A Simple Method for Classifying Genes and a Bootstrap Test for Classifications

Kazuharu Misawa and Fumio Tajima

Department of Biological Sciences, Graduate School of Science, The University of Tokyo, Tokyo, Japan


    Abstract
 TOP
 Abstract
 Introduction
 Theory
 Example
 Discussion
 Acknowledgements
 literature cited
 
A new simple method for classifying genes is proposed based on Klastorin's method. This method classifies genes into monophyletic groups which are made distinct from each other by evolutionary changes. The method is applicable as long as the phylogenetic tree of genes is obtained. There is a fast algorithm for obtaining the classification. A bootstrap test of a classification is also presented. As an example, we classified opsin genes. The classification obtained by this method is the same as the previous classification based on the function of opsins.


    Introduction
 TOP
 Abstract
 Introduction
 Theory
 Example
 Discussion
 Acknowledgements
 literature cited
 
To date, a large number of classifications of genes have been used for molecular sequence analyses. For example, genes are classified into gene families (see Dayhoff 1978Citation ). Virus variants are typed into several groups (Chan et al. 1992Citation ; Okamoto et al. 1992Citation ; Bukh, Purcell, and Miller 1993Citation ; Louwagie et al. 1993Citation ; Ohba et al. 1995Citation ; Seibert et al. 1995Citation ). Samples of human population surveys are also typed into haplogroups to infer human history (Horai et al. 1993Citation ; Torroni et al. 1993Citation ).

In Origin of the Species, Darwin (1859)Citation wrote, "All the forgoing rules and aids and difficulties in classification may be explained, if I do not greatly deceive myself, on the view that Natural System is founded on descent with modification. ..." Many taxonomists agree with his idea, although some arguments remain (see Mayr 1981Citation ; Wiley 1981Citation ).

A large number of methods have been developed for classifying genes. Dayhoff (1978)Citation constructed gene families on the basis of sequence similarity. Some authors typed virus variants by the significance levels of phylogenetic trees (Louwagie et al. 1993Citation ; Ohba et al. 1995Citation ; Seibert et al. 1995Citation ), and some typed them by PCR with type-specific primers (Okamoto et al. 1992Citation ). Torroni et al. (1993)Citation used haplogroups based on certain restriction sites. This variety of methods, however, causes confusion in classifications. A widely applicable method for classifying genes is needed.

Klastorin (1982)Citation proposed a method for classifying hospitals based on the cluster analysis (Sneath and Sokal 1973Citation ). In this paper, we apply Klastorin's (1982)Citation method to the classification of genes by using the phylogenetic tree of genes. This method is applicable as long as the phylogenetic tree is obtained. We also present a test of classifications based on bootstrap resampling (Efron and Tishirani 1991Citation ).


    Theory
 TOP
 Abstract
 Introduction
 Theory
 Example
 Discussion
 Acknowledgements
 literature cited
 
In this paper, a classification refers to a set of groups and a group refers to a set of operational taxonomic units (OTUs). We consider only one-rank classification. Thus, every gene must be included in one of the groups in a classification.

Criterion for Classifying Genes
Klastorin (1982)Citation proposed a method for classifying hospitals by using a dendrogram. A dendrogram is a treelike graph which shows the similarities between OTUs (Sneath and Sokal 1973Citation ). Figure 1 shows a dendrogram. In Klastorin's (1982)Citation method, a group must be defined by a branch, so that the length of a branch represents a group's distinctiveness. In figure 1 , there are four groups, a, b, c, and d.



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Fig. 1.—A dendrogram. a, b, c, and d are groups. l(a), l(b), l(d), and l(e) are the distinctivenesses of these groups

 
If P is a classification, the expected distinctiveness for this classification, E(P), is defined as


(1)
where n(x) is the number of genes in x and l(x) is the distinctiveness of group x (Klastorin 1982Citation , eq. 2 ). Klastorin (1982)Citation suggested that the classification which has the largest expected distinctiveness should be chosen (see Klastorin [1982]Citation for detailed discussion).

