Visualizing spatial population structure with estimated effective migration surfaces

Journal name:
Nature Genetics
Year published:
Published online


Genetic data often exhibit patterns broadly consistent with 'isolation by distance'—a phenomenon where genetic similarity decays with geographic distance. In a heterogeneous habitat, this may occur more quickly in some regions than in others: for example, barriers to gene flow can accelerate differentiation between neighboring groups. We use the concept of 'effective migration' to model the relationship between genetics and geography. In this paradigm, effective migration is low in regions where genetic similarity decays quickly. We present a method to visualize variation in effective migration across a habitat from geographically indexed genetic data. Our approach uses a population genetic model to relate effective migration rates to expected genetic dissimilarities. We illustrate its potential and limitations using simulations and data from elephant, human and Arabidopsis thaliana populations. The resulting visualizations highlight important spatial features of population structure that are difficult to discern using existing methods for summarizing genetic variation.

At a glance


  1. A schematic overview of EEMS (Estimated Effective Migration Surfaces), using African elephant data for illustration.
    Figure 1: A schematic overview of EEMS (Estimated Effective Migration Surfaces), using African elephant data for illustration.

    (ac) Setting up the population grid. (a) Samples are collected at known locations across a two-dimensional habitat; green and orange represent two species of African elephant, forest and savanna, respectively. (b) A dense triangular grid is chosen to span the habitat. (c) Each sample is assigned to the closest deme on the grid. (df) EEMS analysis. (d) Migration rates vary according to a Voronoi tessellation that partitions the habitat into 'cells' with constant migration rate; colors represent relative rates of migration, ranging from low (orange) to high (blue). (e) Each edge has the same migration rate as the cell into which it falls. Cell locations and migration rates are adjusted, using Bayesian inference, so that expected genetic dissimilarities under the EEMS model match observed genetic dissimilarities. (f) The EEMS is a color contour plot produced by averaging draws from the posterior distribution of the migration rates, interpolating between grid points. Here and in all other figures, log(m) denotes the effective migration rate on a log10 scale, relative to the overall migration rate across the habitat. (Thus, log(m) = 1 corresponds to effective migration that is tenfold faster than the average.) The main feature of the EEMS for the African elephant is a barrier of low effective migration that separates the habitats of the two species: forest elephants to the west and savanna elephants to the north, south and east.

  2. Simulations comparing EEMS and PCA.
    Figure 2: Simulations comparing EEMS and PCA.

    For each method, we show results for two migration scenarios—representing uniform migration and a barrier to migration—and three different sampling schemes. (a,b) The true underlying migration rates for the uniform (a) and barrier (b) scenarios; colors represent relative migration rates. (c) The three sampling schemes used; the size of the circle at each node is proportional to the number of individuals sampled at that location, and locations are color-coded to facilitate cross-referencing the EEMS and PCA results. (d) PCA results. (e) EEMS results. In contrast to PCA, EEMS is robust to sampling scheme and shows clear qualitative differences between the estimated effective migration rates under the two scenarios, reflecting the underlying simulation truth.

  3. Simulations illustrate that EEMS infers effective migration rates rather than actual steady-state migration rates.
    Figure 3: Simulations illustrate that EEMS infers effective migration rates rather than actual steady-state migration rates.

    (a) Individuals have uniform migration rates, but the central area of the habitat has a lower population density (demes in this region have fewer individuals, as represented by smaller circles in gray). Thus, fewer migrants are exchanged per generation in the central area, producing an effective barrier to gene flow that is reflected in the EEMS. (b) A simple population split scenario: migration was initially uniform, but at some time in the past a complete barrier to migration arose in the central area (represented by dashed edges). Under this scenario, the groups on either side of the central region have diverged, which creates a barrier in the EEMS.

  4. EEMS analysis of African elephant data.
    Figure 4: EEMS analysis of African elephant data.

    (a) African elephant samples are collected from two species in five biogeographic regions: the forest elephant (in green) inhabits the western and central regions, and the savanna elephant (in orange) inhabits the northern, eastern and southern regions. (b) Estimated effective migration rates for forest and savanna samples analyzed jointly. (c,d) Estimated effective migration rates for savanna (c) and forest (d) samples analyzed separately.

  5. EEMS analysis of human population structure in Western Europe and in sub-Saharan Africa.
    Figure 5: EEMS analysis of human population structure in Western Europe and in sub-Saharan Africa.

    (a) Effective migration rates in Western Europe, estimated using geo-referenced data from the POPRES (Population Reference Sample) project34. (b) Effective migration rates in sub-Saharan Africa, estimated using geo-referenced data from two previously published studies35, 36. Population abbreviations are defined in the Supplementary Note.

  6. EEMS analysis of A. thaliana data from the RegMap panel.
    Figure 6: EEMS analysis of A. thaliana data from the RegMap panel.

    (a) Estimated effective migration rates in North America, Europe and across the Atlantic Ocean. (b) Estimated effective migration rates within North America. (c) Estimated effective migration rates within Europe.


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Author information


  1. Department of Statistics, The University of Chicago, Chicago, Illinois, USA.

    • Desislava Petkova &
    • Matthew Stephens
  2. Wellcome Trust Centre for Human Genetics, Oxford, UK.

    • Desislava Petkova
  3. Department of Human Genetics, The University of Chicago, Chicago, Illinois, USA.

    • John Novembre &
    • Matthew Stephens


M.S. and J.N. conceived the project. D.P., J.N. and M.S. developed and refined methods. D.P. implemented methods. D.P., J.N. and M.S. wrote the manuscript.

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The authors declare no competing financial interests.

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