Androgens, the main male sex steroids, are the critical factors responsible for the development of the male phenotype during embryogenesis and for the achievement of sexual maturation and puberty. In adulthood, androgens remain essential for the maintenance of male reproductive function and behavior. Androgens, acting through the androgen receptor (AR), regulate male sexual differentiation during development, sperm production beginning from puberty, and maintenance of prostate homeostasis. Several substances present in the environment, now classified as endocrine disruptors (EDCs), strongly interfere with androgen actions in reproductive and non-reproductive tissues. EDCs are a heterogeneous group of xenobiotics which include synthetic chemicals used as industrial solvents/lubricants, plasticizers, additives, agrochemicals, pharmaceutical agents, and polyphenols of plant origin. These compounds are even present in the food as components (polyphenols) or food/water contaminants  rendering the diet as the main route of exposure to EDCs for humans. Although huge amount of literature reports the (anti)estrogenic effects of different EDCs, relatively scarce information is available on the (anti)androgenic effects of EDCs. Here, the effects and mechanism of action of phytochemicals and pesticides and plasticizers as possible modulators of AR activities will be reviewed taking into account that insight derived from principles of endocrinology are required to estimate EDC consequences on endocrine deregulation and disease. Unfortunately, the increased production of new chemicals is associated with their release in the environment and, mainly through the food chain, to their potentially harmful effects on human and wildlife health. One of the most unpredictable and serious consequences of this phenomenon is the potential interference with the endocrine system of these man-made chemicals (or xenobiotics) defined as endocrine disruptors (EDs) or endocrine disrupting chemicals (EDCs). The most recent worldwide accepted definition of EDC came from the World Health Organization that made an authoritative definition of an ED as “an exogenous substance or mixture that alters function(s) of the endocrine system and consequently causes adverse health effects in an intact organism, or its progeny, or (sub)populations .” A recee action  devoid oe action . Indeed,e action .EDCs are a heterogeneous group of xenobiotics –5 which Here, the effects and mechanisms of action of food components and food/water contaminants , acting as modulators of androgen receptor (AR) activities, are reviewed.Androgens are all steroids with 19 carbons  and benign prostate hyperplasia (BPH), increases with age. This increased incidence could be related to the local conversion of T to DHT being 5α-reductase upregulated [d = 2 nM, T Kd = 8 nM) [N-terminal domain , with properties of transcriptional activation function (AF-1); (ii) a DNA binding domain ; (iii) a short hinge region ; and (iv) a C-terminal domain , which contains the ligand binding domain (LBD) and a coactivator binding surface (AF-2)  –23. The  = 8 nM) . The AR N and C termini of AR could prevent inappropriate co-regulator recruitment to the receptor until it is engaged with DNA. Intriguingly, the natural occurring atraric acid inhibits the transactivation of AR and androgen-mediated growth of AR-expressing human PCa cell lines by inhibiting the AR N/C-terminal interaction [The inactive AR is localized in the cytoplasm associated with a heath shock protein (HSP)90 chaperone complex; it undergoes to proteasome-mediated degradation in the absence of ligand , 29. Upoeraction .2+ concentration, and cyclic adenosine monophosphate (cAMP) levels [In addition to this canonical nuclear (or genomic) mechanism of action, AR-dependent, rapid (seconds to minutes) extra-nuclear mechanisms occur upon androgen treatment. These extra-nuclear mechanisms start at the plasma membrane and involve extracellular signal-regulated kinase (ERK), the phosphatidyl-inositol 3-kinase (PI3K)/Akt pathway, G protein coupled receptors (GPCRs), intracellular Ca) levels –37. Thes) levels  that was) levels . Palmito) levels . On the ) levels .As a whole, the pleiotropic effects elicited by androgens are obtained by different signal transduction pathways , whose activation depends on the cellular context of the target cell, the AR intracellular localization , and the ligand itself  .The male reproductive system comprises the paired units consisting of the testis, epididymis and vas deferens, and the penis and scrotum. The prostate, seminal vesicles, and bulbourethral glands are the male reproductive system accessory glands. The male testis have the dual responsibilities for the production and release of the germ cells and for the biosynthesis and secretion of T. Prostate plays an essential role in male reproduction secreting the prostatic fluid (highly responsive to androgens), an essential component of the seminal fluid , 42. TheAndrogens are critical for male sexual differentiation, pubertal development, spermatogenesis, and maintenance of adult secondary sexual characteristics. However, androgens are pleiotropic hormones since they exert biological effects in many different non-reproductive tissues and cell types. Androgens act on the male reproductive tract inducing in utero differentiation and growth of the epididymis, seminal vesicles, and vas deferens. Prostate cell growth, function, and homeostasis are regulated by complex systemic and local mechanisms involving either the action of androgens and growth factors produced by the pituitary or the prostatic stroma . After tIntriguingly, the adult human male produces approximately 45 μg per day of E2, the most active within estrogens, mostly from the aromatization of T in the adipose tissue, brain, bone, breast, blood vessels, liver, and both Sertoli and Leydig cells of the testes. The T aromatization is a critical step for the closure of epiphyseal plate of the bone during puberty, for mineral resorption of the bone, and for brain function including mood and the regulatory feedback of LH production . MechaniPCa is the most frequently diagnosed non-skin malignant tumor and the third leading cause of cancer mortality in men. It is estimated that, in Western countries, about 30% of all men will develop microscopic PCa during their lifetime. PCa consists of glandular epithelial cells from the prostate gland. The tumor usually grows slowly and remains confined to the gland for many years. During this time, the tumor produces little or no symptoms or outward signs. As the cancer advances, however, it can spread beyond the prostate into the surrounding tissues and can metastasize throughout other areas of the body, such as the bones, which is the preferential metastasis site of PCa. Androgens have long been established as playing a role in the causation of PCa . AlthougAlthough the aggressive phase of PCa is androgen-independent, prostate cancerous cells still require AR to survive and proliferate. It may appear as nonsense, but many mechanisms are thought to participate to AR aberrant signaling in PCa in the absence of circulating androgens. Indeed, several AR truncated forms have been discovered in PCa, even if many of them have been also identified in non-cancerous tissues Fig. . ExpressA meta-analysis from 1992  indicated that there had been a decline in semen quality during a period of half a century . AlthougPesticides are defined as substances or mixtures of substances intended for controlling, preventing, destroying, repelling, or attracting any biological organism deemed to be a pest . InsectiEpidemiological studies have identified pesticide application in agriculture and pesticide manufacture as associated with the PCa . The expVIN is a dicarboxymide fungicide, widely used on fruits and vegetables, acting as an AR antagonist in vitro and/or in vivo , 74. Morlevator ani plus bulbocavernosus muscles) in a dose-dependent manner in the Hershberger assay using castrated immature rats treated with T [With the same molecular mechanism, and with almost the same potency as the classical antiandrogenic drug flutamide, the two VIN primary metabolites, M1 and M2  of DHT. Therefore, it has been suggested that GA could act through a mutated AR bearing the point mutation T877A expressed in LNCaP cells [Albeit GA was not reported to have any hormone-like activity, its potential influence on AR-dependent or AR-independent-mediated pathways was recently investigated by cell-based in vitro assays . InteresaP cells .GLYP, a glycine derivate, is the active ingredient of several broad-spectrum herbicide formulations used on multiple food and non-food crops. GLYP kills plants by inhibiting 5-enolpyruvylshikimate-3-phosphate synthase, a key enzyme in the shikimate biosynthetic pathway necessary for the production of the aromatic amino acids, auxin, phytoalexins, folic acid, lignin, plastoquinones, and many other secondary products. The carcinogenic potential of GLYP, and its formulations, is a recent matter of debate at the regulatory and scientific level. Indeed, although IARC classified GLYP as a “probable human carcinogen” (IARC category 2A), due to sufficient evidence of carcinogenicity in animals, limited evidence of carcinogenicity in humans and strong evidence for two carcinogenic mechanisms have been reported and considered relevant for its toxic mechanism of action. On the other hand, EFSA reached opposite conclusions and stated that “classification and labelling for carcinogenesis is not warranted” and “glyphosate is devoid of genotoxic potential.” Such position of EFSA has been deeply criticized due to the fact that they did not gave a relevant importance to data obtained by rodent experimental models, particularly to renal carcinogenicity, as IARC did . HoweverPlasticizers are additives that increase the plasticity or viscosity of a material Fig. . PlasticPlasticizers are not covalently bound to the polymer matrix, thus, the abrasion of the plastic/resin as well as changes of temperature and pH allow plasticizers to migrate in food; therefore, food may contain detectable levels of these compounds.Phthalates, or phthalate esters, such as DBP and DEHP, are commonly used plasticizers, primarily necessary to soften polyvinyl chloride (PVC). High exposure levels to phthalates, in particular to DEHP, are due to their presence in medical devices . PhthalaAlthough the phthalate DEHP does not bind to AR, in utero exposure to phthalates disrupts the differentiation program of androgen-dependent tissues in male rat offspring , 117–120The reproductive tract malformations in the androgen-dependent tissues are similar but not equal to the effects of antiandrogen pesticides such as VIN (see paragraph 5). Phthalates have been shown to reduce testicular T levels in fetal and neonatal male rats . This deConsiderable amounts of BPA (ranging from 0.25 to 1.11 mg/kg) have been found in randomly selected fresh food samples from an area of Southern Italy, probably deriving from plastic irrigation pipes . ConsequExposure to BPA has been associated to a reduced proportions of male births in the populations of a number of countries, increased the risk of cryptorchidism and hypospadias, and reduced semen quality in males suggesting a possible BPA interference with the male reproductive function. However, very few data are available on BPA effects on AR transcriptional activity, while a lack of knowledge is still present on the ability of these compounds to interfere with androgen-dependent extra-nuclear signals , 131. BPExperiments performed in rodent models and human prostate cell lines showed that BPA can influence carcinogenesis, modulate PCa cell proliferation, and for some tumors, stimulate progression. Early life exposure to BPA may increase susceptibility to hormonal carcinogenesis in the prostate gland, possibly by developmentally reprogramming carcinogenic risk . StudiesPhytochemicals are a ubiquitous class of plant secondary metabolites; some are responsible for color and other for organoleptic properties of fruits and vegetables. A “recommended” human diet should warrant a high proportion of energy from fruits and vegetables, therefore providing, among other factors, a huge intake of phytochemicals in general considered “health promoting” by virtue of their antioxidant activity and positively modulating, either directly or indirectly, cellular and tissue redox balance . HoweverNKX3.1 whose expression is associated with a more aggressive phenotype of PCa [ODC) mRNA was inhibited by QRC. The product of ODC gene is the key regulator of the synthesis of polyamines, which are essential for cell proliferation. ODC is critical in cell transformation and suggested to be a proto-oncogene [PSA, KLK2, NKX3.1, and ODC, which play roles in development and progression of PCa. Overall, QRC has the potential to become a chemo-preventive and/or chemotherapeutic agent for PCa.Flavonoids are widely present in fruits, vegetables, and beverages  and in many dietary supplements and herbal remedies Fig. . Quercete of PCa . In addioncogene . It was