-hyperplasia effects of Rosa rugosa polyphenols in rats with hyperplasia of mammary gland[END_REF]. The pathogenesis of MGH is closely related to endocrine disorder, mainly owing to high estrogen release or low progesterone production caused hormones imbalance to increase incomplete differentiation of glandular epithelium, and made the proliferative tissue unredintegration to induce MGH [START_REF] Bennett | Serum oestradiol in women with and without breast disease[END_REF].

Although hormones, such as progesterone, tamoxifen and vitamins, are usually used for treatment of MGH [START_REF] Li | Therapeutic Effects of a Traditional Chinese Medicine Formula Plus Tamoxifen vs. Tamoxifen for the Treatment of Mammary Gland Hyperplasia: A Meta-Analysis of Randomized Trials[END_REF], long-term oral administration of such drugs will cause hormone imbalance disorder and various medication discomforts to aggravate the severity of MGH [START_REF] Kouhestani | Kaempferol attenuates cognitive deficit via regulating oxidative stress and neuroinflammation in an ovariectomized rat model of sporadic dementia[END_REF]. It is important for us to find new drugs with more convenient, effective, and have few side effects to treat MGH. It has been reported that traditional Chinese medicine (TCM) has the protective effects on MGH by possible biological mechanism [START_REF] Li | Mechanisms of traditional Chinese medicine in the treatment of mammary gland hyperplasia[END_REF].

Medicine and Food Homology is regarded as a combination of food and medicine functions, nutritional value, diseases prevention and treatment, and healthcare activities [START_REF] Chien | Gastroprotective potential against indomethacin and safety assessment of the homology of medicine and food formula cuttlebone complex[END_REF]. Yiqi Yangyin complex (YYC), as a medicine and food homology from TCM formula, including Polygonati Rhizoma, Lycii Fructus, Codonopsis pilosula, Rehmanniae Radix, which are used as the common ingredients of the stew soup and famous prescriptions of TCM for replenishing Qi and nourishing Yin, and called as Yi Qi Yang Yin in Chinese. Polysaccharides have been regarded as the main components of the stew soup or water decoction of many herbs for replenishing Qi and nourishing Yin function, which play an important role in exhibiting immunomodulatory activities [START_REF] Yin | Advances in research on immunoregulation of macrophages by plant polysaccharides[END_REF]. Polygonati Rhizoma polysaccharide known as an important active compound, has the potential as a drug or dietary adjuvant for the treatment of atherosclerosis and hyperlipidemia [START_REF] Li | The effect of Polygonatic rhizome on hyperlipoidemia and anti-atherosclerosis[END_REF], which has strong antioxidant, lipid-regulating, anti-inflammatory, and endothelial function improvement effects [START_REF] Wang | Experimental study of Polygonatum polysaccharides on antioxidation of rats[END_REF][START_REF] Zhang | Endothelial progenitor cells: The new target of anti-atherosclerosis drugs[END_REF]. Lycii Fructus is traditionally used in Chinese home cooking, such as tea, soups, porridge, taste sweet, and in the Chinese pharmacopoeia as an aid for vision and longevity to balance the "Yin" and "Yang" of the body [START_REF]Chinese Pharmacopoeia Commission, Pharmacopoeia of the People's Republic of China Part I[END_REF]. The polysaccharide of Lycii Fructus partly decreased the protein expression of HIF-1α and Bax to regulate the production of inflammatory factors through NF-κB signaling pathway [START_REF] Wang | Retinal structure and function preservation by polysaccharides of wolfberry in a mouse model of retinal degeneration[END_REF]. Codonopsis pilosula contained sterol, triterpenes, glycoside, alkaloid, polysaccharide and other components [START_REF] Xin | The inhibitory effect of a polysaccharide from Codonopsis pilosula on tumor growth and metastasis in vitro[END_REF], and its polysaccharides had several biological activities, such as tumor growth prevention [START_REF] Yang | Structural characterization and antitumor activity of a pectic polysaccharide from Codonopsis pilosula[END_REF], immune system modulation and anti-oxidant activity [START_REF] Li | Protective effect of a polysaccharide from stem of Codonopsis pilosula against renal ischemia/reperfusion injury in rats[END_REF]. Rehmanniae Radix has been traditionally known as lowering blood fever, nourish Yin and promoting the body fluids, curing macula, skin rash, nosebleeds and so on. Meanwhile, Rehmanniae Radix could nourish Yin and replenish blood, benefit the essence, and was mainly used to treat anemia, diabetes, tinnitus and heart palpitations [START_REF]Chinese Pharmacopeia Commission, Pharmacopeia of the people's republic of china Beijing[END_REF]. The above four herbs are usually used for Chinese home cooking and TCM to exerting the function of Reinforcing Qi and Nourishing Yin. However, the efficacy and potential biological mechanism of the polysaccharide formed by a mixture of four herbs (YYC) on MGH have not been completely investigated.

Network pharmacology is an emerging discipline based on the effective mapping of unexplored target space of nature products, which become a novel and powerful method by multi-component and multi-target action mode [START_REF] Kibble | Network pharmacology applications to map the unexplored target space and therapeutic potential of natural products[END_REF]. Network pharmacology may combine several pharmacological networks with human disease-related genes by multichannel regulation of signaling pathways and revealing disease-related drug targets [START_REF] Jiang | Integrated analysis of potential pathways by which aloe-emodin induces the apoptosis of colon cancer cells[END_REF]. It has been successfully applied to decipher the bioactive compounds and synergistic mechanisms of the TCM Li-Ru-Kang (LRK) against MGH from the molecular network level [START_REF] Wei | Network pharmacology exploration reveals the bioactive compounds and molecular mechanisms of Li-Ru-Kang against hyperplasia of mammary gland[END_REF]. Chen Y et al found that the integrated analysis of network pharmacology and bioinformatics analysis may be used to reveal the potential targets and the molecular mechanism of essential oil from Rhizoma Curcumae on liver fibrosis [START_REF] Chen | Essential oil from the raw and vinegar-processed Rhizoma Curcumae ameliorate CCl4-induced liver fibrosis: integrating network pharmacology and molecular mechanism evaluation[END_REF]. Tu C et al, also found that inflammatory state-dependent dietary supplement hepatotoxicity responses in normal and diseased rats were investigated by network pharmacology [START_REF] Tu | Network pharmacology oriented study reveals inflammatory state-dependent dietary supplement hepatotoxicity responses in normal and diseased rats[END_REF]. Therefore, network pharmacology may provide new ideas for the potential molecular mechanisms of YYC on MGH.

In this study, the anti-hyperplasia biological mechanism of the polysaccharides from the four herbs and YYC on mice with MGH were identified, and the potential key targets and possible signaling pathways were investigated by network pharmacology-based prediction and verification (Figure 1). The present study may provide a useful reference for exploring the potential mechanism and action pathways of the function food from Reinforcing Qi and Nourishing Yin herbs are helpful for the healthcare of MGH.

Materials and methods

Preparation of YYC and analysis of polysaccharides

Polygonati Rhizoma (Sichuan, China), Lycii Fructus (Qinghai, China), Codonopsis pilosula (Gansu, China), and Rehmanniae Radix (Henan, China) were purchased from Tianfangjian (China) Pharmaceutical Co. LTD. The four raw materials, including Polygonati Rhizoma, Lycii Fructus, Codonopsis pilosula, and Rehmanniae Radix were extracted in boiling water for 2 h and then further extracted in boiling water for 1 h. All collected filtrates were processed with vacuum concentration to obtain extracts. The concentrated extracts were purified with 75% ethanol and contained 30-45% solid contents. The extracts were followed by drying in a 70~100 ℃ oven to obtain the polysaccharides of four raw materials. Finally, the four polysaccharides were mixed to obtain YYC. The proportion of Polygonati Rhizoma, Lycii Fructus, Codonopsis pilosula, and Rehmanniae Radix in YYC were 25% (w/w), 25% (w/w), 25% (w/w), and 25% (w/w), respectively.

Determination of sugar and protein content

The basic physicochemical properties of four polysaccharides and YYC were performed. The total sugar contents of YYC were determined with the phenol sulfuric acid assay [START_REF] Masuko | Carbohydrate analysis by a phenol-sulfuric acid method in microplate format[END_REF]. The protein contents were determined using BCA assay [START_REF] Smith | Measurement of protein using bicinchoninic acid[END_REF]. The molecular weight distributions of YYC were determined by high performance gel-permeation chromatography (HPGPC) [START_REF] Wang | Structure analysis of a neutral polysaccharide isolated from green tea[END_REF].

