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To create and annotate a de novo transcriptome assembly for Antarctic krill a preliminary investigation focusing on the efficiency and quality of already existing strategies for de novo transcriptome assembly of non-model organisms was performed. In a second step, we focused on identifying and applying the best transcriptome assembly strategy to finally explore the gene expression levels across different developmental stages and krill responses to different environmental conditions. At first, separate transcriptome reconstructions using different assembly programs were carried out. A combination of two filtering steps was applied to these results to discard artifacts and improve the assembly quality. Reconstructed transcripts across all assemblers were joined, producing a set of non-redundant representative transcripts. We obtained these results by applying the EvidentialGene pipeline (version 4), which was specifically designed to combine different reconstructions and to eliminate redundant sequences. Finally, we applied another filter to identify redundant or mis-assembled sequences still appearing in the transcriptome.Transcriptome qualityWe checked the quality of our reconstructed transcriptome step by step, starting from the independent de novo assemblies, then evaluating the potential of merging all assemblies into a unique meta-assembly, and finally filtering the transcriptome for redundancy. All these results are summarized in Fig. 1, Tables 1 and 2. The result of our reconstruction strategy was evaluated using different measures: the N50 statistics highlighted an increase in transfrag lengths at each step. Recent benchmarks, such as18, have shown that, while reconstructing the transcriptome of a species, no single approach is uniformly superior: the quality of each result is influenced by a number of factors, both technical (k-mer size, strategy for duplicate resolution) and biological (genome size, presence of contaminants). In our study, we observed that, although a consistent number of sequences was removed through each step of the assembly, merging and filtering procedure, we didn’t encounter any decline in the quality described by the basic statistics of the reconstructed transcripts (Table 1).Figure 1Transcriptome quality assessment results. Results of the first assembly filtering in terms of total number of transcripts.Full size imageTable 1 Quality measures computed at each assembly step, from the independent de novo assembly algorithms (a), after the first filtering process (b) and finally comparing the quality of the EvidentialGene meta-assembly and the final krill transcriptome after the redundancy filter (c).Full size tableTable 2 BUSCO assessment results on independent de novo assemblies from RNA-seq stranded library.Full size tableWe then explored the completeness of the krill transcriptome according to conserved ortholog content using BUSCO (version 4.0.5) comparing our sequences to all the expected single-copy orthologs from the Arthropoda phylum. The results of the BUSCO analyses performed on each independent de novo assembly, on the EvidentialGene reconstruction and the final transcriptome are reported in Table 2. This analysis confirms that our strategy for controlling redundancy did not affect transcriptome completeness: indeed, the fraction of complete single-copy essential genes dropped by 1.8% only, while 123,376 redundant transfrags were discarded.We finally compared our quality assessment results with those from previously released krill transcriptomes (Table 3). Our latest assembly significantly improves all the metrics we have discussed above. While this evidence suggests that our assembly is reasonably close to providing a complete representation of the krill transcriptome, it is more difficult to gauge the amount of redundancy it contains. Specifically, it remains difficult to distinguish between splice variants of a gene and possible paralogous copies. We believe that only the availability of a genome draft will make it possible to reliably discriminate between these two signals.Table 3 Quality statistics of the previously released krill transcriptomes compared to the newly assembled KrillDB2. GenBank accession GFCS00000000.1 refers to the SuperbaSe krill transcriptome reference19.Full size tableFunctional classificationThe assembled fragments were aligned against known protein and nucleotide databases to understand whether they could be linked to specific functions or processes described in other species. The functional annotation analyses showed that 63,903 contigs (42% of the total krill transcriptome) matched at least one protein from the NCBI NR (non-redundant) collection for a total of 98,316 unique proteins, while 62,518 transfrags found homology with a UniProtKB/TREMBL protein sequences (41% of the total), matching a total of 96,005 unique proteins. Furthermore, 22,024 krill transcripts (15% of the total) had significant matches with sequences in the NCBI NT nucleotide database. To classify transcripts by putative function, we performed a GO assignment. Specifically, 2833 GO terms (corresponding to 13,064 genes) were assigned: 1224 of those (corresponding to 11,575 genes) represented molecular functions; 1193 terms (corresponding to 6990 genes) were linked to biological processes; 416 terms (corresponding to 4301 genes) represented cellular components.A case study on the discovery of opsin genesTo evaluate the gene discovery potential of the new assembly, we searched the transcriptome for novel members of the opsin family. Opsins are a group of light sensitive G protein-coupled receptors with seven transmembrane domains. Fourteen genes were annotated as putative opsins, and the conserved domains analysis revealed that all of them possess the distinctive 7 α-helix transmembrane domain structure. The eight previously cloned opsins20 were all represented in KrillDB2 (sequence identity  > 90%; Table S1 Supplementary Material). The other six genes we identified can therefore be considered new putative opsins. Among those, we found four putative rhabdomeric