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 \(c\).
\(|P_{f(4DKP_1)}(2) \setminus P_{f(2WSD_1)}(2)|=60\),
\(|P_{f(2WSD_1)}(2) \setminus P_{f(4DKP_1)}(2)|=128\).
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 1:11001000110100101000010011100101100101001010010001011000110010001101100010101010110110010101010111100001110111000000101011000100100001101110001110101100011100001000100111010001010000100110101101001000101001001100100010001000011101101101010100100010110000001100001100010100101011001001101101010110111101010010010111100011100000001011110100010001000011010
Pair
\(Z_2\)
Length of longest common subsequence
4DKP_1,2WSD_1
188
4
4DKP_1,9JBX_1
180
4
2WSD_1,9JBX_1
132
6
Newick tree
[
4DKP_1:99.19,
[
9JBX_1:66,2WSD_1:66
]:33.19
]
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 \((0.85)(0.8)(\frac{866
}{\log_{20}
866}-\frac{353}{\log_{20}353})=138.\)
Status
Protein1
Protein2
d
d1/2
Query variables
4DKP_1
2WSD_1
180
149.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]