Animal experiments

Eight-week-old female KunMing mice weighing 18-20 g (license number: SCXK2018-0002) were commercially obtained from the Experimental Animal Centre of Guangdong Province. The mice were housed at a controlled room (23±1℃, humidity 60±5%, 12 h day/light). They were acclimated under climate-controlled conditions for 7 days before the experiments began. Mice were randomly divided into seven groups with six mice in each group, including a control group (without treatment), model group (0.5 mg/kg/d estrogen for first [START_REF] Masuko | Carbohydrate analysis by a phenol-sulfuric acid method in microplate format[END_REF] Rehmanniae Radix, at a ratio of 0.25:0.25:0.25:0.25, 67 mg/kg/d). The dose of polysaccharides were used according to the guidelines of the Chinese Pharmacopoeia (2015). Mice except for control group were injected with estrogen (0.5 mg/kg/d) into the muscle of hind leg for consecutive 25 days, and followed with progestogen (5 mg/kg/d) for another 5 days [START_REF] Wang | Anti-Inflammatory Effect of a TCM Formula Li-Ru-Kang in Rats With Hyperplasia of Mammary Gland and the Underlying Biological Mechanisms[END_REF]. For positive group, CP group, RP group, PR group and YYC group, mice were treated with once daily intragastric administration before intramuscular injection for 30 days. For control group and model group, mice were intragastrically administered with equal volume of saline. Mice were sacrificed 24 h after the last polysaccharides administration. The nipple height was firstly detected, and mice were weighed. The blood were collected from eyeball extraction, and then the mice were sacrificed. The mammary glands were immediately removed and fixed 4% paraformaldehyde. The blood was centrifuged at 3000 rpm for 15 min to separate the serum without hemolysis, and then stored at -80℃. This study was approved by the Animal Care and Use Committee of Guangdong University of Technology.

Determination of nipple diameter and organ coefficients

The diameter of the mice's nipple was measured on 1, 7, 15, 30 day (d). The ovary and uterus were collected and weighted at the end of this experiment. The uterus and ovary index were calculated by uterus or ovary weight divided by body weight [START_REF] Wang | The anti-hyperplasia of mammary gland effect of Thladiantha dubia root ethanol extract in rats reduced by estrogen and progestogen[END_REF][START_REF] Li | Anti-hyperplasia Effects of Total Saponins From Phytolaccae Radix in Rats With Mammary Gland Hyperplasia via Inhibition of Proliferation and Induction of Apoptosis[END_REF].

Biochemical analysis and enzyme-linked immunosorbent assay (ELISA)

Blood was collected by eyeball extraction, and then centrifuged at 3000 rpm for 15 min to obtain the serum. The serum was collected and stored at -80 ℃ for hormone assays. The concentrations of E2, P in serum were measured by commercial detection kits (Nanjing Jiancheng Bioengineering Institute, Nanjing, China).

Histological analysis

Fourth inguinal mammary glands (n=6) was obtained for histopathological examination and fixed in 4% paraformaldehyde for 48 h. After processed in a series of graded ethanol and dimethyl benzene, the tissues were embedded in paraffin and cut into 4 µm thick sections, and then stained with hematoxylin and eosin (H&E). Finally, pathological changes were observed by using SZX10 microscope (Olympus Corp., Tokyo, Japan).

Immunofluorescence assays

Each mammary gland tissue block was sectioned at 4 µm on the graded slide. Slices were dried overnight and washed with PBS for 5 min. Sections were blocked with BSA for 1 h at room temperature on a shaker. The samples were incubated at 4℃ overnight with primary antibodies ERα (1:200), or PR (1:200) (Danvers, MA, United States) and incubated at 4℃. After being washed with PBS, the sections were treated with the secondary antibody conjugated with horseradish peroxidase for 1 h and then DAPI was added into slices for nuclear counter-staining for 5 min [START_REF] Wang | Anti-Inflammatory Effect of a TCM Formula Li-Ru-Kang in Rats With Hyperplasia of Mammary Gland and the Underlying Biological Mechanisms[END_REF]. The sections were captured by microscope (Olympus, Tokyo, Japan). The mean integrated optical density (IOD) of these areas was measured by image analysis software Image J.

Immunohistochemistry assay

In situ expression of Bcl-2 and Bax in mammary gland was performed as follows.

Paraffin-embedded sections (4 µm) were dewaxed in xylene, sequentially rehydrated in alcohol and incubated in 3% H 2 O 2 for 20 min. The sections were heated twice in a microwave oven for 5 min in 0.01 M citrate buffer (pH 6.0) for antigen retrieval and followed by overnight incubation at 4℃ with the primary antibodies Bcl-2 (1:50) and Bax (1:50) (Danvers, MA, United States) [START_REF] Li | Anti-hyperplasia Effects of Total Saponins From Phytolaccae Radix in Rats With Mammary Gland Hyperplasia via Inhibition of Proliferation and Induction of Apoptosis[END_REF]. The sections were washed and incubated with the HRP conjugated secondary antibody for 30 min at 37℃. After staining with DAB, the tissue slides were counterstained with hematoxylin, dehydrated with a graded ethanol series, and sealed with neutral gum in the end. The sections were captured by microscope (Olympus, Tokyo, Japan). The mean integrated optical density (IOD) of these areas was measured by software Image J.

Database construction

The chemical constituents from four polysaccharides were obtained from TCMSP.

Known targets of single polysaccharide were collected from Herbal Ingredients' Targets Database (HIT), and the putative targets from these were screened out from Therapeutic Targets Database (TTD) through structural similarity comparison. Gene and protein targets associated with MGH were collected from the Online Mendelian Inheritance in Man (OMIM) database and GeneCards server. The targets of interactive proteins were obtained from Database of Interacting Proteins (DIP) and ID types of the proteins were converted to UniProt IDs. Based on the previous steps, the targets were prepared, namely, drug-related genes and disease targets. The crossed genes were screened by the R software by using the Venn Diagram.

Target protein-protein interaction (PPI) network construction

To provide the scientific and reasonable interpretation of the complex relationships between chemical constituents and targets associated with MGH, network analysis was performed. The single polysaccharide-target network was constructed by using candidate substance and significant targets for MGH. The network was performed by using Cytoscape 3.5.1 software. The topological features of each node in the network were calculated by "Degree", "Betweenness centrality", and "Closeness centrality" ("Degree" values were two fold greater than the median value of all the network nodes, "Betweenness centrality" and "Closeness centrality" value were greater than the median value of all the network nodes). Targets with higher value were screened as the candidates for MGH.

Go gene enrichment analysis and KEGG pathway

To elucidate the function of the four polysaccharides target compounds and its role in signal transduction, the Database for Annotation, Visualization and Integrated Discovery (DAVID) database were used to analyze the GO and KEGG pathway enrichment. The biological processes, cellular components, molecular functions for GO enrichment and the pathways were also described.

Statistical analysis

The data were expressed as the mean values ± standard error of mean (SEM).

Statistical analysis was performed by GraphPad Prism 5.0 software, using student t-tests or one-way analysis of variance (ANOVA). Difference with P-value (P < 0.05) was considered as significance and drew the diagrams.

Results

The physicochemical properties of YYC

These polysaccharides had significant differences in total sugars and proteins. Among 1). The components of Polygonati Rhizoma with molecular weight less than 5× 10 3 Da had 92% of the peak area. The components of Lycii Fructus with molecular weight between 5× 10 3 Da and 4.8 × 10 4 Da had 92% of the peak area. The components of Rehmanniae Radix with molecular weight less than 5× 10 3 Da had 84% of the peak area. The components of Codonopsis pilosula with molecular weight between 5× 10 3 Da and 3 × 10 5 Da had 80% of the peak area. The components of YYC with molecular weight between 5× 10 3 Da and 2 × 10 4 Da had 70% of peak area by HPGPC (Table 1, supplementary material Fig. S1). 2A and Figure 2B). This result suggested that LF, PR and YYC significantly relieved the nipple diameter in mice with MGH.

which
The mammary gland as the target organ for sex hormones is closely related to the endocrine status of the ovary, while the uterus index and ovary index reflect the changes of the uterus and ovary [START_REF] Li | Therapeutic Effects of a Traditional Chinese Medicine Formula Plus Tamoxifen vs. Tamoxifen for the Treatment of Mammary Gland Hyperplasia: A Meta-Analysis of Randomized Trials[END_REF]. (Figure 2C and Figure 2D). Taken together, this result suggested that YYC significantly down-regulated the ovary index and uterine coefficient to relieve the symptom of mice with MGH.

Effect of YYC on the histological changes in mice with MGH

In order to verify whether the YYC could alleviate MGH, HE staining was used to detect the pathological changes of mammary gland tissue. As shown in Figure 3, compared with the control group, there was obvious proliferative lesions in mammary epithelial cell tissue, including lobules hyperplasia, increase of count of acini and ducts, and irregular arrangement and obvious expansion of duct lumen in model group.

Compared with the model group, administration of PRPX and YYC for consecutive 30 days significantly inhibited the typical histological patterns, whereas treatment of CP, RR and PR were also capable to decrease the area of proliferative lesions and counts of mammary acini and ducts in different degrees (Figure 3). This result showed that YYC had therapeutic efficacy on mice with MGH.

YYC modulated the serum biochemical parameters and protein expression of ERα and PR

Estrogen receptors play a critical role in regulating cell proliferation and differentiation in mammary glands, which may be as important nuclear transcription factors activated by E2 or P [START_REF] Jonsson | Support of a bi-faceted role of estrogen receptor beta (ERbeta) in ERalpha-positive breast cancer cells[END_REF]. As shown in Figure 4, compared with control group, the level of Estrogen 2 (E2) was significantly elevated (p < 0.05) in model group.

