Limits...
Statistical properties of pairwise distances between leaves on a random Yule tree.

Sheinman M, Massip F, Arndt PF - PLoS ONE (2015)

Bottom Line: A Yule tree is the result of a branching process with constant birth and death rates.To make our results more useful for realistic scenarios, we explicitly take into account that the leaves of a tree may be incompletely sampled and derive a criterion for poorly sampled phylogenies.We show that our result can account for empirical data, using two families of birds species.

View Article: PubMed Central - PubMed

Affiliation: Max Planck Institute for Molecular Genetics, Berlin, Germany.

ABSTRACT
A Yule tree is the result of a branching process with constant birth and death rates. Such a process serves as an instructive model of many empirical systems, for instance, the evolution of species leading to a phylogenetic tree. However, often in phylogeny the only available information is the pairwise distances between a small fraction of extant species representing the leaves of the tree. In this article we study statistical properties of the pairwise distances in a Yule tree. Using a method based on a recursion, we derive an exact, analytic and compact formula for the expected number of pairs separated by a certain time distance. This number turns out to follow a increasing exponential function. This property of a Yule tree can serve as a simple test for empirical data to be well described by a Yule process. We further use this recursive method to calculate the expected number of the n-most closely related pairs of leaves and the number of cherries separated by a certain time distance. To make our results more useful for realistic scenarios, we explicitly take into account that the leaves of a tree may be incompletely sampled and derive a criterion for poorly sampled phylogenies. We show that our result can account for empirical data, using two families of birds species.

No MeSH data available.


An example of the rooted Yule tree of age T. Filled circles (1, 3, 5, 7 and 8) denote observed leaves.Empty circles (2, 4 and 6) denote survived but not observed leaves. Short horizontal lines denotes an extinction event. Long, dashed horizontal lines denote the origin of the tree, the first branching event and the time of sampling the tree, from top to bottom. After the first branching at time T1 the two resulting subtrees both encompass M1 = M2 = 4 leaves. However, the number of observed leaves is 2 (leaves 1 and 3) for the left subtree and 3 (leaves 5, 7 and 8) for the right one. The thick green line denotes the pairwise evolutionary distance between the two observed leaves 5 and 7. The horizontal dimension is meaningless. In this example for leaf 1 the first closest observed leaf is 3, the second (as well as the third and the fourth) is 5 (or 7 or 8). The tree has two observed cherry pairs: (1, 3) and (7, 8).
© Copyright Policy
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC4380457&req=5

pone.0120206.g002: An example of the rooted Yule tree of age T. Filled circles (1, 3, 5, 7 and 8) denote observed leaves.Empty circles (2, 4 and 6) denote survived but not observed leaves. Short horizontal lines denotes an extinction event. Long, dashed horizontal lines denote the origin of the tree, the first branching event and the time of sampling the tree, from top to bottom. After the first branching at time T1 the two resulting subtrees both encompass M1 = M2 = 4 leaves. However, the number of observed leaves is 2 (leaves 1 and 3) for the left subtree and 3 (leaves 5, 7 and 8) for the right one. The thick green line denotes the pairwise evolutionary distance between the two observed leaves 5 and 7. The horizontal dimension is meaningless. In this example for leaf 1 the first closest observed leaf is 3, the second (as well as the third and the fourth) is 5 (or 7 or 8). The tree has two observed cherry pairs: (1, 3) and (7, 8).

Mentions: A Yule tree is defined as follows [1, 2]. At time t = 0 there is one individual. As time progresses, this individual can branch and give birth to another individual. In an infinitesimally short time interval [t, t+dt], all individuals can give birth to another one, each with the probability λdt. The probability of an individual to die in the same time interval is μdt. We consider an ensemble of trees of age (height) T, referring to all existing individuals at this time as leaves. To make the model more realistic, we assume that due to incomplete sampling (or a short massive extinction event) just before the time T, each leaf is observed with a certain probability 0 ≤ σ ≤ 1. The described process is illustrated in Fig. 2. We assume that the incompleteness of the sampling is random and ignore possible biases due to different sampling schemes [24].