opsins: EsRh7 and EsRh8, with 70% and 59% of amino acid identity to EsRh1a and EsRh4, respectively; EsRh9 and EsRh10 showing high sequence identity (87% and 74%, respectively) to EsRh5. Furthermore, we identified two putative ancestral opsins: a non-visual arthropsin (EsArthropsin), and an onychopsin (EsOnychopsin) with 70% and 49% of sequence identity with crustacean and onychophoran orthologous, respectively. Phylogenetic analysis (Fig. 2) suggested that EsRh7-10 are middle-wavelength-sensitive (MWS) rhabdomeric opsins, and further confirmed EsArthropsin and EsOnychopsin annotation.Figure 2Phylogenetic relationships of Euphausia superba opsins shown as circular cladogram. Colored dots indicate krill opsins: red, previously cloned opsins; green, novel identified opsins. The spectral sensitivities of rhabdomeric opsin clades were inferred from the curated invertebrate-only opsin dataset proposed by DeLeo & Bracken‐Grissom, 2020. Represented opsin classes: LWS, long-wavelenght-sensitive; LSM, long/middle-wavelenght-sensitive; MWS, middle-wavelenght-sensitive; SWS/UV, short/UV-wavelenght-sensitive; ONY, onychopsins; MEL, melanopsins; PER, peropsin; ART, arthropsin. Rectangular phylogram is reported in Fig. S1 (Supplementary Material).Full size imageDifferential expressionThe availability of a new assembly of the krill transcriptome, reconstructed by collecting the largest amount of experimental data available thus far, suggested the possibility of performing a more detailed investigation of differential expression patterns. Therefore, we decided to reanalyze the dataset from Höring et al.21 to assess the possibility of identifying differentially expressed genes that were not detected in the original study due to the use of an older reference transcriptome15.Our design matrix for the model included all the independent factors (season, area and sex) and, in addition, the interaction between area and season, sex and area, sex and season.In total 1741 genes were differentially expressed (DEG) among experimental conditions. They correspond to around 2% of the total reconstructed genes. In the previous work by Höring21, the same samples were quantified against 58,581 contigs15 producing 1654 DEGs. Table 4 summarizes the list of performed contrasts, each one with the number of differentially expressed up and down regulated genes.Table 4 List of contrasts computed with total number of differentially expressed genes and numbers of up- and downregulated genes.Full size table1195 DEGs were identified in the comparison between summer and winter specimens: 1078 were up-regulated and 117 down-regulated. In addition, 396 of such DEGs had some form of functional annotation. In general, these results are in accordance with the discussion by Höring21, which found that seasonal differences are predominant compared to regional ones. A summary of the DEGs is listed in Table 5. Complete tables of differentially expressed genes are downloadable on KrillDB2 (Fig. 3c; https://krilldb2.bio.unipd.it/, Section “Differentially Expressed Genes (DEGs)”).Table 5 List of biologically relevant DEGs identified, starting from those already described by Höring et al.35.Full size tableFigure 3Blast search section. The new search box for sequence searches (a) with an example of a BLAST search (highlighted in yellow) and the corresponding results (b). By clicking on each target identifier, the user will be redirected to that specific transcript page, where new sections have been added, as shown in Fig. 6.Full size imageSummer versus winterWe selected a series of genes among seasonal DEGs according to what has been already described in the literature. Höring et al.21 previously identified and described 35 relevant DEGs involved in seasonal physiology and behavior: we recovered the same gene signature in our analysis by comparing summer to winter samples. The majority of these DEGs appear to be involved in the development of cuticles (chitin synthase, carbohydrate sulfotransferase 11), lipid metabolism (fatty acid synthase 2, enoyl-CoA ligase), reproduction (vitellogenin, hematopoietic prostaglandin D synthase), metabolism of different hormones (type 1 iodothyronine deiodinase) and in the circadian clock (cryptochrome). Our results also include DEGs that were involved in the moult cycle of krill in other studies16. Specifically, we identified a larger group of genes involved in the different stages of the cuticle developmental process (peritrophin-A domain, calcified cuticle protein, glycosyltransferase 8-domain containing protein 1, collagen alpha 1, glutamine-fructose 6 phosphate), including proteins such as cuticle protein-3,6,19.8, early cuticle protein, pupal cuticle protein, endocuticle structural glycoprotein, chitinase-3 and chitinase-4, the latter representing a group of chitinase which have been shown to be expressed predominantly in gut tissue during larval and/or adult stages in other arthropods and are proposed to be involved in the digestion of chitin-containing substrates22. Finally, in addition to trypsin and crustin 4 (immune-related gene, essential in early pre-moult stage when krill still have a soft cuticle to protect them from pathogen attack, as seen by Seear et al.16), we also identified crustin-1,2,3,5 and 7. All the reported genes were up-regulated in summer, the period in which growth takes place and krill moult regularly.Cuticle development genes were also identified as differentially expressed in the analysis of the interaction of multiple factors, between male samples coming from South Georgia and female specimens coming from the area of Bransfield Strait-South Orkney (considered as a unique area since they are placed at similar latitudes). Strikingly, we also identified a pro-resilin gene, whose role in many insects consists in providing efficient energy storage, being up-regulated in South Georgia male specimens.Interaction effectsA number of relevant DEGs were found among specific regional and seasonal factors interactions. For instance, by comparing krill samples coming from South Georgia in summer and individuals sampled in Bransfield Strait-South Orkney in