Compared with model group, the secretion of E2 was significantly decreased in RPXP, CP, RR, LF, PR and YYC group (p < 0.05). However, there were no significant differences in the level of progesterone (P) between all groups (Figure 4A).

To investigate the effect of YYC on Estrogen receptors, the levels of ERα and PR were detected by immunofluorescence assay. The fluorescence intensity of ERα (p < 0.05) and PR (p < 0.01) in model group were significantly elevated in comparison with control group (Figure 4B and Figure 4C). Compared with model group, the fluorescence intensity of ERα was markedly inhibited in all group (p < 0.05), while the fluorescence intensity of P was significantly decreased in RPXP (p < 0.01), LF (p < 0.05), PR (p < 0.05) and YYC (p < 0.05) group (Figure 4B and Figure 4C). This result illustrated that LF, PR and YYC could regulate the estrogen receptors on the mice with MGH-induced by estrogen and progestogen.

Effect of YYC on expression of BCL-2 and Bax in mice with MGH

Apoptosis plays an important role in maintaining tissue homeostasis and cancer prevention [START_REF] Fulda | Targeting apoptosis for anticancer therapy[END_REF]. In order to verify the effect of YYC on apoptosis in MGH, the apoptotic protein Bax and anti-apoptotic protein BCL- 5A and Figure 5B). This result suggested that YYC might be involved in regulating the symptom of mice with MGH via apoptosis. 6).

Network construction and analysis of target protein-protein interaction

To study the target of combined polysaccharides affecting MGH, the cytoscape software was used to predict the target protein interaction. The target protein-protein interaction network was constructed (Figure 7). Through the analysis of topological parameters, the 16 direct targets of Polygonati Rhizoma included heat shock protein

The GO enrichment and KEGG signaling pathway of YYC

The targets were analyzed for GO biological functions and KEGG signaling pathway enrichments. As shown in Figure 8A, the top 10 highly enriched GO biological processes, molecular functions and cellular components were found (p-value < 0.05).

The results showed that the GO functions of target protein molecules were mainly enzyme binding, hormone response process, and mitochondrial activity. The top 3 enrichments in Biological Process category were response to hormone, response to steroid hormone and intracellular steroid hormone receptor. The top 3 enrichments in Molecular Function category were enzyme binding, protein binding, estrogen response element binding; in Cell Component category, the top 3 enrichments were mitochondrial envelope, mitochondrial membrane, mitochondrial outer membrane.

The KEGG pathway analysis was further performed to identify the signaling pathway that the predicted targets may be participated. The bubble map of KEGG pathway showed that the core targets based on top 10 enrichment pathway (Figure 8B). The top 10 involved in signaling pathways were mainly related to hormone signaling pathway, such as estrogen signaling pathway, thyroid hormone signaling pathway, apoptosis. We further constructed the target-component-pathway network diagram of action mechanism of YYC (Figure 8C). This result showed that each active compound could act on multiple targets. Notably, various targets on MGH were significantly enrichment on estrogen signaling pathway, which includes the target of ER, BCL-2, HSP90, EGFR. The KEGG pathway of polysaccharides target MGH showed that the above targets might be activated by polysaccharides intervention, and then influenced EGFR tyrosine kinase inhibitor resistance signaling pathway, apoptosis-multiple species signaling pathway, hepatitis B signaling pathway, proteoglycans in cancer signaling pathway, PI3K-Akt signaling pathway. The induction of estrogen signaling pathway and apoptosis play an important role in the treatment of MGH with drugs (Supplementary Figure 2). Therefore, the two significant enrichment pathways were in accordance with the animal experimental verification.

Discussion

The sex hormones were mainly secreted from mammary gland, and its imbalanced secretion inducing endocrine disorders was considered as main cause to lead to MGH [START_REF] Liu | Systems approaches and polypharmacology for drug discovery from herbal medicines: an example using licorice[END_REF][START_REF] You | Modulatory effect of fermented papaya extracts on mammary gland hyperplasia induced by estrogen and progestin in female rats[END_REF]. MGH, a common disease in middle-age women has severe cancerous tendency to cause higher risk of mammary gland cancer [START_REF] Zhang | Modulation of expression of p16 and her2 in rat breast tissues of mammary hyperplasia model by external use of rupifang extract[END_REF]. At present, the surgery and medication were referred as the main treatment of MGH, whereas its adverse side effects severely impact quality of life [START_REF] Henry | Endocrine therapy toxicity: management options[END_REF]. Hence, it is crucial to find few side effects drugs improvement of estrogen-induced endocrine disorders to prevent the prevalence and progression of MGH. As an alternative program to traditional therapeutic interventions, Chinese herbal products are getting more and more attention to deal with estrogen-related health issues. YYC, a part of medicine and food homology formula, was traditionally used for Chinese home cooking to exert the function of Reinforcing Qi and Nourishing Yin. However, the molecular mechanism of YYC for MGH has been yet unclear. Thus, in our study, the underlying biological mechanism of YYC against MGH was explored.

To study the molecular mechanism for YYC on MGH in more detail, a mouse model of MGH was successfully constructed. Compared with model group, the level of E2 could be significantly reduced in YYC group in mice, but the level of P has no significant difference with control group. This result suggested that the absolute or relative increase of E2 or P deficiency could lead to imbalance of E2/P ratio to cause excessive proliferation of mammary glandular parenchyma. ER and PR could specially bind to E2 and P. ER is a protein molecule with two subtypes of ERα and ERβ, which combines with E2 to form a hormone receptor complex [START_REF] Higa | Sex hormone receptor repertoire in breast cancer[END_REF]. PR as the regulatory protein of E2 is accompanied by ER. E2 could promote expression of ER and PR in breast epithelial cells, and increase the sensitivity of mammary gland tissue to hormones-induced MGH [START_REF] Mehta | Differential roles of ERalpha and ERbeta in normal and neoplastic development in the mouse mammary gland[END_REF]. Meanwhile, the proliferating cells promote the synthesis of ER and PR through the hormone receptor system, forming a pathological Apoptosis is the primary mode of cell death involved in development and homeostasis [START_REF] Fuchs | Programmed cell death in animal development and disease[END_REF]. BCL-2 is a well-known inhibitor of apoptosis to induce tumorigenesis [START_REF] Vaux | Bcl-2 gene promotes haemopoietic cell survival and cooperates with c-myc to immortalize pre-B cells[END_REF], while Bax is a pro-apoptotic protein. Both of them are generally considered as important molecular proteins for apoptosis [START_REF] Zhang | Emerging roles of lipids in BCL-2 family-regulated apoptosis[END_REF]. BCL-2 could block MOMP to antagonize Bax and then prevent apoptosis [START_REF] Eisenberg-Lerner | Life and death partners: apoptosis, autophagy and the cross-talk between them[END_REF]. It has been reported that Tongru Sanjie decoction could regulate the expression of BCL-2 to inhibit MGH, thereby blocking malignant transformation of mammary gland. Also, the excessive expression of BCL-2 in many B-cell malignancies could induce its growth and proliferation [START_REF] Roberts | Substantial susceptibility of chronic lymphocytic leukemia to BCL2 inhibition: results of a phase I study of navitoclax in patients with relapsed or refractory disease[END_REF]. Apoptosis could cause diseases, such as hyperplasia in the peripheral lymphoid organs, which accelerates autoimmune disease and tumorigenesis [START_REF] Nagata | Fas-induced apoptosis, and diseases caused by its abnormality[END_REF]. Here, we analysed apoptosis by immunohistochemical assay and found that the polysaccharides from Polygonati Rhizoma, Lycii Fructus, Codonopsis pilosula, Rehmanniae Radix and YYC significantly inhibited the level of anti-apoptotic factor BCL-2, whereas only YYC markedly elevated the pro-apoptotic factor Bax expression.

