Reconstructing evolutionary trees by maximum parsimony....

Reconstructing evolutionary trees by maximum parsimony. Suppose we manage to sequence a particular gene across a whole bunch of different species. For concreteness, say there are $n$ species, and the sequences are strings of length $k$ over alphabet $\Sigma=\{A, C, G, T\} .$ How can we use this information to reconstruct the evolutionary history of these species? Evolutionary history is commonly represented by a tree whose leaves are the different species, whose root is their common ancestor, and whose internal branches represent speciation events (that is, moments when a new species broke off from an existing one). Thus we need to find the following: $\bullet$An evolutionary tree with the given species at the leaves. $\bullet$For each internal node, a string of length $k:$ the gene sequence for that particular ancestor. For each possible tree $T,$ annotated with sequences $s(u) \in \Sigma^{k}$ at each of its nodes $u,$ we can assign a score based on the principle of parsimony: fewer mutations are more likely. \[ \operatorname{score}(T)=\sum_{(u, v) \in E(T)}(\text { number of positions on which } s(u) \text { and } s(v)\text { disagree) } \] Finding the highest-score tree is a difficult problem. Here we will consider just a small part of it: suppose we know the structure of the tree, and we want to fill in the sequences $s(u)$ of the internal nodes $u .$ Here's an example with $k=4$ and $n=5:$ (a) In this particular example, there are several maximum parsimony reconstructions of the internal node sequences. Find one of them. (b) Give an efficient (in terms of $n$ and $k$ ) algorithm for this task. (Hint: Even though the sequences might be long, you can do just one position at a time.)

Reconstructing evolutionary trees by maximum parsimony. Suppose we manage to sequence a particular gene across a whole bunch of different species. For concreteness, say there are $n$ species, and the sequences are strings of length $k$ over alphabet $\Sigma=\{A, C, G, T\} .$ How can we use this information to reconstruct the evolutionary history of these species? Evolutionary history is commonly represented by a tree whose leaves are the different species, whose root is their common ancestor, and whose internal branches represent speciation events (that is, moments when a new species broke off from an existing one). Thus we need to find the following:
$\bullet$An evolutionary tree with the given species at the leaves.
$\bullet$For each internal node, a string of length $k:$ the gene sequence for that particular ancestor.
For each possible tree $T,$ annotated with sequences $s(u) \in \Sigma^{k}$ at each of its nodes $u,$ we can assign a score based on the principle of parsimony: fewer mutations are more likely.
\[
\operatorname{score}(T)=\sum_{(u, v) \in E(T)}(\text { number of positions on which } s(u) \text { and } s(v)\text { disagree) }
\]
Finding the highest-score tree is a difficult problem. Here we will consider just a small part of it: suppose we know the structure of the tree, and we want to fill in the sequences $s(u)$ of the internal nodes $u .$ Here's an example with $k=4$ and $n=5:$
(a) In this particular example, there are several maximum parsimony reconstructions of the internal node sequences. Find one of them.
(b) Give an efficient (in terms of $n$ and $k$ ) algorithm for this task. (Hint: Even though the sequences might be long, you can do just one position at a time.)

Key Concepts

Fitch Algorithm

The Fitch algorithm is a dynamic programming technique used to solve the maximum parsimony problem for a single character. It involves a postorder pass through the tree where, for each node, one constructs a set of possible states that could minimize the number of changes along the branches. The final state assignment is then obtained by making choices that are consistent with these sets, leading to an overall minimal mutation score.

Dynamic Programming on Trees

Dynamic programming is a method used to solve problems by breaking them down into subproblems and solving each of these only once. When applied to phylogenetic trees, it allows the problem to be solved recursively by considering each node independently for each position in the gene sequence. This reduces the computation time by exploiting the independence of sites and the tree structure.

Maximum Parsimony

Maximum parsimony is a criterion in evolutionary biology used to infer phylogenies by choosing the tree and ancestral character states that require the smallest number of evolutionary changes. It is based on the principle of Occam's razor, positing that the simplest explanation, with the fewest assumptions or changes, is preferred. In reconstructing evolutionary histories, the method scores a tree by summing the differences between states at connected nodes and seeks to minimize this score.

Phylogenetic Trees

Phylogenetic trees are graphical representations of the evolutionary history and relationships among species. In these trees, the leaves represent current species, internal nodes represent common ancestors, and branches correspond to evolutionary relationships. Understanding the structure of these trees is crucial for mapping genetic data onto historical evolutionary events.

Ancestral State Reconstruction

Ancestral state reconstruction refers to the process of inferring the characteristics (such as gene sequences) of ancestral species from the observed traits of their descendants. In the context of maximum parsimony, the goal is to assign states to the internal nodes of a given phylogenetic tree in such a way that the total number of state changes across the tree is minimized.

Transcript

00:01 For this question, the scenario described with the stack of papers can basically be summarized as follows.

00:07 So we have some sequence, a, b, c, any permutation of these three letters.

00:16 And then for each transition, one of the letters gets selected and moved to the left.

00:24 So for example, b may be selected with probability b, in which case it's moved to the left.

00:34 So it transitions to b, a, c.

00:43 And x sub n represents the sequence after the nth transition.

00:51 So, for example, if this happened to be the third transition, then this sequence here would be x sub 3.

01:00 So for part a, we're told that the initial state is abc, and we're asked to determine the state space for this chain.

01:19 So if all letters of a, b, and c have non -zero probabilities of being selected to be the one moved to the left, then the state space is every permutation that's possible.

01:33 So we have a, b, c, a, c, b, a, c, b, a, c.

02:06 So that's part a.

02:07 And then for part b, we're told, again, if this is the initial state, what are the possibilities for the first state? the first transition.

02:22 So there's only three things that can happen here.

02:25 A will be selected, or b will be selected, or c will be selected.

02:36 So if a is selected, it remains on the left, and we end up with the same state.

02:43 If b is selected, it gets shifted to the left, and we end up with b, a, c.

02:49 And if c gets selected to the far left, and we end up with c, a, b.

02:57 These are the only three possible states for x, sub 1.

03:04 And the question also mentions is every arrangement listed in a possible.

03:09 So obviously not...