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 \(c\).
\(|P_{f(5YPY_1)}(2) \setminus P_{f(6FAN_1)}(2)|=68\),
\(|P_{f(6FAN_1)}(2) \setminus P_{f(5YPY_1)}(2)|=32\).
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 1:10100000001110101000111100101111001101011101110000000111110000010011001001001101000000101100111110
Pair
\(Z_2\)
Length of longest common subsequence
5YPY_1,6FAN_1
100
3
5YPY_1,5EHM_1
187
3
6FAN_1,5EHM_1
165
3
Newick tree
[
5EHM_1:97.63,
[
5YPY_1:50,6FAN_1:50
]:47.63
]
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 \((0.85)(0.8)(\frac{161
}{\log_{20}
161}-\frac{63}{\log_{20}63})=33.5\)
Status
Protein1
Protein2
d
d1/2
Query variables
5YPY_1
6FAN_1
44
34
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]