Let us examine an example in figure 1 . There are two possible classifications: P = {a, d} and Q = {b, c, d}. From equation (1) , we obtain E(P) = n(a)l(a) + n(d)l(d) = 1.0, and E(Q) = n(b)l(b) + n(c)l(c) + n(d)l(c) = 0.8. Thus, P is the classification to be chosen.

We can classify genes by using Klastorin's (1982)Citation method with the phylogenetic tree of genes instead of the dendrogram, because the phylogenetic tree expresses the evolutionary history of genes. The distinctiveness of a group corresponds to the number of evolutionary changes occurring on the ancestral lineage.

Algorithm for Classifying Genes
We denote the classification with the largest expected distinctiveness by G({alpha}), where {alpha} is the phylogenetic tree of the genes to be classified. Klastorin (1982)Citation developed an algorithm to obtain G({alpha}). We developed a new algorithm to obtain G({alpha}) which is simpler than Klastorin's (1982)Citation algorithm.

Before we present the algorithm, we must note that E has a favorable feature. Namely, when X, Y, and Z are classifications and Z = X {cup} Y, we have

(2)
This equation can be proved as follows: Consider that X consists of x1, x2, ... , xk, and Y consists of y1, y2, ... , yh. From equation (1) , we obtain E(X) = {Sigma}ki = 1 n(xi)l(xi), and E(Y) = {Sigma}hj = 1 n(yj)l(yj). Since Z = X {cup} Y = {x1, x2, ... , xk, y1, y2, ... , yh}, we obtain E(Z) = {Sigma}ki = 1 n(xi)l(xi) + {Sigma}hj = 1 n(yj)l(yj) = E(X) + E(Y).

Because of equation (2) , there are only two candidates for G({alpha}) when G(ß) and G({gamma}) are given, where ß and {gamma} are subtrees of {alpha}. One candidate is C({alpha}), the classification in which all genes are classified into one group. The other candidate is G(ß) {cup} G({gamma}). Thus, we can find G({alpha}) by using C({alpha}), G(ß), and G({gamma}) as follows:

  1.  Algorithm G({alpha} ){
  2.   if (N({alpha}) > 1) {
  3.    divide {alpha} into ß and {gamma} at the oldest branching point of {alpha};
  4.    if (E[C({alpha})] > E[G(ß)] + E[G({gamma})]) return C({alpha});
  5.    else return G(ß) {cup} G({gamma});
  6.   } else return C({alpha});
  7.  }

where N({alpha}) is the number of genes in tree {alpha}. Although Klastorin's (1982)Citation algorithm consists of two types of search procedures, while our algorithm consists of one type of recursive procedure, both are equivalent and yield the same classification. It is important to note that this algorithm takes a short time. Each branch length and each number of members are evaluated only once. The number of comparisons is proportional to the number of branches. The number of branches is also proportional to the number of genes. Thus, the time this algorithm takes is proportional to the number of genes.

We implemented this algorithm as a Java program that reconstructs the phylogenetic tree of genes from the sequence data by using the neighbor-joining method (Saitou and Nei 1987Citation ) and obtains the classification of genes. This program is available from the authors on request.

Let us see how this algorithm works by examining figure 2 , which shows hypothetical examples of phylogenetic trees. {alpha} in figure 2a is a phylogenetic tree. ß and {gamma} in figure 2b are the subtrees of {alpha}. {delta} and {epsilon} in figure 2c are the subtrees of ß. Figure 2 also shows five possible groups, a, b, c, d, and e. The branch lengths of these groups are also shown in figure 2 , as l(a) = 1, l(b) = 4, l(c) = 7, l(d) = 3, and l(e) = 2. Figure 2d shows possible classifications. {alpha} has three possible classifications: P = {a}, Q = {b, c}, and R = {d, e, c}. ß has two possible classifications: S = {b} and T = {d, e}. {gamma}, {delta}, and {epsilon} each have one possible classification: U = {c}, V = {d}, and W = {e}, respectively.



