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Parikh vectors
9EAU_1 7KYJ_1 1NRG_1 Letter Amino acid
36 28 16 A Alanine
17 15 24 E Glutamic acid
29 12 21 G Glycine
7 7 7 M Methionine
11 22 21 R Arginine
15 3 6 N Asparagine
17 4 4 I Isoleucine
4 6 7 W Tryptophan
17 1 6 C Cysteine
13 3 11 Q Glutamine
21 20 23 L Leucine
27 0 14 K Lycine
28 15 19 P Proline
46 10 14 V Valine
22 14 12 D Aspartic acid
13 7 6 H Histidine
18 12 14 F Phenylalanine
39 6 16 S Serine
39 6 11 T Threonine
19 3 9 Y Tyrosine 

9EAU_1|Chains A, D[auth E], G[auth I], J[auth M]|Spike glycoprotein E1|Ross river virus (STRAIN T48) (11032)
>7KYJ_1|Chains A, B|Acetyltransferase PA3944|Pseudomonas aeruginosa (287)
>1NRG_1|Chain A|pyridoxine 5'-phosphate oxidase|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 \(w=f(c)\)
9EAU , Knot 186 438 0.86 40 231 417 
YEHTATIPNVVGFPYKAHIERNGFSPMTLQLEVVETSLEPTLNLEYITCEYKTVVPSPFIKCCGTSECSSKEQPDYQCKVYTGVYPFMWGGAYCFCDSENTQLSEAYVDRSDVCKHDHASAYKAHTASLKATIRISYGTINQTTEAFVNGEHAVNVGGSKFIFGPISTAWSPFDNKIVVYKDDVYNQDFPPYGSGQPGRFGDIQSRTVESKDLYANTALKLSRPSPGVVHVPYTQTPSGFKYWLKEKGSSLNTKAPFGCKIKTNPVRAMDCAVGSIPVSMDIPDSAFTRVVDAPAVTDLSCQVVVCTHSSDFGGVATLSYKTDKPGDCAVHSHSNVATLQEATVKVKTAGKVTVHFSTASASPAFKVSVCDAKTTCTAACEPPKDHIVPYGASHNNQVFPDMSGTAMTWVQRLASGLGGLALIAVVVLVLVTCITMRR
7KYJ , Knot 85 194 0.77 38 125 184 
GHMNANLPPSAISELHGPRLLLRAWRDSDREAFAEMCADPQVMEFFPSVLDRAQSDALVDRVQAHFAERGYGPWALELPGEAAFIGFTGLFDVTMDVHFAPTVEIGWRLAPAYWGRGLAREAAETALDFAFERLRLPEVVAFTTPPNRRSWGLMERLGMRRDPAEDFDHPLLAADHPMRRHILYRVDAARWAER
1NRG , Knot 118 261 0.83 40 184 252 
MTCWLRGVTATFGRPAEWPGYLSHLCGRSAAMDLGPMRKSYRGDREAFEETHLTSLDPVKQFAAWFEEAVQCPDIGEANAMCLATCTRDGKPSARMLLLKGFGKDGFRFFTNFESRKGKELDSNPFASLVFYWEPLNRQVRVEGPVKKLPEEEAECYFHSRPKSSQIGAVVSHQSSVIPDREYLRKKNEELEQLYQDQEVPKPKSWGGYVLYPQVMEFWQGQTNRLHDRIVFRRGLPTGDSPLGPMTHRGEEDWLYERLAP

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 \(c\).

\(|P_{f(9EAU_1)}(2) \setminus P_{f(7KYJ_1)}(2)|=157\), \(|P_{f(7KYJ_1)}(2) \setminus P_{f(9EAU_1)}(2)|=51\). 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 1:000010110111110010100011011010101100010101010010000001110111000100000000010000010011011111110010000000100101000010000010100100101010101001010000011101001101110011111100110110001110000100001110101011011010000100001010011010010111101100001011001100010010001111001000110110011101110101100110011011110010001110000001111101000000110011000001101001010100110101010010101110101001000001100110001110110000011101010110110011011111111111111110010100
Pair \(Z_2\) Length of longest common subsequence
9EAU_1,7KYJ_1 208 3
9EAU_1,1NRG_1 187 3
7KYJ_1,1NRG_1 159 6

Newick tree

 
[
	9EAU_1:10.55,
	[
		1NRG_1:79.5,7KYJ_1:79.5
	]:25.05
]

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 \((0.85)(0.8)(\frac{632 }{\log_{20} 632}-\frac{194}{\log_{20}194})=124.\)
Status Protein1 Protein2 d d1/2
Query variables 9EAU_1 7KYJ_1 161 115
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]