winter, we found genes up-regulated in summer in South Georgia related to reproductive activities, such as doublesex and mab-3 related transcription factor. The latter is a transcription factor crucial for sex determination and sexual differentiation, which was already described in other arthropods23. Since no differentially expressed gene related to reproduction was found by Höring et al.21 in the same comparisons, this suggests that the new krill transcriptome improves the power to identify new expression patterns and characterize the krill samples.Finally, the comparison between male individuals from the Lazarev Sea and female specimens from the Bransfield Strait-South Orkney showed additional DEGs involved in reproduction, such as ovochymase 2, usually highly expressed in female adults or eggs, serine protease and a trypsin-like gene. In particular, trypsin-like genes are usually thought to be digestive serine proteases, but previous works suggested that they can play other roles24; many trypsins show female or male-specific expression patterns and have been found exclusively expressed in males, as in our analysis, suggesting that they play a role in the reproductive processes.The simultaneous presence of differentially expressed genes involved in different steps of the krill moulting cycle, in the reproductive process and in sexual maturation that appear to be differentially expressed in the same comparisons is in accordance with what was already observed in krill25 and other krill species26. In particular, there is evidence of a strong relation between the krill moulting process and its growth and sexual maturation during the year, which supports and confirms the reliability of our results in terms of genes involved in such krill life cycle steps.Identification of microRNA PrecursorsAlthough microRNAs play a key role in the regulation of gene expression and in many important biological processes, such as development or cell differentiation, there is still no information about microRNAs in krill species.Here we performed an investigation to test whether the new transcriptome could also include sequences with a significant homology to known mature microRNAs.In total we identified 261 krill transcripts whose sequences are highly similar to 644 known microRNAs from other species. 306 sequences were linked to at least one GO term, matching 54 krill transcripts (Table S2, Supplementary Material). Among them, we identified 5 putative microRNAs involved with changes in cellular metabolism (age-dependent general metabolic decline—GO:0001321, GO:0001323), as well as changes in the state or activity of cells (age-dependent response to oxidative stress—GO:0001306, GO:0001322, GO:0001324), 35 microRNAs involved in interleukin activity and production. We found 26 putative microRNAs likely involved in ecdysteroidogenesis (specifically GO:0042768), a process resulting in the production of ecdysteroids, moulting and sex hormones found in many arthropods. In addition, we found a microRNA involved in fused antrum stage (GO:0048165) which appears to be related in other species to oogenesis. We also identified 27 microRNAs related to rhombomere morphogenesis, formation and development (GO:0021661, GO:0021663, GO:0021570). These functions have been linked to the development of portions of the central nervous system in vertebrates, which share the same structure of those found in arthropod brains. Lastly, 26 krill sequences showed high similarity with 2 mature microRNA related to the formation of tectum (GO:0043676), which represents in arthropods and, specifically, crustaceans, the part of the brain acting as visual center.KrillDB2 web InterfaceThe KrillDB website has been redesigned to include the new version of the transcriptome assembly. Figures 3, 4, 5 and 6 collect images taken from the new main sections of the database. The integrated full-text search engine allows the user to search for a transcript ID, gene ID, GO term, a microRNA ID or any other free-form query. Results of full-text searches are now organized into several separate tables, each representing a different data source or biological aspect (Fig. 5). Results of GO term searches are summarized in a table reporting the related genes with corresponding domain or microRNA match and associated description. Both gene and transcript-centric pages have been extended with two new sections: “Orthology” and “Expression” (Fig. 6). The Orthology section summarizes the list of orthologous sequences coming from the OMA analysis, each one with the species it belongs to and the identity score.Figure 4Differential Expression section. The new section collecting all differentially expressed genes tables (a) with an example of the corresponding result for a selected contrast (b).Full size imageFigure 5New search engine of KrillDB2. Example of the results of a full-text search on KrillDB2.Full size imageFigure 6Additional sections in gene and transcript pages. The new sections in the gene-centric page show a table listing the orthologous sequences with their belonging species and the identity score (a), a visualization of the gene structure as estimated by Lace software (d) and a boxplot coming from Expression Atlas analyses (c). Both Orthology and Expression sections are integrated also in the transcript-centric page. When a transcript is annotated as a putative microRNA, a “Predicted Hairpin” section displays a visualization of the hairpin predicted secondary structure and tables showing the alignment length, the HHMMiR score and the list of mature microRNAs matching (b).Full size imageThe “Expression” section shows a barplot representing abundances estimates obtained from Salmon. An additional section, called “Gene Structure” (Fig. 6), was added to the gene page on the basis of the results coming from the SuperTranscript analysis. Specifically, we modified the STViewer.py Python script (from Lace), optimizing and adapting it to our own data and database structure, in order to produce a visualization of each gene with its transcripts. Since Lace relies on the construction of a single directed splice graph and it is not