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    Functional differences between TSHR alleles associate with variation in spawning season in Atlantic herring

    AnimalsTissue samples for expression analysis were collected on August 24 and September 9 2016 from a population of spring-spawning herring kept in captivity at University of Bergen; the rearing of captive herring was approved by the Norwegian national animal ethics committee (Forsøksdyrutvalget FOTS ID-5072). The tissue samples used for ATAC-sequencing were collected at Hästskär on June 19 2019 from wild spring-spawning Atlantic herring in the Baltic Sea, which do not require ethical permission.Genome scan and genetic diversity at the TSHR locusWe used a 2 × 2 contingency X2 test to estimate the extent of SNP allele frequency differences between seven spring- and seven autumn-spawning herring populations from the Northeast Atlantic (Supplementary Table 2), and thus, identify the major genomic regions associated with seasonal reproduction. The SNP allele frequencies were generated in a previous study16 and were derived from Pool-seq data. For the X2 test, we formed two groups, spring and autumn spawners, and summed the reads supporting the reference and the alternative alleles for the pools included in each group.To characterize genetic diversity at the TSHR locus, we calculated nucleotide diversity (π) and Tajima’s D for the same seven spring- and seven autumn-spawning Atlantic herring populations used in the genome scan (n = ~41–100 per pool) (Supplementary Table 2). The whole-genome re-sequence data of these pools were previously reported by Han et al.16 (for details of the pools used here see Supplementary Table 2). Unbiased nucleotide diversity π and Tajima’s D were calculated for each pool using the program PoPoolation 1.2.236, which accounts for the truncated allele frequency spectrum of pooled data. In brief, a pileup file of chromosome 15 was generated from each pool BAM file using samtools v.1.1037,38. Indels and 5 bp around indels were removed to exclude spurious SNPs due to misalignments around indels. To minimize biases in the π and Tajima’s D calculations, which are sensitive to sequencing errors and coverage fluctuations39, the coverage of each pileup file was subsampled without replacement to a uniform value based on a per-pool coverage distribution (the target coverage corresponded to the 5th percentile of the coverage frequency distribution, which was used as the minimum coverage allowed for an SNP to be included in the analysis) (Supplementary Table 2). We also calculated the diversity parameters but skipping the coverage subsampling step and obtained very similar results with both approaches (Supplementary Fig. 5), thus, we decided to keep working with the subsampled datasets as coverage subsampling is recommended by the software developers36. To exclude spurious SNPs associated with repetitive sequences and copy number variants, we applied a maximum coverage filter equivalent to the per-pool 99th percentile of the coverage frequency distribution (Supplementary Table 2). A minimum base quality of 20, a minimum mapping quality of 20, and a minor allele count of 2 were required to retain high quality SNPs for further analysis. Both, π and Tajima’s D statistics were calculated using a sliding window approach with a window size of 10 kb and a step size of 2 kb (the selected window-step combination offered a good genomic resolution while reducing the noise from single SNPs, after testing windows of 5, 10, 20, 40, 50, and 100 kb for non-overlapping and overlapping windows with a step size equivalent to 20% of the window size). Only windows with a coverage fraction ≥ 0.5 were included in the computations. In addition, we estimated the effective allele frequency difference, or delta allele frequency (dAF), between spring and autumn spawning groups at the TSHR locus using the formula dAF = abs(mean(spring pools)−mean(autumn pools)). For each of the diversity parameters, we assessed whether the mean differences between sets of SNPs within chr 15: 8.85–8.95 Mb (215 SNPs) and outside (214 635 SNPs) the TSHR region were statistically significant among spring- and autumn-spawning groups using a Wilcoxon test. Data postprocessing, statistical tests, and plotting were performed in the R environment40 (for specific parameters used in PoPoolation, see the associated code to this publication).Identification of the 5.2 kb structural variantSequences spanning the entire TSHR gene plus 10 kb upstream and downstream from two reference assemblies, ASM96633v115 and Ch_v2.0.218, were aligned using BLAST41 and the output was subsequently processed with a custom R script40. Repeats were annotated by CENSOR21 for the region harboring the 5.2 kb structural variant. To validate the structural variant, long-range PCR was performed with genomic DNA from two spring- and autumn-spawning Atlantic herring in a 20 μL reaction containing 0.8 mM dNTPs, 0.3 μM each of the forward and reverse 5.2kb-confirm primers (Supplementary Table 1) and 0.75 U PrimeSTAR GXL DNA Polymerase (TaKaRa) following the program: 95 °C for 3 min, 35 cycles of 98 °C for 10 s, 58 °C for 20 s and 68 °C for 2 min 30 s, and a final extension of 10 min at 68 °C.ATAC-seq analysisBSH and brain without BSH were dissected from two spring-spawning herring caught in the Baltic Sea and transported to the lab on dry ice, then kept at –80 °C before nuclei isolation. ATAC-seq libraries were constructed according to the Omni-ATAC protocol with minor modifications42. Briefly, tissue