oncogene . QRC hasGenistein is the most abundant isoflavone in soybeans. It exhibited diverse biological activities, among these, its anticancer effects are most noteworthy . ThroughBesides GEN, soy isoflavones consist of several types of other components, such as daidzein, the less abundant glycitein, and the metabolite equol. Daidzein is metabolized in the intestine to equol at relatively low or high levels dependent upon several biological, dietary, and presumably environmental factors. S-equol has been shown to have a modest affinity for binding to ERβ, and little affinity for ERα. Furthermore, equol  can act as an antiandrogen. Equol’s antiandrogen activity is unique as it has been demonstrated that equol does not bind AR, but specifically binds DHT with high affinity preventing the binding of AR to DHT . Howevercis isomers. High concentration of cis isomers were also observed in human serum and prostate tissue [trans to cis form. It has been demonstrated that serum and prostate levels of lycopene in patients with PCa were significantly lower than their age-matched controls suggesting that these patients lack the ability to isomerize dietary lycopene and therefore do not absorb it efficiently [Carotenoids are tetraterpenoid organic pigments that are naturally occurring in the chloroplasts and chromoplasts of plants and some other photosynthetic organisms like algae, some bacteria, and some types of fungi Fig. . Like foe tissue , suggesticiently .Fig. 5ChAlthough risk assessments have been historically conducted on a chemical-by-chemical basis, regulatory agencies are beginning to consider cumulative risk of chemicals. The effects of mixtures of chemicals like the ubiquitous phthalates and plasticizers are of concern since humans are exposed at the same time to multiple compounds .50 for hypospadias or epididymal agenesis. The binary mixtures included chemicals with different mechanism of action such as AR antagonists , phthalate esters , a phthalate ester plus an AR antagonist , and a phthalate ester plus a dioxin . The data obtained confirmed the prediction that each chemical by itself would induce no or few malformations, but any binary mixture had led to about 50% of the males turning out to be malformed. In the same study, it has been also conducted a combinatorial mixture study exposing pregnant rats to either seven  or ten  different antiandrogens. The complex mixture experimental data have shown that these chemicals elicit antiandrogenic effects at two different sites in the androgen signaling pathway . Overall, it was demonstrated that chemicals acting via disparate mechanisms display cumulative, dose-additive effects when present in combination.Binary mixture studies were performed in rats during pregnancy exposed at dosage levels equivalent to approximately one half of the EDIn another recent study , conductBoth epidemiology studies and animal models sustain the idea that specific EDCs may influence the development or progression of male reproductive endocrine disorders including PCa , 171. InAs a whole, the combined effect of EDCs on androgen-dependent gene expression and, more general, on animal physiology is very complex because many EDCs can act as modulator of AR or estrogen receptors leading to the activation and the interaction of multiple signaling pathways, and in turn, EDCs can affect reproduction and development by more than one mechanism. Moreover, the evidence that AR mutant gain the ability to utilize some EDCs  as an agonist enlarge the effect of these substances. In spite of the huge number of studies evaluating the antiandrogenic properties of EDCs, only androgen metabolism and AR or estrogen receptors transcriptional activity have been taken into consideration, while a lack of knowledge is still present on the ability of these compounds to interfere with steroid-dependent extra-nuclear signals. Since the alteration of androgen signaling can induce a variety of endocrine disruptive responses, further studies are required to identify the downstream targets of EDC-modulated AR signaling, in order to elucidate their specific impact on male health.


Antireflection (AR) at the interface between the air and incident window material is paramount to boost the performance of photovoltaic devices. 3D nanostructures have attracted tremendous interest to reduce reflection, while the structure is vulnerable to the harsh outdoor environment. Thus the AR film with improved mechanical property is desirable in an industrial application. Herein, a scalable production of flexible AR films is proposed with microsized structures by roll‐to‐roll imprinting process, which possesses hydrophobic property and much improved robustness. The AR films can be potentially used for a wide range of photovoltaic devices whether based on rigid or flexible substrates. As a demonstration, the AR films are integrated with commercial Si‐based triple‐junction thin film solar cells. The AR film works as an effective tool to control the light travel path and utilize the light inward more efficiently by exciting hybrid optical modes, which results in a broadband and omnidirectional enhanced performance. In addition, the nanofabrication methods are costly and time consuming with a scale limitation. Unlike the nanostructured textures, microstructures can be produced more efficiently and effectively scatter the incoming light, and in return reduce the surface reflectivity. In this work, we demonstrated the plastic AR films with microstructures fabricated by roll‐to‐roll (R2R) process with large‐area continuous production capability. The microstructures are realized on ethylene‐tetrafluoroethylene (ETFE) films, which are being widely used in flexible PV industry as the polymer front sheet to encapsulate solar cell modules due to its excellent UV resistance, high transmittance, and outstanding tensile strength.2Figurey‐axis. For the subsequent comparison, the nanopillar structure was also transferred from the nanostructured nickel mold through planar hot embossing process with the pitch size, height, and diameter of 1 µm, 500 and 500 nm, respectively   of 97° ,−1. The SEM images  partially covered with the trapezoid AR structure. At the area covered by the trapezoid prisms, one can clearly see the significant AR effect improvement. The normal incident reflectance spectra are measured on the devices covered with flat and trapezoid AR films as shown in Figure JSC of three subcells by integrating Air Mass (AM) 1.5G spectrum–2 for the device with flat film, while it is 7.2, 6.80, and 6.53 mA cm–2 for the device with AR film. This indicates that the AR film benefits a higher improvement for the top cell, which utilizes comparative short wavelength.As a demonstration, the trapezoid AR films were integrated with commercial Si‐based triple‐junction thin film solar cells (Xunlight (Kunshan) Co., Ltd).FigureE|) distributions of the triple‐junction solar cells at the wavelengths of 642, 811, and 933 nm, respectively, where the guidelines in Figure The optical light field behaviors are verified by electromagnetic simulation via finite‐difference‐time‐domain (FDTD) method. The simulation for the structures of solar cells laminated with flat and AR film were depicted in Figure S3 (Supporting Information), where the thickness of flat film and the flat part in the trapezoid AR film are reduced in order to save computational resource without losing accuracy. z‐axis  can be found in the flat device  instead of straight across the cells. The constructive modes formed along the horizontal direction were defined as Bloch modes resulted from the periodical structure of the AR film ,A series of finger patterns along the e Figure a1–a3, ore Figure a2,a3. Wiyoz and xoz planes. As the power conversion efficiency (PCE) and the internal resistance of solar cells were related to cell size and conductivity.2) are used in the performance evaluations. The small area devices are cut from the same large size solar cell in order to minimize the performance variation. FigureJ–V) curves of the devices recorded at 0°  and 60° of incident light (in yoz plane) under AM 1.5G solar simulator irradiation, where the current densities are calculated using the device active area, regardless of the change of projection area under different incident angles. A series of J–V measurements were further performed at different incident angles varying from 0° to 60° with a 10° interval in yoz plane.yoz plane, one can find an 11.5% enhancement of PCE compared with its flat counterpart.It is important to understand that the angle of solar irradiation changes over the time during a day in practical operation. Therefore, omnidirectional light trapping ability of AR film is critical. Due to the anisotropic characteristic of trapezoid structure, the incident light angle dependent performances are characterized in both xoz plane due to the anisotropic trapezoid geometry. Under 60° incident angle in xoz plane, the PCE value (3.70%) with AR film is slightly lower than that with flat window layer, which could be attributed to a higher reflection at a specific angle. As a result, it is preferable to have the incident irradiation move in yoz plane. This behavior is adoptable for practical application because one can have the motion trajectory of the sun approximately fall into the vertical plane with respect to the solar panel.More intriguingly, we found an angle dependence of the AR film  when applied in yoz plane) is corresponding to 12 pm to 4 h away from 12 o'clock (from 8:00 am to 16:00 pm).EDaily is the daily energy output, Poutput is the power output at designated daytime, and 0.667 h is the time interval corresponding to 10° incident angle change. The daily energy output of 0.5540 kW h m−2 is achieved with the presence of AR film, which outperforms the planar counterpart by 5.5%. Although the PCE improvement with AR film is not as high as that with previous nanostructured 3D AR films,Figure 3In summary, we have demonstrated large‐area flexible AR films with microprism structures by a versatile R2R imprinting process. Large‐area microscale trapezoid structure can be easily transferred through roller molding and hot pressing fabrication resulting in reduced cost and prolonged the service life of the mold. The AR films delivered broadband suppression of light reflectance together with improved hydrophobic property that can be easily integrated with flexible and rigid solar cells. The solar cell with the presence of AR film demonstrates a 5.5% energy generation enhancement over the device with flat encapsulation film. The AR film shows attractive application prospects with much enhanced mechanical strength and can be easily utilized on many commercial solar panels.4Fabrication of AR Films with R2R Process: The AR films are fabricated by an R2R imprinting equipment, as shown in Figure S6a (Supporting Information). A copper‐coated steel roller mold  was made by CNC lathe technology, whose surface had triangular prisms microstructure (pitch = 25 µm and height = 5 µm). The roller mold was used as the driver roller, where a heater was inserted into its core. The microstructure was transferred from the rollers to ETFE films at 120 °C with a pressure of 0.5 MPa. After cooling down, the polymer with microstructure demolds from the roller. In order to realize the trapezoid prism structure, a hot pressing process was subsequently carried out in a homemade vessel with a heater , and triple‐junction thin film solar cell (provided by Xunlight (Kunshan) Co. Ltd) were stacked together in order. They were equipped with a double‐chamber setup as illustrated in Figure 2 gas with the pressure of 0.2 MPa. The overall applied pressure was 0.3 MPa, plus the pressure of 0.1 MPa from down chamber. The temperature increased from 120 to 160 °C and kept for 15 min. Finally, the solar cell encapsulated with ETFE film was cooled to room temperature.Numerical Simulation: FDTD method was used to investigate the absorption properties of the devices with or without AR film. The thickness of flat film and the flat part in the trapezoid AR film were reduced in order to save computational resource. A plane wave light source with wavelengths ranging from 300 to 1000 nm was utilized. The unit cell of the compound device was set as the simulation region utilizing Bloch boundaries in the x‐ and y‐axis a Perfectly Matched Layer (PML) boundaries in the z‐axis at a normal incident angle. The optical data used in this simulation were obtained from Palik's Handbook of Optical ConstantsCharacterization: The SEM images were taken using cold field emission scanning electronic microscopy  at 3.0 kV. The reflectance spectra were measured by a fiber‐optic spectrometer . The contact angle of the polymer film was measured by a Kruss Kontaktwinkle DSA100 setup. The J–V characteristic of solar cells was carried out under 25 °C using an Xe lamp solar simulator  coupled with an air mass 1.5 global (AM 1.5G) filter. The external QE measurements of the solar cell were gained by QEX10 .The authors declare no conflict of interest.SupplementaryClick here for additional data file.


