NEFF measures sequence diversity within MSAs, which are crucial for extracting correlated mutation information and essential for biological investigations like contact map and structure prediction. NEFF has shown a strong correlation with prediction accuracy in models such as AlphaFold.

For an MSA, NEFF can be formulated as:

\[ \left( \frac{1}{\sqrt{L}} \right) \sum_{n=1}^{N} \frac{1}{1 + \sum_{m=1, m \neq n}^{N} I[S_{m,n} \geq thr]} \]

where \(L\) is the number of residues in the sequence, \(N\) is the number of sequences in an MSA, \(S_{m,n}\) is the sequence identity between \(m\)-th and \(n\)-th sequences, \(thr\) is the threshold cutoff to determine whether two sequences are similar or not, and \(I\) is the inversion bracket, meaning that \(I[S_{m,n} \geq thr]\) equals 1 if \(S_{m,n} \geq thr\) and 0 otherwise.

Note that \(\frac{1}{\sqrt{L}}\) is used as a normalization factor here.

Generally, one can see NEFF simply as a normalized summation of sequence weights for all sequences in an MSA. If the number of sequences (including itself) similar to sequence \(i\) is \(n_i\), then its sequence weight is \(\frac{1}{n_i}\). This approach for calculating NEFF has been widely used in various contact and structure prediction tools, as demonstrated in references [1-6].

References

1. Morcos, F., et al. "Direct-coupling analysis of residue coevolution captures native contacts across many protein families." Proceedings of the National Academy of Sciences 108.49 (2011): E1293-E1301.
2. Simkovic, F., et al. "ConKit: a Python interface to contact predictions." Bioinformatics 33.14 (2017): 2209-2211.
3. Wu, Q., et al. "Analysis of several key factors affecting DCA-based contact prediction in metagenome coevolution." Bioinformatics 35.14 (2019): 2497-2503.
4. Zhang, J., et al. "DeepMSA: constructing deep multiple sequence alignment to improve contact prediction and fold-recognition for distant-homology proteins." Bioinformatics 36.5 (2020).
5. Liu, Y., et al. "Protein contact prediction using metagenome sequence data improves fold recognition." Bioinformatics 37.12 (2021): 1770-1776.
6. Li, Y., et al. "TripletRes: fragment-free protein structure prediction using triplet transformers." Bioinformatics 37.22-23 (2021): 4101-4107.

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