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Optimized ancestral state reconstruction using Sankoff parsimony
 José C Clemente^{1}Email author,
 Kazuho Ikeo^{1},
 Gabriel Valiente^{2} and
 Takashi Gojobori^{1}Email author
https://doi.org/10.1186/147121051051
© Clemente et al; licensee BioMed Central Ltd. 2009
 Received: 27 November 2008
 Accepted: 07 February 2009
 Published: 07 February 2009
Abstract
Background
Parsimony methods are widely used in molecular evolution to estimate the most plausible phylogeny for a set of characters. Sankoff parsimony determines the minimum number of changes required in a given phylogeny when a cost is associated to transitions between character states. Although optimizations exist to reduce the computations in the number of taxa, the original algorithm takes time O(n^{2}) in the number of states, making it impractical for large values of n.
Results
In this study we introduce an optimization of Sankoff parsimony for the reconstruction of ancestral states when ultrametric or additive cost matrices are used. We analyzed its performance for randomly generated matrices, JukesCantor and Kimura's twoparameter models of DNA evolution, and in the reconstruction of elongation factor1α and ancestral metabolic states of a group of eukaryotes, showing that in all cases the execution time is significantly less than with the original implementation.
Conclusion
The algorithms here presented provide a fast computation of Sankoff parsimony for a given phylogeny. Problems where the number of states is large, such as reconstruction of ancestral metabolism, are particularly adequate for this optimization. Since we are reducing the computations required to calculate the parsimony cost of a single tree, our method can be combined with optimizations in the number of taxa that aim at finding the most parsimonious tree.
Keywords
 Cost Tree
 Character State
 Branch Length
 Ancestral State
 Cost Matrix
Background
Reconstruction of ancestral states aims at discovering the conformation of past proteins, genes or whole genomes from extant species data. This approach has been successfully utilized to reconstruct ancestral steroid receptors [1], mitochondrial DNA [2], antiviral RNase [3], or fluorescent proteins [4]. In a similar fashion, several studies have hypothesized on the evolution of hormonereceptor complexes [5], composition of ancestral genomes [6], thermostability of extinct proteins [7], properties of ancestral promoters [8], expansion of human segmental duplications [9], or ancestral codon usage [10].
Parsimony, maximum likelihood or bayesian approaches are commonly utilized to infer ancestral states. Parsimony was originally introduced by Edwards and CavalliSforza [11], but its application to reconstruct ancestral characters was first described by Fitch [12]. Sankoff later proposed a modification to take into account different rates of change between states [13, 14]. The popularity of likelihood methods in phylogenetics is mostly due to the optimization proposed by Felsenstein [15]. Yang described its application to infer ancestral sequences [16]. Bayesian approaches have also gained favor thanks to their combined used with Markov Chain Monte Carlo (MCMC) methods. Huelsenbeck and Bollback have proposed an algorithm for bayesian ancestral reconstruction [17].
Each of these approaches has advantages and weaknesses, and it is passionately debated which of them is more accurate. Parsimony is known for being biased when the rate of change per branch is high and tends to reconstruct the wrong tree due to long branch attraction [18], while likelihood does not suffer from these problems [19, 20]. On the other hand, when the characters under study evolve at nonuniform rates over time, maximum likelihood and bayesian methods have been shown to be inconsistent and perform worse than parsimomy [21].
Regardless of the preferred method, the computational complexity of ancestral reconstruction algorithms is high and optimizations are required to work with large number of sequences. In the particular case of parsimony, an algorithm to reduce the the number of computations has been previously proposed [22]. Goloboff has introduced diverse optimization strategies [23, 24], and Ronquist [25] has further improved some of the previous works. All these optimizations aim at reducing the calculations in the number of taxa when looking for the most parsimonious tree, that is, when looking for the tree in the search space that minimizes the number of changes. Nevertheless, no correct optimization in the number of states of the weighted parsimony algorithm proposed by Sankoff is known. Wheeler and Nixon proposed an optimization [26] later proved incorrect by Swofford and Siddall [27].
In this paper, we present a twofold optimization of Sankoff parsimony. Our algorithm reduces the number of operations in the number of states needed to calculate the parsimony cost of a given phylogeny, as well as the time required to reconstruct the ancestral states. This optimization can be utilized when the cost matrix for transitions between character states is either ultrametric or additive, and it reduces the original O(n^{2}) operations required with n states per node and character. While the optimization is moderate when the number of states is small, as in the case of nucleotides or amino acids, the optimization is more effective the larger the number of states, with an 8fold reduction in running time in the case of metabolic enzymes. The algorithms here presented were originally developed precisely to obtain fast reconstructions of ancestral metabolism, motivated by the recent interest in obtaining phylogenetic signal from metabolic data [28–33].