Our findings provide a molecular justification for the traditional use of AR in treating CRC, and a pharmacological basis for developing AR-derived modern anti-CRC agent(s).Colorectal cancer (CRC) is a leading cause of cancer-related morbidity and mortality worldwide. Signal transducer and activator of transcription 3 (STAT3) signaling is constantly activated in CRC, and has been proposed as a pathogenic factor and a therapeutic target of CRC. Ampelopsis Radix (AR), a traditional Chinese medicinal herb, possesses low toxicity and has long been used clinically for the treatment of cancers including CRC. Some constituents of AR have been reported to exert anti-cancer properties by targeting STAT3. However, the anti-CRC mode and mechanism of action of AR have not been fully elucidated. Here, we investigated the involvement of STAT3 signaling in the anti-CRC effects of AR. Results showed that AR reduced cell viability, induced cell apoptosis, and suppressed cell migration and invasion in human HCT-116 and SW480 CRC cells. Mechanistic studies showed that AR potently suppressed STAT3 and Src phosphorylation, and inhibited STAT3 nuclear localization in cultured CRC cells. AR also downregulated the expression of STAT3 target genes Mcl-1, Bcl-xL, and MMP-2 that are involved in cell survival and mobility. Moreover, the cytotoxic effect of AR was diminished by overexpressing STAT3C, a persistent active variant of STAT3. In conclusion, AR exerted anti-CRC effects  Colorectal cancer (CRC) is one of the prevalent cancer types worldwide, with an estimated 1.4 million new cases diagnosed each year . AlthougAmpelopsis japonica  written 2,000 years ago. It has been traditionally used to treat various diseases such as Yongzhong , Luoli , Changfeng , that are nowadays diagnosed as cancers including CRC. In modern clinical Chinese medicine practice, AR is commonly used for treating CRC, and no AR-related toxicity was observed (Ampelopsis Radix (AR), a traditional Chinese medicinal herb, is the dried root tuber of japonica . AR was observed . Pharmacobserved . Some coobserved , catechiobserved , and epiobserved , have beobserved . However3--2,5-diphenyltetrazolium bromide (MTT) and DMSO were purchased from Sigma Chemicals Ltd. . Antibodies against GAPDH and SP1 were purchased from Santa Cruz Biotechnology , goat anti-rabbit IgG, goat anti-mouse IgG and protein marker were supplied by Bio-Rad , and the other antibodies were obtained from Cell Signaling Technology . All materials for cell culture were obtained from Life Technologies Inc. .AR, originated from Jiangsu province, China, was purchased from the Chinese Medicine Clinic of the Hong Kong Baptist University, and authenticated in accordance with the corresponding monograph in the 2015 edition of CP by the corresponding author. Voucher specimen of AR (No. 20151201) was deposited at the School of Chinese Medicine, Hong Kong Baptist University.Supplementary Figure S1). The contents of these three compounds in AR were 0.23, 1.25, and 0.18 g/kg, respectively.Preparation of AR extract: powdered AR was reflux-extracted twice with 30% ethanol  for 2 h each. Combined extracts were filtered after cooling and then concentrated under reduced pressure to remove the solvent. The powdered extract (yield: 1.71%) was obtained by lyophilizing of the concentrated samples with a Virtis Freeze Dryer . HPLC analysis showed that gallic acid, catechin, and epicatechin were in the extract (we name the extract AR hereafter) . Cells were cultured in DMEM  supplemented with 5% heat-inactivate USA) and 1% penicillin/streptomycin at 37°C in a humidified 5% CO50 value of AR against each cell line was calculated using the software Origin 4.0 .The cytotoxic effect of AR on CRC (HCT-116 and SW480) cells and normal human colon mucosal epithelium (NCM460) cells were determined by the MTT assay as described previously . HCT-1165 cells/well) and allowed to adhere overnight. Apoptotic effects of AR  on CRC cells were evaluated by AnnexinV/PI double staining according to the manufacturer’s instructions. Briefly, cells were harvested after 48-h treatment and 1 × 105 cells were then incubated in 100 μL labeling solution  in darkness at room temperature for 15 min. After that, 400 μL of 1X binding buffer was added to stop the staining reaction. Flow cytometric analyses were performed on a FACSCaliburTM  utilizing 10,000 events.HCT-116 and SW480 cells were seeded on 6-well plates  according to the manufacturer’s instruction (5 cells in 0.5 mL of DMEM-0.1% BSA with AR (25 μg/ml) or vehicle were placed in the upper chamber. Invaded cells (cells on the lower surface of the membrane) in five microscope areas (200× magnification) were counted and imaged by a microscope .Cell invasion was determined by using BD BioCoattruction . 0.75 mLMatrix metalloproteinase-2 (MMP-2) activity was measured by gelatin zymography. Cells were treated for 24 h with AR  or vehicle in serum-free DMEM medium, and then the supernatant was collected and measured as described previously . The gelHCT-116 and SW480 cells were seeded as described in Section “Apoptosis Assay.” After incubation for 24 h, AR-treated  and vehicle-treated cells were collected. The cytoplasmic and nuclear extracts were prepared as described previously .HCT-116 and SW480 cells were seeded and treated as described in Section “Apoptosis Assay.” Cell extracts were prepared, electrophoresed under denaturing conditions. The proteins were transferred onto polyvinyldifluoridine membranes. The membranes were washed in TBS containing 0.05% (v/v) Tween-20 and incubated over night at 4°C with corresponding primary antibodies. The membranes were then incubated with secondary antibodies and signals were detected by ECL detection reagents  .The constitutive activated STAT3 expression construct STAT3-C Flag pRc/CMV was obtained from Addgene (USA). Transfection of STAT3 plasmid into HCT-116 or SW480 cells were conducted by using Lipofectamine 2000  following manufacturer’s protocol . Cells wp < 0.05 was considered statistically significant.Results were presented as mean ± SD. Data were analyzed by one-way ANOVA followed by the Dunnett’s multiple comparisons. Statistical analyses were carried out using GraphPad Prism version 5.0 . Figure 1A, AR decreased HCT-116 and SW480 cell viabilities in both time- and dose-dependent manners, with IC50 values of 204.0 and 183.9 μg/mL after 24-h treatment, and 123.5 and 128.5 μg/mL after 48-h treatment, respectively. Moreover, the cytotoxicity AR in NCM460 cells was less potent than that in both CRC cell lines. To determine whether AR induced cell apoptosis, the Annexin V/PI double staining assay was used. As shown in Figure 1B, AR dose-dependently induced apoptosis in both HCT-116 and SW480 cells. A substantial induction of poly (ADP-ribose) polymerase (PARP), caspase-3, caspase-7, caspase-8, and caspase-9 cleavages in AR-treated HCT-116 and SW480 cells further confirmed the pro-apoptotic effect of AR .The cytotoxicity of AR was examined in two human CRC cell lines (HCT-116 and SW480) and a normal human colon mucosal epithelium (NCM460) cell line by using the MTT assay. As shown in Figure 2A, after 48 h treatment, AR at the concentration of 25 μg/mL, although did not affect cell viability , significantly inhibited the migratory ability of both HCT-116 and SW480 cells. Likewise, a significant reduction of the cell invasiveness was observed in the AR treatment groups .Wound healing assay and transwell invasion assay were performed to determine the effects of AR on CRC cell migration and invasion, respectively. As shown in Figure 3A). Both Src and JAK2 are the upstream tyrosine kinases of STAT3, and they have been found to be constitutively phosphorylated/activated in various cancers including CRC  but not phospho-JAK2 (Y1007/1008) (data not shown) in both HCT-116 and SW480 cells, suggesting that AR inhibited Src-STAT3 signaling in CRC cells.Immunoblotting assay was employed to determine the effects of AR on the phosphorylation/activation of Src and STAT3 in CRC cells. It was found that the levels of phosphorylated STAT3 at the tyrosine 705 (Tyr705) site were reduced in a dose-dependent manner after a 24-h treatment with AR, but the treatment did not affect total STAT3 expression, in both HCT-116 and SW480 cells .STAT3 dimerizes upon tyrosine phosphorylation at site 705, which leads to its nuclear translocation. Inhibition of STAT3 phosphorylation should suppress nuclear localization of STAT3. As expected, the protein levels of STAT3 in nuclear fractions of both HCT-116 and SW480 cells were significantly reduced by AR in a dose-dependent manner . In SW480 cells, the cytotoxic effect of 100 μg/mL of AR was also significantly reduced by STAT3C overexpression.To determine the role of STAT3 inhibition in the anti-CRC effects of AR, HCT-116 and SW480 cells were transiently transfected with an empty vector or a plasmid containing STAT3C, an oncogenic mutant of STAT3 that is constitutively activated without tyrosine phosphorylation . The eff50 of AR water extract is as high as 50 g/kg (Currently available chemotherapeutics against CRC have various disadvantages such as high price, toxicity, and low efficacy . These d 50 g/kg . However 50 g/kg .1, however, some of them were aborted because of severe adverse effects (in vitro, suggesting that the multi-component and multi-target AR may have greater clinical significance compared to specific STAT3 inhibitors.Previous studies have shown that some compounds occurring in AR exert anti-cancer effects by inhibiting STAT3 signaling . In this effects . AR has in vitro, and these effects are partially due to the inhibition of STAT3 signaling. These findings provide a justification for the clinical application of AR at molecular levels. Together with ancient records, our data also suggest that AR and AR-derived compounds have the potential to be developed as modern alternative and/or complimentary agents for CRC management.In summary, we demonstrated that AR reduced cell viability, induced cell apoptosis and suppressed cell migration and invasion in human CRC cells. Mechanistic studies showed that AR decreased STAT3 and Src phosphorylation, reduced STAT3 nuclear localization, and downregulated the expression STAT3 target genes. Moreover, overactivation of STAT3 decreased the cytotoxic effect of AR in CRC cells. Our results indicate that AR has anti-CRC effects TS, J-XB, Y-JC, and X-NW performed the majority experiments. X-QF, TL, HG, and P-LZ participated in several experiments. TS and Z-LY interpreted the data. TS and YW drafted the manuscript. Z-LY supervised the study and finalized the manuscript. All authors have read and approved the final manuscript.