The phylogeny of Neoaves, the largest clade of extant birds, has remained unclear despite intense study. The difficulty associated with resolving the early branches in Neoaves is likely driven by the rapid radiation of this group. However, conflicts among studies may be exacerbated by the hypothesis that relationships are sensitive to the data type analyzed. For example, analyses of coding exons typically yield trees that place Strisores (nightjars and allies) sister to the remaining Neoaves, while analyses of non-coding data typically yield trees where Mirandornites (flamingos and grebes) is the sister of the remaining Neoaves. Our understanding of data type effects is hampered by the fact that previous analyses have used different taxa, loci, and types of non-coding data. Herein, we provide strong corroboration of the data type effects hypothesis for Neoaves by comparing trees based on coding and non-coding data derived from the same taxa and gene regions. A simple analytical method known to minimize biases due to base composition (coding nucleotides as purines and pyrimidines) resulted in coding exon data with increased congruence to the non-coding topology using concatenated analyses. These results improve our understanding of the resolution of neoavian phylogeny and point to a challenge - data type effects - that is likely to be an important factor in phylogenetic analyses of birds (and many other taxonomic groups). Using our results, we provide a summary phylogeny that identifies well-corroborated relationships and highlights specific nodes where future efforts should focus.

Many people, including Horadam, have studied the numbers $W_n$, satisfying the recurrence relation $W_n=u W_{n-1}+v W_{n-2}$ ($n\ge 2$) with $W_0=0$ and $W_1=1$. In this paper, we study the $p$-numerical semigroups of the triple $(W_i,W_{i+2},W_{i+k})$ for integers $i,k(\ge 3)$. For a nonnegative integer $p$, the $p$-numerical semigroup $S_p$ is defined as the set of integers whose nonnegative integral linear combinations of given positive integers $a_1,a_2,\dots,a_\kappa$ with $\gcd(a_1,a_2,\dots,a_\kappa)=1$ are expressed in more than $p$ ways. When $p=0$, $S=S_0$ is the original numerical semigroup. The largest element and the cardinality of $\mathbb N_0\backslash S_p$ are called the $p$-Frobenius number and the $p$-genus, respectively.

Phylogenomic analyses have revolutionized the study of biodiversity, but they have revealed that estimated tree topologies can depend, at least in part, on the subset of the genome that is analyzed. For example, estimates of trees for avian orders differ if protein coding or non-coding data are analyzed. The bird tree is a good study system because the historical signal for relationships among orders is very weak, which should permit subtle non-historical signals to be identified, while monophyly of orders is strongly corroborated, allowing identification of strong non-historical signals. Hydrophobic amino acids in mitochondrially-encoded proteins, which are expected to be found in transmembrane helices, have been hypothesized to be associated with non-historical signals. We tested this hypothesis by comparing the evolution of transmembrane helices and extramembrane segments of mitochondrial proteins from 420 bird species, sampled from most avian orders. We estimated amino acids exchangeabilities for both structural environments and assessed the performance of phylogenetic analysis using each data type. We compared those relative exchangeabilities with values calculated using a substitution dataset for transmembrane helices from a variety of sampled set of nuclear- and mitochondrially-encoded proteins, allowing us to compare the bird-specific mitochondrial models with a general model of transmembrane protein evolution. To complement our amino acid analyses, we examined the impact of protein structure on patterns of nucleotide evolution. Models of transmembrane and extramembrane sequence evolution for amino acids and nucleotides exhibited striking differences, but there was no evidence for strong topological data type effects. However, incorporating protein structure into analyses of mitochondrially-encoded proteins improved model fit. Thus, we believe that considering protein structure will improve analyses of mitogenomic data, both in birds and in other taxa.

Phylogenomics, the use of large datasets to examine phylogeny, has revolutionized the study of evolutionary relationships. However, genome-scale data have not been able to resolve all relationships in the tree of life; this could reflect, at least in part, the poor-fit of the models used to analyze heterogeneous datasets. Some of the heterogeneity may reflect the different patterns of selection on proteins based on their structures. To test that hypothesis, we developed a pipeline to divide phylogenomic protein datasets into subsets based on secondary structure and relative solvent accessibility. We then tested whether amino acids in different structural environments had distinct signals for the topology of the deepest branches in the metazoan tree. The most striking difference in phylogenetic signal reflected relative solvent accessibility; analyses of exposed sites (on the surface of proteins) yielded a tree that placed ctenophores sister to all other animals whereas sites buried inside proteins yielded a tree with a sponge-ctenophore clade. These differences in phylogenetic signal were not ameliorated when we repeated our analyses using the CAT model, a mixture model that is often used for analyses of protein datasets. In fact, the heterogeneous CAT model resulted in several rearrangements that are unlikely to represent evolutionary history. However, analyses conducted after recoding amino acids to limit the impact of deviations from compositional stationarity increased the congruence in the estimates of phylogeny for exposed and buried sites; after recoding amino acids both trees supported placement of ctenophores sister to all other animals. These results provide striking evidence that it is necessary to achieve a better understanding of the constraints due to protein structure to improve phylogenetic estimation.