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Fig. 2.—Phylogenetic tree and classifications. a, Phylogenetic tree {alpha}. a, b, c, d, and e are groups. l(a), l(b), l(c), l(d), and l(e) are the distinctivenesses of these groups. b, Subtrees of {alpha}. ß and {gamma} are subtrees of {alpha}. c, Subtrees of ß. {delta} and {epsilon} are subtrees of ß. d, Possible classifications of {alpha}, ß, {gamma}, {delta}, and {epsilon}

 
Because N({alpha}) = 3, {alpha} is divided into subtrees in line 3. The subtrees of {alpha} are ß and {gamma}, as shown in figure 2b. Then, we must obtain G(ß) and G({gamma}) which are used in line 4. First, let us obtain G(ß) by using this algorithm. Because N(ß) = 2, ß is divided into two trees in line 3. The subtrees of ß are {delta} and {epsilon}, as shown in figure 2c. In addition, in this case, we have to obtain G({delta}) and G({epsilon}). Because N({delta}) = N({epsilon}) = 1, line 6 yields G({delta}) = C({delta}) = V and G({epsilon}) = C({epsilon}) = W. Now, we return to obtaining G(ß). Since E[S] = 8 and E[T] = E[V] + E[W] = 5, line 4 yields G(ß) = S. Next, let us obtain G({gamma}) by using this algorithm. Because N({gamma}) = 1, line 6 yields G({gamma}) = C({gamma}) = U. Finally, we return to obtaining G({alpha}). Since E[P] = 3 and E[S] + E[U] = 15, line 5 yields G({alpha}) by G({alpha}) = Q = S {cup} U.

Bootstrap Test
On the basis of bootstrap resampling (Efron and Tishirani 1991Citation ), we present a test of classifications. The test consists of two procedures: (1) reconstructing the phylogenetic tree by using the resampled sequences, and (2) classifying the genes into groups. In order to calculate the bootstrap probability of a certain group (Pg), we repeat these procedures many times (say, 1,000 times) and count the number of cases in which the group is selected in the classification with the largest E.


    Example
 TOP
 Abstract
 Introduction
 Theory
 Example
 Discussion
 Acknowledgements
 literature cited
 
As an example, the classification of opsin genes was obtained by using this method. The sources of the DNA sequences are given in table 1 . There were 31 opsin genes from different vertebrates. The root of vertebrate opsin was their last common ancestor, which was determined by using opsin from Drosophila melanogaster as an outgroup.


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Table 1 Accession Numbers of Opsin Genes

 
The absorption color of each opsin is also indicated. Multiple alignments were conducted with CLUSTAL W (Thompson, Higgins, and Gibson 1994Citation ) and corrected by eye. Gapped sites were excluded. The alignment is available from the authors on request.

The evolutionary distances were estimated based on amino acid sequence comparison using Poisson model (Zuckerkandl and Pauling 1965Citation ). The phylogenetic tree was reconstructed using neighbor-joining method (Saitou and Nei 1987Citation ), and the classification was obtained using our method. Bootstrap tests were conducted for phylogenetic relationship (P) and for groups (Pg). Bootstrap resampling was repeated 1,000 times for each test.

Figure 3 shows the phylogenetic tree of opsin genes and their classification. The asterisks indicate the branches which are significantly supported (P > 95%). The thick lines correspond to the ancestral branches of the groups. The numbers above the branches are Pg > 10% of the groups, which correspond to the branches.



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Fig. 3.—Phylogenetic relationship and classification of opsin. R, G, B, V, UV, and RH indicate red opsin, green opsin, blue opsin, violet opsin, ultraviolet opsin, and rhodopsin, respectively. L, S, M1, and M2 are groups. Bold lines represent the ancestral branches of these groups. Highly supported clusters (P > 95%) are indicated by asterisks. The numbers above branches are Pg (%) > 10% of the groups, which correspond to the branches

 
The opsin classification has four groups. The groups obtained here are the same as those obtained by Okano et al. (1992)Citation . The groups L, S, and M1/M2 correspond to long, short, and two kinds of middle-wavelength pigment groups, respectively (Okano et al. 1992Citation ). Group L is highly supported, groups S and M2 are supported with Pg > 80%, and group M1 is supported with a low bootstrap value (Pg < 50%).


