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Parikh vectors
5SPU_1 3SXC_1 6WKB_1 Letter Amino acid
16 7 7 N Asparagine
6 17 24 E Glutamic acid
19 28 33 L Leucine
11 12 20 S Serine
5 17 15 P Proline
22 31 35 V Valine
15 17 32 A Alanine
2 21 20 R Arginine
6 9 23 I Isoleucine
13 13 27 K Lycine
3 15 18 Q Glutamine
2 10 6 M Methionine
0 6 4 W Tryptophan
7 13 15 Y Tyrosine
5 18 12 F Phenylalanine
6 16 19 T Threonine
9 14 30 D Aspartic acid
3 4 6 C Cysteine
13 15 36 G Glycine
6 10 13 H Histidine 

5SPU_1|Chains A, B|Non-structural protein 3|Severe acute respiratory syndrome coronavirus 2 (2697049)
>3SXC_1|Chain A|Histo-blood group ABO system transferase|Homo sapiens (9606)
>6WKB_1|Chains A, B|S-adenosylmethionine synthase isoform type-2|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 \(w=f(c)\)
5SPU , Knot 80 169 0.81 38 119 162 
SMVNSFSGYLKLTDNVYIKNADIVEEAKKVKPTVVVNAANVYLKHGGGVAGALNKATNNAMQVESDDYIATNGPLKVGGSCVLSGHNLAKHCLHVVGPNVNKGEDIQLLKSAYENFNQHEVLLAPLLSAGIFGADPIHSLRVCVDTVRTNVYLAVFDKNLYDKLVSSFL
3SXC , Knot 132 293 0.85 40 198 285 
FMVSLPRMVYPQPKVLTPCRKDVLVVTPWLAPIVWEGTFNIDILNEQFRLQNTTIGLTVFAIKKYVAFLKLFLETAEKHFMVGHRVHYYVFTDQPAAVPRVTLGTGRQLSVLEVGAYKRWQDVSMRRMEMISDFCERRFLSEVDYLVCVDVDMEFRDHVGVEILTPLFGTLHPSFYGSSREAFTYERRPQSQAYIPKDEGDFYYMGAFFGGSVQEVQRLTRACHQAMMVDQANGIEAVWHDESHLNKYLLRHKPTKVLSPEYLWDQQLLGWPAVLRKLRFTAVPKNHQAVRNP
6WKB , Knot 171 395 0.86 40 223 385 
MNGQLNGFHEAFIEEGTFLFTSESVGEGHPDKICDQISDAVLDAHLQQDPDAKVACETVAKTGMILLAGEITSRAAVDYQKVVREAVKHIGYDDSSKGFDYKTCNVLVALEQQSPDIAQGVHLDRNEEDIGAGDQGLMFGYATDETEECMPLTIVLAHKLNAKLAELRRNGTLPWLRPDSKTQVTVQYMQDRGAVLPIRVHTIVISVQHDEEVCLDEMRDALKEKVIKAVVPAKYLDEDTIYHLQPSGRFVIGGPQGDAGLTGRKIIVDTYGGWGAHGGGAFSGKDYTKVDRSAAYAARWVAKSLVKGGLCRRVLVQVSYAIGVSHPLSISIFHYGTSQKSERELLEIVKKNFDLRPGVIVRDLDLKKPIYQRTAAYGHFGRDSFPWEVPKKLKY

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 \(c\).

\(|P_{f(5SPU_1)}(2) \setminus P_{f(3SXC_1)}(2)|=51\), \(|P_{f(3SXC_1)}(2) \setminus P_{f(5SPU_1)}(2)|=130\). 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 1:0110010101010001010010110010010101110110101001111111100100011010000011001110111001101001100010111101001001011001000100001111111011111101100101010010001011110001000110011
Pair \(Z_2\) Length of longest common subsequence
5SPU_1,3SXC_1 181 4
5SPU_1,6WKB_1 174 4
3SXC_1,6WKB_1 159 4

Newick tree

 
[
	5SPU_1:91.64,
	[
		6WKB_1:79.5,3SXC_1:79.5
	]:12.14
]

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 \((0.85)(0.8)(\frac{462 }{\log_{20} 462}-\frac{169}{\log_{20}169})=86.2\)
Status Protein1 Protein2 d d1/2
Query variables 5SPU_1 3SXC_1 115 87.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]