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Parikh vectors
1UOH_1 6NOE_1 7MJN_1 Letter Amino acid
9 0 26 H Histidine
1 6 78 F Phenylalanine
10 12 95 T Threonine
2 3 12 W Tryptophan
19 18 51 E Glutamic acid
12 7 105 S Serine
7 4 42 R Arginine
5 3 30 C Cysteine
7 4 66 Q Glutamine
27 9 107 L Leucine
16 14 58 K Lycine
6 3 54 Y Tyrosine
13 15 95 V Valine
28 6 78 A Alanine
12 7 87 N Asparagine
11 6 61 D Aspartic acid
18 8 94 G Glycine
11 6 71 I Isoleucine
6 3 11 M Methionine
6 3 67 P Proline 

1UOH_1|Chain A|26S PROTEASOME NON-ATPASE REGULATORY SUBUNIT 10|HOMO SAPIENS (9606)
>6NOE_1|Chain A|Cellular retinoic acid-binding protein 2|Homo sapiens (9606)
>7MJN_1|Chain A[auth B]|Spike glycoprotein|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
1UOH , Knot 100 226 0.80 40 143 216 
MEGCVSNLMVCNLAYSGKLEELKESILADKSLATRTDQDSRTALHWACSAGHTEIVEFLLQLGVPVNDKDDAGWSPLHIAASAGRDEIVKALLGKGAQVNAVNQNGCTPLHYAASKNRHEIAVMLLEGGANPDAKDHYEATAMHRAAAKGNLKMIHILLYYKASTNIQDTEGNTPLHLACDEERVEEAKLLVSQGASIYIENKEEKTPLQVAKGGLGLILKRMVEG
6NOE , Knot 67 137 0.80 38 111 132 
PNFSGNWKIIRSENFEELLKVLGVNVMLRKIAVAAASKYAVEIKQEGDTFYIKVSTTVYTTEDNFKVGEEFEEQTVDGRPCKSLVKWESENKMVCEQKLLKGEGPKTSWTKELTNDGELIETMTADDVVCTQVFVRE
7MJN , Knot 453 1288 0.84 40 329 1102 
MFVFLVLLPLVSSQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHVSGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLGVYYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPINLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTYGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQDVNCTEVPVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSPGSASSVASQSIIAYTMSLGAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSPIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGPALQIPFPMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTPSALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGRSLEVLFQGPGHHHHHHHHSAWSHPQFEKGGGSGGGGSGGSAWSHPQFEK

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 \(c\).

\(|P_{f(1UOH_1)}(2) \setminus P_{f(6NOE_1)}(2)|=92\), \(|P_{f(6NOE_1)}(2) \setminus P_{f(1UOH_1)}(2)|=60\). 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 1:1010100111001100101001000111000110000000001101100110001101110111110000011101101110110001101111011010110001001100110000001111110111010100000101100111010101101110001000100001001101100000100101110011010100000001101101111111001101
Pair \(Z_2\) Length of longest common subsequence
1UOH_1,6NOE_1 152 4
1UOH_1,7MJN_1 212 4
6NOE_1,7MJN_1 236 3

Newick tree

 
[
	7MJN_1:12.85,
	[
		1UOH_1:76,6NOE_1:76
	]:45.85
]

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 \((0.85)(0.8)(\frac{363 }{\log_{20} 363}-\frac{137}{\log_{20}137})=68.7\)
Status Protein1 Protein2 d d1/2
Query variables 1UOH_1 6NOE_1 85 68.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]