able to compute it for complex clusters with more than 30 splicing variants, this section is available for a selection of genes only.The new KrillDB2 release includes completely updated transcript and gene identifiers. However, the user searching for a retired ID is automatically redirected to the page describing the newest definition of the appropriate transcript or gene.The KrillDB2 homepage now includes two additional sections: one is represented by the possibility to perform a BLAST search (Fig. 3). Any nucleotide or protein sequence (query) can be aligned against krill sequences stored in the database. Results are summarized in a table containing information about the krill transcripts (target) that matched with the user’s query, and the e-value corresponding to the alignment. The other new section, called “Differentially Expressed Genes”, allows the user to browse all the tables listing the genes that were found to be differentially expressed among the conditions we have described above (Fig. 4). A drop-down menu gives access to the different comparisons; DEG tables list for each gene its log fold-change, p- and FDR values as estimated by edgeR. Moreover, each gene is linked to a functional description (if available) inferred from sequence homology searches.Information about krill transcripts showing homology with an annotated microRNA is available in the “Predicted Hairpin” (Fig. 6). It contains a summary table with details about the hairpin length and the similarity score (as estimated by HHMMiR), followed by full listing of all the corresponding mature microRNAs (including links to their miRBase page). In addition, an image displaying the predicted secondary structure of the hairpin is included (computed by the “fornac” visualization software from the ViennaRNA suite). More

Lehtonen12 presents three simple models with the same broad structure: a single mutant individual with divergent mating behaviour arises in a population of ‘residents’ that all play the same strategy, and the success of that mutant is then followed (Figs. 1, 2). Specifically, Lehtonen investigates the fitness benefits of increased mating for mutant males in comparison to mutant females. Two important parameters can be varied: (i) the degree of anisogamy (defined here as the ratio of sperm number to egg number), which captures how divergent males and females are in the size (and thus number) of gametes they produce, and (ii) the efficiency of fertilisation, which determines how easily gametes can find and fuse with each other. If fertilisation is highly efficient, then gametes of the less numerous type will achieve nearly full fertilisation; on the other hand, inefficient fertilisation can result in gametes of both sexes going unfertilised.Fig. 2: Structure of the three models of Lehtonen12, showing differences in mating behaviour between resident males (green), resident females (blue) and mutant males and females (both yellow).For illustration, we suppose that females produce four eggs each and males produce eight sperm (the anisogamy ratio in nature is typically much higher). In Model 1, resident individuals spawn monogamously in a ‘nest’ (black outline), whereas mutant males and females can bring additional partners to their nest to spawn in a group. In Model 2, resident individuals divide their gametes equally among m spawning groups, each consisting of m individuals of each sex (shown here with m = 2). Mutant males and females instead divide their gametes among a larger or smaller number of groups, mmutant (shown here with mmutant = 4). In Model 3, there is a further sex asymmetry in addition to anisogamy: Fertilisation takes place inside the female’s body. Resident individuals mate with m partners (shown here with m = 2), whereas mutant males and females mate with a larger or smaller number of partners, mmutant (shown here with mmutant = 4).Full size imageIn the first two models, fertilisation is external and no assumptions are made about pre-existing differences between the sexes apart from the number of gametes they produce. In other words, males and females are identical except that males produce sperm in greater numbers than females produce eggs. In Model 1, resident individuals are assumed to mate monogamously, whereas a mutant can monopolise multiple partners of the opposite sex (Fig. 2). Importantly, both male and female mutants can bring additional partners back to their ‘nest’ to spawn in a group. When fertilisation is highly efficient, females can fertilise all of their eggs by bringing back a single male, and there is simply no benefit (in this model) of seeking further partners (Fig. 1A). In contrast, anisogamy means that males always produce at least some gametes in excess, and thus can benefit from seeking additional mates. When fertilisation is inefficient, however, both sexes benefit from increasing the concentration of opposite-sex gametes at their ‘nest’ (Fig. 1B). This latter benefit is sex-symmetric, whereas the former continues to apply only to males. As a consequence, the Bateman gradients are always steeper for males than for females (Fig. 1A, B), confirming Bateman’s argument.Model 2 similarly assumes external fertilisation, but in this case the resident males and females meet in groups consisting of m individuals of each sex (Fig. 2). Fertilisation occurs via group spawning. It is assumed that each resident individual divides its gametes evenly across M groups, whereas mutant individuals can instead spread their gametes over a larger or smaller number of groups (note that the author assumes that M = m, but this assumption could be relaxed without undermining the core argument). Spreading gametes out across a larger number of spawning groups does not increase the concentration of opposite-sex gametes they encounter (Fig. 2). However, a mutant that spreads its gametes more widely reduces the density of its own gametes across those groups in which it spawns. This in turn results in there being more opposite-sex gametes for each gamete of the mutant’s sex in those groups. For example, in Fig. 2, mutant males spawn in twice as many groups as resident