Lycii Fructus contains three bioactive compounds, including carotenoids, phenolics, and polysaccharides, and the polysaccharide is as the most important and highest component [START_REF] Kulczyriski | Goji berry (Lycium barbarum): Composition and health effects-a review[END_REF]. Lycii Fructus polysaccharides could up-regulate anti-apoptotic protein BCL-2 in lens epithelial cells and increase ratio of BCL-2 to BAX (pro-apoptotic protein) in the lens mainly by its anti-oxidative effects [START_REF] Wang | The regulation of LBP (lycium barbarum polysaccharide, LBP) on the expression of apoptosis-related genes Bcl-2 and Bax in SD rat LEC (lens epithelial cells, LEC) induced by oxidative injuries[END_REF]. Polygonati Rhizoma consists of saponins and polysaccharides and exerts its pharmacological activity, such as immune promotion, antiaging, antifatigue, blood glucose regulation and lipid regulation [START_REF] Yang | Muscle Fatigue-Alleviating Effects of a Prescription Composed of Polygonati Rhizoma and Notoginseng Radix et Rhizoma[END_REF]. Codonopsis pilosula contains polysaccharides, sesquiterpenes, saponins, and phytosterols [START_REF] He | The genus codonopsis (campanulaceae): a review of phytochemistry, bioactivity and quality control[END_REF], and the polysaccharides are active compounds to exert multiple functions, including antitumor, antimicrobial, antioxidant, and immunoenhancing properties [START_REF] Deng | Polysaccharide from Radix Codonopsis has beneficial effects on the maintenance of T-cell balance in mice[END_REF][START_REF] Qin | Study of the selenizing codonopsis pilosula polysaccharides protects RAW264.7 cells from hydrogen peroxide-induced injury[END_REF]. The main components of Rehmanniae Radix are polysaccharides, triterpenoid saponins, iridoids, ionones, phenylglycol glycosides, phenolic acids, and lignans [START_REF] Zhang | Three new 2,2'-difurylketone derivatives and two new chromones from the Rehmanniae Radix Praeparata[END_REF], and the polysaccharides are major active components to exhibit anticancer, anti-aging, antioxidant, and immunomodulatory activities [START_REF] Zhou | Structural characterization and immunomodulatory activities of two polysaccharides from Rehmanniae Radix Praeparata[END_REF][START_REF] Li | Effects of nine steamed nine sundried rehmanniae radix praeparata polysaccharides on apoptosis of Lewis lung cancer bearing mice[END_REF]. The above literature suggested that the main anti-hyperplasia potential may be attributed to the high content of polysaccharides in [58]. Estradiol promotes epithelial cell proliferation in the uterine endometrium and mammary glands starting in puberty [START_REF] Koos | Putting physiology back into estrogens' mechanism of action[END_REF]. The increase of E2 and persistent lack of P promoted the mammary gland excessive hyperplasia and incomplete repairment [START_REF] Schindler | Dydrogesterone and other progestins in benign breast disease: an overview[END_REF]. The predicted targets ER and PR were in accordance with the animal experimental results. STAT3 could interact with polypeptide receptor to regulate extracellular signals [START_REF] You | The role of STAT3 in autophagy[END_REF]. BCL-2 as an anti-apoptotic protein is the key regulator in intrinsic apoptosis pathway [START_REF] Singh | Regulation of apoptosis in health and disease: the balancing act of BCL-2 family proteins[END_REF]. It has been reported that the post-treatment of curcumin has an effect against myocardial ischemia and reperfusion to activate JAK2/STAT3 pathway by down-regulation of Caspase3 and up-regulation of BCL-2 [START_REF] Duan | The effects of curcumin post-treatment against myocardial ischemia and reperfusion by activation of the JAK2/STAT3 signaling pathway[END_REF][START_REF] He | Curcumin, inflammation, and chronic diseases: how are they linked?[END_REF]. STAT3 activated BCL-2 to inhibit autophagy, or inhibition of STAT3 could cause autophagy [START_REF] Feng | Metformin promotes autophagy and apoptosis in esophageal squamous cell carcinoma by downregulating Stat3 signaling[END_REF]. Autophagy has been function as a tumor suppressive mechanism to remove or mitigate harmful stimuli, including oxidative stress, inflammation [START_REF] You | The role of STAT3 in autophagy[END_REF]. Inhibition of autophagy resulted in an accumulation of toxic proteins and mitochondrial dysfunction to trigger apoptosis [START_REF] Epelman | Origin and functions of tissue macrophages[END_REF].

These results demonstrated that YYC played an important role in regulating MGH by apoptosis pathway. This is consistent with our animal experimental results. Taken together, our results provides preliminary evidence that YYC may induce apoptosis and Estrogen signaling pathway, which further verified the synergetic effects of YYC on MGH (Figure 9). However, there are some limitations, which only verified the key molecules significantly enrichment on estrogen signaling pathway and apoptosis, but several targets have not been yet detected in this study. Additionally, although potential active compounds were predicted by network pharmacology, and our animal experimental results displayed the combination of four polysaccharides in equal proportion (YYC) may had better synergistic effect than polysaccharide alone on MGH, the composition and mechanism of these active ingredients are still unclear. In future study, we will pay more attention to the composition of the active components of polysaccharides in YYC and crosstalk of multi-pathway.

FIGURE 1

 1 FIGURE 1 The biological mechanisms of MGH. The establishment of animal experimental model and network pharmacology-based computational predictions.

FIGURE 2

 2 FIGURE 2 Effects of polysaccharides on the nipple diameters in MGH. (A) The pathological features of mammary gland tissues in different group. (B) Diameter of nipples in different group. (C) The pathological features of uterus and ovary. (D) The ovary and uterus coefficient in

FIGURE 3

 3 FIGURE 3 Analysis of histopathological for mammary gland tissue with polysaccharides. Control group (Control), Model group (Model), Positive group (Rupixiao Pian, RPXP), Codonopsis pilosula (CP), Rehmanniae Radix (RR), Polygonati Rhizoma (PR), Lycii Fructus (LF), Yiqi Yangyin Complex (YYC). (Arrow indicated hyperplasia of ductal epithelial cells.)

FIGURE 4

 4 FIGURE 4 Effect of polysaccharides on sex hormones levels and the expression of ERα and PR in mice's mammary gland tissue. (A) Sex hormones level for E2 and P in serum. (B) Immunofluorescence analysis for ERα and PR in mammary gland tissue. Red signals represents PR expression, green signals represents ERα expression, blue signals represents nuclei. (C) The quantification of immunofluorescence signals in different groups. Data are expressed as the mean ± SEM. Control group (Control), Model group (Model), Positive group (Rupixiao Pian, RPXP),

FIGURE 5

 5 FIGURE 5 Effects of polysaccharides on level of BCL-2 and Bax in mammary gland with MGH. (A) Immunohistochemical analysis of BCL-2 and Bax in different group. (B) The quantification of immunohistochemistry signals in different groups. Control group (Control), Model group (Model),

FIGURE 6

 6 FIGURE 6 The targets of affecting MGH for four polysaccharides. A-D: Venn diagram of the candidate targets in four polysaccharides (Polygonati Rhizoma (A), Codonopsis pilosula (B), Lycii Fructus (C), Rehmanniae Radix (D)) and MGH; Note: Purple circle represents targets for MGH; yellow circle represents targets for four single polysaccharide.

  HSP 90-alpha (HSP90AA1), signal transducer and activator of transcription 3 (STAT3), estrogen receptor (ESR1), BCL2, cyclin-dependent kinases1 (CDK1), cyclinB1 (CCNB1), SH2 domain-containing transforming protein C1 (SHC1), Bcl-2-like protein 1 (BCL2L1), vascular endothlial growth factor (VEGFA), progesterone receptor (PGR), salt overly sensitive 1 (SOS1), growth factor receptor-bound protein 2 (GRB2), cell Division Cycle 25A (CDC25A), protein kinase C delta type (PRKCD), protein kinase C alpha (PRKCA), PPP2CA. The 21 direct targets of Codonopsis pilosula included steroid receptor coactivator (SRC), EGFR, HSP90AA1, BCL2, STAT3, ESR1, BCL2L1, VEGFA, SHC1, caspase 3 (CASP3), GRB2, PRKCD, fibroblast growth factor 2 (FGF2), CDK1, breast cancer anti-estrogen resistance 1 (BCAR1) , PRKCA, CCNB1, PGR, SOS1, fibroblast growth factor 1 (FGF1), Cell Division Cycle 37 (CDC37). The 23 direct targets of Lycii Fructus included SRC, endothlial growth factor receptor (EGFR), HSP90AA1, BCL2, STAT3, BCL2L1, VEGFA, CASP3, Src homology 2 domain containing (SHC1), BAX, GRB2, PRKCD, CDK1, FGF2, PRKCA, BCAR1, CCNB1, PGR, FGF1, CDC37, SOS1, caspase 7 (CASP7), caspase 1 (CASP1). The 12 direct targets of Rehmanniae Radix included SRC, EGFR, BCL2, VEGFA, STAT3, BCL2L1, FGF1, FGF2, KDR, CDK1, BAX, CCNB1 (Degree >10).

FIGURE 7

 7 FIGURE 7 Analysis of the target network of four polysaccharides influencing MGH. A-D: Polygonati Rhizoma (A), Codonopsis pilosula (B), Lycii Fructus (C), Rehmanniae Radix (D) influencing MGH for the analysis of target network. Note: Red circle represents high combination degree with surrounding targets; blue circle represents low combination degree with surrounding targets.