was homogenized in 2 ml homogenization buffer (5 mM CaCl2, 3 mM Mg(Ac)2, 10 mM Tris-HCl (pH = 7.8), 0.017 mM PMSF, 0.17 mM ß-mercaptoethanol, 320 mM Sucrose, 0.1 mM EDTA and 0.1% NP-40) with a Dounce homogenizer on ice. 400 μl suspension was transferred to a 2 ml tube for the density gradient centrifugation with different concentrations of Iodixanol solution. After centrifugation, a 200 μl fraction containing the nuclei band was collected, stained with Trypan blue and counted with a Countess II Automated Cell Counter (Thermo Fisher Scientific). An aliquot of 100,000–200,000 nuclei was used as input in a 50 μl transposition reaction containing 2 X TD buffer and 100 nM assembled Tn5 transposase for a 30-min incubation at 37 °C. Tagmented DNA was purified with a Zymo clean kit (Zymo Research). Purified DNA was used for an initial pre-amplification for 5 cycles, and the additional amplification cycle was determined by qPCR based on the “R vs Cycle Number” plot43. Amplified libraries were purified with a Zymo clean kit again, and library concentrations and qualities were evaluated using the 2200 TapeStation System (Agilent Technologies).ATAC-seq was performed with a MiniSeq High Output Kit (150 cycles) on a MiniSeq instrument (Illumina) and 7–9 million reads were generated for each ATAC-seq library. Quality control, trimming, mapping, and peak calling of the sequenced reads were conducted following the ENCODE ATAC-seq pipeline (https://www.encodeproject.org/atac-seq/). The trimmed reads were aligned to the Atlantic herring reference genome (Ch_v2.0.2)18 with Bowtie244 and the mapping rate was 85–95%. Duplicate reads, reads with low mapping quality and those aligned to the mitochondria genome were removed. The remaining reads (4–5 million) were subjected to peak calling by MACS245, where 22–32 K peaks were called. Sequenced library qualities were further evaluated by calculating the TSS enrichment score and checking the library complexity with the Non-Redundant Fraction (NRF) and PCR Bottlenecking Coefficients (PBC1 and 2). Finally, conserved peaks between two biological replicates were identified by evaluating the irreproducible discovery rate (IDR).Genotyping of six differentiated variants and haplotype analysisAll six genetic variants, including the 5.2 kb structural variant, two non-coding SNPs, two missense SNPs and the copy number variant of C-terminal 22aa repeat, were genotyped in 45 spring-, 67 autumn-spawning Atlantic herring and 13 Pacific herring. TaqMan Custom SNP assays were performed to genotype the four SNPs in 5 μl reactions with a template of 20 ng genomic DNA (ThermoFisher Scientific). Copy number of the C-terminal 22aa repeat was determined by the PCR product size generated with geno22aa primers. Genotyping of the 5.2 kb structural variant was performed in a PCR reaction containing two forward primers (geno5.2kb-1F and geno5.2kb-2F) and one reverse primer (geno5.2kb-R), which generated PCR products with different sizes between spring and autumn spawners. All the primers used for genotyping are listed in Supplementary Table 1.Tissue expression profiles by quantitative PCRTotal RNA was prepared from gonad, heart, spleen, kidney, gills, intestine, hypothalamus and saccus vasculosus (BSH), and brain without BSH (brain) of six adult spring-spawning Atlantic herring using RNeasy Mini Kit (Qiagen). RNA was then reverse transcribed into cDNA with a High-Capacity cDNA Reverse Transcription Kit (ThermoFisher Scientific). TaqMan Gene Expression assay (ThermoFisher Scientific) containing 0.3 μM primers and 0.25 μM TaqMan probe (Integrated DNA Technologies) was performed to compare the relative expression levels of TSHR among different tissues. qPCR with SYBR Green chemistry was used for TSHB and DIO2 in a 10 μl reaction of SYBR Green PCR Master Mix (ThermoFisher Scientific) and 0.3 μM primers, with a program composed of an initial denaturation for 10 min at 95 °C followed by 40 cycles of 95 °C for 15 s and 60 °C for 1 min. Ct values were first normalized to the housekeeping gene ACTIN, then the average expression for each gene in the gonad was assumed to be 1 for the subsequent calculation of the relative expression in other tissues.Plasmid constructsThe coding sequence for the herring single-chain TSH (scTSH) was designed following a strategy previously used for mammalian gonadotropins46 that contained an in-frame fusion of the cDNA sequences (5′–3′) of herring TSH beta subunit (NCBI: XM_012836756.1) and alpha subunit (NCBI: XM_012822755.1) linked by six histidines and then the C-terminal peptide of the hCG beta subunit. The designed sequence should generate a protein with a size of 30.6 kDa. Both scTSH and spring herring TSHR cDNA sequences were synthesized in vitro and cloned in the expression vector pcDNA3.1 by Genscript (Leiden, Netherlands). pcDNA3.1 plasmid expressing human TSHR was kindly provided by Drs. Gilbert Vassart and Sabine Costagliola (Université libre de Bruxelles, Belgium). Then, the spring herring TSHR and human TSHR plasmids were used as templates for site-directed mutagenesis to generate constructs coding for different mutant herring or human TSHRs. Plasmids for the dual-luciferase assay, including pGL4.29[luc2P/CRE/Hygro] containing cAMP response elements (CREs) to drive the transcription of luciferase gene luc2P and pRL-TK monitoring the transfection efficiency, were purchased from Promega. Five ng of each plasmid was used to transform the XL1-Blue competent cells (Agilent), plasmid DNA was subsequently extracted from 200 ml overnight culture of a single transformant