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 \(c\).
\(|P_{f(2VYE_1)}(2) \setminus P_{f(1WPS_1)}(2)|=149\),
\(|P_{f(1WPS_1)}(2) \setminus P_{f(2VYE_1)}(2)|=33\).
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 1:1001100011100101001111111101011011000111001001100011011101100101101101010111100100111100100110011011010001011000011001100100110010000001011100100011010000001110010011100000101100001010111011001001001100001111110101100111101100110000001111010101001110110101010100100101010011010111101001110100010101001010000100001111111001011010100000000010010001011100101111110010001000000011100100010100010111110000000000000011011110000111101011110000011010001001011111
Pair
\(Z_2\)
Length of longest common subsequence
2VYE_1,1WPS_1
182
4
2VYE_1,4JHV_1
154
4
1WPS_1,4JHV_1
180
3
Newick tree
[
1WPS_1:94.57,
[
2VYE_1:77,4JHV_1:77
]:17.57
]
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 \((0.85)(0.8)(\frac{601
}{\log_{20}
601}-\frac{147}{\log_{20}147})=131.\)
Status
Protein1
Protein2
d
d1/2
Query variables
2VYE_1
1WPS_1
165
108
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]