The Splitting Operator on Edges

As we have seen, the splitting operator is a special function that uses the Kronecker product to increase the dimension of the phylogenetic tensor space in a way that represents the splitting of one ancestral edge into two descendant edges. Recall that the splitting operator acts directly on an individual unit vector as follows,

δ_·ei=ei⊗ei, (3.29)

where·is the matrix product and⊗is the Kronecker product.

The unit column vector, ei, corresponds to the state, i, in an individual character sequence representing a single taxon. It follows that ei⊗ei represents the combination of states, ii, at a given site in two character sequences representing two taxa. Immediately after a speciation

event, when no time has elapsed, the two resultant character sequences must be identical. Hence, the splitting operator follows our intuition on speciation.

Let’s now look at a basic two-taxon clock-like tree with the splitting operator acting on the tree, shown in Figure 3.2 below.

π δ

Figure 3.2: Two-taxon clock-like tree with the splitting operator, δ, and the initial probability distribution,π.

The tree starts at the top, with the initial probability distribution, π, at the root. At the root there is only a single taxon present. The initial probability distribution is the phylogenetic tensor for the single root taxon when no time has elapsed. The number of elements in the initial probability distribution will be equal to the number of states in the state space.

We will choose the initial probability distribution to be

π= " p0(0) p1(0) # = " ₁ 2 1 2 # , (3.30)

the stationary distribution. The stationary distribution is invariant under the action of any transition matrix from the chosen model on a single edge. In other words, ifπis the stationary distribution then eQt_{·π=π for any time,t, and for anyQfrom the binary symmetric model.} The stationary distribution is the equilibrium distribution, which cannot be left once reached. Consequently, the stationary distribution represents the long run probability distribution for a single taxon. For an n-taxon phylogenetic tensor, the initial probability distribution can be recovered from the marginal probability distributions for each taxon.

The vertical line down from the root is the edge representing the evolution of the single taxon over time from the root. Eventually this single taxon splits into two new descendant taxa. The dotted line represents the stage in time where the splitting operator is used to represent the splitting of the single ancestral taxon into two descendant taxa. Over time the two new taxa diverge from each other, represented by two edges which move further apart through time.

After placing the splitting operator wherever a splitting event occurs on a tree, we can push each splitting operator through the tree. After pushing the splitting operators through edges towards the root, it is equivalent to think of a single edge as a collection of edges being forced to remain identical, with the number of edges equal to the number of descendant taxa. When one splitting operator is pushed back, two edges become identical. The number of identical edges is equal to one plus the number of splitting operators below it which have been pushed back above the edge. In Figure 3.3 below is an example of the action the splitting operator performs on a single taxon, transforming it into a two-taxon tree.

π M Mδ′ _M′′ pushing throughδ −−−−−−−−−−−→ δ_·π M M′ _M′′

Figure 3.3: Action of the splitting operator on a single taxon, transforming it into a two-taxon tree. M, M′_{, M}′′ _{and M refer to the Markov matrices on those edges. π is the probability}

distribution at the root.

Algebraically, the above scenario is given by (M′_⊗M′′_{)·(δ·M)·π→ (M}′_⊗M′′_{)· M ·δ·π,}

where the edge labels are the transition matrices for the edges in question.

The consequence is that the splitting operator now acts further up on the tree or network, in this case at the root. If the root is further up the tree or network, we can repeat this process as many times as necessary until the splitting operator acts directly on the root. Likewise, we can perform this process on every splitting operator on the tree or network. Every splitting operator would then act directly on the root.

Mplays the role of implementing correlated changes. By pushing back the splitting operator, we now have a rate matrix which models multiple identical edges being forced to remain identical. An intriguing question that arises from this result is “what happens when this rate matrix is applied to multiple adjacent non-identical edges?”. Sumner et al. [2012c] discussed the issue of whether the rate matrices arising from pushing back the splitting operator can be used to model convergence on parts of a network which are isolated from splitting processes.