MATHEMATICAL STUDIES ON THE EVOLUTIONARY CHANGE OF DNA SEQUENCES

With the aim of understanding the mechanism of molecular evolution, mathematical problems on the evolutionary change of DNA sequences are studied. The problems studied and the results obtained are as follows: (1) Estimation of evolutionary distance between nucleotide sequences. Studying the pattern of nucleotide substitution for the case of unequal substitution rates, a new mathematical formula for estimating the average number of nucleotide substitutions per site between two homologous DNA sequences is developed. It is shown that this formula has a wider applicability than currently available formulae. A statistical method for estimating the number of nucleotide changes due to deletion and insertion is also developed. (2) Biases of the estimates of nucleotide substitutions obtained by the restriction enzyme method. The deviation of the estimate of nucleotide substitutions obtained by the restriction enzyme method from the true value is investigated theoretically. It is shown that the amount of the deviation depends on the nucleotides in the recognition sequence of the restriction enzyme used, unequal rates of substitution among different nucleotides, and nucleotide frequences, but the primary factor is the unequal rates of nucleotide substitution. When many different kinds of enzymes are used, however, the amount of average deviation is generally small. (3) Distribution of restriction fragment lengths. To see the effect of undetectable restriction fragments and fragment differences on the estimate of nucleotide differences, the theoretical distribution of fragment lengths is studied. This distribution depends on the type of restriction enzymes used as well as on the relative frequencies of four nucleotides. It is shown that undetectability of small fragments or fragment differences gives a serious underestimate of nucleotide substitutions when the length-difference method of estimation is used, but the extent of underestimation is small when the site-difference method is used. (4) Evolutionary relationships of DNA sequences in finite populations. A mathematical theory on the expected evolutionary relationships among DNA sequences (nucleons) randomly chosen from the same or different populations is developed under the assumption that the evolutionary change of nucleons is determined solely by mutation and random genetic drift. . . . (Author's abstract exceeds stipulated maximum length. Discontinued here with permission of author). UMI ^

THEORETICAL STUDIES ON THE METHODS OF RECONSTRUCTING PHYLOGENETIC TREES FROM DNA SEQUENCE DATA (MOLECULAR EVOLUTION, HOMINOID EVOLUTION, COMPUTER SIMULATION)

(1) A mathematical theory for computing the probabilities of various nucleotide configurations is developed, and the probability of obtaining the correct phylogenetic tree (model tree) from sequence data is evaluated for six phylogenetic tree-making methods (UPGMA, distance Wagner method, transformed distance method, Fitch-Margoliash's method, maximum parsimony method, and compatibility method). The number of nucleotides (m*) necessary to obtain the correct tree with a probability of 95% is estimated with special reference to the human, chimpanzee, and gorilla divergence. m* is at least 4,200, but the availability of outgroup species greatly reduces m* for all methods except UPGMA. m* increases if transitions occur more frequently than transversions as in the case of mitochondrial DNA. (2) A new tree-making method called the neighbor-joining method is proposed. This method is applicable either for distance data or character state data. Computer simulation has shown that the neighbor-joining method is generally better than UPGMA, Farris' method, Li's method, and modified Farris method on recovering the true topology when distance data are used. A related method, the simultaneous partitioning method, is also discussed. (3) The maximum likelihood (ML) method for phylogeny reconstruction under the assumption of both constant and varying evolutionary rates is studied, and a new algorithm for obtaining the ML tree is presented. This method gives a tree similar to that obtained by UPGMA when constant evolutionary rate is assumed, whereas it gives a tree similar to that obtained by the maximum parsimony tree and the neighbor-joining method when varying evolutionary rate is assumed. ^

Assessment of the effect on statistical power of regression model misspecification by using techniques of mathematical statistics and simulation study

Objectives. This paper seeks to assess the effect on statistical power of regression model misspecification in a variety of situations. ^ Methods and results. The effect of misspecification in regression can be approximated by evaluating the correlation between the correct specification and the misspecification of the outcome variable (Harris 2010).In this paper, three misspecified models (linear, categorical and fractional polynomial) were considered. In the first section, the mathematical method of calculating the correlation between correct and misspecified models with simple mathematical forms was derived and demonstrated. In the second section, data from the National Health and Nutrition Examination Survey (NHANES 2007-2008) were used to examine such correlations. Our study shows that comparing to linear or categorical models, the fractional polynomial models, with the higher correlations, provided a better approximation of the true relationship, which was illustrated by LOESS regression. In the third section, we present the results of simulation studies that demonstrate overall misspecification in regression can produce marked decreases in power with small sample sizes. However, the categorical model had greatest power, ranging from 0.877 to 0.936 depending on sample size and outcome variable used. The power of fractional polynomial model was close to that of linear model, which ranged from 0.69 to 0.83, and appeared to be affected by the increased degrees of freedom of this model.^ Conclusion. Correlations between alternative model specifications can be used to provide a good approximation of the effect on statistical power of misspecification when the sample size is large. When model specifications have known simple mathematical forms, such correlations can be calculated mathematically. Actual public health data from NHANES 2007-2008 were used as examples to demonstrate the situations with unknown or complex correct model specification. Simulation of power for misspecified models confirmed the results based on correlation methods but also illustrated the effect of model degrees of freedom on power.^