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 \(c\).
\(|P_{f(6PED_1)}(2) \setminus P_{f(8UXJ_1)}(2)|=89\),
\(|P_{f(8UXJ_1)}(2) \setminus P_{f(6PED_1)}(2)|=63\).
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 1:11000000000011110101110011011010110111001001100011110110111101111010101101011101111011110110100010101110010010000101011100110111101000101111011011011001100110111111001001100111111011010111011010000100110110001101001100011000101101000
Pair
\(Z_2\)
Length of longest common subsequence
6PED_1,8UXJ_1
152
4
6PED_1,1VLW_1
146
6
8UXJ_1,1VLW_1
138
6
Newick tree
[
6PED_1:76.26,
[
1VLW_1:69,8UXJ_1:69
]:7.26
]
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 \((0.85)(0.8)(\frac{503
}{\log_{20}
503}-\frac{233}{\log_{20}233})=77.6\)
Status
Protein1
Protein2
d
d1/2
Query variables
6PED_1
8UXJ_1
93
89
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]