FIGURE 8

 8 FIGURE 8 GO and KEGG enrichment analysis of YYC. (A) The GO enrichment analysis of potential targets including molecular function, cellular component, biological process. (B) The enrichment analysis of KEGG signaling pathways. The size of point indicates the number of genes in the pathway and the color of point corresponds to p-value ranges. (C) The network diagram of YYC with target-compound-pathway analysis.

  proliferation cycle. Here, our animal study showed that the polysaccharides from Polygonati Rhizoma, Lycii Fructus and YYC significantly improved the expression of ERα and inhibited production of PR by immunohistochemical assay in MGH mice model, indicating that the polysaccharides from Polygonati Rhizoma and Lycii Fructus in YYC played an important role in relieving MGH. Previous study showed that Polygonati Rhizoma polysaccharide has potential anti-inflammatory effect on 12-O-tetradecanoylphorbol-acetate (TPA)-induced inflammatory in mice[START_REF] Xian | Effect of Rhizoma Polygonati on 12-O-tetradecanoylphorbol-acetate-induced ear edemainmice[END_REF]. Lycii Fructus inhibits growth of ER + human breast cancer cells (MCF-7 cell line) by altering E2 metabolism[START_REF] Li | Lycium barbarum inhibits growth of estrogen receptor positive human breast cancer cells by favorably altering estradiol metabolism[END_REF]. Interestingly, KEGG pathway analyses by network pharmacology showed that two targets ER and PR were directly involved in the enrichment on Estrogen signaling pathway. This pathway could transmit the signal to specific functional modules, such as apoptosis, carcinogenesis, cell proliferation, differentiation/development and inflammation, which are mainly initiated through estrogen or estrogen chemicals binding to ERs[START_REF] Kiyama | Estrogenic endocrine disruptors: Molecular mechanisms of action[END_REF]. ERs will transducer signals by PI3K, MAPK/ERK and NF-κB signaling pathway[START_REF] Rahal | Paracrine-acting adiponectin promotes mammary epithelial differentiation and synergizes with genistein to enhance transcriptional response to estrogen receptor beta signaling[END_REF], and then influence each other by crosstalk or bypassing at the intracellular level, or deliver signals to different cells or tissues by the secretion of hormones and growth factors to cause completely different types of functional outcome[START_REF] Kiyama | Estrogenic endocrine disruptors: Molecular mechanisms of action[END_REF]. Taken together, these results suggested that YYC may not only alleviate the symptoms of MGH, but also further inhibit MGH development to carcinogenesis by crosstalk of various pathways.

FIGURE 9

 9 FIGURE 9 The schema of synergistic mechanism of YYC on MGH

  the content of total sugar of Polygonati Rhizoma, Lycii Fructus, Codonopsis pilosula, Rehmanniae Radix and YYC respectively 22.9%, 16.21%, 19.2%, 18.4%, 22.31% (Table 1). The protein content of Polygonati Rhizoma, Lycii Fructus, Codonopsis pilosula, Rehmanniae Radix and YYC are respectively 4.28%, 17.67%, 22.19%, 10.68%, 9.72% (Table

Table 1

 1 Physicochemical composition of Polygonati Rhizoma, Lycii Fructus, Codonopsis pilosula,

	Rehmanniae Radix and YYC
	Samples	Total sugar %	Protein %	Molecular Weights (Da)
	Polygonati Rhizoma	22.9%	4.28	< 5× 10 3 Da
	Lycii Fructus	16.21%	17.67	5× 10 3 Da -4.8 × 10 4 Da
	Codonopsis pilosula	19.2%	22.19	5× 10 3 Da -3 × 10 5 Da
	Rehmanniae Radix	18.4%	10.68	< 5× 10 3 Da
	YYC	22.31%	9.72	5× 10 3 Da -2 × 10 4 Da
	After 14 or 30 days' administration of polysaccharides, the nipple diameters were
	obviously suppressed in RPXP (p < 0.05), LF (p < 0.05), PR (p < 0.05) and YYC
	group (p < 0.05) (Figure

3.2 YYC improved the nipple diameter and organ coefficients of mice with MGH

In order to assess the therapeutic efficacy of YYC on MGH, a MGH model-induced by estrogen and progesterone in mice was firstly established. The efficacy of YYC on MGH within 30 days was observed. As shown in Figure

2

, there was no significant difference with nipple diameter between the model group and YYC group in 7 d.

  Compared with the control group, ovary and uterine coefficient were significantly increased in model group. Compared with the

	model group, the ovary index was markedly inhibited in RPXP (p <0.01), CP (p
	<0.05), LF (p <0.01), PR (p <0.01) and YYC group (p <0.001), and uterine coefficient

was significantly reduced in RPXP (p <0.01), LF (p <0.05) and YYC group (p <0.05)

  Lycii Fructus polysaccharide, Rehmanniae Radix polysaccharide are respectively 41, 76, 63, 84. Finally, the Venn diagram was constructed by predicted targets for MGH and targets of four polysaccharides. The targets of affecting MGH for Polygonati

	3.6 The targets of four polysaccharides affecting MGH were predicted by network
	pharmacology
	A total of 3684 targets for MGH were firstly predicted by the GeneCards server.
	Meanwhile, the targets of four polysaccharides were predicted by
	Swiss-target-prediction server. After the duplicate genes have been removed, the
	targets for Polygonati Rhizoma polysaccharide, Codonopsis pilosula polysaccharide,

Control group (Control), Model group (Model), Positive group (Rupixiao Pian, RPXP), Codonopsis pilosula group (CP), Rehmanniae Radix group (RR), Polygonati Rhizoma group (PR), Lycii Fructus group (LF), Yiqi Yangyin Complex group (YYC). Data are expressed as the mean± SEM. # represents MGH model group vs control group ( # p < 0.05, ## p < 0.01); * indicated significant difference in polysaccharide group vs MGH model group (* p < 0.05, ** p < 0.01). ns indicated no significant difference. Rhizoma polysaccharide, Codonopsis pilosula polysaccharide, Lycii Fructus polysaccharide, Rehmanniae Radix polysaccharide were 26, 46, 38, 46 (Figure

  YYC, and the synergistic effect of YYC ingridents probably own to their different polysaccharides interacted to different targets with a synergistic way, and reduce the side effects or enhance the pharmacological potency. Here, it is boldly speculated, although the four polysaccharides alone could not completely affect the expression of related apoptotic factors, the combination of four polysaccharides in equal proportion (YYC) might exhibit synergy effect on MGH by inducing apoptosis. In addition, YYC used for this study has strict quality control to test the polysaccharides content and molecular weight distribution from the herbal sources, extraction, and final product for assuring the stability of every batches, referenced by the standard of the YYC related product successfully launched in market by Infinitus (China) Co., LTD. GO and KEGG pathway analyses showed that YYC could be significantly enriched on estrogen signaling pathway and apoptosis pathway. Interestingly, BCL-2 was as Lycii Fructus polysaccharide and Polygonati Rhizoma polysaccharide affecting on MGH were respectively 16, 21, 23, and 12. To further elucidate the relevant targets of four single polysaccharides, the relevant targets were chosen for analysis. Here, ER, PR, STAT3 and BCL-2 were as our interesting targets for four single polysaccharides. Estrogens are sex steroid hormones, which could regulate menstrual cycle and reproduction, cholesterol mobilization, development of mammary gland and sexual organs, and control of inflammation

common target between estrogen signaling pathway and apoptosis pathway by network pharmacology, indicating that crosstalk of different pathways regulate MGH development. Hence, the above result demonstrated that polysaccharides from YYC may activate multiple mechanisms of action to regulate MGH. To further enrich animal experimental results, the potential targets of the four single polysaccharides on MGH were predicted by network pharmacology. The results showed that the direct targets of Polygonati Rhizoma polysaccharide, Codonopsis pilosula polysaccharide,

Acknowledgments

The current research work was supported financially by National Natural Science Foundation of China (21272043), Guangdong Provincial Department of Science and Technology (2015KQNCX024). National Science Foundation of China (NSFC) (22178070), Project of Rejuvenation of science and technology for Mongolia (2021CG0029).

Availability of data and materials

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Authors' contributions

ZY and DZ conceived the experiments and organized the manuscript. QX wrote the manuscripts. QX and WB performed the experiments and analyzed the data. YZ revised the manuscript. YF performed bioinformatics analysis. WY, ZY, YY, TJ, JZ analyzed the data, LL, ZL contributed reagents and materials. All authors read and approved the final manuscript.

Consent for publication

Not applicable

Conflicts of interest

The authors declare that they have no conflicts of interests.

Ethical statement

The study was approved by the Animal Care and Use Committee of Guangdong University of Technology.

Supplementary FIGURE 1 The spectra of four polysaccharides and YYC were performed by high-performance gel permeation chromatography (HPGPC). Polygonati


Lorenzo Nasi

Marie-Christine Firpo
email: firpo@lptp.polytechnique.fr

Enhanced confinement with increased extent of the low magnetic shear region in tokamak plasmas

Keywords: numbers: 52.55.Fa, 05.45.+a, 52.25.Gj

The Hamiltonian representation of magnetic field lines enables to study their confinement properties in tokamaks through the use of symplectic maps such as the symmetric tokamap and its bounded version, the latter being introduced here. In this time-independent purely magnetic framework, we observed the drastic improvement on the confinement of magnetic field lines produced by the local vanishing of the shear profile. This amounts to a non-twist condition that notably acts in the same way the safety profile being (non-strictly) monotonic or having reversed-shear. We single out the effect of the amount of flatness of the safety profile in the vicinity of its zero shear point. All other things being equal, the beneficial effect of the vanishing of the shear profile is shown to be increased if the radial extent of the low-shear region is increased. To be specific, the low-shear region induces the formation of a belt of robust KAM tori acting as an internal transport barrier whose width is all the larger as the extent of the low-shear region is broad.