clone using an EndoFree plasmid Maxi Kit (Qiagen).Cell cultureChinese hamster ovary (CHO) (ATCC CCL-61) and human embryonic kidney 293 (HEK293) (ATCC CRL-1573) cells were maintained in DMEM supplemented with 5% (CHO) or 10% FBS (HEK293), 100 U/ml penicillin, 100 μg/ml streptomycin and 292 μg/ml l-Glutamine (ThermoFisher Scientific) at 37 °C with 5% CO2. Epithelioma Papulosum Cyprini (EPC) cells (ATCC CRL-2872) were cultured in EMEM (Sigma) supplemented with 10% FBS, 100 U/ml penicillin, 100 μg/ml streptomycin, 292 μg/ml l-Glutamine and 1 mM Sodium Pyruvate (ThermoFisher Scientific) at 26 °C with 5% CO2.Production of recombinant herring scTSHCHO cells were transfected with the scTSH expression plasmid using Lipofectamine 3000 (Invitrogen), stable clones were subsequently selected with 500 μg/ml G418 (Invitrogen) and screened for producing scTSH by western blot using a polyclonal antisera against the sea bass alpha subunit47. A positive clone was expanded in 225 cm2 cell culture flasks (Corning) in culture medium containing 5% FBS until confluence, then the cells were maintained in serum-free DMEM for hormone production for 7 days at 25 °C48. After 7 days, culture medium containing scTSH or without (negative control) was centrifuged at 15000 x g for 15 min and concentrated by ultrafiltration using Centricon Plus-70 / Ultracel PL-30 (Merck Millipore Ltd.). Then, western blotting was performed to confirm TSH production. Concentrated medium containing herring scTSH was denatured at 94 °C for 5 min in 0.1% SDS and 50 mM 2-mercaptoethanol, and then treated with 2.5 units of peptide-N-glycosidase F (Roche Diagnostics) at 37 °C for 2 h in 20 mM sodium phosphate with 0.5% Nonidet P-40, pH 7.5. All samples were run in 12% SDS-PAGE in the reducing condition and transferred to a PVDF membrane (Immobilon P; Millipore Corp.), then blocked overnight with 5% skimmed milk at 4 °C. After blocking, the membrane was incubated with polyclonal antisera against the sea bass alpha subunit (dilution 1:2000) for 90 min at room temperature, washed, and then further incubated with 1:25000 goat anti-rabbit immunoglobulin G (IgG) horseradish peroxidase conjugate (Bio-Rad Laboratories) for 60 min at room temperature. Immunodetection was performed by chemiluminescence with a Pierce ECL Plus Western Blotting Substrate kit (ThermoFisher Scientific).Cell surface expressionA Rhotag (MNGTEGPNFYVPFSNKTGVVYEE) was inserted at the N-terminus of herring TSHR for flow cytometry analysis of receptor cell surface expression. Anti-Rhotag polyclonal antibody was kindly provided by Drs. Gilbert Vassart and Sabine Costagliola (Université Libre de Bruxelles, Brussels, Belgium). PBS containing 1% BSA and 0.05% sodium azide was prepared as the flow cytometry (FCM) buffer for the washing and antibody incubation steps. 2.2 × 106 EPC cells were seeded in a 100 mm poly-d-Lysine-treated petri dish the day before transfection. Each dish was transfected with 10 μg TSHR or empty pcDNA3.1 expression plasmid using 20 μl jetPRIME transfection reagent in 500 μl jetPRIME transfection buffer (Polyplus transfection). Cells were harvested 24 h after transfection, then washed once in cold PBS and fixed in 2% PFA for 10 min at room temperature. After fixation, cells were washed three times with FCM buffer, then incubated with anti-Rhotag antibody or FCM buffer (negative control) for 1 h at room temperature. Cells were washed again with FCM buffer three times and stained with Alexa Fluor 488-labeled chicken anti-mouse IgG (H + L) antibody (1:200 dilution, ThermoFisher Scientific) or FCM buffer (negative control) for 45 min in the dark. After the fluorescent staining, cells were washed three times and resuspended in FCM buffer before analysis on a CytoFLEX instrument (Beckman Coulter). A minimum of 100,000 events was recorded for each sample, fluorescence intensities of negative control and cells transfected with empty pcDNA3.1 plasmid were used as the background for gating strategy. Cell surface expression was represented by the mean fluorescence intensity of the positively stained cell population.Dual-luciferase reporter assayEPC or HEK293 cells were plated in a 48-well plate at a density of 1 × 105 cells/well the day before transfection. A total of 250 ng plasmid mixture containing pGL4.29[luc2P/CRE/Hygro], TSHR expression plasmid (or empty pcDNA3.1) and pRL-TK with the ratio of 20:5:1 was prepared to transfect each well of cells using jetPRIME transfection reagent (Polyplus). Medium was replaced by fresh medium containing 10% FBS (TSH-induced condition) or serum-free medium (constitutive activity condition) 4 h after transfection. On day three, cells were treated with serum-free medium containing different dilutions of the concentrated scTSH medium for 4 h (TSH-induced condition) or directly subjected to the luminescence measurement without TSH induction (constitutive activity condition). Luminescence was measured using a Dual-Luciferase Reporter assay (Promega) on an Infinite M200 Microplate Reader (Tecan Group Ltd., Switzerland), and luciferase activity was represented as the ratio of firefly (pGL4.29[luc2P/CRE/Hygro]) to Renilla (pRL-TK) luminescence.5′-RACE to identify the herring DIO2 TSSTotal RNA was prepared from brain of a spring-spawning Atlantic herring using the RNeasy Mini Kit (Qiagen). Six μg of the isolated RNA was used for 5′-RACE with a FirstChoiceTM RLM-RACE Kit (ThermoFisher Scientific). One μl cDNA or Outer RACE PCR product was used as PCR template in a 20 μL reaction containing 0.8 mM dNTPs, 0.3 μM of each forward and reverse primer (Supplementary Table 1) and 0.75 U PrimeSTAR GXL DNA Polymerase (TaKaRa). Amplification was carried out with an initial denaturation of 3 min at 95 °C, followed by 35 cycles of 98 °C for 10 s, 58 °C for 20 s and 68 °C for 40 s, and a final extension of 10 min at 68 °C. The final 5′ RACE product was sequenced at Eurofins Genomics (Ebersberg, Germany).Sequence conservation analysisGenomic sequences covering the TSHR locus were extracted from Ensembl Genome Browser for Atlantic herring and 11 other fish species, including Amazon molly (Poecilia formosa), denticle herring (Denticeps clupeoides), goldfish (Carassius auratus), guppy (Poecilia reticulata), Neolamprologus brichardi, Japanese medaka (Oryzias latipes), northern pike (Esox lucius), orange clownfish (Amphiprion percula), spotted gar (Lepisosteus oculatus), three-spined stickleback (Gasterosteus aculeatus) and spotted green pufferfish (Tetraodon nigroviridis). The extracted sequences were firstly aligned using progressiveCactus49,50, and a subsequent alignment was generated using the hal2maf program from halTools51 with Atlantic herring assembly (Ch_v2.0.2)18 as the co-ordinate backbone. This alignment was used for the downstream phastCons score calculation by running phyloFit24 and phastCons25 from the PHAST package with default parameters. Peaks were called by grouping signals with a minimum phastCons score of 0.2 within 500 bp region.Structure modeling of human and herring TSHRsIn order to explore the possible interactions of the variant residues with other receptor interacting proteins and to study intramolecular interactions, we built a structural homology model for the herring TSHR (herrTSHR) complexed with herring TSH and Gs-protein. The TSHR hinge region that harbors the Q370H substitution and the C-terminus containing the 22aa repeat were excluded from the homology model due to the lack of structural templates for these regions. The homology model was constructed by using the following structural templates of evolutionarily related class A GPCRs: (i) the leucine-rich repeat domain (LRRD) complexed with hormone was modeled based on the solved FSHR LRRD – FSH complex structure (Protein Data Bank (PDB) ID: 4AY9)52,53, this part of model included herring TSHR Cys33 – Asn296 and fragments of the hinge region Gln297 – Thr312 and Ser393 – Ile421; (ii) the available structural complex of β2-adrenoreceptor with Gs-protein (PDB ID: 3SN6)54 was used as the template to model the seven-transmembrane helix domain (7TMD) of herring TSHR in the active conformation; (iii) the extracellular loop 2 (ECL2) was built by using the ECL2 of μ-opioid receptor (PDB ID: 6DDE)55. To prepare the template for herring TSHR modeling, the fused T4-lysozyme and bound ligand of β2-adrenoreceptor were deleted, the ECL1 and ECL3 loops were adjusted manually to the loop length of herring TSHR. Due to the lack of third intracellular loop (ICL3) in the β2-adrenoreceptor structure, amino acid residues of herring TSHR ICL3 were manually added to the template. Since herring TSHR does not have the TMH5 proline, which is highly conserved among all class A GPCRs and responsible for the helical kinks and bulges within this region56, we assumed a rather regular (stretched) helix conformation for the herring TSHR TMH5 and therefore replaced the kinked β2-adrenoreceptor TMH5 template with a regular α-helix. Moreover, the ECL2 template was substituted with μOR ECL2 structure because of its higher sequence similarity with herring TSHR in this region. Finally, amino acid residues of this chimeric 7TMD template and FSHR N-terminus were mutated to the corresponding spring herring TSHR residues and sequence of the heterodimeric FSH ligand was substituted by the herring TSH. All homology models were generated by using SYBYL-X 2.0 (Certara, NJ, US). The 7TMD structure was then fused with FSHR N-terminus at position 421. The assembled complex was subsequently optimized by the energy minimization under constrained backbone atoms (the AMBER F99 force field was used), followed by a 2 ns molecular dynamics simulation (MD) of the side chains. The entire TSHR complex was energetically minimized without any constraints until converging at a termination gradient of 0.05 kcal/mol*Å. Next, for autumn herrTSHR modeling, the spring TSHR sequence was substituted with autumn TSHR sequence. For humTSHR, the spring TSHR sequence was substituted with human TSHR, and the herring TSH ligand was replaced by the bovine TSH sequence. Both complex models were energetically minimized until converging at a termination gradient of 0.05 kcal/mol*Å.To investigate the microenvironment around the L471M mutation at TMH2 position 2.51, local short MD’s of 4 ns on Met4712.51 (spring herrTSHR), Leu471 (autumn herrTSHR) or Phe461 (humTSHR) and its surrounding amino acids were performed. During MD simulations, backbone atoms of the entire complexes as well as all side chains, except residues at positions 1.47, 1.51, 1.54, 2.48, 2.52, and 2.55 that form the hydrophobic patch around position 2.51, were constrained.Statistics and reproducibilityResults were presented as the mean + SD (standard deviation) calculated from at least four biological replicates for each experiment, and at least two independent experiments were conducted for each assay. Unpaired two-tailed Student’s t test was performed to calculate the P-values and means were judged as statistically significant when P ≤ 0.05.Reporting summaryFurther information on research design is available in the Nature Research Reporting Summary linked to this article. More

