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 \(c\).
\(|P_{f(2FOL_1)}(2) \setminus P_{f(7UAX_1)}(2)|=75\),
\(|P_{f(7UAX_1)}(2) \setminus P_{f(2FOL_1)}(2)|=87\).
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 1:11000000000011110101000011011111001110001110110000000010011101010010101001010110011000100100000011011111001000001001001100100010001001111000010000110000100110011111100010010010001101110100011
Pair
\(Z_2\)
Length of longest common subsequence
2FOL_1,7UAX_1
162
3
2FOL_1,6COC_1
179
3
7UAX_1,6COC_1
171
4
Newick tree
[
6COC_1:89.59,
[
2FOL_1:81,7UAX_1:81
]:8.59
]
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 \((0.85)(0.8)(\frac{425
}{\log_{20}
425}-\frac{191}{\log_{20}191})=68.9\)
Status
Protein1
Protein2
d
d1/2
Query variables
2FOL_1
7UAX_1
86
78.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]