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 \(c\).
\(|P_{f(5SSS_1)}(2) \setminus P_{f(5XVH_1)}(2)|=55\),
\(|P_{f(5XVH_1)}(2) \setminus P_{f(5SSS_1)}(2)|=97\).
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 1:110111111100111100111101110111100001001111100010011011110001110001100001011001001111010110111111101000111010010100101010100011000100100110010010100000001011000
Pair
\(Z_2\)
Length of longest common subsequence
5SSS_1,5XVH_1
152
5
5SSS_1,1ECS_1
160
3
5XVH_1,1ECS_1
168
3
Newick tree
[
1ECS_1:83.93,
[
5SSS_1:76,5XVH_1:76
]:7.93
]
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 \((0.85)(0.8)(\frac{465
}{\log_{20}
465}-\frac{159}{\log_{20}159})=90.3\)
Status
Protein1
Protein2
d
d1/2
Query variables
5SSS_1
5XVH_1
112
84.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]