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    Multi-year presence of humpback whales in the Atlantic sector of the Southern Ocean but not during El Niño

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    Pairwise interact-and-imitate dynamics

    The modelConsider a unit-mass population of agents who repeatedly interact in pairs to play a symmetric stage game. The set of strategies available to each agent is finite and denoted by (S equiv {1, ldots , n}). A population state is a vector (x in X equiv {x in {mathbb{R}}^n_+: sum _{i in S} x_i = 1}), with (x_i) the fraction of the population playing strategy (i in S). Payoffs are described by a function (F: S times S rightarrow {mathbb{R}}), where F(i, j) is the payoff received by an agent playing strategy i when the opponent plays strategy j. As a shorthand, we refer to an undirected pair of individuals, one playing i and the other playing j, as an ij pair. The set of all possible undirected pairs is denoted by (mathscr {P}).The interaction structure is modeled as a function (p : X times mathscr {P} rightarrow left[ 0, 1/2 right] ) subject to (sum _{ij in mathscr {P}} p_{ij}(x)=1/2) (since the mass of pairs is half the mass of agents), with (p_{ij}(x)) indicating the mass of ij pairs formed in state x. Note that the mass of ij pairs can never exceed (min {x_i,x_j}), that is, (p_{ij}(x) le min {x_i,x_j}) for all x. We assume that p is continuous in X, and that (p_{ij}(x) > 0) if and only if (x_i > 0) and (x_j > 0 )—meaning that the probability of an ij pair being formed is strictly positive if and only if strategies i and j are played by someone. In the case of uniform random matching, (p_{ii} = x_i^2/2) and (p_{ij} = x_i x_j) for any i and (j ne i).The revision protocol is modeled as a function (phi : X times S times S rightarrow [-1,1]), where (phi _{ij}(x) in [-1,1]) is the probability that an ij pair will turn into an ii pair minus the probability that it will turn into a jj pair, conditional on the population state being x and an ij pair being formed. We assume that (phi ) is continuous in X. We note that by construction (phi _{ij}=-phi _{ji}) for all (i,j in S), and hence (phi _{ii}=0) for all (i in S). Our main assumption on the revision protocol is the following, which is met, among others, by pairwise proportional imitative and imitate-if-better rules22.
    Assumption 1

