Author: Martin R. Smith

# Analysing landscapes of phylogenetic trees

#### Martin R. Smith

Landscapes of trees are mappings of tree space that are contoured according to some optimality criterion – often, but not necessarily, a tree’s score under a phylogenetic reconstruction technique (Bastert et al., 2002). Detecting “islands” or “terraces” of trees can illuminate the nature of the space of optimal trees and thus inform tree search strategy (Maddison, 1991; Sanderson et al., 2011).

For simplicity (and to avoid scoring trees against a dataset), this example uses a tree’s balance (measured using the total cophenetic index) as its score (Mir et al., 2013). We assume that mappings have already been shown to be adequate (Smith, 2022).

A landscape is most simply visualized by colouring each tree according to its score:

    # Load required libraries
    library("TreeTools", quietly = TRUE)
    library("TreeDist")

    # Generate a set of trees
    trees <- as.phylo(as.TreeNumber(BalancedTree(16)) + 0:100 - 15, 16)

    # Create a 2D mapping
    distances <- ClusteringInfoDist(trees)
    mapping <- cmdscale(distances, 2)

    # Score trees according to their balance
    scores <- TotalCopheneticIndex(trees)

    # Normalize scores
    scoreMax <- TCIContext(trees[[1]])[["maximum"]]
    scoreMin <- TCIContext(trees[[1]])[["minimum"]]
    scores <- scores - scoreMin
    scores <- scores / (scoreMax - scoreMin)

    # Generate colour palette
    col <- colorRamp(c("orange", "blue"))(scores)
    rgbCol <- rgb(col, maxColorValue = 255)

    # Plot trees, coloured by their score
    plot(
      mapping,
      asp = 1, # Preserve aspect ratio - do not distort distances
      ann = FALSE, axes = FALSE, # Don't label axes: dimensions are meaningless
      col = rgbCol, # Colour trees by score
      pch = 16 # Plotting character: Filled circle
    )

    # Add a legend
    PlotTools::SpectrumLegend(
      "left",
      title = "Tree balance",
      palette = rgb(colorRamp(c("orange", "blue"))(0:100 / 100) / 255),
      legend = floor(seq(scoreMax, scoreMin, length.out = 6)),
      bty = "n"
    )

A more sophisticated output can be produced using contours, interpolating between adjacent trees. This example uses a simple inverse distance weighting function for interpolation; more sophisticated techniques such as kriging or (in continuous tree spaces) the use of geodesics (Khodaei et al., 2022) can produce even better results.

    # Use an inverse distance weighting to interpolate between measured points
    Predict <- function (x, y) {
      Distance <- function (a, b) {
        apply(a, 2, function (pt) sqrt(colSums((pt - b) ^ 2)))
      }
      predXY <- rbind(x, y)
      dists <- Distance(t(mapping), predXY)
      invDist <- 1 / dists
      weightings <- invDist / rowSums(invDist)

      # Return:
      colSums(scores * t(weightings))
    }

    # Generate grid for contour plot
    resolution <- 32
    xLim <- range(mapping[, 1]) * 1.1
    yLim <- range(mapping[, 2]) * 1.11
    x <- seq(xLim[1], xLim[2], length.out = resolution)
    y <- seq(yLim[1], yLim[2], length.out = resolution)
    z <- outer(x, y, Predict) # Predicted values for each grid square

    # Plot
    filled.contour(
      x, y, z,
      asp = 1, # Preserve aspect ratio - do not distort distances
      ann = FALSE, axes = FALSE, # Don't label axes: dimensions are meaningless
      plot.axes = {points(mapping, xpd = NA)} # Use filled.contour coordinates
    )

A variety of R add-on packages facilitate three-dimensional plots.

    if (requireNamespace("plotly", quietly = TRUE)) {
      library("plotly", quietly = TRUE)
      fig <- plot_ly(x = x, y = y, z = z)
      fig <- fig %>% add_surface()
      fig
    } else {
      print("Run `install.packages('plotly')` to view this output")
    }

    ## [1] "Run `install.packages('plotly')` to view this output"

(Use the mouse to reorient)

# References

Bastert, O., Rockmore, D., Stadler, P. F., & Tinhofer, G. (2002). Landscapes on spaces of trees. Applied Mathematics and Computation, 131(2-3), 439–459. https://doi.org/10.1016/S0096-3003(01)00164-3
Khodaei, M., Owen, M., & Beerli, P. (2022). Geodesics to characterize the phylogenetic landscape. bioR\(\chi\)iv. https://doi.org/10.1101/2022.05.11.491507
Maddison, D. R. (1991). The discovery and importance of multiple islands of most-parsimonious trees. Systematic Biology, 40(3), 315–328. https://doi.org/10.1093/sysbio/40.3.315
Mir, A., Rosselló, F., & Rotger, L. A. (2013). A new balance index for phylogenetic trees. Mathematical Biosciences, 241(1), 125–136. https://doi.org/10.1016/j.mbs.2012.10.005
Sanderson, M. J., McMahon, M. M., & Steel, M. (2011). Terraces in Phylogenetic Tree Space. Science, 333(6041), 448–450. https://doi.org/10.1126/science.1206357
Smith, M. R. (2022). Robust analysis of phylogenetic tree space. Systematic Biology, 71(5), 1255–1270. https://doi.org/10.1093/sysbio/syab100