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Free radicals production by toxicity of arsenic (Ar) is most important in the nephrotoxicity. There is accumulating evidence that zinc (Zn), has anti-oxidant properties.The aim of present study was to evaluate protective and ameliorative effects of Zn against Ar-induced nephrotoxicity in rat pups during gestation and lactation.Twenty-four adult pregnant wistar rats were randomly divided into four groups (n = 6). Group one was given vehicle only. Group two received Zn (ZnSO4) at 20 mg/kg/d. Group three received Ar at 5 mg/kg/d as sodium meta-arsenite. Group four received Ar + Zn at the same dose that mentioned in groups of two and three. At the end of the study, 24 hours after the last treatment, samples were killed with overdose of sodium pentobarbital and kidneys were harvested for measuring malondialdehyde (MDA), glutathione (GSH) and histopathological assessment.The MDA level in kidney was increased in the Ar group, which was decreased after Zn administration in the Ar + Zn group. The GSH level in kidney was decreased in the Ar group, which were increased after Zn administration in the Ar + Zn group. Also, the histopathological changes which were detected in the Ar group attenuated after Zn consumption.Our findings suggested that administration of Zn during gestation and lactation could have protective and prevent effect in Ar-induced oxidative stress in kidney tissue. In this experimental study, we found that Zn as an antioxidant agent and ability to cross of placenta, can protect kidney against Ar induced nephrotoxicity during gestation and lactation. The main mechanism of Zn in renoprotective effects was inhibition of oxidative stress by amelioration of lipid peroxidation produced as well as elevation of GSH.Arsenic (Ar) is a natural component of the earth’s crust and is widely distributed throughout the environment in the air, water, and land . It is slike miscarriage and birth defects . The speZinc is the main source of antioxidants. Two mechanisms have been mentioned for that; first, the protection of sulfhydryl groups against oxidation and second, the inhibition of the production of reactive oxygen’s  by transThe purpose of the present study is to investigate the protective effects of zinc consumption against Ar induced-nephrotoxicity in offspring during the gestation and lactation.Adult female Wistar rats (250-300 g) were used from laboratory animal research center, Sari, Iran in this study. They were kept in the laboratory under constant conditions of temperature (23±2ºC) and in a reverse 12 hours dark/light cycle (lights on at 20:00 pm) for at least 1 week before and through the experimental work. After this period, they were mated with males. A sperm-positive vaginal smear test was taken to detect the first day of pregnancy. From this time, rats were housed individually in cages and maintained under the supervision according to the guidelines of the university’s animal care codes.®) prepared in distilled water  Control group, which received distilled water; (ІІ) Zinc (Zn) treated group, which received Zn (ZnSO4) at 20 mg/kg/d prepared in distilled water ,25;  ArThe collected samples (1 kidney from each pups) were abstergent by PBS in order to thoroughly cleaned of blood, then were immediately frozen and stored in a -80°C freezer for biochemical analysis. Two enzymes were assessed. First, malondialdehyde (MDA)  and secoThe obtained samples were thoroughly cleaned of blood, and then were immediately fixed in 10% (w/v) buffered formalin for 24 hours, embedded in paraffin. Five-micrometer serial sections were prepared from the paraffin-embedded blocks using by microtome. For histopathological assessment, some tissue sections were deparaffinized with xylene, stained with hematoxylin and eosin (H&E), and also periodic acid–Schiff (PAS) and studied by light microscopy  to assess the histopathological changes.Tubular damage (in 5 fields/each section) was scored by using the percentage of cortical tubules that showing epithelial necrosis: 0 index; <25% damage was assigned 1; 25%-50% damage was assigned 2; 50%-75% damage was assigned 3 and >75% damage was assigned a 4 index. The means of tubular necrosis was the blebbing of apical membranes, loss of the proximal tubular brush border, tubular epithelial cell detachment from the basement membrane or intraluminal aggregation of cells and proteins ,30. All The research followed the tenets of the Declaration of Helsinki. The research was approved by ethical committee of Mazandaran University of Medical Sciences. Prior to the experiment, the protocols were confirmed to be in accordance with the guidelines of Animal Ethics Committee of Mazandaran University of Medical Sciences .Statistical analysis was carried out in SPSS . Results were presented as mean values (±SD). The Kolmogorov–Smirnov test was used in order to evaluate the normality of the data. The Tukey׳s multiple comparison tests and the analysis of the variance were used to compare each two groups and data among the groups, respectively. A value of P < 0.05 was considered significant.P < 0.05). Also, in comparison with control and Zn groups, total birth numbers and infants dead in Ar group had significant decreased and increased, respectively  compared to other groups. The MDA levels in the Ar + Zn group were significantly lower than those of the Ar group  in GSH level compared to the other groups. We found the GSH levels significantly increased (P < 0.05) in Ar+Zn group compared to Ar group (MDA levels in the Ar group was a significant elevation (Ar group .P <0.001).Results of histopathological examination following PAS staining are shown in The main findings of the current study showed that administration of zinc during gestation and lactation to mother, can attenuate histopathological changes, inflammation, and lipid peroxidation in pups. Meanwhile, it improves antioxidant capacity in kidney tissue against Ar-induced nephrotoxicity.Kidney with excretion of methylated species of Ar is considered as a target organ in Ar toxicity . So, chrEndogenous antioxidants prevent cellular oxidative damage caused by free radicals. Studies have shown that the total antioxidant capacity and GSH levels in kidney tissues were significantly decreased after Ar administration compared to control groups . Also, lOur results showed that exposure of Ar during gestation and lactation increased MDA and decreased GSH levels whereas administration of zinc attenuated in the treatment group with Ar. Ar due to binds to the glomerular membrane, it can be lipophilicity and increases lipid peroxidation . Also, pZinc has antioxidant properties and by absorbing free radicals, reduces lipid peroxidation level can prevention of oxidative stress . On the Although the majority of zinc absorption is through the mouth and proximal intestine, studies revealed that placental zinc transport is by microvillus border membranes and cultured syncytiotrophoblasts . TherefoFurthermore, the quality of the maternal diet has a dominant effect on embryonic and fetal development and a significant moderating effect has on the expression of developmental toxicity of some agents .In this study, difference was observed in total weight in rat pups that exposed to Ar; this result was in contrast with Pineda et al studies , who repHistopathological findings showed less damage in the group that received zinc concurrent with Ar in compared to Ar group. The reduced degree of pyknosis of renal epithelium and mild dilation of the bowman capsule and renal tubules in Ar plus zinc group again implied the nephroprotective effect of zinc, but compared with Ar group was not significant. Perhaps it could be due to lack of evolution is completely all at birth. New nephrons in the kidney cortex margin, their maturity gained in a few days after birth. At this time, the amount of Ar that passes through breast milk is relatively insignificant . Zinc adOur results support that zinc – a good source of antioxidant – can markedly attenuate the indicators of the Ar-induced nephrotoxicity during gestation and lactation; so it can be recommended as a dietary supplement to reduce the side effects of synthetic pyrethroid in during pregnancy.DN and FT; study design, data gathering and data interpretation, preparation of manuscript and final revision. ARK and AE; consultant of study. MZ; performed the biochemical tests. All authors read and approved the paper.None declaredThis study was supported by Student Research Committee and Molecular and Cell Biology Research Center of Mazandaran University of Medical sciences (Grant No: 95/21).