    For every (x in X), (phi _{ij}(x) > 0) if (F(i,j) > F(j,i)).
    In what follows we consider a dynamical system in continuous time with state space X, characterized by the following equation of motion.

    Definition 1

    (Pairwise interact-and-imitate dynamics—PIID) For every (x in X) and every (i in S):$$begin{aligned} dot{x}_i = sum _{j in S} p_{ij}(x) phi _{ij}(x). end{aligned}$$
    (1)

    Main findingsGlobal asymptotic convergenceIn any purely imitative dynamics, if (x_i(t)=0), then (x_i(t^{prime})=0) for every (t^{prime} > t). This implies that we cannot hope for global asymptotic convergence in a strict sense. Thus, to assess convergence towards a certain state x in a meaningful way, we restrict our attention to those states where all strategies that have positive frequency in x have positive frequency as well. We denote by (X_x) the set of states whose support contains the support of x.

    Definition 2

    (Supremacy) Strategy (iin S) is supreme if (F(i,j) >F(j,i)) for every (j in S setminus {i}).
    We note that under PIID, the concept of supremacy is closely related to that of asymmetry33,34, in that (F(i,j) > F(j,i)) implies that agents can only switch from strategy j to strategy i.

    Proposition 1

    If (i in S) is a supreme strategy, then state (x^* equiv left{ x in X : x_i = 1 right} ) is globally asymptotically stable for the dynamical system with state space (X_{x^*}) and PIID as equation of motion.
    Relation to replicator dynamicsTo further characterize the dynamics induced by the pairwise interact-and-imitate protocol, we make two additional assumptions. First, matching is uniformly random, meaning that everyone in the population has the same probability of interacting with everyone else; formally, (p_{ii} = x_i^2/2) and (p_{ij} = x_i x_j) for all i and (j ne i). Second, the probability that an agent has to imitate the opponent is proportional to the difference in their payoffs if the opponent’s payoff exceeds her own, and is zero otherwise. As a consequence, (phi _{ij} = F(i,j) – F(j,i)) up to a proportionality factor. Let

    (F left( i, x right) :=sum _j x_j F left( i, j right) ),

    (F left( x, i right) :=sum _j x_j F left( j, i right) ), and

    ( F left( x, x right) :=sum _i sum _j x_i x_j F left( i, j right) ).

    Under these assumptions, at any point in time, the motion of (x_i) is described by:$$begin{aligned} dot{x}_i&= sum _{j ne i} x_j x_i left[ F left( i, j right) – F left( j, i right) right] = x_i sum _{j} x_j left[ F left( i, j right) – F left( j, i right) right] nonumber \&= x_i left[ F left( i, x right) – F left( x, i right) right] , end{aligned}$$
    (2)
    which is a modified replicator equation. According to (2), for every strategy i chosen by one or more agents in the population, the rate of growth of the fraction of i-players, (dot{x}_i / x_i), equals the difference between the expected payoff from playing i in state x and the average payoff received by those who are matched against an agent playing i. In contrast, under standard replicator dynamics35, the fraction of agents playing i varies depending on the excess payoff of i with respect to the current average payoff in the whole population, i.e., (dot{x}_i = x_i left[ F left( i, x right) – F left( x, x right) right] ).A noteworthy feature of replicator dynamics is that they are always payoff monotone: for any (i,j in S), the proportions of agents playing i and j grow at rates that are ordered in the same way as the expected payoffs from the two strategies36. In the case of PIID, this result fails.

