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 \(c\).
\(|P_{f(1RNM_1)}(2) \setminus P_{f(5ZWE_1)}(2)|=55\),
\(|P_{f(5ZWE_1)}(2) \setminus P_{f(1RNM_1)}(2)|=120\).
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 1:0001110100001000001100000000110000100000011001100011010110000011000100000000001010000001000010010000010001111001010111010101
Pair
\(Z_2\)
Length of longest common subsequence
1RNM_1,5ZWE_1
175
3
1RNM_1,5BNP_1
181
4
5ZWE_1,5BNP_1
164
4
Newick tree
[
1RNM_1:91.23,
[
5ZWE_1:82,5BNP_1:82
]:9.23
]
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 \((0.85)(0.8)(\frac{395
}{\log_{20}
395}-\frac{124}{\log_{20}124})=82.1\)
Status
Protein1
Protein2
d
d1/2
Query variables
1RNM_1
5ZWE_1
104
76
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]