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Brownian motion

Brownian motion over a phylogenetic tree is a key model used in evolutionary biology to describe how a continuous trait evolves over time along the branches of a tree representing the evolutionary relationships among species. It assumes that trait evolution follows a random walk process, governed by Brownian motion, where changes in the trait value occur in a way that is uncorrelated over time but with a variance that increases linearly with time.

1. Brownian Motion Basics

Brownian motion is a mathematical model used to describe random processes where the position of a particle or variable (here, a biological trait) evolves randomly over time. In this context:

  • The trait's change at any time is a normally distributed random variable.
  • The expected change over time is zero, but the variance of the trait increases with time.

Mathematically, for a trait value $X(t)$ at time $t$, starting with $X(0)$ at the root of the tree (the common ancestor), we have:

$$X(t) = X(0) + \epsilon(t)$$

where $\epsilon(t) \sim N(0, \sigma^2 t)$, meaning that the change in the trait value follows a normal distribution with mean 0 and variance proportional to $\sigma^2 t$, where $\sigma^2$ is the rate of Brownian motion.

2. Phylogenetic Tree and Time Evolution

In a phylogenetic tree:

  • The tree’s branches represent evolutionary time.
  • The tips of the tree correspond to species or taxa.
  • The nodes represent common ancestors of the species at the tips.

Brownian motion over a phylogenetic tree assumes that as lineages diverge from a common ancestor, their trait values evolve independently along different branches of the tree. The longer the time between divergences, the greater the expected variance in the trait values.

3. Brownian Motion on a Phylogenetic Tree

In the context of a phylogenetic tree, Brownian motion works as follows:

3.1 Root Value

At the root of the tree (the common ancestor of all species), a trait value is assumed to have an initial state, typically denoted as $X(0)$. This value could be treated as known or unknown, depending on the application.

3.2 Trait Evolution Along Branches

As evolution proceeds along the branches of the tree, the trait value for each lineage evolves according to Brownian motion. For a branch of length $t$, the trait value $X(t)$ at the end of the branch is modeled as:

$$X(t) = X(0) + \epsilon(t)$$

where $\epsilon(t) \sim N(0, \sigma^2 t)$ is a normally distributed random variable with variance proportional to $t$, the length of the branch. Here, $\sigma^2$ is the evolutionary rate parameter controlling how quickly the trait diverges over time.

3.3 Trait Evolution at the Tips

The species at the tips of the tree each inherit trait values that have evolved along their respective lineages. If two species split from a common ancestor at time $t_{split}$, the trait value for each species will have evolved independently from the trait value at that split point.

For example, if two species diverged from a common ancestor at time $t_{split}$, then their trait values at the tips $X_A$ and $X_B$ will be distributed as:

$$X_A = X_{split} + \epsilon_A$$ $$X_B = X_{split} + \epsilon_B$$

where $X_{split}$ is the trait value at the time of the split, and $\epsilon_A$ and $\epsilon_B$ are independent random changes following $N(0, \sigma^2 t_A)$ and $N(0, \sigma^2 t_B)$, where $t_A$ and $t_B$ are the times since the split for each lineage.

4. Covariance Between Species

A key feature of Brownian motion on a phylogenetic tree is that the covariance between trait values of species at the tips is determined by the shared evolutionary history. The closer two species are on the tree (i.e., the more recent their common ancestor), the more correlated their trait values will be.

The covariance between the trait values of two species $A$ and $B$ can be calculated as:

$$\text{Cov}(X_A, X_B) = \sigma^2 \cdot t_{shared}$$

where $t_{shared}$ is the amount of evolutionary time they share (i.e., the time along the path from the root to their most recent common ancestor).

For instance, if two species share a long common evolutionary history, their trait values will have a high covariance, reflecting that much of their trait evolution was shared before they diverged.

5. Mathematical Representation in a Tree

The evolution of a continuous trait over a phylogenetic tree is often represented using a variance-covariance matrix $V$, where each entry $V_{ij}$ corresponds to the covariance between the trait values of species $i$ and $j$. This matrix is built from the branch lengths of the tree and the shared evolutionary history of the species.

The covariance matrix $V$ for a phylogenetic tree can be constructed as follows:

  • For two species $i$ and $j$, the entry $V_{ij} = \sigma^2 \cdot t_{shared}$, where $t_{shared}$ is the shared evolutionary time between them.
  • For the variance of a species $i$, $V_{ii} = \sigma^2 \cdot t_i$, where $t_i$ is the total evolutionary time from the root to the species.

The covariance matrix $V$ captures the evolutionary relationships and how the trait covaries between species.

6. Applications of Brownian Motion Over Phylogenetic Trees

  • Trait Evolution Modeling: Brownian motion on a phylogenetic tree is widely used to model how traits such as body size, metabolic rate, or other continuous characteristics evolve across species. It provides a framework to study evolutionary trends, variation, and how traits diverge.

  • Phylogenetic Comparative Methods: These methods use the Brownian motion model to test hypotheses about trait evolution. For example, they can test whether a trait evolves under neutral drift or under more complex models like directional selection or stabilizing selection.

  • Estimation of Evolutionary Rates: The parameter $\sigma^2$, the rate of trait evolution, can be estimated from observed trait data at the tips of a phylogenetic tree. This helps quantify the rate at which traits evolve over time across species.

  • Correlation of Traits: By using the covariance matrix derived from Brownian motion, researchers can estimate correlations between traits across species, correcting for the shared evolutionary history.

7. Limitations of Brownian Motion on Phylogenetic Trees

  • Simplicity: The Brownian motion model assumes that trait evolution is a random process with constant variance over time. In reality, evolution is often more complex, with processes such as selection, adaptation, or constraints that may cause deviations from this model.

  • No Directionality: Brownian motion does not account for trends or biases in trait evolution. For example, if a trait is evolving under directional selection, a model with a trend (such as an Ornstein-Uhlenbeck process) may be more appropriate.

  • Branch Length Sensitivity: The model is sensitive to the branch lengths of the phylogenetic tree, which represent evolutionary time. Incorrect branch lengths can lead to biased estimates of evolutionary rates or trait correlations.

Conclusion

Brownian motion over a phylogenetic tree is a fundamental model in evolutionary biology that describes the random evolution of continuous traits across species. It assumes that changes in traits follow a random walk with variance increasing over time, and it uses the structure of the phylogenetic tree to account for shared evolutionary history. While simple, it forms the basis of many phylogenetic comparative methods and provides valuable insights into how traits evolve across related species.