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Phylogenetic Analysis

The study of evolutionary relationships among organisms based on their genetic similarities and differences, utilizing genomic data to construct phylogenetic trees.
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The statement of the theorem

Let S1,S2,…,SNS_1, S_2, \dots, S_N be the sequences of NN species, each of length LL. Define the pairwise distance d(Si,Sj)d(S_i, S_j) using a substitution model M\mathcal{M} (e.g., Jukes-Cantor or Kimura 2-parameter) based on the observed differences DijD_{ij} at each site ll: d(Si,Sj)=1L∑l=1L(1−12∑c∈{A,T,C,G}pc(l))d(S_i, S_j) = \frac{1}{L} \sum_{l=1}^{L} \left(1 - \frac{1}{2} \sum_{c \in \{A, T, C, G\}} p_{c}(l) \right) where pc(l)p_{c}(l) is the probability of character cc at site ll under model M\mathcal{M}. The phylogenetic tree T\mathcal{T} is then optimized by maximizing the likelihood function L(T∣S1,…,SN)=∏i=1NL(Si∣T)\mathcal{L}(\mathcal{T} | S_1, \dots, S_N) = \prod_{i=1}^{N} \mathcal{L}(S_i | \mathcal{T}), typically using methods like Maximum Likelihood or Neighbor-Joining.