males and thereby halve the density of their own sperm in each group. The resulting egg-to-sperm ratio of (frac{4}{6}=frac{2}{3}) is more favourable than the ratio of (frac{4}{8}=frac{1}{2}) that the resident males experience. Mutant females can similarly increase local sperm-to-egg ratios by spreading their eggs over more groups. However, in contrast to males, this only leads to fitness benefit if fertilisation is inefficient, and even then the benefit to females is very modest (scarcely perceptible in Fig. 1D). Gamete spreading reduces wasteful competition among the mutants’ own gametes for fertilisation. Such ‘local’ gamete competition, like gamete competition more generally, is stronger among sperm than among eggs because sperm are more numerous under anisogamy13,14. Consequently, as in Model 1, Bateman gradients are always steeper in males (Fig. 1C, D). Recall that the results of the above models emerge in the absence of any assumptions beyond the sex difference in the number of gametes produced.The third and final model allows for a further pre-existing difference between the sexes in addition to anisogamy: internal fertilisation, which is common and widespread in animals (Fig. 2)15. Each female is assumed to mate with m males, while each male divides his gametes evenly among m females. As in the previous two models, males benefit more than females from additional matings under most conditions. However, in the particular case where fertilisation is highly inefficient and the ratio of sperm to eggs is not too large, the pattern can theoretically reverse, such that female Bateman gradients exceed their male counterparts (Fig. 1F). The reason is that the effects of gamete concentration are asymmetric under internal fertilisation: Multiple mating by a female increases the local concentration of sperm its eggs experience, whereas a male’s multiple mating does not increase the concentration of eggs around its sperm (Fig. 2). Under conditions of severe sperm limitation—due to both weak anisogamy and highly inefficient fertilisation—this can lead to females benefitting more from additional matings than males (Fig. 1F). Although intriguing, it is unclear whether this finding has any empirical relevance, as sperm limitation is probably rarely severe in internal fertilisers. Under more realistic conditions of moderate to high fertilisation rates, sex differences in the degree of local gamete competition once again become decisive, and male Bateman gradients exceed their female counterparts (Fig. 1E). More

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Eco-evolutionary model on spatial graphsWe establish an individual-based model (IBM) where individuals are structured over a trait space and a graph representing a landscape. For the sake of simplicity, we consider the case of asexual reproduction and haploid genetics29. Individuals die, reproduce, mutate and migrate in a stochastic fashion, which together results in macroscopic properties. The formulation of the stochastic IBM allows an analytical description of the dynamics at the population level, which links emergent properties to the elementary processes that generate them.The trait space ({{{{{{{mathcal{X}}}}}}}}subseteq {{mathbb{R}}}^{d}) is continuous and can be split into a neutral trait space ({{{{{{{mathcal{U}}}}}}}}) and an adaptive trait space ({{{{{{{mathcal{S}}}}}}}}). We refer to neutral traits (uin {{{{{{{mathcal{U}}}}}}}}) as traits that are not under selection, in contrast to adaptive traits (sin {{{{{{{mathcal{S}}}}}}}}), which experience selection. The graph denoted by G is composed of a set of vertices {v1,v2,…,vM} that correspond to habitat patches (suitable geographical areas), and a set of edges that constrain the movement of individuals between the habitat patches. We use the original measure of genetic differentiation for quantitative traits QST (standing for Q-statistics) in the case of haploid populations45,46. We denote the neutral trait value of the kth individual on vi as ({u}_{k}^{(i)}), the number of individuals on vi as N(i), the mean neutral trait on vi as ({overline{u}}^{(i)}), and the mean neutral trait in the metapopulation as (overline{u}). It follows that we quantify neutral differentiation QST,u as$${Q}_{ST,u}={sigma }_{B,u}^{2}/({sigma }_{B,u}^{2}+{sigma }_{W,u}^{2})$$
(1)
where ({sigma }_{B,u}^{2}={mathbb{E}}[frac{1}{M}{sum }_{i}{left({overline{u}}^{(i)}-overline{u}right)}^{2}]) denotes the expected neutral trait variance between the vertices and ({sigma }_{W,u}^{2}=frac{1}{M}mathop{sum }nolimits_{i}^{M}{mathbb{E}}left[frac{1}{{N}^{(i)}}{sum }_{k}{left({u}_{k}^{(i)}-{overline{u}}^{(i)}right)}^{2}right]) denotes the average expected neutral trait variance within vertices. We similarly quantify adaptive differentiation QST,s.Following the Gillespie update rule47, individuals with trait ({x}_{k}in {{{{{{{mathcal{X}}}}}}}}) on vertex vi are randomly selected to give birth at rate b(i)(xk) and die at rate d(N(i)) = N(i)/K, where K is the local carrying capacity. The definition of d therefore captures competition, which is proportional to the number of individuals on a vertex and does not depend on the individuals’ traits (we relax this assumption later on). The offspring resulting from a birth event inherits the parental traits, which can independently be affected by mutations with probability μ. A mutated trait differs from the parental trait by a random change that follows a normal distribution with variance ({sigma }_{mu }^{2}) (corresponding to the continuum of alleles model48). The offspring can further migrate to neighbouring vertices by executing a simple random walk on G with probability m. A schematic overview of the two different settings considered is provided in Fig. 1. Under the setting with no selection, individuals are only characterised by neutral traits so that ({{{{{{{mathcal{X}}}}}}}}={{{{{{{mathcal{U}}}}}}}}). For individuals on a vertex with trait xk ≡ uk we define b(i)(xk) ≡ b, so that the birth rate is constant. This ensures that neutral traits do not provide any selective advantage. Under the setting with heterogeneous selection, each vertex of the graph vi is labelled by a habitat type with environmental condition Θi that specifies the optimal adaptive trait value on vi. It follows that, for individuals with traits ({x}_{k}=({u}_{k},{s}_{k})in {{{{{{{mathcal{U}}}}}}}}times {{{{{{{mathcal{S}}}}}}}}) on vi, we define$${b}^{(i)}({x}_{k})equiv {b}^{(i)}({s}_{k})=b(1-p{({s}_{k}-{{{Theta }}}_{i})}^{2})$$
