Philippa Kaur

Two databases including yield, management, and weather data for maize (n = 17,013) and soybean (n = 24,848) involving US crop performance trials conducted in 28 states between 2016 to 2018 for maize and between 2014 to 2018 for soybean, were developed (Fig. 1). Crop yield and management data were obtained from publicly available variety performance trials which are typically performed yearly in several locations across each state (see methods for more information). Final databases were separated in training (80% of database) and testing (20% of database) datasets using stratified sampling by year, use of irrigation, and soil type. For each crop, an extreme gradient boosting (XGBoost, see methods for more information) algorithm to estimate yield based on soil type and weather conditions (E), seed traits (G) and management practices (M) was developed (see variables listed in Tables S1 and S2 for maize and soybean, respectively, and data science workflow in Fig. S1).Figure 1Locations where maize and soybean trials were performed during the examined period. The map was developed in ArcGIS Pro 2.8.0 (https://www.esri.com).Full size imageThe developed algorithms exhibited a high degree of accuracy when estimating yield in independent datasets (test dataset not used for model calibration) (Fig. 2). For maize, the root mean square error (RMSE) and mean absolute error (MAE) was a respective 4.7 and 3.6% of the dataset average yield (13,340 kg/ha). For soybean, the respective RMSE and MAE was 6.4 and 4.9% of the dataset average yield (4153 kg/ha). As is evident in the graphs (Fig. 2), estimated yields exhibited a high degree of correlation with actual yields for both algorithms in the independent datasets. For maize and soybean, 72.3 and 60% of cases in the test dataset deviated less than 5% from actual yields, respectively. Maximum deviation for maize and soybean reached 43 and 70%, respectively. Data points with deviations greater than 15% from actual yield were 1.5% in maize and 3.6% in soybean databases. These results suggest that the developed algorithms can accurately estimate maize and soybean yields utilizing database-generated information involving reported environmental, seed genetic, and crop management variables.Figure 2Actual versus algorithm-derived maize (left) and soybean (right) yield in test datasets. Black solid line indicates y = x, red short-dashed lines, black dashed lines, and red long-dashed lines indicate ± 5, 10, and 15% deviation from the y = x line. RMSE, root mean square error; MAE, mean absolute error; r2, coefficient of determination; n = number of observations. Each observation corresponds to a yield of an individual cropping system in a specific environment (location-year).Full size imageIn contrast to statistical models, ML algorithms can be complex, and the effect of single independent variables may not obvious. However, accumulated local effects (ALE) plots14 can aid the understanding and visualization of important and possibly correlated features in ML algorithms. For both crops, indicatively important variables included sowing date, seeding rate, nitrogen fertilizer (for maize), row spacing (for soybean) and June to September cumulative precipitation (Fig. 3). Across the entire region and for both crops, the algorithm-derived trends suggest that above average yields occur in late April to early May sowing dates, but sharply decrease thereafter. Similar responses have been observed in many regional studies across the US for both, maize15,16,17,18 and soybean19. Similarly, simulated yield curves due to increasing seeding rate are in close agreement with previous maize20,21 and soybean22 studies. The maize algorithm has captured the increasing yield due to increasing N fertilizer rate. The soybean algorithm suggests that narrower row spacing resulted in above average yield compared to wider spacing. Such response has been observed in many regions across the US23. Season cumulative precipitation between 400 and 700 mm resulted in above average yields for both crops.Figure 3Accumulated local effect plots for maize sowing date (A), seeding rate (B), Nitrogen fertilizer rate (C), and cumulative precipitation between June and September (mm) (D), and soybean sowing date (E), seeding rate (F), row spacing (G), and cumulative precipitation between June and September (mm) (H).Full size imageThe responses in the ALE plots (Fig. 3) suggest that these algorithms have captured the general expected average responses for important single features. Nevertheless, our databases include hundreds of locations with diverse environments across the US and site-specific crop responses which may vary due to components of the G × E × M interaction. We argue that, instead of examining a single or low-order management interactions, site-specific evaluation of complex high order interactions (a.k.a. cropping