Nrp1 knockout. We then reassessed the model for the original OR transgene with which the model was developed, using the same publicly available mouse strains. We discovered that glomerular positions do not undergo the simple anterior shift that has been reported in the conditional Nrp1 knockout for this OR transgene. Taken together, our findings do not support the Nrp1 model for the anterior-posterior patterning of glomerular positions in the olfactory bulb.It is known since 1996 that mouse odorant receptors (ORs) are involved in determining the positions of the sites of coalescence of axons of olfactory sensory neurons (OSNs)—the thousands of glomeruli in the olfactory bulb. But the molecular and cellular mechanisms of OR-mediated axonal coalescence into glomeruli remain unclear. A model was proposed in 2006–2009 whereby OR-derived cAMP signals, rather than direct action of OR molecules, determine the target destinations (glomeruli) of OSNs in the bulb. This model hypothesizes that OR-derived cAMP signals determine the expression levels of neuropilin 1 (Nrp1) in OSN axon termini; that levels of Nrp1 in glomeruli form a gradient from anterior-low to posterior-high throughout the bulb; and that these Nrp1 levels mechanistically determine anterior-posterior patterning of glomeruli. Here, we describe the first independent evaluation of the Nrp1 model since it was formulated a decade ago. We tested the model for the well-characterized mouse OR M71 using our gene-targeted mouse strains, which are publicly available. In contradiction to the model, we observed a variety of configurations for the M71 glomeruli in the conditional  In the mouse, each olfactory sensory neuron expresses one of ∼1100 odorant receptor genes. The odorant receptor determines to which odorants the neuron responds physiologically and in which glomerulus of the olfactory bulb its axon terminates. A model was proposed 10 years ago whereby intracellular signals derived from the odorant receptor determine the level of neuropilin 1, which in turn determines the position of the glomerulus along the anterior-posterior axis of the bulb. We provide the first test of this model, for a well-characterized odorant receptor and separately with the original mouse strains that led to the formulation of the model. Our results do not support the neuropilin 1 model of anterior-posterior patterning of glomerular positions in the olfactory bulb.Nrp1 knockout  genes  determinknockout  with a tknockout . Agonistknockout .Nrp1 knockout has on the lateral glomerulus in mice that express rat OR I7 from a transgenic mouse MOR23 promoter.Curiously, when formulating the Nrp1 model, data were reported only for the projection sites (glomeruli) in the medial aspect of the bulb . It is w+  or belt . Taken together, our results pose a challenge to the Nrp1 model  as #6707, official strain name STOCK Nrp1<tm1.1Mom>/MomJ. The design of this targeted mutation is similar to the published Nrp1 floxed strain . Nesting, bedding, and enrichment were provided as nestpak, Datesand Grade 6 . Mouse experiments were carried out in accordance with guidelines of the National Institutes of Health and the German Animal Welfare Act, European Communities Council Directive 2010/63/EU, and the institutional ethical and animal welfare guidelines of the Max Planck Institute of Biophysics and the Max Planck Research Unit for Neurogenetics. Approval came from the IACUC of The Rockefeller University ; the Regierungspräsidium Darmstadt (Germany); and the Veterinäramt of the City of Frankfurt (Germany).Mice were maintained in specific pathogen–free conditions in individually ventilated cages of the Tecniplast green line. Mice received Mice  were anesthetized by intraperitoneal injection of ketamine HCl and xylazine , and perfused intracardially with ice-cold PBS followed by 4% paraformaldehyde in PBS. The brain was immersed in 15% sucrose in PBS and 30% sucrose in PBS, each overnight at 4ºC on a shaker. After cryoprotection, the brain was trimmed and frozen in optimal cutting temperature compound (Tissue-Tek) on dry ice in ethanol. Serial horizontal sections encompassing the olfactory bulbs were generated with a Leica CM3050 S cryostat, set at 12-µm thickness. The sections were washed with 1× PBS and blocked with 10% normal donkey serum and 0.3% Triton X-100 in 1× PBS for 1 h at room temperature. After the blocking step, sections were incubated in 3% bovine serum albumin and 0.3% Triton X-100 in 1× PBS overnight at 4˚C with primary antibodies: rabbit anti-Adcy3  and goat anti-Nrp1 . Sections were then incubated for 1.5 h at room temperature with secondary antibodies: donkey anti-rabbit IgG Alexa Fluor 647  for rabbit anti-Adcy3, and donkey anti-goat IgG Alexa Fluor 546  for goat anti-Nrp1. Nuclear staining was done with 4′,6-diamidino-2-phenylindole  after the washing steps. Sections were imaged with a Zeiss LSM 710 confocal microscope .Samples were processed according to the iDISCO protocol . Primaryz-stacks were captured at an x,y resolution of 1.02 µm per pixel and 5 µm per z-plane, resulting in voxels of ∼5 µm3. Stacks were captured starting 50 µm below the cutting plane until a depth of 150 µm, thus spanning 100 µm in the z-axis. Stacks of images were processed using custom Python, Matlab, and ImageJ scripts provided by TissueVision. The 3D reconstruction and measurements were performed with Amira 6 .Serial block-face two-photon tomography was performed with a TissueCyte 1000 scanner  equipped with a Zeiss 20× 1.0-NA objective and a Ti:Sapphire laser . Methods were as described in Nrp1 knockout is the population of OSNs that express the OR gene M71 (M71+ OSNs), also known as Olfr151. The choice of this OR gene is motivated by extensive knowledge that has been collected over the past 15 years, including the characterization of odorous ligands for M71  axons typically coalesce into a single glomerulus (sometimes into two or three glomeruli) posteriorly each in the medial and lateral aspects of the olfactory bulb at PD21 , homozygous for the gene-targeted M71-IRES-Cre allele (MC)  , and hemele (MC) .+ OSNs express Cre and become devoid of Nrp1; moreover, they become GFP+ by virtue of Cre-mediated excision of a loxP-flanked segment of the reporter transgene that results in permanent GFP expression. Examples of whole mounts of bulbs of triple-mutant n5247MCZ mice are shown for PD14 and PD21  is the most common: it consists of a single small ectopic anterior glomerulus and a single large ectopic dorsal glomerulus and is exemplified in the bulbs shown in There was substantial variability, however, in the configurations of sites of coalescence (glomeruli) of GFPive ages A: 10 bul– M71+ OSNs innervate these ectopic anterior glomeruli.We have described a linear relationship between the number of OSNs that express a given OR gene and the total volume of the corresponding glomeruli in the bulbs . We founNrp1 knockout and the substantial variability are not consistent with the Nrp1 model for anterior-posterior patterning of glomeruli  and in six bulbs of triple-mutant n5247MCZ mice (Nrp1–) also at PD21. Methods of image acquisition, registration, and image processing have been described in detail , homozygous for M71-IRES-Cre (MC), and hemizygous or homozygous for the Z/EG transgene (Z); it is analogous to the triple-mutant n5247MCZ strain but with our Nrp1 floxed allele. As in triple-mutant n5247MCZ mice, ectopic glomeruli also exist in triple-mutant NMCZ mice, both the large ectopic dorsal glomerulus , homozygous for M71-IRES-tauRFP2 (MR) , compound heterozygous at the M71 locus , and hemizygous or homozygous for the Z/EG reporter (Z). In these quadruple-mutant mice, OSNs that express the M71-IRES-tauRFP2 allele are red fluorescent protein–positive (RFP+) and Nrp1+, and OSNs that express the M71-IRES-Cre allele are GFP+ (via expression of the Z/EG reporter) and Nrp1–.We generated a novel floxed allele of d allele . We thenomerulus A and theomerulus B. The trFP2 (MR) , and hemD), the most frequent configuration , there is a single small ectopic anterior glomerulus anteriorly  has the same glomeruli as configuration V plus a single lateral glomerulus, at a position that is typical for the lateral M71 glomerulus (F). Altogether, in 65% of bulbs , the RFP+ axons  coalesce into glomeruli at positions that are typical for M71 glomeruli, and they mix in these glomeruli with GFP+ axons .The numbers of labeled OSNs in the triple-mutant and quadruple-mutant mice are comparable to those in M71-IRES-tauGFP mice at PD21 : n5247MC+ neurons can be examined easily and specifically in mice of the gene-targeted Gucy1b2-IRES-tauGFP strain: neurons that express Gucy1b2 coexpress GFP, and the expression level is high enough to visualize the signal in glomeruli by the intrinsic fluorescence of GFP  shifting anteriorly at PD14, we observed multiple glomeruli formed by the coalescence of axons expressing rat OR I7 from a transgenic mouse MOR23 promoter. When viewed in a merged bulb after 3D construction, together these medial glomeruli occupied a broad domain of the bulb that can best be characterized as a sector that is anchored posteriorly on the tightly clustered medial glomeruli in the transgenic mice with Nrp1. This sector radiates anteriorly and ventrally along various vectors and is not consistent with a simple shift along an anterior-posterior “axis” of the bulb; at best, the vectors have an anterior component. This sector may reflect the domain of the bulb within which these axons, now devoid of Nrp1, are allowed to navigate. We then proceeded to flip these bulbs and examined the labeled glomeruli in the lateral aspect, which were not documented in During the course of this project, we realized that we needed to take a step back and seek to reproduce the core original findings with which the Nrp1 model was formulated . With puNrp1 knockout in one particular population of OSNs. In this context, it must be kept in mind that the glomeruli that are formed by the coalescence of axons expressing rat OR I7 from a transgenic mouse MOR23 promoter are novel and artificial: they do not exist in wild-type, nontransgenic mice.At PD21, the scattering of glomeruli appears to have increased, with glomeruli radiating further within the sector  and the belt . We have not examined mice at older ages. There may thus be a phenomenon of dynamic instability when mechanisms of axonal coalescence of OSNs are perturbed by a conditional Possible explanations for the discrepancy in our findings with + glomeruli, which are Adcy3– and Nrp1–. We found that there are Nrp1+ and Nrp1– glomeruli in these posterior domains, thus breaking the link between Adcy3 and Nrp1 expression, and constituting numerous exceptions to the notion that posterior glomeruli are Nrp1+. Moreover, there is substantial heterogeneity within these posterior domains: some glomeruli adjacent to Gucy1b2+ glomeruli are Nrp1+ (and Adcy3+), and others are Nrp1– . Serial block-face two-photon tomography  were completely excluded from the anteromedial region … and the ventral region”  reported data that are overlapping and consistent with our data, interpretation, and conclusions.