    Proposition 2

    Pairwise-Interact-and-Imitate dynamics need not satisfy payoff monotonicity.
    To verify this, it is sufficient to consider any symmetric (2 times 2) game where (F left( i, j right) > F left( j, i right) ) but (F left( j, x right) > F left( i, x right) ) for some (x in X), meaning that i is the supreme strategy but j yields a higher expected payoff in state x. See Fig. 1 for an example where, in the case of uniform random matching, the above inequalities hold for any x; if strategies are updated according to the interact-and-imitate protocol, then this game only admits switches from i to j, therefore violating payoff monotonicity. Proposition 2 can have important consequences, including the survival of pure strategies that are strictly dominated.Survival of strictly dominated strategiesAn recurring topic in evolutionary game theory is to what extent does support exist for the idea that strictly dominated strategies will not be played. It has been shown that if strategy i does not survive the iterated elimination of pure strategies strictly dominated by other pure strategies, then the fraction of the population playing i will converge to zero in all payoff monotone dynamics37,38. This result does not hold in our case, as PIID is not payoff monotone.More precisely, under PIID, a strictly dominated strategy may be supreme and, therefore, not only survive but even end up being adopted by the whole population. This suggests that from an evolutionary perspective, support for the elimination of dominated strategies may be weaker than is often thought. Our result contributes to the literature on the conditions under which evolutionary dynamics fail to eliminate strictly dominated strategies in some games, examining a case which has not yet been studied39.To see that a strictly dominated strategy may be supreme, consider the simple example shown in Fig. 1. Here each agent has a strictly dominant strategy to play A; however, since the payoff from playing B against A exceeds that from playing A against B, strategy B is supreme. Thus, by Proposition 1, the population state in which all agents choose B is globally asymptotically stable.Figure 1A game where the supreme strategy is strictly dominated.Full size imageFigure 1 can also be used to comment on the relation between a supreme strategy and an evolutionary stable strategy, which is a widely used concept in evolutionary game theory40,41. Indeed, while B is the supreme strategy, A is the unique evolutionary stable strategy because it is strictly dominant. However, if F(B, A) were reduced below 2, holding everything else constant, then B would become both supreme and evolutionary stable. We therefore conclude that no particular relation holds between evolutionary stability and supremacy: neither one property implies the other, nor are they incompatible.ApplicationsHaving obtained general results for the class of finite symmetric games, we now restrict the discussion to the evolution of behavior in social dilemmas. We show that if the conditions of Proposition 1 are met, then inefficient conventions emerge in the Prisoner’s Dilemma, Stag Hunt, Minimum Effort, and Hawk–Dove games. Furthermore, this result holds both without and with the assumption that agents interact assortatively.Ineffectiveness of assortmentConsider the (2 times 2) game represented in Fig. 2. If (c > a > d > b), then mutual cooperation is Pareto superior to mutual defection but agents have a dominant strategy to defect. The resulting stage game is the Prisoner’s Dilemma, whose unique Nash equilibrium is (B, B). Moreover, since (F (B,A) > F(A,B)), B is the supreme strategy and the population state in which all agents defect is globally asymptotically stable.We stress that defection emerges in the long run for every matching rule satisfying our assumptions, and therefore also in the case of assortative interactions. Assortment reflects the tendency of similar people to clump together, and can play an important role in the evolution of cooperation42,43,44,45. Intuitively, when agents meet assortatively, the risk of cooperating in a social dilemma may be offset by a higher probability of playing against other cooperators. However, under PIID, this is not the case: the decision whether to adopt a strategy or not is independent of expected payoffs, and like-with-like interactions have no effect except to reduce the frequency of switches from A to B.Figure 2A (2 times 2) stage game.Full size imageEmergence of the maximin conventionIf (a > c > b), (a > d) and (d > b), then the game in Fig. 2 becomes a Stag Hunt game, which contrasts risky cooperation and safe individualism. The payoffs are such that both (left( A, Aright) ) and (left( B, Bright) ) are strict Nash equilibria, that (left( A, Aright) ) is Pareto superior to (left( B, Bright) ), and that B is the maximin strategy, i.e., the strategy which maximizes the minimum payoff an agent could possibly receive. We also assume that (a + c ne c + d), so that one of A and B is risk dominant46. If (a + b > c + d), then A (Stag) is both payoff and risk dominant. When the opposite inequality holds, the risk dominant strategy is B (Hare).Since (F (B,A) > F(A,B)), B is supreme independently of whether or not it is risk dominant to cooperate. This can result in large inefficiencies because, in the long run, the process will converge to the state in which all agents play the riskless strategy regardless of how rewarding social coordination is. As in the case of the Prisoner’s Dilemma, this holds for all matching rules satisfying our assumptions.Evolution of effort exertionIn a minimum effort game, agents simultaneously choose a strategy i, usually interpreted as a costly effort level, from a finite subset S of ({mathbb{R}}). An agent’s payoff depends on her own effort and on the minimum effort in the pair:$$begin{aligned} F left( i, j right) = alpha min left{ i, j right} – beta i , end{aligned}$$where (beta > 0) and (alpha > beta ) are the cost and benefit of effort, respectively. From a strategic viewpoint, this game can be seen as an extension of the Stag Hunt to cases where there are more than two actions. The best response to a choice of j by the opponent is to choose j as well, and coordinating on any common effort level gives a Nash equilibrium. Nash outcomes can be Pareto-ranked, with the highest-effort equilibrium being the best possible outcome for all agents. Thus, choosing a high i is rationalizable and potentially rewarding but may also result in a waste of effort.Under PIID, any (i > j) implies (phi _{ij} < 0) by Assumption 1, meaning that agents will tend to imitate the opponent when the opponent’s effort is lower than their own. The supreme strategy is therefore to exert as little effort as possible, and the population state in which all agents choose the minimum effort level is the unique globally asymptotically stable state.Emergence of aggressive behaviorConsider again the payoff matrix shown in Fig. 2. If (c > a > b > d), then the stage game is a Hawk–Dove game, which is often used to model the evolution of aggressive and sharing behaviors. Interactions can be framed as disputes over a contested resource. When two Doves (who play A) meet, they share the resource equally, whereas two Hawks (who play B) engage in a fight and suffer a cost. Moreover, when a Dove meets a Hawk, the latter takes the entire prize. Again we have that (F (A,B) < F(B,A)), implying that B is the supreme strategy and that the state where all agents play Hawk is the sole asymptotically stable state.The inefficiency that characterizes the (B, B) equilibrium in the Hawk–Dove game arises from the cost that Hawks impose on one another. This can be viewed as stemming from the fact that neither agent owns the resource prior to the interaction or cares about property. A way to overcome this problem may be to introduce a strategy associated with respect for ownership rights, the Bourgeois, who behaves as a Dove or Hawk depending on whether or not the opponent owns the resource41. If we make the standard assumption that each member of a pair has a probability of 1/2 to be an owner, then in all interactions where a Bourgeois is involved there is a 50 percent chance that she will behave hawkishly (i.e., fight for control over the resource) and a 50 percent chance that she will act as a Dove.Let R and C denote the agent chosen as row and column player, respectively, and let (omega _R) and (omega _C) be the states of the world in which R and C owns the resource. The payoffs of the resulting Hawk–Dove–Bourgeois game are shown in Fig. 3. If agents behave as expected payoff maximizers, then All Bourgeois can be singled out as the unique asymptotically stable state. Under PIID, this is not so; depending on who owns the resource, an agent playing C against an opponent playing B may either fight or avoid conflict and let the opponent have the prize. It is easy to see that (F left( C, B mid omega _R right) = F left( B,C mid omega _C right) = d), meaning that the payoff from playing C against B, conditional on owning the resource, equals the payoff from playing B against C conditional on not being an owner. In contrast, the payoff from playing C against B, conditional on not owning the resource, is always worse than that of the opponent, i.e., (F left( C, B mid omega _C right) = b < c = F left( B, C mid omega _R right) ). Thus, in every state of the world, B (Hawk) yields a payoff that is greater or equal to that from C (Bourgeois). Moreover, since (F left( B,A right) > F left( A, B right) ) in both states of the world, strategy B is weakly supreme by Definition 4, and play unfolds as an escalation of hawkishness and fights.Figure 3The Hawk–Dove–Bourgeois game.Full size image More