In the rest of this paper we will review the original Sankoff algorithm, describe our optimization, and analyze its performance for both randomly generated data and biologically wellknown cost matrices for nucleotides, amino acids, and metabolic enzymes.
Methods
Original Sankoff Parsimony
Sankoff parsimony [13, 14] counts the number of evolutionary changes for a specific site in a phylogenetic tree, assuming a set of n character states i = 1, ..., n (for instance, 4 nucleotides or 20 amino acids) for which a cost matrix C = (c_{ ij }) of changes between states is given. Each node p of the phylogeny has assigned a cost vector, S^{(p)}, which contains the minimal evolutionary cost ${S}_{i}^{(p)}$ for each of the character states. If node p is assigned state i, the quantity ${S}_{i}^{(p)}$ reflects the minimum cost of events (state changes) from p to the root of the tree.
Equation (1) states that the cost of being in character state i for node p is the cost of moving from character state i to j in child q (c_{ ij }) plus the cost of having reached state j at node q from the leaves (${S}_{j}^{(q)}$). Character j is selected to minimize this sum, with the same procedure being applied to character k in child r. Algorithm 1 presents the original implementation of Sankoff parsimony to calculate the cost vector of all nodes in a tree for a single character.
Algorithm 1 (Original Sankoff algorithm: Up phase). A procedure call Sankoff_Up(T, C, S) calculates the cost vector S^{(p)}of all nodes p of the phylogeny T, given a cost matrix C = (c_{ ij }).
procedure Sankoff_Up(T, C, S)
for all nodes p of T in postorder do
if p is a leaf then
for all i in 1, ..., n do
if state i observed at leaf p then
${S}_{i}^{(p)}$ ← 0
else
${S}_{i}^{(p)}$ ← ∞
else
{q, r} ← children of p
for all i in 1, ..., n do
${S}_{i}^{(p)}$ ← cost(q, i) + cost(r, i)
function cost(x, i)
min ← ∞
for all j in 1, ..., n do
if c_{ ij }+ ${S}_{j}^{(x)}$ < min then
min ← c_{ ij }+ ${S}_{j}^{(x)}$
return min
Algorithm 2 (Original Sankoff algorithm: Down phase). A procedure call Sankoff_Down(x, T, C, S, S_{ anc }) calculates the ancestral states ${S}_{anc}^{(p)}$of all nodes p of the phylogeny T, given the root x of T, a cost matrix C = (c_{ ij }) of transition costs between states, and the cost vectors S^{(p)}for all nodes p of T as calculated by Sankoff_Up(T, C, S).
procedure Sankoff_Down(x, T, C, S, S_{ anc })
${S}_{anc}^{(x)}\leftarrow \mathrm{arg}{\mathrm{min}}_{i}{S}_{i}^{(x)}$
for all j in ${S}_{anc}^{(x)}$do
for all child y of x do
Sankoff_Down(j, y, T, C, S, S_{ anc })
procedure Sankoff_Down(i, x, T, C, S, S_{ anc })
min_states(i, x, C, S, S_{ anc })
for all j in ${S}_{anc}^{(x)}$do
for all child y of x do
Sankoff_Down(j, y, T, C, S, S_{ anc })
procedure min_states(i, x, C, S, S_{ anc })
min ← ∞
for all j in 1, ..., n do
if x = root(T) then
trans_cost ← ${S}_{j}^{(x)}$
else
trans_cost ← c_{ ij }+ ${S}_{j}^{(x)}$
if trans_cost < min then
min ← trans_cost
${S}_{anc}^{(x)}$ ← {j}
else if trans_cost = min then
${S}_{anc}^{(x)}$ ← ${S}_{anc}^{(x)}$ ∪ {j}
Optimized Sankoff Parsimony
Our optimization is based on the observation that not all transition costs between character states need to be computed if the cost matrix is ultrametric or additive.
Definition 1. A cost matrix is ultrametric if for every three indices i, j, k, one of the three following inequalities holds (three point condition):

c_{ ij }≤ c_{ ik }= c_{ jk }

c_{ ik }≤ c_{ ij }= c_{ jk }

c_{ jk }≤ c_{ ij }= c_{ ik }
Definition 2. A cost matrix is additive if for every four indices i, j, k, ℓ, one of the three following inequalities holds (four point condition):

c_{ ij }+ c_{k ℓ}≤ c_{ ik }+ c_{j ℓ}= c_{i ℓ}+ c_{ jk }

c_{ ik }+ c_{j ℓ}≤ c_{ ij }+ c_{k ℓ}= c_{i ℓ}+ c_{ jk }

c_{i ℓ}+ c_{ jk }≤ c_{ ij }+ c_{k ℓ}= c_{ ik }+ c_{j ℓ}
Notice that ultrametric matrices are also additive, since they satisfy the four point condition [35].
