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 \(c\).
\(|P_{f(1ADL_1)}(2) \setminus P_{f(1IKQ_1)}(2)|=28\),
\(|P_{f(1IKQ_1)}(2) \setminus P_{f(1ADL_1)}(2)|=187\).
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 1:00111101011000010001001111110001111101011101010110100000100001010111010010100001001101011111010010100001000001001110011011000010001
Pair
\(Z_2\)
Length of longest common subsequence
1ADL_1,1IKQ_1
215
3
1ADL_1,3GPO_1
140
3
1IKQ_1,3GPO_1
185
4
Newick tree
[
1IKQ_1:10.51,
[
1ADL_1:70,3GPO_1:70
]:38.51
]
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 \((0.85)(0.8)(\frac{744
}{\log_{20}
744}-\frac{131}{\log_{20}131})=174.\)
Status
Protein1
Protein2
d
d1/2
Query variables
1ADL_1
1IKQ_1
220
133
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]