    Discussion
 TOP
 Abstract
 Introduction
 Theory
 Example
 Discussion
 Acknowledgements
 literature cited
 
In this paper, a simple method for classifying genes was developed based on Klastorin's (1982)Citation method. Our method classifies genes into groups by using phylogenetic trees and is applicable as long as a phylogenetic tree is obtained. To obtain a reliable classification, however, the phylogenetic tree must be reconstructed as correctly as possible.

Our method classifies genes by using the branch lengths of phylogenetic trees. On the other hand, Dayhoff (1978)Citation classified genes based on the similarity between two genes. Note that the similarity between two genes is the proportion of shared characters, which are categorized into two types: (1) the ones which were inherited from their ancestor and (2) the ones which arose through parallel substitutions. Since parallel substitutions would make the classification less informative (Farris 1979Citation ), our method would be more suitable for gene classification than would Dayhoff's (1978)Citation method.

Although our method classifies genes by relative branch lengths of their phylogenetic tree, there might be some methods which classify genes using absolute branch lengths. Such methods, however, would depend on the genes to be classified, since the branch lengths depend on the evolutionary rates of genes. For example, fast-evolving genes would be classified into a large number of small groups, and slow-evolving genes would be classified into a small number of large groups. Therefore, the classification methods based on relative branch lengths might be more appropriate than those based on absolute branch lengths.

We presented a simple algorithm for obtaining the classification. It is worth noting that the number of possible classifications can be very large. For example, the number of possible classifications for 10 genes is 115,975, and that for 20 genes is larger than 5 x 1013. If hundreds of genes must be classified, an exhaustive search is almost impossible. On the other hand, the time our algorithm takes is proportional to the number of OTUs. Thus, our algorithm might have a tremendous advantage in the classification of a large number of genes.

The opsin classification obtained here was based on the phylogenetic tree which was reconstructed through sequence comparison. Although gene functions were not used, the classification obtained here is the same as the previous classification based on light absorption wavelengths. This agreement comes from the fact that the changes in absorption wavelength are caused by the amino acid changes (see Yokoyama 1997Citation and references therein). This result suggests that classifications can identify the functions of genes in some cases.

We developed a test for classification on the basis of the bootstrap method, because it requires less statistical assumptions (Efron and Tishirani 1991Citation ). The bootstrap method is widely used in statistical analyses. Felsenstein (1985)Citation developed a bootstrap test for topology estimation. Dopazo (1994)Citation obtained branch length errors by using bootstrap resampling. Our test is a compromise of these methods and is suitable for testing the reliability of classifications, because classification is affected by both tree topologies and branch lengths.

It is worth noting that the reliability of a classification depends on the sampling. From figure 3 , we can see that group M1 is not supported with a high bootstrap value. This is probably because the branch leading to M92037 is long. In group M1, only M92037 is sampled from tetrapods. If orthologous genes of M92037 are found in other tetrapods, this branch will be divided into shorter branches, and the group M1 will be supported more strongly. This suggests that the reliability of a classification depends on the sampling. To obtain the classification using our method, it might be better to sample genes from various organisms.


    Acknowledgements
 TOP
 Abstract
 Introduction
 Theory
 Example
 Discussion
 Acknowledgements
 literature cited
 
We thank three anonymous reviewers for their valuable suggestions and comments.


    Footnotes
 
Masami Hasegawa, Reviewing Editor

1 Keywords: molecular classification molecular phylogeny bootstrap test opsin classification Back

2 Address for correspondence and reprints: Fumio Tajima, Department of Biological Sciences, Graduate School of Science, The University of Tokyo, Hongo, Bunkyo-Ku, Tokyo 113-0033, Japan. E-mail: ftajima{at}biol.s.u-tokyo.ac.jp Back


    literature cited
 TOP
 Abstract
 Introduction
 Theory
 Example
 Discussion
 Acknowledgements
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Accepted for publication August 15, 2000.





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