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 \(c\).
\(|P_{f(8SLW_1)}(2) \setminus P_{f(6YFO_1)}(2)|=239\),
\(|P_{f(6YFO_1)}(2) \setminus P_{f(8SLW_1)}(2)|=4\).
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 1:10000000010111100001101100101101110010101110100000110110011101110111001011110111011100011110011001100111011000111110011011110011011000011000001011111110100110001101101000011001100100011101100010011110110110100111010101000100000101100010111101110101001001001100111111111011110111111101011110100001000110001010111010011001110101101001000100010011101110100000001000100101111100101100011010101000010011001110001011011100110110110010100100010100111011000000101011111100000111111110100100110011001001100010010001110011100111010000001100111111100000110011111001111111100110010010001011001001000011101100000000001111110000010000010110010101111001101110011110110101110111110011110001101000110010100100100101000110001101001001101110111011111001001111110111101110111111100111101011101101001010111101110010011100000011001010100010000111111111110001110110110011110111101011011110001110111100110011111111011111011000111010010101110011001010111011100100101000101111000100101010111111110111100111101111110001011010101110100001110001011111111110010111001100010000001000110110001101001000011001000000000011000100101110011110000001001000100011110010001111100000000100001101000001011110000
Pair
\(Z_2\)
Length of longest common subsequence
8SLW_1,6YFO_1
243
4
8SLW_1,5NXM_1
195
4
6YFO_1,5NXM_1
168
4
Newick tree
[
8SLW_1:11.58,
[
5NXM_1:84,6YFO_1:84
]:33.58
]
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 \((0.85)(0.8)(\frac{1300
}{\log_{20}
1300}-\frac{131}{\log_{20}131})=314.\)
Status
Protein1
Protein2
d
d1/2
Query variables
8SLW_1
6YFO_1
393
217.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]