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    High-throughput 16S rRNA gene sequencing of the microbial community associated with palm oil mill effluents of two oil processing systems

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    Red light, green light: both signal ‘go’ to deadly algae

    Green and red lighting might be good for migratory birds and sea turtles, but could have undesirable effects if marine algae are present. Credit: Getty

    Ecology
    24 June 2021
    Red light, green light: both signal ‘go’ to deadly algae

    Artificial lighting thought to be more wildlife-friendly than white light could encourage algal blooms.

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    Green or red lights in seaside areas have been proposed as alternatives to white light to protect wildlife. But new experiments show that exposure to red or green light at night boosts the growth of some ocean algae — including species known to rob waters of oxygen.Little is known about the impact of artificial light on marine life, even though many brightly lit cities are coastal. To address that knowledge gap, Sofie Spatharis at the University of Glasgow, UK, and her colleagues exposed a mix of microscopic marine algae collected from Scottish waters to standard white light. They also exposed the mixture to red and green lights, which have been proposed to minimize impacts on sea turtles and migratory seabirds, respectively.The team found that all light colours enhanced growth of the microalgae mix. Red light had the most pronounced effect, doubling the number of cells produced. The proportions of species in the mixture also shifted: both red and green light especially favoured growth of harmful species in the Skeletonema genus, which form dense blooms that are deadly to fish.

    Proc. R. Soc. B (2021)

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