Definitions

# Neighbor-joining

In bioinformatics, neighbor-joining is a bottom-up clustering method used for the construction of phylogenetic trees. Usually used for trees based on DNA or protein sequence data, the algorithm requires knowledge of the distance between each pair of taxa (e.g. species or sequences) in the tree.

## The algorithm

Neighbor-joining is an iterative algorithm. Each iteration consists of the following steps:

1. Based on the current distance matrix calculate the matrix Q (explained below).
2. Find the pair of taxa in Q with the lowest value. Create a node on the tree that joins these two taxa (i.e. join the closest neighbors, as the algorithm name implies).
3. Calculate the distance of each of the taxa in the pair to this new node.
4. Calculate the distance of all taxa outside of this pair to the new node.
5. Start the algorithm again, considering the pair of joined neighbors as a single taxon and using the distances calculated in the previous step.

### The Q-matrix

Based on a distance matrix relating r taxa, calculate Q as follows:

$Q\left(i,j\right)=\left(r-2\right)d\left(i,j\right)-sum_\left\{k=1\right\}^r d\left(i,k\right) - sum_\left\{k=1\right\}^r d\left(j,k\right)$

d(i,j) is the distance between taxa i and j.

For example, if we have four taxa (A, B, C, D) and the following distance matrix:

A B C D
A
B 7
C 11 6
D 14 9 7

We obtain the following values for the Q matrix:

A B C D
A
B −40
C −34 −34
D −34 −34 −40

In the example above, two pairs of taxa have the lowest value, namely −40. We can select either of them for the second step of the algorithm. We follow the example assuming that we joined taxa A and B together.

### Distance of the pair members to the new node

For each neighbor in the pair just joined, use the following formula to calculate to the new node (f and g are the paired taxa and u is the newly generated node):

$d\left(f,u\right)=frac\left\{1\right\}\left\{2\right\}d\left(f,g\right)+frac\left\{1\right\}\left\{2\left(r-2\right)\right\} left \left[sum_\left\{k=1\right\}^r d\left(f,k\right) - sum_\left\{k=1\right\}^r d\left(g,k\right) right \right] quad$

In the example above, this formula would give a distance of 6 between A and the new node. It would also give and a distance of 1 between B and the new node.

### Distance of the other taxa to the new node

For each taxon not considered in the previous step, we calculate the distance to the new node as follows:

$d\left(u,k\right)=frac\left\{1\right\}\left\{2\right\} \left[d\left(f,k\right)-d\left(f,u\right)\right] + frac\left\{1\right\}\left\{2\right\} \left[d\left(g,k\right)-d\left(g,u\right)\right]$

where u is the new node, k is the node for which we want to calculate the distance and f and g are the members of the pair just joined.

Following the example, the distance between C and the new node is 5. Also, the distance between the new node and D is 8.

### The next recursion step

From the steps above, the following matrix will result (AB acting as a new taxon):

AB C D
AB
C 5
D 8 7

We can start the procedure anew taking this matrix as the original distance matrix. In our example, it suffices to do one more step of the recursion to obtain the complete tree.

## Pros and cons of the NJ method

Neighbor-joining is based on the minimum-evolution criterion for phylogenetic trees, i.e. the topology that gives the least total branch length is preferred at each step of the algorithm. However, neighbor-joining may not find the true tree topology with least total branch length because it is a greedy algorithm that constructs the tree in a step-wise fashion. Even though it is sub-optimal in this sense, it has been extensively tested and usually finds a tree that is quite close to the optimal tree. Nevertheless, it has been largely superseded in phylogenetics by methods that do not rely on distance measures and offer superior accuracy under most conditions.

The main virtue of neighbor-joining relative to these other methods is its computational efficiency. That is, neighbor-joining is a polynomial-time algorithm. It can be used on very large data sets for which other means of phylogenetic analysis (e.g. minimum evolution, maximum parsimony, maximum likelihood) are computationally prohibitive. Unlike the UPGMA algorithm for phylogenetic tree reconstruction, neighbor-joining does not assume that all lineages evolve at the same rate (molecular clock hypothesis) and produces an unrooted tree. Rooted trees can be created by using an outgroup and the root can then effectively be placed on the point in the tree where the edge from the outgroup connects.

Furthermore, neighbor-joining is statistically consistent under many models of evolution. Hence, given data of sufficient length, neighbor-joining will reconstruct the true tree with high probability.