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 \(c\).
\(|P_{f(2IJX_1)}(2) \setminus P_{f(2GRR_1)}(2)|=88\),
\(|P_{f(2GRR_1)}(2) \setminus P_{f(2IJX_1)}(2)|=64\).
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 1:1011000101100110111011001110100001011110010101101011001100001000101110000110110110000110110001001110111000001010010100001101010101010100011001100100100011101000011101010001010100001110010100000000010010011010010001010110001101110101110100111101
Pair
\(Z_2\)
Length of longest common subsequence
2IJX_1,2GRR_1
152
3
2IJX_1,4TWV_1
140
4
2GRR_1,4TWV_1
156
3
Newick tree
[
2GRR_1:79.20,
[
2IJX_1:70,4TWV_1:70
]:9.20
]
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 \((0.85)(0.8)(\frac{405
}{\log_{20}
405}-\frac{161}{\log_{20}161})=72.8\)
Status
Protein1
Protein2
d
d1/2
Query variables
2IJX_1
2GRR_1
91
75.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]