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 \(c\).
\(|P_{f(6NJD_1)}(2) \setminus P_{f(2YCF_1)}(2)|=26\),
\(|P_{f(2YCF_1)}(2) \setminus P_{f(6NJD_1)}(2)|=148\).
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 1:1111010111001010010101010001001010100001111000011111001001001000010000010111010001011100101001010101000100101010000111100001111100100100100001000001011101011
Pair
\(Z_2\)
Length of longest common subsequence
6NJD_1,2YCF_1
174
3
6NJD_1,6UVX_1
176
3
2YCF_1,6UVX_1
168
3
Newick tree
[
6NJD_1:88.63,
[
2YCF_1:84,6UVX_1:84
]:4.63
]
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 \((0.85)(0.8)(\frac{479
}{\log_{20}
479}-\frac{157}{\log_{20}157})=94.8\)
Status
Protein1
Protein2
d
d1/2
Query variables
6NJD_1
2YCF_1
125
78
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]