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 \(c\).
\(|P_{f(7OZH_1)}(2) \setminus P_{f(8XRR_1)}(2)|=32\),
\(|P_{f(8XRR_1)}(2) \setminus P_{f(7OZH_1)}(2)|=153\).
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 1:10111011001001111110000011101110000111011000001110111011000000100111111011011100010111011110111000011000011100111001110100010011000100000101
Pair
\(Z_2\)
Length of longest common subsequence
7OZH_1,8XRR_1
185
3
7OZH_1,6SVI_1
134
3
8XRR_1,6SVI_1
185
4
Newick tree
[
8XRR_1:99.55,
[
7OZH_1:67,6SVI_1:67
]:32.55
]
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 \((0.85)(0.8)(\frac{496
}{\log_{20}
496}-\frac{140}{\log_{20}140})=105.\)
Status
Protein1
Protein2
d
d1/2
Query variables
7OZH_1
8XRR_1
138
92
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]