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Parikh vectors
2WWV_1 6PTU_1 5ZIA_1 Letter Amino acid
6 15 9 Q Glutamine
8 18 10 K Lycine
3 12 24 T Threonine
1 7 10 Y Tyrosine
2 17 6 N Asparagine
0 0 4 C Cysteine
12 15 3 E Glutamic acid
3 19 9 H Histidine
5 17 4 I Isoleucine
16 18 13 L Leucine
0 10 6 W Tryptophan
1 12 6 F Phenylalanine
6 16 25 V Valine
16 16 14 A Alanine
3 12 6 R Arginine
3 14 8 D Aspartic acid
6 16 22 G Glycine
7 9 3 M Methionine
0 6 15 P Proline
5 14 29 S Serine 

2WWV_1|Chains A, B, C|N\,N'-DIACETYLCHITOBIOSE-SPECIFIC PHOSPHOTRANSFERASE ENZYME IIA COMPONENT|ESCHERICHIA COLI (562)
>6PTU_1|Chains A, B, C, D|Class D Carbapenemase OXA-48|Klebsiella pneumoniae (573)
>5ZIA_1|Chains A, D, G, J[auth H], M[auth K], P[auth O]|Heavy chain of CBTAU-24.1 antibody|Homo sapiens (9606)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
2WWV , Knot 50 103 0.75 34 75 99 
AEELEEVVMGLIINSGQARSLAYAALKQAKQGDFAAAKAMMDQSRMALNEAHLVQTKLIEGDAGEGKMKVSLVLVEAQLHLMTSMLARELITELIELHEKLKA
6PTU , Knot 118 263 0.83 38 172 252 
MKHHHHHHMHHHHHHENLYFQGWQENKSWNAHFTEHKSQGVVVLWNENKQQGFTNNLKRANQAFLPASTFKIPNSLIALDLGVVKDEHQVFKWDGQTRDIATWNRDHNLITAMKYSVVPVYQEFARQIGEARMSKMLHAFDYGNEDISGNVDSFWLDGGIRISATEQISFLRKLYHNKLHVSERSQRIVKQAMLTEANGDYIIRAKTGYSTRIEPKIGWWVGWVELDDNVWFFAMNMDMPTSDGLGLRQAITKEVLKQEKIIP
5ZIA , Knot 97 226 0.77 40 146 210 
QVQLVQSGAEVKKPGASVKVSCTASGYTFTGYYLHWVRQAPGQGLEWMGWVNPRSGGTSYPPKFQGRVTMTRDTSINTAYMDLTWLTSDDTAVYYCAVGRIPDVTAFDIWGQGTPVTVSSASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKRVGSHHHHHH

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 \(c\).

\(|P_{f(2WWV_1)}(2) \setminus P_{f(6PTU_1)}(2)|=32\), \(|P_{f(6PTU_1)}(2) \setminus P_{f(2WWV_1)}(2)|=129\). 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 1:1001001111111001010011011100100101111011100001110010110001101011010101011110101011001110011001101000101
Pair \(Z_2\) Length of longest common subsequence
2WWV_1,6PTU_1 161 3
2WWV_1,5ZIA_1 159 3
6PTU_1,5ZIA_1 174 6

Newick tree

 
[
	6PTU_1:85.20,
	[
		2WWV_1:79.5,5ZIA_1:79.5
	]:5.70
]

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 \((0.85)(0.8)(\frac{366 }{\log_{20} 366}-\frac{103}{\log_{20}103})=81.0\)
Status Protein1 Protein2 d d1/2
Query variables 2WWV_1 6PTU_1 102 70
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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