Consider a simple example where the cost matrix C has c_{ ii }= 0 and c_{ ij }= k for all i ≠ j. When calculating the cost ${S}_{i}^{(p)}$ in Equation (1), we can substitute min_{ j }(c_{ ij }+ ${S}_{j}^{(q)}$) for min(${S}_{i}^{(q)}$, k + min_{j≠i}${S}_{j}^{(q)}$), and similarly for the other child r of p. In the more general case of ultrametric or additive cost matrices, we can efficiently represent them with a unique rooted weighted cost tree T_{ C }using UPGMA [36] or neighborjoining [37] respectively. The length of the path between any two leaves i, j in T_{ C }corresponds to the cost c_{ ij }. For ultrametric matrices, consider any set of leaves L = {a, b, ...} in the tree that have the same last common ancestor, lca(L). By definition, lca(L) is equidistant to any leaf in L, and all leaves in L are at the same distance d to each other (which is double the distance from the leaf to lca(L)). For any two leaves a, b in L we can then simplify the expression min_{ L }(c_{ ab }+ ${S}_{L}^{(q)}$) as d + min ${S}_{L}^{(q)}$. Therefore, given the cost tree T_{ C }obtained by UPGMA from the ultrametric matrix, for each state i we only need to compute the minimum costs at the last common ancestor of that state and any other, that is, the inner nodes in the path from i to the root of T_{ C }. With additive matrices, since the distance from an inner node to its descendant leaves can vary, we need to take into consideration the specific length of each branch when calculating the minimum. Therefore, in our algorithm each cost vector S^{ p }is replaced by a cost tree ${T}_{C}^{(p)}$, whose inner nodes will contain the value that minimizes. c_{ ij }+ ${S}_{L}^{(q)}$ for all descendant leaves L. Algorithm 3 presents the optimized version of Sankoff parsimony for the calculations from the leaves to the root of the phylogeny.
Algorithm 3 (Optimized Sankoff algorithm: Up phase). A procedure call Opt_Sankoff_Up(T, T_{ C }, S) calculates the cost vector S^{(p)}of all nodes p of the phylogeny T, given a cost tree T_{ C }representing an ultrametric or additive cost matrix C = (c_{ ij }).
procedure Opt_Sankoff_Up(T, T_{ C }, S)
for all nodes p of T in postorder do
if p is a leaf then
for all i in 1, ..., n do
if state i observed at leaf p then
${S}_{i}^{(p)}$ ← 0
else
${S}_{i}^{(p)}$ ← ∞
update(p, i, ${S}_{i}^{(p)}$, T_{ C })
else
{q, r} ← children of p
for all i in 1, ..., n do
${S}_{i}^{(p)}$ ← cost(q, i) + cost(r, i)
update(p, i, ${S}_{i}^{(p)}$, T_{ C })
procedure update(x, i, v, T_{ C })
n ← leaf of ${T}_{C}^{(x)}$corresponding to state i
cost(n) ← v
min_tags(n) ← {i}
cost ← branch length between n and its parent in ${T}_{C}^{(x)}$
repeat
n ← parent of n in ${T}_{C}^{(x)}$
if v + cost <cost(n) then
cost(n) ← v + cost
min_tags(n) ← {i}
else if v + cost = cost(n) then
min_tags(n) ← min_tags(n) ∪ {i}
if n ≠ root of ${T}_{C}^{(x)}$then
cost ← cost + branch length between n and its parent in ${T}_{C}^{(x)}$
until n = root of ${T}_{C}^{(x)}$
function cost(x, i)
n ← leaf of ${T}_{C}^{(x)}$corresponding to state i
min ← cost(n)
cost ← branch length between n and its parent in ${T}_{C}^{(x)}$
repeat
n ← parent of n in ${T}_{C}^{(x)}$
if cost + cost(n) < min then
min ← cost + cost(n)
if n ≠ root of ${T}_{C}^{(x)}$then
cost ← cost + branch length between n and its parent in ${T}_{C}^{(x)}$
until n = root of ${T}_{C}^{(x)}$
return min
Algorithm 3 utilizes a cost tree ${T}_{C}^{(p)}$ with the same topology as T_{ C }for each node p, and where each node is annotated with the minimum value corresponding to c_{ ij }+ ${S}_{L}^{(q)}$, as implemented in the function cost. The function update saves each of these cost trees by moving from the leaves to the root and storing minimum values in the nodes (cost(n)), as well as the leaf responsible for the value stored in the node (min_tags(n)), which will be later used to optimize the reconstruction of ancestral states. The complexity of Algorithm 3 depends on the internal path length of T_{ C }. The worst case would be a degenerate tree with linear structure, in which case the complexity for n states is (n^{2}  n)/2 per node and character [[38], §2.3.4.5]. Notice that in practice this will be a rare case, and most cost trees have a more favourable topology to our optimization (as will be shown in the Results section), while the original algorithm takes O(n^{2}) time no matter the cost matrix used.
