Below, we first introduce the relevant models, and then describe the three tools that implement them.
The birth-death process
The birth-death process is a well-characterized mathematical model [16], used in biology in general, and in evolution in particular [17, 18]. In the context of host tree evolution, the birth-death process over a pre-defined time interval can be described by the following properties; see Figure 1a: (i) A single lineage starts evolving at the start of the interval. (ii) At any time, a lineage evolves independently of all other lineages. (iii) For a lineage, births occur at rate λ and deaths occur at rate μ. A birth creates two new independent child lineages replacing the parental lineage. A death stops the evolution of the lineage. (iv) When a lineage reaches the end of the interval, its evolution stops. The result can be depicted as a bifurcating tree, where edges represent lineages and vertices represent births and lineage ends. Usually, lineages not reaching the end of the interval are pruned away so that leaves of the resulting host tree comprise only extant lineages.
A special instance of the birth-death process is the pure birth process (or Yule process), with μ=0. The birth-death process has been used to model speciations and extinctions in species evolution; see, e.g., [18]. An extension where each leaf of the host tree, when reaching the end of the time interval, has a fixed probability of being removed from the tree has been proposed [18]. This could be used to model, for instance, species sampling or mass extinction events.
The duplication-loss model
The birth-death process has also been used to model gene evolution, an idea originating from Nei et al. [19]. The duplication-loss (DL) model [6, 8] is a canonical generalization of the birth-death process onto a bifurcating host tree, S, instead of a single time interval. The tree S is dated so that vertices have divergence times (and edges thus time intervals). The root of S has an additional incoming “stem edge”, representing how long before the root the DL process started. The process can be described by the following properties: (i) The process starts with a single guest lineage at the start of the stem edge. (ii) Over any edge e in S, guest lineages evolve independently according to a birth-death process with duplication rate λ and loss rate μ. If a duplication occurs, the guest lineage splits into two child lineages that continue to evolve in e. If a loss occurs, the lineage terminates. (iii) When a guest lineage reaches an interior vertex x in S (i.e., a speciation in a species tree), the lineage splits into two lineages: one evolving in the first child edge of x, and the other evolving in the second child edge of x. (iv) The process continues recursively to the leaves of S where the process stops. The result is a bifurcating guest tree embedded in S. Lineages that failed to reach the leaves of the S are usually pruned away so as to produce a pruned guest tree.
The DL model was developed to model gene family evolution, but may be used also to model other tree-in-tree evolutionary processes, such as allopatric and sympatric species evolution in area trees [7]. In the current model, λ and μ are homogenous over the host tree. Support for, for instance, varying λ and μ over host tree edges, may be included in the future.
The duplication-loss-transfer model
The duplication-loss-transfer (DLT) model [10] is an extension of the DL model to include lateral gene transfer (LGT; also known as horizontal gene transfer, HGT), an important evolutionary process in, for example, bacteria. The DLT model is analogous to the DL model, save for the addition of LGT events, see Figure 1b. More specifically, over any edge in the host tree, guest lineages are—apart from duplications and losses—also exposed to LGT events at rate τ, in the same vein as rates λ and μ. If a guest lineage has an LGT event in host edge e, the lineage will split into two child lineages: one copy will continue evolving in e, whereas the other lineage is instantaneously transferred to a uniformly selected contemporaneous host edge f ≠ e. The transferred lineage continues its evolution in f.
Again, while originally developed for gene evolution [10], the DLT model may be used for other tree-in-tree processes, such as modeling biogeography or parasite co-evolution with host-switches. In the current model, τ is homogenous over the host tree. It is possible that future versions will refine on this, for instance to allow increased transfer rates between more closely related host edges.
Relaxed clock models
The notion of a relaxed evolutionary clock for substitutions dates back to Gillespie [20], who, after showing the inadequacy of the molecular clock hypothesis, discussed variants of relaxing it. The essential idea is to multiply the clock-like edge times of a tree with different edge rates provided by a model to achieve relaxed edge lengths (Figure 1c). Two main variants of relaxed clock models exist.
In the first variant, uncorrelated models[11, 21-23], substitution rates on different edges of the tree are modeled typically as independent and identically distributed (IID) variables. Several underlying distributions have been suggested and evaluated, including the log-normal, gamma and exponential distributions.
In the second variant, autocorrelated models [22-26], a gradual rate evolution of the tree is modeled from the root to the leaves, so that rates are autocorrelated over paths in the tree. Different flavors exist, such as using log-normal, Ormstein-Uhlenbeck or CIR distributions for the process.
A different approach to uncorrelated clock relaxation specific to gene trees was taken by Rasmussen and Kellis [27]. This model assumes that the gene tree is reconciled with a species tree. Here the species tree has an individual gamma distribution associated with each species edge, yielding a “species-specific” factor. A gene lineage will derive its rate from the species edges it passes over, multiplied by a “gene family-specific” factor to account for rate heterogeneity across gene families.
We now present the tools that correspond to the models introduced above.
The HostTreeGen tool
HostTreeGen generates a bifurcating host tree over a specified time interval using a birth-death process; see Figure 1a. The user provides the parameters (λ, μ) and the time interval for the model. A uniform sampling probability for each leaf of the resulting pruned tree can be applied. It is also possible to incur limits for the minimum/maximum number of leaves in the tree, enforcing an immediate branching at start, sampling of desired pruned tree sizes from a specified list, etc. The standard output consists of both the unpruned and pruned version of the tree (in Newick format), and files with auxiliary information on node counts, time span, etc.
The GuestTreeGen tool
GuestTreeGen generates a dated guest tree over a dated, clock-like host tree using either the DL model or the DLT model (see Figure 1b). The user provides the host tree and parameters (λ, μ, τ) for the model. Similar to above, a guest leaf sampling probability can be incurred, along with various options such as minimum/maximum number of guest leaves in total or per host leaf, and sampling the desired size of the pruned guest tree from a list. The output is a reconciled guest tree (in Newick format), both in its unpruned and pruned form, along with a file detailing its reconciliation with the host tree. Auxiliary information is also produced, such as counts of the different event types, and maximum likelihood estimates of rates based on the unpruned tree and the number of events, computed in accordance with [28].
The BranchRelaxer tool
BranchRelaxer applies a relaxed clock model to a tree, typically a guest tree; see Figure 1c. It generates rates for the edges of the tree, and outputs the tree with relaxed edge lengths. Several clock models are implemented: (i) A molecular clock. (ii) Uncorrelated IID models with the following underlying distributions: expontential, gamma, log-normal, normal, and uniform rates [11, 21-23]. (iii) Autocorrelated models include an exponential model [26], two variants of a log-normal model [23, 24], and the CIR process [22]. (iv) The host tree-guided model of [27], which as input requires the guest tree to be relaxed, the host tree with assigned edge rate distributions, and a reconciliation of the two trees. Currently, this model does not support LGT events. (v) An empirical model that samples rates uniformly with replacement from a predefined list derived from, for instance, an MCMC analysis.
For all models, it is possible to require edge rates to be within a specified range.