Introduction

In order to approach ignition, it is required that the present hot tokamak plasmas have a better energy confinement. The energy confinement time is commonly limited by deleterious small-scale turbulent transport. Because internal transport barriers (ITB) [START_REF] Wolf | Internal transport barriers in tokamak plasmas[END_REF][START_REF] Connor | A review of internal transport barriers physics for steady-state operation of tokamaks[END_REF][START_REF] Tala | Physics of internal transport barriers[END_REF] reduce or even quench the turbulent transport within the inner plasma region improving energy confinement in the plasma core which should yield high fusion gains, they have become an essential ingredient of advanced scenarios in which the tokamak could operate as an economic steady-state fusion reactor.

The physics of the triggering and sustainment of ITB as well as the explanation of their very nature are still subjects of intensive investigations [START_REF] Tala | Physics of internal transport barriers[END_REF]. Besides an increasingly detailed empirical and experimental knowledge, their theoretical modeling is still incomplete. It is commonly admitted that two ingredients should control their formation, namely the E×B flow shear and the magnetic shear through the safety factor (q) profile. An increased number of works have recently reported the importance of the q-profile. In particular, Eriksson et al. [START_REF] Eriksson | Discharges in the jet tokamak where the safety factor profile is identified as the critical factor for triggering internal transport barriers[END_REF] demonstrated experimentally in the JET tokamak the critical role of the safety factor profile in the formation of ITB by using it as a control parameter in discharges where other parameters, including the E × B flows, were kept the same. They obtained an ITB for a non-monotonic q profile (i.e. with reversed shear) but not with a monotone q. Actually these reversed shear scenarios have been recognized as favorable experimental conditions to obtain ITB whereas the ITB's locations have been mostly related to the low shear zones.

In this article, we wish to study the appearance and robustness of ITB within a purely magnetic and static (time-independent) framework following a line of approach notably initiated by Balescu and coworkers. This simplified, yet relevant, approach is guided by the fact that, in a first approximation, charged particles in a tokamak follow the magnetic field lines so that transport properties for the laters transfer to the formers. We shall focus on the influence of the behaviour of the q-profile in the vicinity of its vanishing shear region on the appearance and robustness of ITB. To be more specific, we wish to single out the role and importance of the amount of flatness of the q-profile in the vicinity of its minimum (for the reversed shear case) or in the vicinity of its inflexion point (in the case of a non-strictly monotonic profile).

In Section 2, the Hamiltonian formulation of magnetic field lines is recalled and symplectic maps are introduced. In order to capture more accurately the magnetic field lines, we shall implement the symmetric tokamap and its bounded version, instead of the standard tokamap. In Section 3, we shall study numerically the influence of the flatness of the q-profile, in the vicinity of a given inflexion or minimum point, on the overall magnetic field lines stochasticity. To do so, we shall consider a family of polynomial winding number profiles. In Section 4, we eventually consider a sample of tokamak-oriented realistic q-profiles with reversed shear. This brings another evidence on the favorable role played by an increased extension of the low-shear region around the minimum of the q-profile. We conclude in Section 5.

Hamiltonian formulation for magnetic field lines and maps

Hamiltonian framework

In order to investigate the role of the q-profile on the existence and robustness of ITBs, we shall use the Hamiltonian structure of the magnetic field lines. We shall use in the following traditional generalized toroidal coordinates ψ, θ, ζ where ψ is the flux coordinate (so that ψ = 0 and ψ = 1 correspond to the magnetic axis and to the the plasma boundary, respectively), θ is the poloidal angle and ζ the toroidal angle. From Maxwell's equation ∇.B = 0, the "equations of motion" for magnetic field lines [START_REF] Hazeltine | Plasma Confinement[END_REF] in a tokamak are

dψ dζ = - ∂F ∂θ , dθ dζ = ∂F ∂ψ (1)
where F (ψ, θ, ζ) is the poloidal flux. Equations ( 1) have a Hamiltonian structure: the toroidal angle ζ plays the role of "time", F is the Hamiltonian and θ and ψ are canonically conjugated coordinates. In order to consider perturbations from the ideal i.e. integrable case, where F = F 0 (ψ) and magnetic surfaces are nested tori wound around the magnetic axis, let us introduce a stochasticity parameter, K, such that the Hamiltonian F is given by the sum of an unperturbed term F 0 and a perturbation δF (ψ, θ, ζ) :

F = F 0 (ψ) + KδF (ψ, θ, ζ). (2)
As K increases, magnetic surfaces may no longer exist and the problem of their existence is formally equivalent to that of the integrability of a Hamiltonian with one-and-a-half degrees of freedom. In realistic tokamaks, these magnetic perturbations may result e.g. from coil imperfections and/or internal instabilities (tearing). Within the above formalism, the safety factor q(ψ) is simply related to the unperturbed Hamiltonian since q(ψ) = 1/W (ψ) where W (ψ) ≡ dF 0 /dψ is the winding number. Therefore q(ψ), the ratio of the number of toroidal turns per poloidal turn, is the inverse of the unperturbed Hamiltonian. Moreover, s = d ln q/d ln ψ is the shear profile.

Maps

2.2.1. Tokamap Instead of performing a time consuming integration of the field line differential equations (1), we shall use a simplified but relevant approach, namely the Poincaré map associated to a given poloidal cross-section [START_REF] Abdullaev | Construction of Mappings for Hamiltonian Systems and Their Applications[END_REF]. It is defined from the intersection points of a magnetic field line starting at position (ψ 0 , θ 0 ) with some given poloidal section ζ = cst. The intersection point after ν toroidal turns is denoted by (ψ ν , θ ν ). The map we shall be interested in, the tokamap, was introduced by Balescu and coworkers [START_REF] Balescu | Tokamap : A hamiltonian twist map for magnetic field lines in toroidal geometry[END_REF][START_REF] Misguich | Noble internal transport barriers and radial subdiffusion of toroidal magnetic lines[END_REF][START_REF] Weyssow | Hamiltonian map for guiding centres in a perturbed toroidal magnetic geometry[END_REF]. Its implicit form is

ψ ν+1 = ψ ν -K ψ ν+1 1 + ψ ν+1 sin θ ν (3)
θ ν+1 = θ ν + W (ψ ν+1 ) - K 2π 1 (1 + ψ ν+1 ) 2 cos θ ν .
Explicitly, we have

ψ ν+1 = 1 2 P (ψ ν , θ ν ) + [P (ψ ν , θ ν )] 2 + 4ψ ν (4)
θ ν+1 = θ ν + W (ψ ν+1 ) - K 2π 1 (1 + ψ ν+1 ) 2 cos θ ν (5)
where

P (ψ, θ) = ψ -1 -K sin θ. ( 6
)
This map is an area-preserving map which is compatible with the toroidal geometry [START_REF] Balescu | Tokamap : A hamiltonian twist map for magnetic field lines in toroidal geometry[END_REF] due to the following important physical properties : (i) the toroidal flux ψ is always positive (this is necessary since ψ ∼ r 2 where r is the minor radius of the tokamak) and (ii) the magnetic axis is invariant (ψ = 0 is mapped to ψ = 0). It has proved to be an important model since it correctly describes the qualitative features of the magnetic field lines stochasticity known in tokamak physics.

2.2.2. Symmetric tokamap Some studies have recently pointed some limitations of this model. It actually turns out that despite a qualitative agreement with the actual magnetic field behaviour obtained from the continuous Hamiltonian system formed by Eqs. ( 1) and ( 2), the tokamap is far less accurate than its symmetric version as proposed by Abdullaev [START_REF] Abdullaev | On mapping models of field lines in a stochastic magnetic field[END_REF][START_REF] Abdullaev | Construction of Mappings for Hamiltonian Systems and Their Applications[END_REF]. We shall therefore implement in this study the symmetric tokamap instead of the standard one. The symmetric tokamap very closely describes the continuous Hamiltonian system (2) with

KδF (ψ, θ, ζ) = - K (2π) 2 ψ 1 + ψ M n=-M cos(θ -nζ), (7)
and 2M + 1 ≫ 1, from which it is derived. Its implicit form is given by

Ψ ν = ψ ν - K 2π Ψ ν 1 + ψ ν sin θ ν Θ ν = θ ν - K 2π 1 (1 + Ψ ν ) 2 cos θ ν Θ ν+1 = Θ ν + 2πW (Ψ ν ) (8)
Ψ ν+1 = Ψ ν ψ ν+1 = Ψ ν+1 - K 2π Ψ ν+1 1 + Ψ ν+1 sin θ ν+1 θ ν+1 = Θ ν+1 - K 2π 1 (1 + Ψ ν+1 ) 2 cos θ ν+1
where K is the stochasticity parameter and the auxiliary variables Θ and Ψ are determined iteratively by the map from a given starting point (θ 0 , ψ 0 ). Contrarily to the tokamap, a numerical approximation is necessary in order to solve the last equation of the symmetric tokamap [START_REF] Misguich | Noble internal transport barriers and radial subdiffusion of toroidal magnetic lines[END_REF] with respect to variable θ ν+1 , but this is an affordable task with usual interpolation methods such as Newton's method or the Brent's algorithm [START_REF] Brent | Algorithms for Minimization Without Derivatives[END_REF] that we used here.

