Nucleotide Substitution Models
Substitution along a phylogenetic branch is modelled as a continuous-time Markov chain on ${A, C, G, T}$. The GTR model specifies an instantaneous rate matrix $Q$ with off-diagonal entries $q_{ij} = \pi_j r_{ij}$, giving transition probabilities:
\[P(t) = e^{Qt}\]Simpler nested models (JC69, HKY85) fix some $r_{ij}$ to be equal, reducing parameter count at the cost of biological realism.
dN/dS (Ka/Ks) Ratio
The ratio of non-synonymous to synonymous substitution rates measures selective pressure on a coding gene:
| Ratio | Interpretation |
|---|---|
| $\omega < 1$ | Purifying (negative) selection |
| $\omega = 1$ | Neutral evolution |
| $\omega > 1$ | Positive (adaptive) selection |
$d_N$ and $d_S$ are estimated by counting synonymous and non-synonymous differences per site, corrected for multiple hits using the Jukes-Cantor formula.
Neutrality Tests
The McDonald-Kreitman test compares the ratio of fixed to polymorphic sites at synonymous ($S$) and non-synonymous ($N$) classes:
\[\text{Neutrality Index} = \frac{P_N / P_S}{D_N / D_S}\]Departure from $1$ detected via a $2\times2$ Fisher’s exact test signals non-neutral evolution. Under neutrality, the ratio should equal $1$.
Synteny and Whole-Genome Alignment
Synteny blocks — conserved gene order across species — are identified by chaining pairwise alignments. The scoring of a collinear chain of anchors $(x_i, y_i)$ uses a gap penalty $\gamma$:
\[\text{Score} = \sum_i s_i - \sum_k \gamma(\Delta x_k, \Delta y_k)\]where $s_i$ is the anchor alignment score and $\gamma$ penalizes inversions and transpositions.