A,B) can be decomposed into its symmetric counterparts by envisioning and investigating the payoff tables (A and B) that constitute the asymmetric game, as two independent, single population, symmetric games. We reveal several surprising formal relationships between an asymmetric two-population game and its symmetric single population counterparts, which facilitate a convenient analysis of the original asymmetric game due to the dimensionality reduction of the decomposition. The main finding reveals that if  is a Nash equilibrium of an asymmetric game , this implies that y is a Nash equilibrium of the symmetric counterpart game determined by payoff table A, and x is a Nash equilibrium of the symmetric counterpart game determined by payoff table B. Also the reverse holds and combinations of Nash equilibria of the counterpart games form Nash equilibria of the asymmetric game. We illustrate how these formal relationships aid in identifying and analysing the Nash structure of asymmetric games, by examining the evolutionary dynamics of the simpler counterpart games in several canonical examples.We introduce new theoretical insights into two-population asymmetric games allowing for an elegant symmetric decomposition into two single population symmetric games. Specifically, we show how an asymmetric bimatrix game ( For instance, Walsh et al. introduce an empirical game theory methodology  that allows for analysing multiagent interactions in complex multiagent games3. This method has been extended by others and been applied e.g. in continuous double auctions, variants of poker and multi-robot systems9. Similar evolutionary methods have been applied to the modelling of human cooperation, language, and complex social dilemma’s16. Though these evolutionary methods have been very useful in providing insights into the type and form of interactions in such systems, the underlying Nash structure, and evolutionary dynamics, the analysis is limited to symmetric situations, i.e., players or agents can be interchanged and have access to the same strategy set, in other words there are no different roles for the various agents involved in the interactions (e.g. a seller vs a buyer in an auction). As such this method is not directly applicable to asymmetric situations in which the players can choose strategies from different sets of actions, with asymmetric payoff structures. Many interesting multiagent scenarios involve asymmetric interactions though, examples include simple games from game theory such as e.g. the Ultimatum Game or the Battle of the Sexes and more complex board games that can involve various roles such as Scotland Yard, but also trading on the internet for instance can be considered asymmetric.We are interested in analysing the Nash structure and evolutionary dynamics of strategic interactions in multi-agent systems. Traditionally, such interactions have been studied using single population replicator dynamics models, which are limited to symmetric situations, i.e., players have access to the same set of strategies and the payoff structure is symmetric as wellThere exist approaches that deal with asymmetry in multiagent interactions, but they usually propose to transform the asymmetric game into a symmetric game, with new strategy sets and payoff structure, which then can be analysed again in the context of symmetric games. This is indeed a feasible approach, but not easily scalable to the complex interactions mentioned before, nor is it practical or intuitive to construct a new symmetric game before the asymmetric one can be analysed in full. The approach we take in this paper does not require constructing a new game and is theoretically underpinned, revealing some new interesting insights in the relation between the Nash structure of symmetric and asymmetric games.18. As such it can be a very useful tool to analyse the Nash structure and dynamics of several interacting agents in a multiagent system. However, when dealing with asymmetric games the analysis quickly becomes tedious, as in this case we have a coupled system of replicator equations, and changes in the behaviour of one agent immediately change the dynamics in the linked replicator equation describing the behaviour of the other agent, and vice versa. This paper sheds new light on asymmetric games, and reveals a number of theorems, previously unknown, that allow for a more elegant analysis of asymmetric multiagent games. The major innovation is that we decouple asymmetric games in their symmetric counterparts, which can be studied in a symmetric fashion using symmetric replicator dynamics. The Nash equilibria of these symmetric counterparts are formally related to the Nash equilibria of the original asymmetric game, and as such provide us with a means to analyse the asymmetric game using its symmetric counterparts. Note that we do not consider asymmetric replicator dynamics in which both intra-species (within a population) and inter-species interactions (between different populations) take place19, but we only consider inter-species interactions in which two different roles interact, i.e., truly asymmetric games20.Analysing multiagent interactions using evolutionary dynamics, or replicator dynamics, provides not only valuable insights into the (Nash) equilibria and their stability properties, but also sheds light on the behaviour trajectories of the involved agents and the basins of attraction of the equilibrium landscapex strategies (player 1) and the y strategies (player 2) of a mixed Nash equilibrium of full support in the original asymmetric game, also constitute Nash equilibria in the symmetric counterpart games. The symmetric counterpart of player 1 (x) is defined on the payoff of player 2 and vice versa. We prove that for full support strategies, Nash equilibria of the asymmetric game are pairwise combinations of Nash equilibria of the two symmetric counterparts. Then, we show that this property stands without the assumption of full support as well. Though this analysis does not allow us to visualise the evolutionary dynamics of the asymmetric game itself, it does allow us to identify its Nash equilibria by investigating the evolutionary dynamics of the counterparts. As such we can easily distinguish Nash equilibria from other restpoints in the asymmetric game and get an understanding of its underlying Nash structure.One of our main findings is that the The paper is structured as follows: we first describe related work, then we continue with introducing essential game theoretic concepts. Subsequently, we present the main contributions and we illustrate the strengths of the theory by carrying out an evolutionary analysis on four canonical examples. Finally, we discuss the implications and provide a deeper understanding of the theoretical results.21. This assumes that players of these games are always fundamentally attached to one role and never need to know/understand how to play as the other player. In many cases, though, a player may want to know how to play as either player. For example, a good chess player should know how to play as white or black. This reasoning inspired the role-based symmetrization of asymmetric games22.The most straightforward and classical approach to asymmetric games is to treat agents as evolving separately: one population per player, where each agent in a population interacts by playing against agent(s) from the other population(s), i.e. co-evolution23.The role-based symmetrization of an arbitrary bimatrix game defines a new (extensive-form) game where before choosing actions the role of the two players are decided by uniform random chance. If two roles are available, an agent is assigned one specific role with probability n! (n being the number of agents) times larger due to the number of permutations of player role assignments. There are two-population variants that formulate the problem slightly differently: a new matrix that encapsulates both players’ utilities assigns 0 utility to combinations of roles that are not in one-to-one correspondence with players24. This too, however, results in an unnecessarily larger  matrix.This single-population model forces the players to be general: able to devise a strategy for each role, which may unnecessarily complicate algorithms that compute strategies for such players. In general, the payoff matrix in the resulting role-based symmetrization is 25. Similarly here, replicator dynamics and their properties are derived by transforming the payoff matrix into a larger symmetric matrix.Lastly, there are approaches that have structured asymmetry, that arises due to ecological constraints such as locality in a network and genotype/genetic relationships between population memberssymmetric counterpart dynamics rather than using standard dynamics on a symmetrised game. Therefore, the traditional role interpretation as well as any method that enlarges the game for the purpose of obtaining symmetry is unnecessarily complex for our purposes. Consequently, we consider the original co-evolutionary interpretation, and derive new  strategy space mappings.Our primary motivation is to enable analysis techniques for asymmetric games. However, we do this by introducing new 27. We briefly specify definitions of Normal Form Games and solution concepts such as Nash Equilibrium in a single population game and in a two-population game. Furthermore, we introduce the Replicator Dynamics (RD) equations for single and two population games and briefly discuss the concept of Evolutionary Stable Strategies (ESS) introduced by Smith and Price in 197330.In this section we concisely outline (evolutionary) game theoretic concepts necessary to understand the remainder of the paperDefinition. A two-player Normal Form Game (NFG) G is a 4-tuple G = , with pure strategy sets S1and S2for player 1, respectively player 2, and corresponding payoff tables A and B. Both players choose their pure strategies  simultaneously.A, B), which gives the payoff for the row player in A, and the column player in B  solution concept can be used to study what players will reasonably choose to do.In case x, y) ∈ ΔS1 × ΔS2, where ΔS1, ΔS2 are the sets of mixed strategies, that is, distributions over the pure strategy sets or action sets. The strategy x (respectively y) is represented as a vector in xTAy and xTBy is the payoff associated with player 2. A strategy profile  now forms a NE if no single player can do better by unilaterally switching to a different strategy. In other words, each strategy in a NE is a best response against all other strategies in that equilibrium. Formally we have,We denote a strategy profile of the two players by the tuple  is a Nash equilibrium, iff the following holds:NE for the set of Nash equilibria of the game G = . Furthermore, a Nash equilibrium is said to be pure if only one strategy of the strategy set is played and we will say that it is completely mixed if all pure strategies are played with a non-zero probability.In the following, we will write A is necessary to define the game (note that this definition only makes sense when |S1| = |S2| = n). In this case, the payoff received by the player is xTAx and the following definition describes the Nash equilibrium:In evolutionary game theory, games are often considered with a single population. In other words, a player is playing against itself and only a single payoff table Definition. In a single population game, a strategy x is a Nash equilibrium, iff the following holds:x ∈ NE(A).In this single population case, we will write that 32. In their most basic form they correspond to the biological selection principle, i.e. survival of the fittest. More specifically the selection replicator dynamic mechanism is expressed as follows:Replicator Dynamics in essence are a system of differential equations that describe how a population of pure strategies, or replicators, evolve through timei. This strategy is inherited by all the offspring of the replicator. xi represents the density of strategy i in the population, A is the payoff matrix which describes the different payoff values each individual replicator receives when interacting with other replicators in the population. The state of the population x can be described as a probability vector x =  which expresses the different densities of all the different types of replicators in the population. Hence (Ax)i is the payoff which replicator i receives in a population with state x and xTAx describes the average payoff in the population. The support Ix of a strategy is the set of actions (or pure strategies) that are played with a non-zero probability Ix = {i |xi > 0}.Each replicator represents one (pure) strategy offspring, if it scores lower than average its presence in the population will diminish and potentially approach extinction. The population remains in the simplex (∑ixi = 1) since ∑i(dxi)/(dt) = 0.In essence this equation compares the payoff a strategy receives with the average payoff of the entire population. If the strategy scores better than average it will be able to replicate 32 (as introduced in the previous section), though this can be extended to multi-population games as well as defined in the next section33. Imagine a population of simple agents playing