(2)
where p is the selection strength41. This ensures that the maximum birth rate on vi is attained for sk = Θi, which results in a differential advantage that acts as an evolutionary stabilising force. In the following we consider two habitat types denoted by I and II with symmetric environmental conditions θI and θII, so that Θi ∈ {θI, θII} and θII = − θI = θ, where θ can be viewed as the habitat heterogeneity41.Fig. 1: Graphical representation of the structure of individuals in the eco-evolutionary model.a Setting with no selection, where individuals are characterised by a set of neutral traits (uin {{{{{{{mathcal{U}}}}}}}}). The scatter plots represent a projection of the first two components of u for the individuals present on the designated vertices at time t = 1000, obtained from one simulation of the IBM. b Setting with heterogeneous selection. In this setting, individuals are additionally characterised by adaptive traits (sin {{{{{{{mathcal{S}}}}}}}}). Blue vertices favour the optimal adaptive trait value θI, while red vertices favour θII. The scatter plots represent a projection of the first component of u and s for the individuals present on the designated vertices at time t = 1000, obtained from one simulation. The majority of individuals are locally well-adapted and have an adaptive trait close to the optimal value, but some maladaptive individuals originating from neighbouring vertices are also present. m = 0.05.Full size imageDeterministic approximation of the population dynamics under no selectionThe model can be formulated as a measure-valued point process (30 and Supplementary Note). Under this formalism, we demonstrate in the Supplementary Note how the population size and the trait dynamics show a deterministic behaviour when a stabilising force dampens the stochastic fluctuations. This makes it possible to express the dynamics of the macroscopic properties with deterministic differential equations, connecting emergent patterns to the processes that generate them. In particular, in the setting of no selection, competition stabilises the population size fluctuations, and the dynamics can be considered deterministic and expressed as$${partial }_{t}{N}_{t}^{(i)}={N}_{t}^{(i)}left[b(1-m)-frac{{N}_{t}^{(i)}}{K}right]+mbmathop{sum}limits_{jne i}frac{{a}_{i,j}}{{d}_{j}}{N}_{t}^{(j)}$$
(3)
where (A={({a}_{i,j})}_{1le i,jle M}) is the adjacency matrix of the graph G and D = (d1,d2,…,dM) is a vector containing the degree of each vertex (number of edges incident to the vertex). The first term on the right-hand side corresponds to logistic growth, which accounts for birth and death events of non-migrating individuals. The second term captures the gains due to migrations, which depend on the graph topology. Assuming that all vertices with the same degree have an equivalent position on the graph, corresponding to a “mean field” approach (see Methods), one can obtain a closed-form solution from Eq. (3) (see Eq. (12)), which shows that the average population size (overline{N}) scales with ({langle sqrt{k}rangle }^{2}/langle krangle), where 〈k〉 is the average vertex degree and (langle sqrt{k}rangle) is the average square-rooted vertex degree. The quantity ({langle sqrt{k}rangle }^{2}/langle krangle), denoted as hd, relates to the homogeneity in vertex degree of the graph and can therefore be viewed as a measure negatively associated with heterogeneity in connectivity. Simulations of the IBM illustrate that hd can explain differences in population size for complex graph topologies with varying migration regimes (Fig. 2a for graphs with M = 7 vertices and Supplementary Fig. 1a for M = 9). This analytical result is connected to theoretical work on reaction-diffusion processes49 and highlights that irregular graphs (graphs whose vertices do not have the same degree) result in unbalanced migration fluxes that affect the ecological balance between births and deaths. Highly connected vertices present an oversaturated carrying capacity (N(i)  > bK, see Methods), increasing local competition and lowering total population size compared with regular graphs (Fig. 2a). Because populations with small sizes experience more drift (31 and Supplementary Fig. 2), this result indicates that graph topology affects neutral differentiation not only through population isolation, but also by affecting population dynamics.Fig. 2: Effect of and hd on average population size (overline{N}) and neutral differentiation QST,u in the setting with no selection.a Response of (overline{N}) to homogeneity in degree ({h}_{d}={langle sqrt{k}rangle }^{2}/langle krangle) for all undirected connected graphs with M = 7 vertices and m = 0.5. b Response of QST,u to average path length for similar simulations obtained with m = 0.01. c Response of QST,u to homogeneity in degree hd for the same data. In a, b, and c, each dot represents average results from 5 replicate simulations of the IBM, the colour scale corresponds to the proportion of the graphs with similar x and y-axis values (graph density), and the blue line corresponds to a linear fit. d Standardized effect of hd and on QST,u, obtained from multivariate regression models independently fitted on similar data obtained for m = 0.01 and m = 0.5. The contributions of and hd to QST,u are alike for low migration regimes. Error bars show 95% confidence intervals. Analogous results on graphs with M = 9 vertices are presented in Supplementary Fig. 1 and all regression details can be found in Supplementary Table 2.Full size imageNonetheless, the stochasticity of the processes at the individual level can propagate to the population level and substantially affect the macroscopic properties. In particular, neutral differentiation emerges from the stochastic fluctuations of the populations’ neutral trait distribution. These fluctuations complicate an analytical underpinning of the dynamics, and in this case simulations of IBM offer a straightforward approach to evaluate the level of neutral differentiation.Effect of graph topology on neutral differentiation under no selectionWe study a setting with no selection and investigate the effect of the graph topology on neutral differentiation. When migration is limited, individuals’ traits are coherent on each vertex but stochastic drift at the population level generates neutral differentiation between the vertices. Migration attenuates neutral differentiation because it has a correlative effect on local trait distributions. Following21,22,26, we expect that the intensity of the correlative effect depends on the average path length of the graph 〈l〉, defined as the average shortest path between all pairs of vertices50. For a constant number of vertices, 〈l〉 is strictly related to the mean betweenness centrality and quantifies the graph connectivity50. High 〈l〉 implies low connectivity and greater isolation of populations, and hence we expect that graphs with high 〈l〉 are associated with high differentiation levels. We consider various graphs with an identical number of vertices and run simulations of the IBM to obtain the neutral differentiation level QST,u attained after a time long enough to discard transient dynamics (see Methods). We then interpret the discrepancies in QST,u across the simulations by relating them to the underlying graph topologies.We observe strong differences in QST,u across graphs for varying m, and find that 〈l〉 explains at least 55% of the variation in QST,u across all graphs with M = 7 vertices for (Fig. 2b). Nonetheless, some specific graphs, such as the star graph, present higher levels of QST,u than expected by their average path length. To explain this discrepancy, we explore the effect of homogeneity in vertex degree hd, as we showed in Eq. (12) that it decreases population size, which should in turn increase QST,u by intensifying stochastic drift. We find that hd explains 57% of the variation for low m (Fig. 2c). However, the fit remains similar after correcting for differences in population size (see Supplementary Table 1), indicating that irregular graphs structurally amplify the isolation of populations. Unbalanced migration fluxes lead central vertices to host more individuals than allowed by their carrying capacity. This causes increased competition that results in a higher death rate, so that migrants have a lower probability of further spreading their trait. Highly connected vertices therefore behave as bottlenecks, increasing the isolation of peripheral vertices and consequently amplifying QST,u.We then evaluate the concurrent effect of 〈l〉 and hd on QST,u with a multivariate regression model that we fit independently for low and high migration regimes (Fig. 2d). The multivariate regression model explains at least 70% of the variation in QST,u for the migration regimes considered and for graphs with M = 7 vertices (see Supplementary Table 2 for details). Moreover, we find that 〈l〉 and hd have akin contributions to neutral differentiation for low m, but the effect of 〈l〉 increases for higher migration regimes while the effect of hd decreases. To ensure that these conclusions can be generalised to larger graphs, we conduct the same analysis on a subset of graphs with M = 9 vertices and find congruent results (Supplementary Fig. 1). In the absence of selection and with competitive interactions, graphs with a high average path length 〈l〉 and low homogeneity in vertex degree hd, or similarly graphs with low connectivity and high heterogeneity in connectivity, show high levels of neutral differentiation.Deterministic approximation of the population dynamics and adaptation under heterogeneous selectionWe next consider heterogeneous selection and investigate the response of adaptive differentiation to the spatial distribution of habitat types, denoted as the Θ-spatial distribution. Adaptive differentiation emerges from local adaptation, but migration destabilises adaptation as a result of the influx of maladaptive migrants. We expect that higher connectivity between vertices of similar habitat type increases the level of adaptive differentiation, because it increases the proportion of well-adapted migrants. Local adaptation can be investigated by approximating the stochastic dynamics of the trait distribution with a deterministic partial differential equation (PDE). We demonstrate under mean-field assumption how the deterministic approximation can be reduced to an equivalent two-habitat model. We analyse the reduced model with the theory of adaptive dynamics36,41 and find a critical migration threshold m⋆ that determines local adaptation. m⋆ depends on a quantity coined the habitat assortativity rΘ, and we demonstrate with numerical simulations that rΘ determines the overall adaptive differentiation level QST,s reached at steady state in the deterministic approximation.Heterogeneous selection, captured by the dependence of the birth rate on Θi, generates a stabilising force that dampens the stochastic fluctuations of the adaptive trait distribution. The dynamics of the adaptive trait distribution consequently shows a deterministic behavior and we demonstrate in the Supplementary Note and Supplementary Figs. 3 and 4 that the number of individuals on vi with traits (sin {{Omega }}subset {{{{{{{mathcal{S}}}}}}}}) can be approximated by the quantity ∫Ωn(i)(s)ds, where n(i) is a continuous function solution of the PDE$${partial }_{t}{n}_{t}^{(i)}(s)= , {n}_{t}^{(i)}(s)left[{b}^{(i)}(s)(1-m)-frac{1}{K}{int}_{{{{{{{{mathcal{S}}}}}}}}}{n}_{t}^{(i)}({{{{{{{bf{s}}}}}}}})d{{{{{{{bf{s}}}}}}}}right]\ +mmathop{sum}limits_{jne i}{b}_{j}(s)frac{{a}_{i,j}}{{d}_{j}}{n}_{t}^{(j)}(s)+frac{1}{2}mu {sigma }_{mu }^{2}{{{Delta }}}_{s}left[{b}^{(i)}(s){n}_{t}^{(i)}(s)right]$$
(4)
