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Parikh vectors
2GBS_1 9KUK_1 8XYH_1 Letter Amino acid
7 2 60 D Aspartic acid
2 1 30 C Cysteine
9 4 96 G Glycine
5 3 74 I Isoleucine
10 1 106 L Leucine
5 1 11 M Methionine
5 3 15 W Tryptophan
7 3 42 R Arginine
2 2 90 N Asparagine
3 3 64 Q Glutamine
10 3 50 E Glutamic acid
9 2 18 H Histidine
12 3 58 K Lycine
2 2 76 F Phenylalanine
18 6 76 A Alanine
5 2 55 Y Tyrosine
7 4 66 P Proline
8 2 106 S Serine
5 3 95 T Threonine
14 6 94 V Valine 

2GBS_1|Chain A|Hypothetical protein Rpa0253|Rhodopseudomonas palustris (258594)
>9KUK_11|Chains K, X|Cytochrome c oxidase subunit 7B, mitochondrial|Bos taurus (9913)
>8XYH_1|Chains A, B, C|Spike glycoprotein|Coronavirinae (693995)
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 \(w=f(c)\)
2GBS , Knot 74 145 0.84 40 117 139 
VAYWLVKSEPSVWSWDQQVAKGAAGEAWTGVRNHSAKLHMVAMRRGDRAFYYHSNEGKEIVGIAEIIREAYPDPTDASGKFVCVDIKADKPLKTPVTLAAVKAEPRLADMALMKYSRLSVQPVTAEEWKLVCKMGGLLEHHHHHH
9KUK , Knot 34 56 0.81 40 52 54 
IHQKRAPDFHDKYGNAVLASGATFCVAVWVYMATQIGIEWNPSPVGRVTPKEWREQ
8XYH , Knot 452 1282 0.84 40 331 1099 
MFVFLVLLPLVSSQCVNLTTRTQLPPAYTNSSTRGVYYPDKVFRSSVLHLTQDLFLPFFSNVTWFHAIHVSGTNGIKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLGVYYHRNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPINLVRDLPPGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITDAVDCSLDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATTFASVYAWNRKRISNCVADYSVLYNSTSFSTFKCYGVSPTKLNDLCFTNVYADSFVVRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFHPTNGVGYQPYRVVVLSFELLNAPATVCGPKKSTNLIKNKCVNFNFNGLTGTGVLTESNKKFLPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNASNQVAVLYQDVNCTEVPVAIHANQLTPTWRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSRSVASQSIIAYTMSLGAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAVEQDKNTQEVFAQVKQIYKTPQIKDFGGFNFSQILPDPSKPSKRSPIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGPALQIPFPMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTPSALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLSDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQYIKWPWYIWGGGSGGGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGGGSAWSHPQFEKGGGSGGGSGGSSAWSHPQFEK

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 \(c\).

\(|P_{f(2GBS_1)}(2) \setminus P_{f(9KUK_1)}(2)|=95\), \(|P_{f(9KUK_1)}(2) \setminus P_{f(2GBS_1)}(2)|=30\). 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 1:1101110001011010001101111011011000010101111001001100000010011111011001010100101011010101001100110111101010110111100001010110100101100111110000000
Pair \(Z_2\) Length of longest common subsequence
2GBS_1,9KUK_1 125 3
2GBS_1,8XYH_1 250 4
9KUK_1,8XYH_1 285 3

Newick tree

 
[
	8XYH_1:15.50,
	[
		2GBS_1:62.5,9KUK_1:62.5
	]:88.00
]

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 \((0.85)(0.8)(\frac{201 }{\log_{20} 201}-\frac{56}{\log_{20}56})=48.8\)
Status Protein1 Protein2 d d1/2
Query variables 2GBS_1 9KUK_1 60 41
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]