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 \(c\).
\(|P_{f(9BNV_1)}(2) \setminus P_{f(7NBR_1)}(2)|=0\),
\(|P_{f(7NBR_1)}(2) \setminus P_{f(9BNV_1)}(2)|=1\).
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 1:011001111010101011010010001011110011001001100000110100001110000001110110101011100100011010100101001000110101100101110001010110001101010101011010010111010000101001001011011011001010101111000010111000010101111101111010011100100010010111100000110000101111101001111101010100110011010011101110001011011000011010
Pair
\(Z_2\)
Length of longest common subsequence
9BNV_1,7NBR_1
1
172
9BNV_1,1NLI_1
178
3
7NBR_1,1NLI_1
179
3
Newick tree
[
1NLI_1:10.05,
[
9BNV_1:0.5,7NBR_1:0.5
]:10.55
]
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 \((0.85)(0.8)(\frac{612
}{\log_{20}
612}-\frac{306}{\log_{20}306})=85.3\)
Status
Protein1
Protein2
d
d1/2
Query variables
9BNV_1
7NBR_1
2
2
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]