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 \(c\).
\(|P_{f(5AEQ_1)}(2) \setminus P_{f(1JHF_1)}(2)|=74\),
\(|P_{f(1JHF_1)}(2) \setminus P_{f(5AEQ_1)}(2)|=72\).
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 1:1100000101011001000001000010010011100010101011110010001111101001101110110000010101001101101000101000101110100100010100001101101100111001011000001000100111110001010101001001001010110110100111
Pair
\(Z_2\)
Length of longest common subsequence
5AEQ_1,1JHF_1
146
3
5AEQ_1,4FVY_1
177
4
1JHF_1,4FVY_1
189
3
Newick tree
[
4FVY_1:96.94,
[
5AEQ_1:73,1JHF_1:73
]:23.94
]
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 \((0.85)(0.8)(\frac{392
}{\log_{20}
392}-\frac{190}{\log_{20}190})=59.9\)
Status
Protein1
Protein2
d
d1/2
Query variables
5AEQ_1
1JHF_1
73
71.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]