Ancestral Reconstruction by Asymmetric Wagner Parsimony over Continuous Characters and Squared Parsimony over Distributions

Contemporary inferences about evolution occasionally involve analyzing infinitely large feature spaces, requiring specific algorithmic techniques. We consider parsimony analysis over numerical characters, where knowing the feature values at terminal taxa allows one to infer ancestral features, namely, by minimizing the total number of changes on the edges using continuous-valued distance measures. In particular, we show that ancestral reconstruction is possible in linear time for both an asymmetric linear distance measure (Wagner parsimony) over continuous-valued characters, and a quadratic distance measure over finite distributions. The former can be used to analyze gene content evolution with asymmetric gain and loss penalties, and the latter to reconstruct ancestral diversity of regulatory sequence motifs and multi-allele loci. As an example of employing asymmetric Wagner parsimony, we examine gene content evolution within Archaea.