Distance-Based Genome Rearrangement Phylogeny

Abstract

Evolution operates on whole genomes through direct rearrangements of genes, such as inversions, transpositions, and inverted transpositions, as well as through operations, such as duplications, losses, and transfers, that also affect the gene content of the genomes. Because these events are rare relative to nucleotide substitutions, gene order data offer the possibility of resolving ancient branches in the tree of life; the combination of gene order data with sequence data also has the potential to provide more robust phylogenetic reconstructions, since each can elucidate evolution at different time scales. Distance corrections greatly improve the accuracy of phylogeny reconstructions from DNA sequences, enabling distance-based methods to approach the accuracy of the more elaborate methods based on parsimony or likelihood at a fraction of the computational cost. This paper focuses on developing distance correction methods for phylogeny reconstruction from whole genomes. The main question we investigate is how to estimate evolutionary histories from whole genomes with equal gene content, and we present a technique, the empirically derived estimator (EDE), that we have developed for this purpose. We study the use of EDE on whole genomes with identical gene content, and we explore the accuracy of phylogenies inferred using EDE with the neighbor joining and minimum evolution methods under a wide range of model conditions. Our study shows that tree reconstruction under these two methods is much more accurate when based on EDE distances than when based on other distances previously suggested for whole genomes.

Appendix

EDE: The Empirically Derived Estimator

EDE produces the best results under all model conditions, even when the evolutionary model is exclusively transpositions. For details about the mathematical derivation of this technique, see Moret et al. (2001).

EDE is based on inverting a function for the expected minimum inversion distance produced by a sequence of random inversions. Theoretical approaches (i.e., actually trying to analytically solve the expected inversion distance produced by k random inversions) proved to be quite difficult, and so we studied this under simulation. Our initial studies showed little difference in the behavior under 120 genes (typical for chloroplasts) and 37 genes (typical of mitochondria) and, in particular, suggested that it should be possible to express the normalized expected inversion distance as a function of the normalized number of random inversions. Therefore, we attempted to define a simple function f(k/n) that approximates E[dINV (G₀, G _k )/n] well, for k the number of random inversions, n the number of genes, G₀ the initial genome, and G _k the result of applying k random inversions to G₀.

The function f should have the following properties.

• 1 0 ≤ f (x) ≤ x, since the inversion distance is always less than or equal to the actual number of inversions.

• 2 lim_x→∞ f (x) ≈ 1, as our simulations show the normalized expected inversion distance is close to 1, when a large number of random inversions is applied.

• 3 f ‘(0) = 1, since a single random inversion always produces a genome that is inversion distance 1 away.

• 4 f⁻¹(y) is defined for all y [0, 1].

We used nf (x) to estimate E[d_INV(G _nx , G₀)], the expected inversion distance after nx inversions are applied. The nonlinear formula

$$ f{(x)}={({\rm{a}}x^{2} +{\rm{b}}x) /(x^{2} +{\rm{c}}x +{\rm{b}})} $$

(1)

satisfies constraints (2) and (4).

We tried several different values for the constant a, and observed in our experiments that setting a = 1 produced the best results in subsequent phylogeny reconstructions using neighbor joining, for all values of n (the number of genes). The estimation of the constants b and c then amounts to a least-squares nonlinear regression; using simulated data we obtained b =0.5956 and c =0.4577. However, with this setting for a, b, and c, the formula does not satisfy the first constraint. Hence, we modified the formula to ensure that constraint (1) holds, and obtained

$$ f(x)={\rm{min}}\{x,({\rm{a}}x^{2} +{\rm{b}}x) /(x^{2} +{\rm{c}}x +{\rm{b}})\} $$

(2)

The inverse of f is given by the formula

$$ \eqalign{f^{-1}(\rm d) = &\rm{max} \{\rm{d},(-(\rm{b-cd})+((\rm{b -cd})^2 \cr & +4\rm{bd}{({1-\rm{d}}))^{1/2}})/(2(1 -\rm{d}))\}} $$

(3)

Using the function f given above, we can thus define EDE, a method of moments estimator, as follows.

• Step 1: Given genomes G and G′, compute the inversion distance d.

• Step 2: Return n f ⁻¹(d/n), where n is the number of genes, as the estimate of the actual number of rearrangement events.

Since the function f is directly invertible, this allows us to estimate distances efficiently.

Theorem 1 (Moret et al. 2001). Let m be the number of genomes and let n be the number of genes. We can compute the pairwise EDE distance between every pair of genomes in O(nm²) time. If the inversion distance matrix is already computed, then we can compute the EDE distance matrix in O(m²) time.