Statistical properties of pairwise distances between leaves on a random Yule tree.

Sheinman M, Massip F, Arndt PF - PLoS ONE (2015)

An example of the rooted Yule tree of age T. Filled circles (1, 3, 5, 7 and 8) denote observed leaves.Empty circles (2, 4 and 6) denote survived but not observed leaves. Short horizontal lines denotes an extinction event. Long, dashed horizontal lines denote the origin of the tree, the first branching event and the time of sampling the tree, from top to bottom. After the first branching at time T1 the two resulting subtrees both encompass M1 = M2 = 4 leaves. However, the number of observed leaves is 2 (leaves 1 and 3) for the left subtree and 3 (leaves 5, 7 and 8) for the right one. The thick green line denotes the pairwise evolutionary distance between the two observed leaves 5 and 7. The horizontal dimension is meaningless. In this example for leaf 1 the first closest observed leaf is 3, the second (as well as the third and the fourth) is 5 (or 7 or 8). The tree has two observed cherry pairs: (1, 3) and (7, 8).
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4380457&req=5

pone.0120206.g002: An example of the rooted Yule tree of age T. Filled circles (1, 3, 5, 7 and 8) denote observed leaves.Empty circles (2, 4 and 6) denote survived but not observed leaves. Short horizontal lines denotes an extinction event. Long, dashed horizontal lines denote the origin of the tree, the first branching event and the time of sampling the tree, from top to bottom. After the first branching at time T1 the two resulting subtrees both encompass M1 = M2 = 4 leaves. However, the number of observed leaves is 2 (leaves 1 and 3) for the left subtree and 3 (leaves 5, 7 and 8) for the right one. The thick green line denotes the pairwise evolutionary distance between the two observed leaves 5 and 7. The horizontal dimension is meaningless. In this example for leaf 1 the first closest observed leaf is 3, the second (as well as the third and the fourth) is 5 (or 7 or 8). The tree has two observed cherry pairs: (1, 3) and (7, 8).
Mentions: A Yule tree is defined as follows [1, 2]. At time t = 0 there is one individual. As time progresses, this individual can branch and give birth to another individual. In an infinitesimally short time interval [t, t+dt], all individuals can give birth to another one, each with the probability λdt. The probability of an individual to die in the same time interval is μdt. We consider an ensemble of trees of age (height) T, referring to all existing individuals at this time as leaves. To make the model more realistic, we assume that due to incomplete sampling (or a short massive extinction event) just before the time T, each leaf is observed with a certain probability 0 ≤ σ ≤ 1. The described process is illustrated in Fig. 2. We assume that the incompleteness of the sampling is random and ignore possible biases due to different sampling schemes [24].

Bottom Line: A Yule tree is the result of a branching process with constant birth and death rates.To make our results more useful for realistic scenarios, we explicitly take into account that the leaves of a tree may be incompletely sampled and derive a criterion for poorly sampled phylogenies.We show that our result can account for empirical data, using two families of birds species.

View Article: PubMed Central - PubMed

Affiliation: Max Planck Institute for Molecular Genetics, Berlin, Germany.

ABSTRACT
A Yule tree is the result of a branching process with constant birth and death rates. Such a process serves as an instructive model of many empirical systems, for instance, the evolution of species leading to a phylogenetic tree. However, often in phylogeny the only available information is the pairwise distances between a small fraction of extant species representing the leaves of the tree. In this article we study statistical properties of the pairwise distances in a Yule tree. Using a method based on a recursion, we derive an exact, analytic and compact formula for the expected number of pairs separated by a certain time distance. This number turns out to follow a increasing exponential function. This property of a Yule tree can serve as a simple test for empirical data to be well described by a Yule process. We further use this recursive method to calculate the expected number of the n-most closely related pairs of leaves and the number of cherries separated by a certain time distance. To make our results more useful for realistic scenarios, we explicitly take into account that the leaves of a tree may be incompletely sampled and derive a criterion for poorly sampled phylogenies. We show that our result can account for empirical data, using two families of birds species.

No MeSH data available.