systems) can reveal yield differences that current research approaches cannot fully explore and quantify. For example, sowing date exerts a well-known impact on maize and soybean yield. For each crop separately, by creating a hypothetical cropping system (a single combination of all management and traits in Tables S1 and S2) in a randomly chosen field in south central Wisconsin (latitude = 43.34, longitude = -89.38), and by applying the developed algorithms, we can generate estimates of maize and soybean yield. For that specific field and cropping system (out of the vast number of management combinations a farmer can choose from), maize yield with May 1st sowing was 711 kg/ha greater (6% increase) than June sowing (Fig. 4A). By creating scenarios with 256 background cropping system choices (Table S3), the resultant algorithm-derived yield estimate difference for the same sowing date contrast (averaged across varying cropping systems) was smaller but still positive (3% increase), although the range of possible yield differences was wider (Fig. 4B). However, when comparing, instead of averaging, the estimated yield potential among the simulated cropping systems, a 2903 kg/ha yield difference (25% difference) was observed (Fig. 4C). Interestingly, when focusing on the early sown fields that were expected to exhibit the greatest yield, the same yield difference was observed (Fig. 4D). This result shows that sub-optimal background management can mitigate the beneficial effect of early sowing (Table S4).Figure 4Maize yield difference (in kg/ha and percentage) due to sowing date (May 1st vs. June 1st) for a single identical background cropping system (A), maize yield difference due to sowing date when averaged across 256 (3 years × 256 cropping systems = 768 year-specific yields) (B), maize yield variability in each of the 256 cropping systems (C), and maize yield variability in each of the 128 cropping systems with early sowing (D). Soybean yield difference due to sowing date (May 1st vs June 1st) for a single identical background cropping system (E), soybean yield difference due to sowing date when averaged across 128 (5 years × 128 cropping systems = 640 year-specific yields) (F), soybean yield in each of the 128 cropping systems (G) and soybean yield variability due in each of the 64 cropping systems with early sowing (H). Within each panel, the horizontal red and grey lines indicate the boxplot with maximum and minimum yield, respectively. In the left four panels, boxes delimit first and third quartiles; solid lines inside boxes indicate median and green triangles indicate means. Upper and lower whiskers extend to maximum and minimum yields. Each maize and soybean cropping system is a respective 8-way and a 7-way interaction of management practices in a randomly chosen field in Wisconsin, USA (Table S3 and S5, respectively).Full size imageIn the case of soybean, a May 1st sowing resulted in greater yield (588 kg/ha; a 14% increase) than a June 1st in the single background cropping system (Fig. 4E). The result was consistent when yield differences due to sowing date were averaged across 128 background cropping system choices (Table S5) (Fig. 4F). Similar to what was observed in maize, among all cropping systems, yield varied by 1704 kg/ha (44% difference) (Fig. 4G). When focusing only on the early sown fields, a 1181 kg/ha yield difference (27% yield increase) was observed (Fig. 4H). In agreement with maize, this result highlights the importance of accounting for sub-optimal background management which can mitigate the beneficial effect of early sowing (Table S6).We note here the ability of farmers to change management practices can be limited due to an equipment constraint (e.g., change planter unit row width) or simply impossible (e.g., change the previous year’s crop). Thus, recommended management practices that were evaluated in studies that used specific background management may not be applicable in some instances. The benefits of the foregoing approach, which involves extensive up-to-date agronomic datasets and high-level computational programing, can have important and immediate implications in future agricultural trials. Our approach allows for more precise examination of complex management interactions in specific environments (soil type and growing season weather) across the US (region covered in Fig. 1). The ability to extract single management practice information (even across cropping systems) is also possible by utilizing ALE plots, or by calculation of the frequency at which a given level/rate of a management practice appeared among the highest yielding cropping systems (Tables S4 and S6).Among all available 30-d weather variables, many were strongly correlated in both crop databases (Figs. S2 and S3 for maize and soybean, respectively). Models using all 30-d interval variables with r  More

