Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\).
Let \(p_w(n)\) be the cardinality of \(P_w(n)\).
Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).
\(|P_{f(2FWW_1)}(2) \setminus P_{f(8GTR_1)}(2)|=79\),
\(|P_{f(8GTR_1)}(2) \setminus P_{f(2FWW_1)}(2)|=115\).
Let
\(
Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)|
\)
be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:11110011000111010101011011010110110101110110011101001111010100001000001111001110101001011101111010011010001001011110001111110110111010000011111110010111100010010001110010010110001101100000000100111110010101101111011010101001110001000101100011001
Pair
\(Z_2\)
Length of longest common subsequence
2FWW_1,8GTR_1
194
3
2FWW_1,8KBF_1
178
4
8GTR_1,8KBF_1
186
4
Newick tree
[
8GTR_1:96.94,
[
2FWW_1:89,8KBF_1:89
]:7.94
]
Let d be the
Otu--Sayood
distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{637
}{\log_{20}
637}-\frac{245}{\log_{20}245})=110.\)
Status
Protein1
Protein2
d
d1/2
Query variables
2FWW_1
8GTR_1
140
115
Was not able to put for d Was not able to put for d1
In notation analogous to
[Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]