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Parikh vectors
8XUY_1 2WON_1 7OEM_1 Letter Amino acid
61 37 14 P Proline
95 40 10 T Threonine
83 34 24 G Glycine
65 58 9 K Lycine
83 19 11 N Asparagine
61 25 18 D Aspartic acid
61 37 11 Q Glutamine
11 6 6 M Methionine
97 19 10 S Serine
11 19 7 W Tryptophan
55 22 6 Y Tyrosine
88 40 16 V Valine
39 18 17 R Arginine
30 2 5 C Cysteine
47 47 12 E Glutamic acid
23 8 13 H Histidine
73 41 17 I Isoleucine
100 49 25 L Leucine
77 13 8 F Phenylalanine
75 26 7 A Alanine 

8XUY_1|Chains A, B, C|Spike glycoprotein|Severe acute respiratory syndrome coronavirus 2 (2697049)
>2WON_1|Chain A|HIV-1 REVERSE TRANSCRIPTASE|HUMAN IMMUNODEFICIENCY VIRUS 1 (11676)
>7OEM_1|Chain A|N6-adenosine-methyltransferase catalytic subunit|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 \(w=f(c)\)
8XUY , Knot 447 1235 0.86 40 332 1079 
SSQCVMPLFNLITTTQSYTNSFTRGVYYPDKVFRSSVLHLTQDLFLPFFSNVTWFHAISGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVFIKVCEFQFCNDPFLDVYHKNNKSWMESESGVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPIIGRDFPQGFSALEPLVDLPIGINITRFQTLLALNRSYLTPGDSSSGWTAGAADYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNVTNLCPFHEVFNATRFASVYAWNRTRISNCVADYSVLYNFAPFFAFKCYGVSPTKLNDLCFTNVYADSFVIKGNEVSQIAPGQTGNIADYNYKLPDDFTGCVIAWNSNKLDSKHSGNYDYWYRLFRKSKLKPFERDISTEIYQAGNKPCKGKGPNCYFPLQSYGFRPTYGVGHQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTKSNKKFLPFQQFGRDIVDTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQGVNCTEVSVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEYVNNSYECDIPIGAGICASYQTQTKSRGSAGSVASQSIIAYTMSLGAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLKRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKYFGGFNFSQILPDPSKPSKRSPIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGPALQIPFPMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLFSTPSALGKLQDVVNHNAQALNTLVKQLSSKFGAISSVLNDILSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQLELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQYIASSGYIPEAPRDGQAYVRKDGEWVLLSTFLEGTKHHHHHH
2WON , Knot 223 560 0.84 40 255 517 
PISPIETVPVKLKPGMDGPKVKQWPLTEEKIKALVEICTEMEKEGKISKIGPENPYNTPVFAIKKKDSTKWRKLVDFRELNKRTQDFWEVQLGIPHPAGLKKKKSVTVLDVGDAYFSVPLDEDFRKYTAFTIPSINNETPGIRYQYNVLPQGWKGSPAIFQSSMTKILEPFRKQNPDIVIYQYMDDLYVGSDLEIGQHRTKIEELRQHLLRWGLTTPDKKHQKEPPFLWMGYELHPDKWTVQPIVLPEKDSWTVNDIQKLVGKLNWASQIYPGIKVRQLCKLLRGTKALTEVIPLTEEAELELAENREILKEPVHGVYYDPSKDLIAEIQKQGQGQWTYQIYQEPFKNLKTGKYARMRGAHTNDVKQLTEAVQKITTESIVIWGKTPKFKLPIQKETWETWWTEYWQATWIPEWEFVNTPPLVKLWYQLEKEPIVGAETFYVDGAANRETKLGKAGYVTNRGRQKVVTLTDTTNQKTELQAIYLALQDSGLEVNIVTDSQYALGIIQAQPDQSESELVNQIIEQLIKKEKVYLAWVPAHKGIGGNEQVDKLVSAGIRKVL
7OEM , Knot 113 246 0.84 40 167 234 
MGHHHHHHSSGRENLYFQGALTQSVGGDSSADRLFPPQWICCDIRYLDVSILGKFAVVMADPPWDIHMELPYGTLTDDEMRRLNIPVLQDDGFLFLWVTGRAMELGRECLNLWGYERVDEIIWVKTNQLQRIIRTGRTGHWLNHGKEHCLVGVKGNPQGFNQGLDCDVIVAEVRSTSHKPDEIYGMIERLSPGTRKIELFGRPHNVQPNWITLGNQLDGIHLLDPDVVARFKQRYPDGIISKPKNL

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 \(c\).

\(|P_{f(8XUY_1)}(2) \setminus P_{f(2WON_1)}(2)|=103\), \(|P_{f(2WON_1)}(2) \setminus P_{f(8XUY_1)}(2)|=26\). 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 1:00001111101100000000010011001001100011010001111110010110110100100010011111001101100000011011111001000000111100100111010010100011101000000011000011000100001001001111010100101001001110010101010000011110011011011011101111101001001111000010110000110111100011010100111000001010011001101100000010010100110000010101000110110100101100110100110101100001000110001100111111100011010010010100101001110100100111100101100000110010101111000010000010000100110000101100010001001100100101100011100011010011100100111101011011101011000001100001010101101011100000011110011001100001100100101101010011110110110000001111001100001011101001010101000100110001101111001000000011111110100000000010110110001110010111000110000011110010101000111101000010001010100000001110010100010011011110000000011101001000111001111010011101001000011001110010110111100010011011100110100101101111110001110000111110100110111111101111101100101111000110000011100100111010001100101110100110001011001100100011110011001100101101010100110101001000100011011010101011100100011100001010101001101100110111110100111000010011110001010110011110010011100001001011000001101000111111000100110101001000100010000010101101011010110100010010011001000110100110000011001011011001010100010111100110100000000
Pair \(Z_2\) Length of longest common subsequence
8XUY_1,2WON_1 129 5
8XUY_1,7OEM_1 203 6
2WON_1,7OEM_1 180 4

Newick tree

 
[
	7OEM_1:10.31,
	[
		8XUY_1:64.5,2WON_1:64.5
	]:39.81
]

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 \((0.85)(0.8)(\frac{1795 }{\log_{20} 1795}-\frac{560}{\log_{20}560})=307.\)
Status Protein1 Protein2 d d1/2
Query variables 8XUY_1 2WON_1 393 282
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]