the same strategy. Assume that this population is invaded by a different strategy, which is initially played by a small proportion of the total population. If the reproductive success of the new strategy is smaller than the original one, it will not overrule the original strategy and will eventually disappear. In this case we say that the strategy is evolutionary stable (ESS) against this newly appearing strategy. In general, we say a strategy is ESS if it is robust against evolutionary pressure from any appearing mutant replicator not yet present in the population .Originally, an Evolutionary Stable Strategy was introduced in the context of a symmetric single population gameWe have assumed replicators come from a single population, which makes the model only applicable to symmetric games. One can now wonder how the previous introduced equations extend to asymmetric games. Symmetry assumes that strategy sets and corresponding payoffs are the same for all players in the interaction. An example of an asymmetric game is the famous Battle of the Sexes (BoS) game illustrated in Table If we would like to carry out a similar evolutionary analysis as before we will now need two populations, one for each player over its respective strategy set, and we need to use the asymmetric or coupled version of the replicator dynamics, i.e.,Definition.A and B, respectively for player 1 and 2. In case A = BT the equations reduce to the single population model.symmetric counterpart replicator dynamics (SCRD) of asymmetric replicator equations. We consider the two payoff tables A and B as two independent games that are no longer coupled, and in which both players participate. In the first counterpart game all players choose their strategy according to distribution y, the original strategy or replicator distribution for the 2nd population, or player 2, and in the second counterpart game all players choose their strategy according to distribution x, the original strategy or replicator distribution for the 1st population, or player 1. This gives us the following two sets of replicator equations:We now introduce a new concept, the In the results Section we will introduce some remarkable relationships between the equilibria of asymmetric replicator equations and the equilibria of their symmetric counterpart equations, which facilitates, and substantially simplifies, the analysis of the Nash structure of asymmetric games.a2 to be chosen is one minus the first. By means of example we show a directional field plot here for the famous Prisoner’s dilemma game . As can be observed all dynamics are absorbed by the pure Nash equilibrium  in which both players defect.The directional field plot, and corresponding trajectories, are shown in Fig. n elements can be represented by the set of vectors  x1, ..., xn ≥ 0 and ∑ixi = 1. This can be seen to correspond to an n − 1-dimensional structure called a simplex Σn . In many of the figures throughout the paper we use Σ3, projected as an equilateral triangle. For example, consider the single population Rock-Paper-Scissors game, described by the payoff matrix shown in Fig. Unfortunately, we cannot use the same type of plot illustrating the dynamics when we consider more than two strategies. However, if we move to single population games we can easily rely on a simplex plot. In the case of a two population game the situation become tedious as we will discuss later. Specifically, the set of probability distributions over 3 trajectory plot for this game. Each of the corners of the simplex corresponds to one of the pure strategies, i.e., {Rock, Paper, Scissors}. For three strategies in the strategy simplex we then plot a trajectory illustrating the flow of the replicator dynamics. As can be observed from the plot, trajectories of the dynamics cycle around the mixed Nash equilibrium, which is not ESS and not asymptotically stable.The game has one completely mixed Nash equilibrium, being source.In fact, three categories of rest points can be discerned in single population replicator dynamics see Figs , 4 and 5In the following, we first present our main findings, formally relating Nash equilibria in asymmetric 2-player games with the Nash equilibria that can be found in the corresponding counterpart games. We also examine the stability properties of the corresponding rest points of the replicator dynamics in these games. Then we experimentally illustrate these findings in some canonical examples.x, y) ∈ NE (where x and y have the same support), then x ∈ NE(BΤ) and y ∈ NE(A). In addition, we prove that the reverse is true: if x ∈ NE(BΤ) and y ∈ NE(A) (where x and y have the same support) then  ∈ NE. We will prove this result in two steps .In this section, we prove the following result: if . This condition can be weakened by adding dominated strategies to the player having the smallest number of actions . Thus, without loss of generality, the theory will focus on square games. To begin, we state an important well-known property of Nash equilibria, that has been given different names; Gintis calls it fundamental theorem of Nash equilibriaLet the strategy profile  be a Nash equilibrium of an asymmetric normal form game , and denote Iz = {i | zi > 0} the support of a strategy z. Then,This result is widely known. We provide it as it is a basis of our theoretical results and for the sake of completeness.x and y constitute a Nash equilibrium then, by definition zΤAy ≤ xΤAy,∀z. Let us suppose that there exists a z with Iz ⊂ Ix such that zΤAy < xΤAy. Then there is a i ∈ Iz ⊂ Ix satisfying (Ay)i < xΤAy, and we get B follows analogously.◽If Let the strategy x be a Nash equilibrium of a single population game A. Then,The proof is similar to the proof of Property 1.◽This property will be useful in the steps of the proofs that follow. We now present our first main result: a correspondence between the Nash equilibria of full support in the asymmetric game with those of full support in the counterpart games. Theorem 2 subsumes this result and we introduce this simpler version first for the sake of readability.If strategies x and y constitute a Nash equilibrium of an asymmetric normal form game G = , with both xi> 0 and yj> 0 for all i, j (full support), and |S1| = |S2| = n, then it holds that x is a Nash equilibrium of the single population game BTand y is a Nash equilibrium of the single population game A. The reverse is also true.This result follows naturally from Property 1 and is implied by Theorem 2.x and y constitute a full support Nash equilibrium of the asymmetric game . By Property 1 and since x and y have full support, we know that:We start by assuming that yTAy = (Ay)i (since the (Ay)i are equal for all I’s in the vector Ay, so multiplying Ay with yT will yield the same number xTB)i = xTBx (and thus (BTx)i = xTBTx), implying that:From this we also know that This concludes the proof.◽y part of the Nash equilibrium of player 2 of the original asymmetric game, in the symmetric counterpart game determined by payoff table A. And similarly, for the second counterpart game this means that players will play according to the x part of the Nash equilibrium of player 1 of the original asymmetric game, in the symmetric game determined by payoff table B. As such both players consider a symmetric version of the asymmetric game, for which this y component and x component constitute a Nash equilibrium in the two new respective symmetric games.For the first counterpart game this means that the players will use the In essence, these two symmetric counterpart games can be considered as a decomposition of the original asymmetric game, which gives us a means to illustrate in a smaller strategy space where the mixed and pure equilibria are located.A direct consequence of Theorem 1 is the following corollary that gives insights on the geometrical structure of Nash equilibrium,Combinations of Nash equilibria of full support of the games corresponding to the symmetrical counterparts of the original asymmetric game also form Nash equilibria of full support in this asymmetric game.This is a direct consequence of Theorem 1.◽x and y have the same support if the set of played actions Ix = {i | xi > 0} and Iy = {i | yi > 0} are equal.The next theorem explores the case where the equilibrium is not of full support. We prove that the theorem stands if the strategies of both players have the same support. Indeed, the first theorem requires that both players play all actions with a positive probability, here we will only require that they play the actions with the same index with a positive probability. We say that Strategies x and y constitute a Nash equilibrium of an asymmetric game G =  with the same support (i.e. Ix= Iy) if and only if x is a Nash equilibrium of the single population game BT, y is a Nash equilibrium of the single population game A and Ix= Iy.x and y constitute a Nash equilibrium of same support (Ix = Iy) of the asymmetric game . By Property 1, and since x and y have the same support, we know that:We start by assuming that yΤAy = xΤAy and xΤBx = xΤBy (by setting z = y and z′ = x). Then, from the Nash equilibrium condition we can write:y is a Nash equilibrium of BΤ and x is a Nash equilibrium of A.Implying that y is a Nash equilibrium of BΤ and x is a Nash equilibrium of A with Ix = Iy. Then, from Property 2 we have:The proof of the other direction follows similar mechanics and uses Property 2. Let us now assume that yΤAy = xΤAy and xΤBx = xΤBy (by setting z = x and z′ = y). From the Nash equilibrium condition of the single population games we can write:In particular we get Strategies x and y constitute a pure (strict) Nash equilibrium of an asymmetric normal form game G = , with support on the strategy with the same index in their respective strategy sets S1and S2, if and only if, y and x are also pure (strict) Nash equilibria of the counterpart games defined by A,and B,This is a direct consequence of Theorem 2.◽Ix = Iy) from both players. One can work around this condition though by simply permuting the actions of one player in matrix A and B to study all configurations of supports of the same cardinality. To be precise, we need to analyze all the counterpart games defined by AΣ = AΣ and x, y) is a Nash equilibrium then |Ix| = |Iy|34).The theorems can only be used for equilibria in the counterpart games with matching supports  is a locally asymptotically stable equilibrium and a two-species ESS of the asymmetric normal form game G =  with support on the strategy with the same index in their respective strategy sets S1 and S2.x, y) is a strict Nash equilibrium of G. It has been shown that  is a strict Nash equilibrium of G iff it is a two-species ESS27.◽This a direct consequence of Corollary 2. More specifically, from Corollary 2 we know that (9) and concerns Leduc Poker. This algorithm produces normal-form “empirical games” which each correspond to an extensive-form game with a reduced strategy space, using incremental best response learning. Finally, the last asymmetric game illustrates the theorems for a single mixed equilibrium of full support, while its counterpart games have many more equilibria.We will now illustrate how the theoretical links between asymmetric games and their counterpart symmetric replicator dynamics facilitate analysis of asymmetric multiagent games, and provide a convenient tool to get insight into their equilibrium landscape. We do this for several examples. The first example concerns the Battle of the Sexes game to illustrate the intuition behind the results. The second example extends the Battle of the Sexes game with one strategy for one of the players, illustrating the permutation argument of the theorems and how to apply the results in case of a non-square game. The third example is a bimatrix game generated in the context of a multiagent learning algorithm called PSRO  strategy for the other (we will illustrate this in the PSRO-produced game on Leduc Poker).The theorems introduced in the previous section help to overcome this problem, and allow to analyse the evolutionary dynamics of the symmetric counterpart games instead of the asymmetric game itself, revealing the landscape of Nash equilibria, which seriously simplifies the analysis.opera or go to the movies, however, the corresponding payoffs for each are different, expressing the difference in preferences that both players have over their choices.Symmetry assumes that strategy sets and corresponding payoffs are the same for all players in the interaction. An example of an asymmetric game is the Battle of the Sexes (BoS) game illustrated in Table x = y = O (Opera), and the y-axis corresponds to the probability with which the 2nd player plays O (Opera). The blue arrows show the vector field and the black lines are the corresponding trajectories. Note that it is still possible here to capture all of the dynamics in a static plot for the case of 2-player 2-action games, but is generally not possible in games with more than two actions.The Battle of the Sexes has two pure Nash equilibria, which are ESS as well  and ), and one unstable completely mixed Nash equilibrium in which the players play respectively We now use this game to illustrate Theorem 1. If we apply Theorem 1 we know that the first and second