Bounded symmetric tokamap

Another adjustment to the tokamap has also been recently proposed: the bounded tokamap. Whereas the tokamap does not impose any upper boundary limit for ψ, so that it can be directly used in the ergodic divertor problem [START_REF] Abdullaev | Asymptotical and mapping methods in study of ergodic divertor magnetic field in a toroidal system[END_REF] where magnetic field lines may go outside the last magnetic surface (defined as ψ = 1), its bounded version is constructed by requiring that ψ = 1 cannot be crossed. We used the approach developed in Refs. [START_REF] Kuzovkov | Bounded tokamap[END_REF][START_REF] Kuzovkov | Bounded tokamap : a hamiltonian map for magnetic field lines in a tokamak[END_REF] to propose the symmetric version of the bounded tokamap with

KδF (ψ, θ, ζ) = - K (2π) 2 ψ(1 -ψ) M n=-M cos(θ -nζ) (9)
and 2M + 1 ≫ 1. This form of perturbation ensures that not only the magnetic axis, corresponding to ψ = 0, but also the plasma boundary, corresponding to ψ = 1 are invariant under the map and cannot be crossed. Then, applying the same method [START_REF] Abdullaev | On mapping models of field lines in a stochastic magnetic field[END_REF][START_REF] Abdullaev | Construction of Mappings for Hamiltonian Systems and Their Applications[END_REF] as in Sec. 2.2.2, we propose the symmetric version of the bounded tokamak as

Ψ ν = ψ ν - K 2π Ψ ν (1 -Ψ ν ) sin θ ν Θ ν = θ ν - K 2π (1 -2Ψ ν ) cos θ ν Θ ν+1 = Θ ν + 2πW (Ψ ν ) (10)
Ψ ν+1 = Ψ ν ψ ν+1 = Ψ ν+1 - K 2π Ψ ν (1 -Ψ ν ) sin θ ν+1 θ ν+1 = Θ ν+1 - K 2π (1 -2Ψ ν ) cos θ ν+1 .
The first equation of [START_REF] Abdullaev | On mapping models of field lines in a stochastic magnetic field[END_REF] has two roots for Ψ ν and the positive one is retained. This modeling is valid for not too large values of the stochasticity parameter as it is required that K/2π < 1.

Motivations

From the theoretical point of view, the effect of various parameters describing the qprofile, such as the boundary values of the q-profile, the value of the minimum of the qprofile in the reversed shear version (in the so-called rev-tokamap), for Hamiltonian maps have already been considered [START_REF] Balescu | Hamiltonian nontwist map for magnetic field lines with locally reversed shear in toroidal geometry[END_REF][START_REF] Constantinescu | Transport barriers and diffusion phenomena for the magnetic field lines in tokamak[END_REF]. Yet, this has been mainly limited to restricted choices of q profiles. In addition, to the authors' knowledge, an essential point has not been investigated systematically: the effect of the local flatness of the q-profile on (hopefully) realistic models for magnetic field lines in a tokamak. This is in our sense a crucial point since there is strong experimental evidence that the radial extent of the low shear region of safety factor's profile plays an outstanding role on ITBs' appearance and robustness [START_REF] Baranov | On the link between the q-profile and internal transport barriers[END_REF][START_REF] Connor | A review of internal transport barriers physics for steady-state operation of tokamaks[END_REF][START_REF] Tala | Physics of internal transport barriers[END_REF]. Thus, modeling is needed in order to asses one of the "critical physics issues relevant to the extrapolation of ITB regimes to next-step experiments, such as ITER " [START_REF] Litaudon | Internal transport barriers : critical physics issues?[END_REF] : the optimum q-profile. This forms the ultimate perspective of the present work.

Numerical results on the stochasticity of magnetic field lines for polynomial winding number profiles

We are interested in assessing the influence of the extent of the low shear region on magnetic confinement. Since it happens that the Hamiltonian ( 2) is more directly linked to the winding number, since W (ψ) ≡ dF 0 /dψ, than the q-profile, we shall consider it in the following. We are then particularly interested in the influence of the behaviour of the W -profile in the vicinity of some unique ψ 0 such that W ′ (ψ 0 ) = 0. If such a ψ 0 exists, the twist condition for the corresponding map is violated, that is the symmetric tokamap is non-twist. A straightforward way to study such behaviour is to consider the family of polynomial W -profiles :

W (ψ) = -c(ψ -ψ 0 ) n + W 0 ( 11
)
where n is a positive integer representing the degree of degeneracy of the zero ψ 0 of the shear profile ‡ associated to [START_REF] Brent | Algorithms for Minimization Without Derivatives[END_REF]. Although there is an extensive bunch of experimental evidence that the maximum value of the winding number influences ITB formation in the reversed shear case [START_REF] Baranov | On the link between the q-profile and internal transport barriers[END_REF][START_REF] Connor | A review of internal transport barriers physics for steady-state operation of tokamaks[END_REF][START_REF] Tala | Physics of internal transport barriers[END_REF], this will not be our point here and we shall rather impose here on an arbitrary basis W 0 = 5/2π, c = 1.3 and ψ 0 = 0.5 in order to focus on the flatness of the W -profile only.

For every profile [START_REF] Brent | Algorithms for Minimization Without Derivatives[END_REF], with the degree n going from 1 to 8, the phase portrait of the symmetric tokamap (8) has been computed. The results of this iterative process, for ψ in the physical domain [0, 1], are given for the same value K = 3 of the stochasticity parameter in Fig. 1. Several comments are called for. First of all, it is clear from the figure that the case n = 1 is dramatically different from the others. In the case n = 1, the map is twist as W ′ (ψ) does not vanish. The figure exemplifies the dramatic improvement on confinement induced by the non-twist condition that is satisfied when n ≥ 2. This has been discussed recently on a general basis by Rypina et al. [START_REF] Rypina | Robust transport barriers resulting from strong kolmogorov-arnold-moser stability[END_REF]. Secondly, when the non-twist condition is satisfied, the core magnetic field lines are almost integrable. The central non-chaotic region is all the broader in ψ-space as the W -profile is flat around ψ 0 , namely as n is large. Yet this effect saturates rapidly: for n ≥ 6, the phase space portraits are almost indistinguishable. This is not surprising as the corresponding q-profiles are indeed almost identical. Thirdly, there is a perfect progression in the phase space appearances between the odd and even n-exponents as n increases in the non-twist case i.e. for n ≥ 2. This means that the fact that the q-profile has a reversed shear or not is not important in terms of the confinement of the magnetic field lines. What matters is the degree of degeneracy of the zero of the shear profile or, so to speak, the amount of local flatness of the W -profile. To check this explicitly, let us consider the two following profiles

W m (ψ) = -c(ψ -ψ 0 ) 3 + W 0 (12)
W rs (ψ) = -c|(ψψ 0 )| 3 + W 0 (13) ‡ More precisely, ψ 0 is a zero of the shear profile when n ≥ 2. where W m [START_REF] Abdullaev | Asymptotical and mapping methods in study of ergodic divertor magnetic field in a toroidal system[END_REF] and W rs [START_REF] Kuzovkov | Bounded tokamap[END_REF] correspond respectively to a monotonic and to a reversedshear profiles with the same local flatness (n = 3). The results are displayed in figure 2. This shows that the global behaviour of the magnetic field lines is indeed the same in both cases: the outer stochastic region is identical. There are only some limited apparent topological discrepancies in the inner ordered zone. This third point is rather novel since previous studies [START_REF] Balescu | Hamiltonian nontwist map for magnetic field lines with locally reversed shear in toroidal geometry[END_REF] commonly used to focus solely on reversed-shear q-profiles to get the non-twist condition.

We have brought some numerical evidence of the influence on magnetic confinement of the amount of flatness of the q-profile around its extremum. In order to definitely ascertain this phenomenon, we eventually wish to avoid the possible bias induced by using different edge values of W for each n in Fig. 1. It actually happens that increasing the edge value of a polynomial W -profile [START_REF] Brent | Algorithms for Minimization Without Derivatives[END_REF] with given degree n may induce the stochastization of the, otherwise regular, core magnetic field lines. This is illustrated in Fig. 3 for the case n = 2. Therefore, in order to single out the effect of the flatness of the W -profile i.e. of the radial extent of the low-shear region around an extremum, the study shown in Fig. 1 was revised for the polynomial profiles [START_REF] Brent | Algorithms for Minimization Without Derivatives[END_REF] with variable c = c n such that W (0) = W (1) be a given constant for all n. Results are depicted in Fig. 4 for even values of n between 2 and 8. This figure shows that, all other things being equal, the radial extent of the regular magnetic field lines is all the wider as the low shear 8). (left) W m (ψ) is given by ( 12); (right) W rs (ψ) is given by [START_REF] Kuzovkov | Bounded tokamap[END_REF]. Parameters K = 3, W 0 = 5/2π, c = 1.3 and ψ 0 = 0.5.

region around the extremum of the q-profile is broad.