Sampling and genotypingWe collected hair and cheek swab samples from 77 men from geographically and linguistically distinct groups of the Caucasus: Kartvelian speakers from Georgia and Turkey, Northeast Caucasian speakers and Turkic speakers from the Russian Federation and Armenian speakers from Georgia’s southern province of Javakheti, descendants of the families displaced from Mush and Erzurum provinces of eastern Turkey in the early nineteenth century (Table 1, Fig. 1). To maximize the representativeness of the genetic signature of each population, the samples were collected from locals with no ancestors from outside of the respective ethnic/geographic population over the last three generations. DNA was extracted from follicles of 10–12 male chest hairs and cheek swab samples. Extraction was performed using Qiagen DNeasy Blood and Tissue kit, following the manufacturer’s recommendations (Qiagen, Valencia, CA, USA). The DNA samples were genotyped for 693,719 autosomal and 17,678 X-chromosomal SNPs by Family Tree DNA (FTDNA—Gene By Gene, Ltd, Houston, TX, www.familytreedna.com).Table 1 Modern study populations of the Caucasus. Latitude and longitude georeference population hubs.Full size tableFigure 1The distribution of the study populations: averaged centroids of ancient populations (uniquely colored points in the main map, see Table 2 for details) and hubs of the modern Caucasian populations (identified in the inset map, see Table 1 for details). Glacial human refugia extracted from Gavashelishvili and Tarkhnishvili5 are shaded in purple. The map is generated using QGIS Desktop 3.10.6-A Coruña (https://qgis.org).Full size imageOur dataset of modern Caucasian genotypes was supplemented with published 10 modern Mbuti (Supplementary Table S1) and 122 Upper Paleolithic-Mesolithic human genotypes, retrieved as a part of 1240 K dataset from David Reich’s Lab website, Harvard University (https://reich.hms.harvard.edu/downloadable-genotypes-present-day-and-ancient-dna-data-compiled-published-papers; see Supplementary Table S2 for details). The ancient genotypes were selected such that they either dated from the LGM or fell within the glacial refugia identified by Gavashelishvili and Tarkhnishvili5. We did so in order to maximize the genetic signature of potential refugial populations in our analysis. We divided the ancient genotypes into 2000-year-long intervals, and then grouped each of these intervals into geographic units (hereafter ancient populations, Table 2, Fig. 1). The modern and ancient genotypes were merged using PLINK 1.9 (PLINK 1.9: www.cog-genomics.org/plink/1.9/27.Table 2 Ancient study populations. The ancient genotypes are divided into 2000-year-long intervals, and then each of these intervals is grouped into geographic units (i.e. ancient populations). Age, latitude and longitude are averaged across each ancient population (see Supplementary Table S2 for details).Full size tableEthics statementThe research team members, through their contacts in the studied communities, inquired whether locals would voluntarily participate in genetic research that would help clarify the genetic makeup of the Caucasus. A verbal agreement was made with volunteer donors of DNA samples, according to which the results would be communicated, electronically or in hard copy, with participants individually. Participants were informed that, upon the completion of the lab work, the research would be published without mentioning the names of sample donors. Those who agreed provided us with the envelopes containing their chest hairs or cheek swab samples, with the birthplace of their ancestors (last three generations) written on the envelope or a piece of paper. In accordance with the preferences of the sample donors, the agreement was verbal and not written. The envelopes and papers are stored as evidence of voluntary provision of the samples and the related information. Analysis of data was done anonymously, using only location and ethnic information; only the first and third authors of the manuscript had access to names associated with the samples. Therefore, this study was based on noninvasive and nonintrusive sampling (volunteers provided hair and swab samples they collected themselves), and the information destined for open publication does not contain any personal information. The study methodology and the procedure of verbal consent was discussed in detail with and approved by the members of the Ilia State University Commission for Ethical Issues before the field survey started, and the commission decided that formal ethical approval was not needed for conducting this study. This is confirmed in a letter from the commission chairman, a copy of which has been provided to the journal editor as part of the submission process.Genetic affinity and geographyFirst, we measured genetic affinity between the modern Caucasian populations, and between the modern populations and the ancient populations of hunter-gatherers, and then tested whether the genetic affinity between these populations was determined by geographic features. Data were mapped using QGIS Desktop 3.10.6-A Coruña, whereas graphs were created using the “ggplot2” package28 in R version 3.5.229.To evaluate genetic affinities and structure of the modern populations, we used Wright’s fixation index (Fst), inbreeding coefficient, admixture analysis and the principal component analysis (PCA). For these procedures we filtered the raw SNP genotypes in PLINK 1.9, first removing all SNPs with the minor allele frequency  0.3, calculated in windows of 50 bp size and 10 bp steps (–maf 0.05 –indep-pairwise 50 10 0.3). Since all individuals in our dataset possess a single copy of the X-chromosome, we did not expect any differential ploidy bias, and SNPs on the X were treated similarly to those on the autosomes. Fst pairwise values were calculated using the smartpca program of EIGENSOFT30 with default parameters, inbreed: YES, and fstonly: YES. The relationship between the modern populations based on Fst values was visualized by constructing a neighbor-joining tree using the “ape” package31 in R version 3.5.2. The average and standard deviation of the inbreeding coefficient for each population