Theoretical foundation to estimate the relative efficiencies of the Jukes–Cantor+gamma model and the Jukes–Cantor model in obtaining the correct phylogenetic tree

Abstract

This paper deals with the theoretical foundation to estimate the relative efficiency (probability of inferring the true tree) of different nucleotide substitution models. A novel theoretical approach has been developed to estimate the relative efficiency of nucleotide substitution models based on the neighbor-relation method. The theory was developed directly by using the four-point condition. Initially theoretical formulas for the Jukes–Cantor+gamma (JC+Γ) model and the Jukes–Cantor (JC) model were developed to estimate their relative efficiencies. Theoretical formulas were used on several model topologies for both models to obtain a true tree. Extensive simulations were performed on the same set of trees to test the strength of the theoretical approach. Simulation results demonstrated a good agreement with results obtained by the theoretical estimation. Overall the theoretical foundation for estimating model efficiencies is very accurate.

Introduction

A basic process in the evolution of DNA sequences is the change in nucleotides with time. This process deserves detailed consideration because changes in nucleotide sequences are used both for estimating the rate of evolution and for reconstructing the evolutionary history of organisms. As more and more sequences are determined, attempts to refine models seem ever more worthwhile. Better models will lead to more accurate estimations of the evolutionary history of the species concerned and to a better understanding of the forces and mechanisms that affected the evolution of the sequences. For this reason, many models have been proposed for studying this process (Jukes and Cantor, 1969, Kimura, 1980, Kimura, 1981, Lanave et al., 1984, Li, 1997, Swofford et al., 1996). Each of these models has some advantages and disadvantages, and the overall relative efficiency of the models in recovering the correct phylogenetic tree is still controversial.

The distance methods of phylogenetic inference seem to be quite efficient in obtaining the correct tree, compared with maximum parsimony and several other methods. The efficiency of a method is usually studied by using either computer simulation or empirical tests. Gojobori et al. (1982), Tajima and Nei (1984), Zharkikh (1994), and others have conducted extensive simulation studies to compare the efficiencies of various methods for estimating the expected number of nucleotide substitutions between two sequences (δ_ij). On the other hand some researchers conducted theoretical studies to compare the efficiency of various methods for estimating the expected number of nucleotide substitutions between two sequences. Cavalli-Sforza and Edwards (1967) proposed the use of the ordinary least squares, whereas Fitch and Margoliash (1967) used a weighted least squares. Bulmer (1991) implemented the generalized least-squares method, in which the weighted sum of squares and cross products is minimized. On the other hand, Rzhetsky and Nei (1992) used the ordinary least-squares and minimum-evolution methods of phylogenetic inference in the four-taxa case. In the case of more than the four taxa, the neighbor-joining (NJ) method of phylogenetics inference is quite efficient in obtaining the correct tree (Saitou and Nei, 1987). Several computer simulations have been conducted to estimate the performance of the NJ method by using a large number of taxon (Kumar and Gadagkar, 2000, Takahashi and Nei, 2000), but no such theoretical estimation for more than four-taxa.

In this article a new theoretical approach is presented to estimate the efficiency of the nucleotide substitution models. In this paper, the efficiency of a model is defined as the probability of the correct topology inferred when the model is used in the tree inference algorithm. This approach is entirely different from the least-squares approach. A unique theoretical foundation is developed to estimate the efficiency of the nucleotide substitution models directly from the four-point condition (Buneman, 1971) by using simple statistical techniques. Initially the Jukes–Cantor+gamma (JC+Γ) model and the Jukes–Cantor (JC) model are considered. In this analysis, it is assumed that the JC+Γ model is a true model. The primary goal of this work is to establish a theoretical foundation to estimate the relative efficiencies of the JC+Γ model and the JC model in obtaining the correct tree. The neighbor-relation method is used for tree estimation (Buneman, 1971), and then the above two sequence evolution models are inferred by using the four-point condition (Eqs. (8), (9)). Computer simulations have been conducted to evaluate the relative efficiencies of these models, but there is no mathematical framework to study the success of each model. One reason for the lack of such a mathematical study seems to be the mathematical complexity of the problem, even for the case in which a moderate number of DNA or amino acids sequences are used. In this work four sequences are used.

The current work is divided into two parts: establish a theoretical foundation to estimate the relative efficiencies of the JC+Γ model and the JC model in obtaining the true tree, and then simulations are made to assess the accuracy of the theoretical results as well as to compare the efficiency of both models.