counterpart symmetric games can be described by the payoff tables shown in Table In Fig. y = x = x = y = Additionally, we also know that the reverse holds, i.e., if we were given the symmetric counterpart games, we would know that x =   for the second counterpart) and x =   for the second counterpart), which are both ESS, are also Nash equilibria in the counterpart games shown in Table x =  and x =   and y =  for the other counterpart of the game) are Nash in both games, we know that x = y =  and x = y =  are also Nash in the original asymmetric game. This can also be observed in Fig. If we now apply Theorem 2 to the Battle of the Sexes game, then we find that pure strategy Nash equilibria R in which she can choose to listen to a concert on the radio instead of going to the opera or movies with her partner. This game is illustrated in Table In order to illustrate the theorems in a game that is non-square, including permutation of strategies, we extend the Battle of the Sexes game with a third strategy. Specifically, we give the second player a third strategy If we would like to carry out a similar evolutionary analysis as before we need two populations for the asymmetric replicator equations. Note that in this case the strategy sets of both players are different. Using the asymmetric replicator dynamics to plot the evolutionary dynamics quickly becomes complicated since the full dynamical picture is high-dimensional and not faithfully represented by projections to the respective player’s individual strategy simplices. In other words, a static plot of the dynamics for one player does not immediately allow conclusions about equilibria, as it only describes a player’s strategy evolution assuming a fixed  strategy of the other player. Again we can apply the counterpart RD theorems here to remedy this problem and consequently analyse the equilibrium structure in the symmetric counterpart games instead, yielding insight into the equilibrium landscape of the asymmetric game.dummy action D for the first player, in order to make sure that both players have the same number of actions in their strategy set (a requirement to apply the theorems) by just adding −1 for both players playing this strategy, which makes D completely dominated and thus redundant.In Tables x = ,y = ),,y = ),,y = ))} (we use the online banach solver http://banach.lse.ac.uk/ to check that the Nash equilibria we find are correct31).The three Nash equilibria of interest of this asymmetric game are the following, {  and  , so either playing O or M. The second counterpart game also has two Nash equilibria, i.e.,  and  playing either O or M as well. Note there are also two rest points at the faces formed by O and R and O and M, which are not Nash ,y = ) and ,y = ) in the asymmetric game for which we did not find a symmetric counterpart at this stage. Now the permutation of the counterpart games, explained earlier in the findings section, comes into play. Recall that in order to study all configurations of supports of the same cardinal for both players one needs to simply permute the actions of one player in matrix A and B. Let’s have a look at such a permutation, specifically, let’s permute the 2nd and 3rd action for player 2, resulting in Tables We now look for the x = ,y = ), which translates into ,y = ) for the asymmetric game as we permuted actions 2 and 3 for the second player and we need to swap these again. Additionally, we also find ,y = ), which translates into equilibrium ,y = ) for the asymmetric game as we permuted action 2 and 3 for the second player. Now we have found all Nash equilibria of the original asymmetric game.Again we can analyse these counterpart games. Specifically, we find Nash equilibria , , and  for permuted counterpart game 1 Table , and NasSo, also in this case, i.e., when the game is not square and strategies need to be permuted, the theorems are still applicable and allow for analysis of the original asymmetric game.35. Leduc poker has a deck of 6 cards . Each player receives an initial private card, can bet a fixed amount of 2 chips in the first round, 4 chips in the second round (with a maxium of two raises in each round). Before the second round starts, a public card is revealed.Policy Space Response Oracles (PSRO) is a multiagent reinforcement learning process that reduces the strategy space of large extensive-form games via iterative best response computation. PSRO can be seen as a generalized form of fictitious play that produces approximate best responses, with arbitrary distributions over generated responses computed by meta-strategy solvers. PSRO was applied to a commonly-used benchmark problem in artificial intelligence research known as Leduc pokerA, B, C} for player 1, and {D, E, F} for player 2. Each one of these strategies represents a larger strategy in the full extensive-form game of Leduc poker, specifically an approximate best response to a distribution over previous opponent strategies. The game produced here then is truly asymmetric, in the sense that the strategy spaces in the original game are inherently asymmetric since player 1 always starts each round, the strategy spaces are defined by different (mostly unique) betting sequences, and even under perfect equilibrium play there is a slight advantage to player 29. So, both players have significantly different strategy sets. In Tables In Table https://youtu.be/10m0f3iBECc.Again we can now analyse the landscape of equilibria of this game using the introduced theorems. Since the Leduc poker empirical game is asymmetric we need two populations for the asymmetric replicator equations. As mentioned before, analysing and plotting the evolutionary asymmetric replicator dynamics now quickly becomes very tedious as we deal with two simplices, one for each player. More precisely, if we consider a strategy for one player in its corresponding simplex, and that player is adjusting its strategy, will immediately cause the trajectory in the second simplex to change, and vice versa. Consequently, it is not straightforward anymore to analyse the dynamics and equilibrium landscape for both players, as any trajectory in one simplex causes the other simplex to change. A movie illustrates what is meant: we show how the dynamics of player 2 changes in function of player 1. We overlay the simplex of the second player with the simplex of the first player; the yellow dots indicate what the strategy of the first player is. The movie then shows how the dynamics of the second player changes when the yellow dot changes, see In order to facilitate the process of analysing this game we can apply the counterpart RD theorems here to remedy the problem, and consequently analyse the game in the far simpler symmetric counterpart games that will shed light onto the equilibrium landscape of the Leduc Poker empirical game.A and C, which absorbs the entire strategy space. Looking at Fig. D, one pure at strategy F, and one unstable mixed equilibrium at the 1-face formed by strategies D and F. Note there is also a rest point at the face formed by strategies D and E, which is not Nash. Given that there are no mixed equilibria with full support in both games we cannot apply Theorem 1. Using Theorem 2 we now know that we only maintain the two mixed equilibria, i.e.  (CP1) and  (CP2), forming the mixed Nash equilibrium ,y = ) of the asymmetric Leduc poker empirical game. The other equilibria in the second counterpart game can be discarded as candidates for Nash equilibria in the Leduc poker empirical game since they also do not appear for player 1 when we permute the strategies for player 1 (not shown here).In Figs A,B) is illustrated in Table As a final example to illustrate the introduced theory, we examine an asymmetric game, that has one completely mixed equilibrium and several equilibria in its counterpart games. The bimatrix game , p2 = ), (b), p2 = ), ((c) p1 = p1 = , p2 = ), (p1 = A and B and B and C. From these seven equilibria only (a), (b) and (c) are of interest since these are symmetric equilibria in which both players play with the same strategy (or support). Also counterpart game 2 has seven equilibria, i.e., The two symmetric counterpart game each have seven equilibria. Counterpart game 1 Table , has theWe observe that only the completely mixed equilibrium of the asymmetric game, i.e., We can make the same observation for the second counterpart game, and see that (d), (e) and (f) are equilibria in Fig. If we now apply Theorem 1 we know that we can combine the mixed equilibria of full support of both counterpart games into the mixed equilibrium of the original asymmetric game, in which the mixed equilibrium of counterpart game 1, i.e. 6. The predominant approach has been the use of symmetric replicator equations, allowing for a relatively straightforward analysis in symmetric games. Many interesting real-world settings though involve roles or player-types for the different agents that take part in an interaction, and as such are asymmetric in nature. So far, most research has avoided to carry out RD analysis in this type of interactions, by either constructing a new symmetric game, in which the various actions of the different roles are joined together in one population24, or by considering the various roles and strategies as heuristics, grouped in one population as well8. In the latter approach the payoffs due to different player-types are averaged over many samples of the player type resulting in a single average payoff to each player for each entry in the payoff table.Replicator Dynamics have proved to be an excellent tool to analyse the Nash landscape of multiagent interactions and distributed learning in both abstract games and complex systemsThe work presented in this paper takes a different stance by decomposing an asymmetric game into its symmetric counterparts. This method proves to be mathematically simple and elegant, and allows for a straightforward analysis of asymmetric games, without the need for turning the strategy spaces into one simplex or population, but instead allows to keep separate simplices for the involved populations of strategies. Furthermore, the counterpart games allow to get insight in the type and form of interaction of the asymmetric game under study, identifying its equilibrium structure and as such enabling analysis of abstract and empirical games discovered through multiagent learning processes , as was shown in the experimental section.matching support (including permutations of strategies for one of the players), it turns out that also the combination of those equilibria form a Nash equilibrium in the corresponding asymmetric game. In general, the vector field for the evolutionary dynamics of one player is a function of the other player’s strategy, and hence a vector field in one player’s simplex doesn’t carry much information as any equilibria you observe in it are changing with time as the other player is moving too. However, if you position the second player at a Nash equilibrium, it turns out that player one becomes indifferent between his different strategies, and remains stationary under the RD. This gives the unique situation in which the vector field plot for the second player’s simplex is actually meaningful, because the assumption of player one being stationary actually holds (and vice versa). This is what we end up using when establishing the correspondence of the Nash Equilibria in asymmetric and counterpart games, and why the single-simplex plots for the counterpart games are actually meaningful for the asymmetric game - but this is also why they only describe the Nash Equilibria faithfully, but fail to be a valid decomposition of the full asymmetric game away from equilibrium.A deeper counter-intuitive understanding of the theoretical results of this paper is that when identifying Nash equilibria in the counterpart games with 36, many real-world situations, in which one aims to study evolutionary or learning dynamics of several interacting agents, are better modelled by asymmetric games. As such these theoretical findings can facilitate deeper analysis of equilibrium structures in evolutionary asymmetric games relevant to various topics including economic theory, evolutionary biology, empirical game theory, the evolution of cooperation, evolutionary language games and artificial intelligence40.These findings shed new light on asymmetric interactions between multiple agents and provide new insights that facilitate a thorough and convenient analysis of asymmetric games. As pointed out by Veller and Hayward‘although the static game pits the row player against the column player, the evolutionary dynamic pits row players against themselves and column players against themselves’32 . He also indicates that this aspect of an evolutionary dynamic is often misunderstood. The use of our counterpart dynamics supports and illustrates this statement very clearly, showing that in the counterpart games species play games within a population and as such show an intra-species survival of the fittest, which is then combined into an equilibrium of the asymmetric game.Finally, the results of this paper also nicely underpin what is said in H. Gintis’ book on the evolutionary dynamics of asymmetric games, i.e.,