was calculated using “fhat2” estimate of PLINK 1.9. The LD pruned genotypes were used in ADMIXTURE 1.3.032, performed in unsupervised mode in order to infer the population structure from the modern individuals. The number of clusters (k) was varied from 2 to 7 and the fivefold cross-validation error was calculated for each k33. We conducted principal components analysis in the smartpca program of EIGENSOFT30, using default parameters and the lsqproject: YES and numoutlieriter: 0 options. Eigenvectors of principal components were inferred with the modern populations from the Caucasus, while the ancient populations were then projected onto the PCA plots. We also assessed the relatedness between sampled individuals using kinship coefficients estimated by KING34.To quantify genetic affinities between the modern and ancient populations, we used the programs qp3Pop and qpDstat in the ADMIXTOOLS suite (https://github.com/DReichLab35 for f3- and f4-statistics, respectively. f3-statistics of the form f3(X,Y,Outgroup) measure the amount of shared genetic drift of populations X and Y after their divergence from an outgroup. We used an ancient population and a modern Caucasian population for X, Y and Mbuti as an outgroup. f4-statistics of the form f4(Outgroup,Test;X,Y) show if population Test is equally related to X and Y or shares an excess of alleles with either of the two. In the f4-statistic calculation we used Mbuti for Outgroup, a modern population of the Caucasus for Test, and X and Y for contemporaneous ancient populations. This meant that f4  0 indicated higher genetic affinity between the test population and Y.To quantify geographic features, we derived least-cost paths and measured least-cost distances (LCD) between the modern and ancient populations using the Least Cost Path Plugin for QGIS. The computation of LCD considers a friction grid that is a raster map where each cell indicates the relative difficulty (or cost) of moving through that cell. A least-cost path minimizes the sum of frictions of all cells along the path, and this sum is the least-cost distance (LCD). For impedance to human movement and expansion, we used 15 geographic features (Table 3). All gridded geographic features (i.e. raster layers) were resampled to a resolution of 1 km using the nearest-neighbor assignment technique. All possible subsets of the 15 geographic features, that did not cancel out each other, were used to calculate different variables of LCD. We assumed that most human movements occurred during climate warming events when the earth’s surface was not dramatically different from that of today, and hence used the current data of the geographic features.Table 3 Geographic features used in combinations to calculate least-cost distances (LCD) between ancient populations and modern Caucasians.Full size tableLinking genetic affinity and geographyGeneralized additive models (GAMs) were used to fit the outgroup f3-statistic to time and variously calculated LCD between the modern and ancient populations using the “mgcv” package36 in R version 3.5.2. Time between the modern and ancient populations was measured in BP (years before present, defined by convention as years before 1950 CE). We used GAMs because without any assumptions they are able to find nonlinear and non-monotonic relationships. GAMs were fitted using a Gamma family with a log link function. Penalized thin plate regression splines were used to represent all the smooth terms. The restricted maximum likelihood (REML) estimation method was implemented to estimate the smoothing parameter because it is the most robust of the available GAM methods36.Model and variable selection were performed by exploring LCD, time BP and the interaction term. The predictive power of the models was evaluated through a tenfold cross-validation. The cross-validation of many models was handled through R’s parallelization capabilities37,38. The best model was selected by the mean squared error of the cross-validation. Akaike’s Information Criterion (AIC) is generally used as a means for model selection. However, we preferred cross-validation for model selection because AIC a priori assumes that simpler models with the high goodness of fit are more likely to have the higher predictive power, while cross-validation without any a priori assumptions measures the predictive performance of a model by efficiently running model training and testing on the available data.We additionally validated the effect of different subsets of geographic features by assessing the relationship between statistically significant values of f4-statistic (i.e. |Z| > 3) and each subset. The relationship between f4-statistic of the form of f4(Outgroup,Test;X,Y) and geographic features was determined by measuring the agreement between the negative/positive signs of f4-statistic and the difference in LCD (LCD.D) for each pair of contemporaneous ancient populations X and Y. LCD.D was calculated as (LCD1–LCD2), where LCD1 was least-cost distance between the test population and X, and LCD2 was least-cost distance between the test population and Y. LCD.D  0 indicated less least-cost distance between Test and Y. So, the same sign of f4 and LCD.D values indicated agreement between geographic proximity and genetic affinity. We used Cohen’s kappa39 to measure the agreement.In order to test if geographic features (Table 3) accounted for present-day genetic differentiation in the Caucasus, we measured the relationship between Fst and LCD across the modern populations using the Mantel test in the “vegan” package40 in R version 3.5.2. In addition, we checked whether contribution from ancient samples was related to today’s genetic differentiation. To do so, we calculated median of f3-statistic of ancient populations of each geographic grouping (e.g. the following 6 populations made up one group: Balkans 39,950–41,950 BP, Balkans 37,950–39,950 BP, Balkans 31,950–33,950 BP, Balkans 9950–11,950 BP, Balkans 7950–9950 BP, Balkans 5950–7950 BP). Then we measured the manhattan distance of f3 median values of all combinations of the geographic groupings between the modern populations and compared the results to Fst and LCD using the Mantel test. More