Equation (4) is similar to Eq. (3), except that it incorporates an additional term corresponding to mutation processes and that the birth rate is trait-dependent. We show how Eq. (4) can be reduced to an equivalent two-habitat model under mean-field assumption. The mean-field approach differs slightly from the setting with no selection because vertices are labelled with Θi. Here we assume that vertices with similar habitat types have an equivalent position on the graph (see Supplementary Fig. 5 for a graphical representation), so that all vertices with habitat type I are characterised by the identical adaptive trait distribution that we denote by ({overline{n}}^{{{{{{{{bf{I}}}}}}}}}), and are associated with the birth rate ({b}^{{{{{{{{bf{I}}}}}}}}}(s)=b(1-p{(s-{theta }_{{{{{{{{bf{I}}}}}}}}})}^{2})). Let P(I, II) denote the proportion of edges connecting a vertex vi of type II to a vertex vj of type I, and let P(I) denote the proportion of vertices vi of type I. By further assuming that habitats are homogeneously distributed on the graph so that (P({{{{{{{bf{I}}}}}}}})=P({{{{{{{bf{II}}}}}}}})=frac{1}{2}), Eq. (4) transforms into$${partial }_{t}{overline{n}}_{t}^{{{{{{{{bf{I}}}}}}}}}(s)= ,{overline{n}}_{t}^{{{{{{{{bf{I}}}}}}}}}(s)left[{b}^{{{{{{{{bf{I}}}}}}}}}(s)(1-m)-frac{1}{K}{int}_{{{{{{{{mathcal{S}}}}}}}}}{overline{n}}_{t}^{{{{{{{{bf{I}}}}}}}}}({{{{{{{bf{s}}}}}}}})d{{{{{{{bf{s}}}}}}}}right]+frac{1}{2}mu {sigma }_{mu }^{2}({{{Delta }}}_{s}{b}^{{{{{{{{bf{I}}}}}}}}}{overline{n}}_{t}^{{{{{{{{bf{I}}}}}}}}})(s)\ +frac{m}{2},[(1-{r}_{{{Theta }}}){b}^{{{{{{{{bf{II}}}}}}}}}(s){overline{n}}_{t}^{{{{{{{{bf{II}}}}}}}}}(s)+(1+{r}_{{{Theta }}}){b}^{{{{{{{{bf{I}}}}}}}}}(s){overline{n}}_{t}^{{{{{{{{bf{I}}}}}}}}}(t)]$$
(5)
(see Methods), where we define$${r}_{{{Theta }}}=2left(P({{{{{{{bf{I}}}}}}}},{{{{{{{bf{I}}}}}}}})-P({{{{{{{bf{I}}}}}}}},{{{{{{{bf{II}}}}}}}})right)$$
(6)
as the habitat assortativity of the graph, which ranges from −1 to 1. When rΘ = − 1, all edges connect dissimilar habitat types (disassortative graph), while as rΘ tends towards 1 the graph is composed of two clusters of vertices with identical habitat types (assortative graph). Eq. (5) can be analysed with the theory of adaptive dynamics36,38,41, a mathematical framework that provides analytical insights by assuming a “trait substitution process”. Following this assumption, the mutation term in Eq. (5) is omitted and the phenotypic distribution results in a collection of discrete individual types that are gradually replaced by others until evolutionary stability is reached (see Methods and36,38,41 for details). By applying the theory of adaptive dynamics, we find a critical migration rate m⋆$${m}^{star }=frac{1}{(1-{r}_{{{Theta }}})}frac{4p{theta }^{2}}{(1+3p{theta }^{2})}$$
(7)
so that when m  > m⋆, a single type of individual exists with adaptive trait ({s}^{* }=left({theta }_{{{{{{{{bf{II}}}}}}}}}+{theta }_{{{{{{{{bf{I}}}}}}}}}right)/2=0) in the steady-state (see Methods for the derivation of Eq. (7)). In this case, adaptive differentiation QST,s is nil and the average population size is given by (overline{N}=bK{(1-ptheta )}^{2}). In contrast, when m = 0 and/or rΘ = 1, all individuals are locally well-adapted with trait Θi on vi, and it follows that the average population size is higher and equal to (overline{N}=bK), while adaptive differentiation is maximal and equal to ({Q}_{ST,s}={{{{{{{rm{Var}}}}}}}}({{Theta }})/left({{{{{{{rm{Var}}}}}}}}({{Theta }})+0right)=1). When 0  m⋆, implying that individuals become equally fit in all habitats. In this case, the isolation effect of heterogeneous selection is lost and QST,u reaches a similar level as in the setting with no selection for m  > m⋆ (Fig. 5a), although QST,u is slightly higher in the setting with heterogeneous selection due to lower population size ((overline{N}=bK(1-ptheta )) vs. (overline{N}=bK), see section above and Methods). This suggests that rΘ reinforces QST,u, as assortative graphs sustain higher levels of adaptive differentiation (Figs. 3 and 4). Simulations on the path graph with varying Θ-spatial distribution support this conclusion for high migration regimes, but show the opposite relationship under low migration regimes, where the habitat assortativity rΘ decreases QST,u (Fig. 5b). Assortative graphs are composed of large clusters of vertices with similar habitats, within which migrants can circulate without fitness losses. Local neutral trait distributions become more correlated within these clusters, resulting in a decline in QST,u for assortative graphs compared with disassortative graphs. Figure 5b therefore highlights the ambivalent effect of rΘ on QST,u. rΘ reinforces QST,u by favouring adaptive differentiation, but also decreases QST,u by decreasing population isolation within clusters of vertices with the same habitat type.We compare the effect of rΘ on QST,u to the effect of the topology metrics 〈l〉 and hd found in the setting with no selection using multivariate regression analysis on simulation results obtained for different graphs with varying Θ-spatial distribution (Fig. 5d for graphs with M = 7 vertices and Supplementary Fig. 7b for M = 9). The multivariate model explains the discrepancies in QST,u across the simulations for low and high migration regimes (see Supplementary Table 3 for details), and we find that rΘ, 〈l〉, and hd contribute similarly to neutral differentiation. Hence, the effects of rΘ and the topology metrics 〈l〉 and hd add up under heterogeneous selection. A change in sign of the standardized effect of rΘ on QST,s for low and high migration regimes verifies that the ambivalent effect of rΘ on QST,u found on the path graph holds for general graph ensembles. Simulations with trait-dependent competition and simulations on realistic graphs with a continuum of habitat types equally confirm the ambivalent effect of rΘ and further support the complementary effect of 〈l〉 and hd on QST,u (see Supplementary Fig. 8). 〈l〉 and hd therefore drive neutral differentiation with and without heterogeneous selection. rΘ becomes an additional determinant of neutral differentiation under heterogeneous selection. In contrast to the non-ambivalent, positive effect of habitat assortativity on adaptive differentiation, rΘ can amplify or depress neutral differentiation depending on the migration regime considered. More