Finally, we wish to insist on the fact that the above conclusion does not depend on the particular choice of the map. It remains valid when considering the -less accuratestandard tokamap and the bounded version of the symmetric tokamap. In this case, the appearance of internal transport barriers can be seen more easily. This can be shown in Fig. 5. Whereas global chaos can be observed for n = 1, the non-twist effect that takes places for n ≥ 2 dramatically improves confinement. A clear internal transport barrier emerges for n = 2 separating the internal and the outer chaotic regions. Increasing n, that is the extent of the low shear region, induces a drastic reduction of chaos.

Influence of the radial extent of the low-shear region for some realistic reversed shear q-profiles

The previous academic case of polynomial centered W -profiles is not realistic. We wish in this Section to briefly test our results on the beneficial effect of the increased radial extent of the low shear domain on more experimentally compatible q-profiles. We shall here consider reversed shear profiles leading to ITBs. Typical locations of the minimum of the q-profile correspond to a normalized radius of 0.3-0.4 [START_REF] Wolf | Internal transport barriers in tokamak plasmas[END_REF] and to a central q(0) ≃ 3 while q min , the minimum value of q, is about 2. We have considered three q-profiles 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.5 compatible with these realistic data and computed the corresponding phase portraits for the magnetic field lines. The results are given in Fig. 6. It is once again clear from this figure that, for increasing flatness of q about its minimum and for constant K, the extent of the non-chaotic, regular, magnetic field lines in the plasma core is broader. In order to show that this effect is robust, namely map independent, we have computed the phase space portraits for the symmetric bounded tokamap [START_REF] Abdullaev | On mapping models of field lines in a stochastic magnetic field[END_REF] for the same q-profiles. Results are plotted in Fig. 6. We do observe once again that for increasing flatness there is chaos reduction.

A quantitative indicator of the degree of flatness of some W -profile around its extremum located at ψ 0 could be given by f

W (ψ 0 ) ≡ lim ψ→ψ 0 ln |W (ψ) -W (ψ 0 )| / ln |ψ -ψ 0 |.
As shown previously on polynomial profiles, for a given stochasticity parameter, confinement is improved as f W (ψ 0 ) increases. However, f W (ψ 0 ) gives the power of the first non-vanishing term in the Taylor expansion of the W -profile around ψ 0 . Since realistic reversed-shear profiles are generically approximated locally by a parabola, this would yield f W (ψ 0 ) = 2. This has been checked on the above profiles. It is thus necessary to produce a practical indicator of the local amount of low shear. Let us then assume that the reversed shear profile may be locally Taylor expanded as

W (ψ) = W (ψ 0 ) + 1 2 W ′′ (ψ 0 )δψ 2 + O(δψ 3 ), (14)
with δψ ≡ ψψ 0 . Then, for a given W (ψ 0 ), the local flatness of W increases as |W ′′ (ψ 0 )| becomes smaller. This is in agreement with the results given in Fig. 6 where a diminution of W ′′ (ψ 0 ) corresponding to an increased local flatness is observed from left to right. More precisely, one obtains numerically W ′′ (ψ 0 ) = 10.6 for the profile located at the left of Fig. 6, W ′′ (ψ 0 ) = 7.8 for the middle profile and W ′′ (ψ 0 ) = 7.4 for the right one. Finally, let us note that in this Figure as well as in Fig. 5, the stochasticity parameter K, that is proportional to the relative amount of magnetic perturbation [START_REF] Misguich | Noble internal transport barriers and radial subdiffusion of toroidal magnetic lines[END_REF], has been taken unrealistically large. This points to the fact that the effect of low shear is much stronger than usual KAM results on the survival of tori that would be obtained for monotone q-profiles and corresponding twist maps [See e.g. the large scale chaos exhibited in the tokamap phase portrait of Fig. 9 in the work by Misguich et al. [START_REF] Misguich | Noble internal transport barriers and radial subdiffusion of toroidal magnetic lines[END_REF] that is obtained also for K = 6].

Conclusions

The Hamiltonian formalism has been used to account for the behavior of magnetic field lines under some perturbations of given equilibrium safety factor profiles. The corresponding symplectic maps were the symmetric tokamap and its bounded version in which the last magnetic surface cannot be crossed. In this time-independent framework, (top) q-profiles of increasing local flatness around q min ; (middle) corresponding phase space portraits of the symmetric tokamap (8) for 500 iterations and K = 6; (bottom) same as above for the bounded symmetric tokamap [START_REF] Abdullaev | On mapping models of field lines in a stochastic magnetic field[END_REF].

we have shown the drastic improvement on the confinement of magnetic field lines produced by the local vanishing of the shear profile: the existence of some ψ 0 such that q ′ (ψ 0 ) = 0 is a non-twist condition that acts in the same way the safety profile being monotonic or having reversed-shear. All other things being equal, the beneficial effect of this vanishing of the shear profile was shown to be increased if the radial extent of the low-shear region was increased. To be specific, the low-shear region induces the formation of a belt of robust KAM tori acting as an internal transport barrier whose width is all the larger as the extent of the low-shear region is broad. This conclusion has been derived here without any consideration of dynamical stability. It is important to note that there are additional dynamical arguments that favor a reduction of (electrostatic) turbulence for low magnetic shear, since it is well known that the linear growth rates of drift and ballooning type modes vanish at zero shear [START_REF] Connor | Microstability in tokamaks with low magnetic shear[END_REF].

Finally, we wish to conclude by discussing the validity of this simplified purely magnetic approach. An obvious limitation of the model comes from its lack of selfconsistency. In particular, there is no retroaction on the turbulence level. However, we would like to stress that such a simplified picture, being in a Hamiltonian form (due to the magnetic field being divergence-free), inherits from robustness properties w.r.t. fluctuations. This is a general property of Hamiltonian systems and ensures that their long-time simulations are valid, at least in an average sense [START_REF] Steinbrecher | Stability under perturbations of the large time average motion of dynamical systems with conserved phase space volume[END_REF]. In the present model, this means that introducing some small perturbation in the symplectic symmetric tokamap, such a perturbation being e.g. a reduction of the magnetic turbulence [START_REF] Zou | Internal magnetic turbulence measurement in plasma by cross polarization scattering[END_REF] or another perturbation that may reflect the effect of electrostatic turbulence, the essential features of the model are fully preserved.

It is now desirable to improve the connection of the present results with tokamak experiments and complete the modeling to address thermal transport. This would amount to couple the present magnetic approach to some heat transport modeling. Nevertheless, we believe that the benefits of the purely magnetic approach, initiated notably by Balescu, have not been exhausted yet. The role of the value of the qprofile in the low shear zone as well as the role of edge-gradients of the q-profile are other challenging important questions on which we shall be focusing our attention in a shortcoming future.

Figure 1 .

 1 Figure 1. Phase space portraits of the symmetric tokamap (8) for different values of the order of degeneracy n of the profile (11). Parameters are K = 3, W 0 = 5/2π, c = 1.3 and ψ 0 = 0.5.

Figure 2 .

 2 Figure2. Phase space portraits of the symmetric tokamap[START_REF] Misguich | Noble internal transport barriers and radial subdiffusion of toroidal magnetic lines[END_REF]. (left) W m (ψ) is given by (12); (right) W rs (ψ) is given by[START_REF] Kuzovkov | Bounded tokamap[END_REF]. Parameters K = 3, W 0 = 5/2π, c = 1.3 and ψ 0 = 0.5.

Figure 3 .

 3 Figure 3. Phase space portraits of the symmetric tokamap (8) with W (ψ) = 5/2π + c(ψ -.5) 2 and K = 3 for: (left) c = 2, (right) c = 1.

Figure 4 .

 4 Figure 4. (up) Reversed-shear q-profiles given by q(ψ) = 1/[5/2π + c n (ψ -.5) n ] with identical boundary values and increasing central flatness (with n even between 2 and 8) with (down) their corresponding phase space portraits of the symmetric tokamap (8) for K = 3.

Figure 5 .Figure 6 .

 56 Figure 5. Phase space portraits of the bounded symmetric tokamap[START_REF] Abdullaev | On mapping models of field lines in a stochastic magnetic field[END_REF] for different values of the order of degeneracy n (n = 1, 2, 3 and 4) of the profile[START_REF] Brent | Algorithms for Minimization Without Derivatives[END_REF]. Parameters are K = 6, W 0 = 5/2π, c = 1.3 and ψ 0 = 0.5.

Acknowledgments

Fruitful discussions with M. Bécoulet, X. Garbet, P. Hennequin, H. Lütjens and J.-M. Rax are gratefully acknowledged. The authors especially thank G. Steinbrecher for useful communications on tokamap related works.