Regular structures and isothermal theoremFor networks where each node represents a single individual, the isothermal theorem of evolutionary graph theory shows that the fixation probability is the same as the fixation probability of a well-mixed population if the temperature distribution is homogeneous across the whole population1. The temperature of a node defined as the sum over all the weights leads to that node. This theorem extends to structured meta-populations for any migration probability (lambda ): If the underlying structure of the meta-population that connects the patches is a regular network and the local population size is identical in each patch, the temperature of all individuals is identical, regardless of the value of the migration probability. Therefore, the fixation probability in a population with such a structure is the same as the fixation probability in a well-mixed population of the same total population size (N=sum _{j=1}^M N_j), given by ( phi _{mathrm{wm}}^N(r)).Small migration regimeIf the migration probability is small enough such that the time between two subsequent migration events (( sim frac{1}{lambda } )) is much longer than the absorption time within any patch, then at the time of each migration event we may suppose that the meta-population is in a homogeneous configuration22,28. In other words, the low migration regime is an approximation in which we neglect the probability that the meta-population is not in a homogeneous configuration at the time of migration events. We define a homogeneous configuration of the meta-population as a configuration in which in all patches either all individuals are mutants, or all are wild-types.Therefore, instead of having (2^N) states, where N is the population size, the system has only (2^M) states, where M is the number of patches. Thus, we can calculate the fixation probability exactly as in the case of a standard evolutionary graph model where each node represents a single individual but with a modified transition probabilities.In a network with homogeneous patches, in order to increase the number of homogeneous mutant-patches one individual mutant needs to migrate to one of its neighbouring homogeneous wild-type-patches and reaches fixation there. For example if node j is occupied by mutants and one of its neighbouring patches, node k, is occupied by wild-types, the probability that one mutant individual from patch j migrates to patch k and reaches fixation there is (frac{lambda }{mathrm{deg} (j)}phi _{mathrm{wm}}^{N_{k}}(r) ), where (mathrm{deg} (j) ) is the degree of node j to take into account that the mutant can move to different patches. This is analogous to the probability that one mutant in node j replaces one wild-type in node k ,(T^{jrightarrow k}), in the network of individuals.Similarly, if node j is occupied by wild-types and one of its neighbouring patches, node j, is occupied by mutants the probability that one wild-type individual from patch j migrates to patch k and reaches fixation there equals to (frac{lambda }{mathrm{deg} (j)}phi _{mathrm{wm}}^{N_{k}}(1/r) ) where (mathrm{deg} (j) ). Overall, we can move from network of individuals to the network of homogeneous patches by replacing the transition probabilities with the product of migration and fixation probabilities.Two-patch meta-populationThe simplest non-trivial case is the fixation probability in a two-patch meta-population with different local size for small migration probability (lambda ). If the migration probability (lambda ) is very small, a new mutant first needs to take over its own patch and only then the first migrant arrives in the second patch. To be more precise, the time between two migration events has to be much higher than the typical time that it takes for the migrant to take over the patch or go extinct again38. In this case, we can divide the dynamics into two phases: A first phase in which a mutant invades one patch and a second phase in which a homogeneous patch of mutants invades the whole meta-population. Assume a new mutation arises in patch 1. Only if this mutant reaches fixation in patch 1, it also has a chance to reach fixation in patch 2. When patch 1 consists of only mutants and patch 2 consists of only wild-types, there are two possibilities for the ultimate fate of the mutant:

(i)

Eventually, the offspring of one mutant selected from patch 1 for reproduction will migrate to patch 2 and reach fixation there. The wild-type goes extinct. This happens with probability ( frac{N_1 r}{N_1 r+N_2} phi _{mathrm{wm}}^{N_2}(r)).

(ii)

Eventually, the offspring of one wild-type selected from patch 2 for reproduction will migrate to patch 1 and the mutant goes extinct. This occurs with probability ( frac{N_2}{N_1r+N_2} phi _{mathrm{wm}}^{N_1}(tfrac{1}{r})).

Therefore, the probability that a single mutant arising in patch 1 reaches fixation in the entire population is $$begin{aligned} phi _{mathrm{wm}}^{N_1}(r) frac{frac{N_1 r}{N_1 r+N_2} phi _{mathrm{wm}}^{N_2}(r)}{frac{N_1 r}{N_1 r+N_2} phi _{mathrm{wm}}^{N_2}(r)+frac{N_2}{N_1r+N_2} phi _{mathrm{wm}}^{N_1}left( tfrac{1}{r}right) }=phi _{mathrm{wm}}^{N_1}(r) phi _{mathrm{wm}}^{N_2}(r) frac{1 }{ phi _{mathrm{wm}}^{N_2}(r) +frac{N_2}{N_1} frac{1}{r}phi _{mathrm{wm}}^{N_1} left( tfrac{1}{r}right) }. end{aligned}$$
(3a)
Similarly the probability that a mutant arising in patch 2 takes over the whole population equals$$begin{aligned} phi _{mathrm{wm}}^{N_2}(r) phi _{mathrm{wm}}^{N_1}(r) frac{1 }{phi _{mathrm{wm}}^{N_1}(r)+frac{N_1}{N_2} frac{1}{r} phi _{mathrm{wm}}^{N_2}left( tfrac{1}{r}right) }. end{aligned}$$
(3b)
If we assume that the mutant arises in a patch with a probability proportional to the patch size, the average fixation probability (phi _{bullet !!-!!bullet }) in a two patch population for small migration probability is the weighted sum of Eqs. (3a) and (3b),$$begin{aligned} phi _{bullet !!-!!bullet }&= phi _{mathrm{wm}}^{N_1}(r) phi _{mathrm{wm}}^{N_2}(r) nonumber \&quad times left( frac{frac{N_1}{N_1+N_2} }{ phi _{mathrm{wm}}^{N_2}(r) +frac{N_2}{N_1} frac{1}{r}phi _{mathrm{wm}}^{N_1}left( tfrac{1}{r}right) } +frac{frac{N_2}{N_1+N_2} }{ phi _{mathrm{wm}}^{N_1}(r) +frac{N_1}{N_2} frac{1}{r} phi _{mathrm{wm}}^{N_2}left( tfrac{1}{r}right) }right) . end{aligned}$$
(4)
In the case of neutrality, (r=1), we recover (phi _{bullet !!-!!bullet } = frac{1}{N_1+N_2})—the fixation probability in a population of the total size of the two patches. For identical patch sizes, ( N_1=N_2 ), Eq. (4) simplifies to$$begin{aligned} phi _{bullet !!-!!bullet } = left( phi _{mathrm{wm}}^{N_1}(r)right) ^2 frac{1}{phi _{mathrm{wm}}^{N_1}(r)+frac{1}{r} phi _{mathrm{wm}}^{N_1}left( tfrac{1}{r}right) } = phi _{mathrm{wm}}^{2 N_1}(r), end{aligned}$$
(5)
where the simplification to the fixation probability within a single population of size (2N_1) reflects the validity of the isothermal theorem.For (N_1 ne N_2), we approximate Eq. (4) for weak and strong selection. Let us first consider highly advantageous mutants, (r gg 1). In this case, we have (phi _{mathrm{wm}}^{N_1}(r) gg phi _{mathrm{wm}}^{N_1}(tfrac{1}{r})) and thus we can neglect the possibility that a wild-type takes over a mutant patch if patch sizes are sufficiently large. The probability (phi _{bullet !!-!!bullet } ) then becomes a weighted average reflecting patch sizes. For identical patch size (N_1=N_2 = N/2), it reduces to (phi _{bullet !!-!!bullet } approx phi _{mathrm{wm}}^{N_1}(r)=phi _{mathrm{wm}}^{N/2}(r)). In other words, taking over the first patch is sufficient to make fixation in the entire population certain. For patches of very different size, (N_1 gg N_2), we have (N approx N_1) and find (phi _{bullet !!-!! bullet } approx phi _{mathrm{wm}}^{N}(r), ) which implies that fixation is driven by the fixation process in the larger patch, regardless of where the mutant arises. Note that there is a difference between the case of identical patch size and very different patch size . The case of highly disadvantageous mutants, (r ll 1), can be handled in a very similar way.Next, we consider weak selection, (r approx 1). We can approximate the fixation probability as (phi _{mathrm{wm}}^{N}(r^{pm 1}) approx frac{1}{N} pm frac{N-1}{2N} (r-1)). With this, we find$$begin{aligned} phi _{bullet !!-!!bullet } approx frac{1}{N_1+N_2} +frac{1}{2} left( 1 – frac{1}{N_1+N_2} -frac{(N_1-N_2)^2}{(N_1^2+N_2^2)^2} N_1 N_2right) (r-1). end{aligned}$$
(6)
For identical patch size (N_1=N_2 = N/2), this reduces to$$begin{aligned} phi _{bullet !!-!!bullet } approx tfrac{1}{N} +tfrac{N-1}{2N} (r-1), end{aligned}$$
(7)
which is the known result for a single population of size (N=N_1+N_2). When patches have very different size, (N_1 gg N_2) such that (N approx N_1), we recover the same result. Thus, the difference between the fixation probability of a two-patch meta-population with identical patch size and the fixation probability of a two-patch meta-population with very different patch size that we found for highly advantageous mutants is no longer observed for weak selection.When migration probabilities become larger, our approximation is no longer valid and we need to rely on numerical approaches. Figure 2 illustrates the difference between the fixation probability of a two-patch structure meta-population and the equivalent well-mixed population of size (N_1+N_2 ) when migration is low using Eq. (4) and comparing with the numerical approach in Ref.39.While the fixation probability of the two-patch meta-population is very close to the fixation probability of the well-mixed population40, a close inspection reveals an interesting property: For low migration probabilities and (N_1 ne N_2), the two patch structure is a suppressor of selection in the original sense of Lieberman et al.1: For advantageous mutations, (r >1), it decreases the fixation probability, whereas for disadvantageous mutations, (r1) and negative for (r1 ) the minimum fixation probability occurs when the two patch sizes are identical, ( N_1=N_2=N/2 ). Similarly, for fitness values ( r More

