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 \(c\).
\(|P_{f(2CXC_1)}(2) \setminus P_{f(2BEH_1)}(2)|=32\),
\(|P_{f(2BEH_1)}(2) \setminus P_{f(2CXC_1)}(2)|=158\).
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 1:101000101001001011001011010001100000011111001011011101101101100111001011000001001100111110100101000011001110100000111110110010010111001111001110
Pair
\(Z_2\)
Length of longest common subsequence
2CXC_1,2BEH_1
190
3
2CXC_1,8PAI_1
134
3
2BEH_1,8PAI_1
200
3
Newick tree
[
2BEH_1:10.76,
[
2CXC_1:67,8PAI_1:67
]:38.76
]
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 \((0.85)(0.8)(\frac{576
}{\log_{20}
576}-\frac{144}{\log_{20}144})=125.\)
Status
Protein1
Protein2
d
d1/2
Query variables
2CXC_1
2BEH_1
160
105.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]