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 \(c\).
\(|P_{f(1ILG_1)}(2) \setminus P_{f(1QFY_1)}(2)|=92\),
\(|P_{f(1QFY_1)}(2) \setminus P_{f(1ILG_1)}(2)|=86\).
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 1:1001000000100001001111011000001110011010100100010010010111110010011001011000011010010001001010101010010110001110011001101110110100011011101101100100111000101101111010010100110100101001010001000111100111011101001100101000001110110110100111100011001000111010001000010110011110111110010010100000110100101110111001111010
Pair
\(Z_2\)
Length of longest common subsequence
1ILG_1,1QFY_1
178
4
1ILG_1,3ZSG_1
189
3
1QFY_1,3ZSG_1
187
5
Newick tree
[
3ZSG_1:95.61,
[
1ILG_1:89,1QFY_1:89
]:6.61
]
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 \((0.85)(0.8)(\frac{624
}{\log_{20}
624}-\frac{308}{\log_{20}308})=88.0\)
Status
Protein1
Protein2
d
d1/2
Query variables
1ILG_1
1QFY_1
108
108
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]