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 \(c\).
\(|P_{f(8ZTQ_1)}(2) \setminus P_{f(3SIR_1)}(2)|=53\),
\(|P_{f(3SIR_1)}(2) \setminus P_{f(8ZTQ_1)}(2)|=124\).
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 1:110100001000011100001100010000100100101010000100100101010000111101010111011001000110011000010010011000101101001000010010110010001001001011111100110010000000
Pair
\(Z_2\)
Length of longest common subsequence
8ZTQ_1,3SIR_1
177
4
8ZTQ_1,5JEX_1
152
3
3SIR_1,5JEX_1
191
4
Newick tree
[
3SIR_1:96.83,
[
8ZTQ_1:76,5JEX_1:76
]:20.83
]
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 \((0.85)(0.8)(\frac{415
}{\log_{20}
415}-\frac{156}{\log_{20}156})=77.3\)
Status
Protein1
Protein2
d
d1/2
Query variables
8ZTQ_1
3SIR_1
101
79
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]