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NEWS AND VIEWS
08 September 2021

Pollination advantage of rare plants unveiled

An analysis of plant–pollinator interactions reveals that the presence of abundant plant species favours the pollination of rare species. Such asymmetric facilitation might promote the coexistence of species in diverse plant communities.

Marcelo A. Aizen

 ORCID: http://orcid.org/0000-0001-9079-9749

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Marcelo A. Aizen

Marcelo A. Aizen is at the Research Institute for Biodiversity and the Environment (INIBIOMA), National University of Comahue – CONICET, 8400 San Carlos de Bariloche, Río Negro, Argentina, and at the Institute for Advanced Study, Berlin, Germany.

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Species diversification results from the balance between the formation of new species (speciation) and the loss of existing ones (extinction). The tremendous proliferation of different life forms on Earth can be attributed to both high rates of speciation and low rates of extinction. Flowering plants — a group called angiosperms — are one of the most diverse groups of non-mobile organism. There are approximately 352,000 plant species, nearly 90% of which depend, to various extents, on insects and other animals for pollination and seed production1. These animal pollinators have been key to the unstoppable diversification of the angiosperms, starting at least 120 million years ago, with pollinators promoting speciation by acting as potent selection agents for a plethora of flower traits2,3. Pollinators also aid species persistence by enabling pollen transfer between relatively distant individuals in sparse plant populations4. Writing in Nature, Wei et al.5 report that, for plant species that flower at the same time, pollinators mediate interactions that might facilitate species coexistence in diverse plant communities.

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doi: https://doi.org/10.1038/d41586-021-02375-z

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