Algorithm 4 presents the optimized version of the reconstruction of ancestral states using Sankoff parsimony. In its original implementation, for each node we had to consider all posible states looking for the one that minimized c_{ ij }+ ${S}_{j}^{(x)}$. Since we have already saved the minimum transition costs and the leaves responsible for them as shown in Algorithm 3, we can use this information to further speed up computation. The ancestral states for the root r of the phylogeny T are obtained as in the original algorithm. For any inner node x of T, and given that its parent had ancestral state k reconstructed, we only need to move from leaf k to the root of ${T}_{C}^{(x)}$ and keep the state min_tags(n) that has the minimum value cost(n).
Algorithm 4 (Optimized Sankoff algorithm: Down phase). A procedure call Opt_Sankoff_Down(x, T, T_{ C }, S_{ anc }) calculates the ancestral states ${S}_{anc}^{(p)}$of all nodes p of the phylogeny T, given the root x of T and the cost tree ${T}_{C}^{(p)}$for each node p of T as calculated by Opt_Sankoff_Up(T, T_{ C }, S).
procedure Sankoff_Down(x, T, T_{ C }, S_{ anc })
${S}_{anc}^{(x)}\leftarrow \mathrm{arg}{\mathrm{min}}_{i}{S}_{i}^{(x)}$
for all j in ${S}_{anc}^{(x)}$do
for all child y of x do
Opt_Sankoff_Down(j, y, T, T_{ C }, S_{ anc })
procedure Opt_Sankoff_Down(i, x, T, T_{ C }, S_{ anc })
n ← leaf of ${T}_{C}^{(x)}$corresponding to state i
min ← cost(n)
cost ← branch length between n and its parent in ${T}_{C}^{(x)}$
repeat
n ← parent of n in ${T}_{C}^{(x)}$
if cost + cost(n) < min then
min ← cost + cost(n)
${S}_{anc}^{(x)}$ ← {min_tags(n)}
else if cost + cost(n) = min then
${S}_{anc}^{(x)}$ ← ${S}_{anc}^{(x)}$ ∪ {min_tags(n)}
if n ≠ root of ${T}_{C}^{(x)}$then
cost ← cost + branch length between n and its parent in ${T}_{C}^{(x)}$
until n = root of ${T}_{C}^{(x)}$
for all j in S_{ anc }do
for all child y of x do
Opt_Sankoff_Down(j, y, T, T_{ C }, S_{ anc })
Algorithm 4 reduces the number of operations compared to the original implementation, since we do not have to review all states at each node but only traverse from a leaf to the root of the cost tree. The original implementation takes O(n) time, while the complexity of our optimization is again a function of the internal path length of T_{ C }, which in general will be less than n.
Results and Discussion
Alternatively, we could have codified the presence or absence of enzymes in each species, and then perform a reconstruction using maximum parsimony. While this is a commonly used approach, the use of measures of enzymatic similarity has shown a better performance than simple patterns of presence/absence of enzymes in the phylogenetic analysis of metabolism [31], and therefore the election of large state sets is to be preferred. For this particular example, the number of states is composed of 925 reactions annotated to at least one of the twelve species under study, and the number of inner nodes in the cost tree is comparatively small, making our optimization 8fold faster than the original implementation.
Conclusion
The optimization here presented provides a computation of Sankoff parsimony faster than the original algorithm when the cost matrix is ultrametric or additive, even for a small number of character states. Since our approach reduces the execution time needed to calculate the parsimony cost of a single tree, it could be easily combined with optimizations looking for the most parsimonous tree. Our algorithm takes comparatively less time to execute when the number of states is large, and therefore problems such as ancestral metabolism reconstruction could be especially wellsuited for this optimization.
Declarations
Acknowledgements
The authors would like to thank Mercè Llabrés for her comments on an early version of this work, and two anonymous reviewers for critically reading the manuscript and providing valuable suggestions. JC was supported by GrantinAid for JSPS Fellows from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan, No. 20·08086. GV was partially supported by the Spanish DGI project MTM200607773 COMGRIO. KI and TG were partially supported by a grant of the Genome Network Project from MEXT, Japan.
Authors’ Affiliations
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