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Parikh vectors
7KHO_1 3AAU_1 8AXZ_1 Letter Amino acid
48 14 32 A Alanine
14 16 7 N Asparagine
52 25 38 G Glycine
15 3 13 H Histidine
10 2 6 M Methionine
13 10 15 Y Tyrosine
39 17 35 V Valine
28 6 30 D Aspartic acid
41 14 33 L Leucine
21 3 12 F Phenylalanine
27 2 20 R Arginine
3 12 6 C Cysteine
20 10 18 Q Glutamine
23 15 23 I Isoleucine
24 14 27 K Lycine
34 10 19 T Threonine
9 4 4 W Tryptophan
30 4 24 E Glutamic acid
13 8 15 P Proline
26 34 21 S Serine 

7KHO_1|Chains A, B, C, D|Amine oxidase|Pseudomonas putida S16 (1042876)
>3AAU_1|Chain A|Cationic trypsin|Bos taurus (9913)
>8AXZ_1|Chain A|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 \(w=f(c)\)
7KHO , Knot 197 490 0.83 40 234 446 
MSDKTKTNEGFSRRSFIGSAAVVTAGVAGLGAIDAASATQKTNRASTVKGGFDYDVVVVGGGFAGATAARECGLQGYRTLLLEARSRLGGRTFTSRFAGQEIEFGGAWVHWLQPHVWAEMQRYGLGVVEDPLTNLDKTLIMYNDGSVESISPDEFGKNIRIAFEKLCHDAWEVFPRPHEPMFTERARELDKSSVLDRIKTLGLSRLQQAQINSYMALYAGETTDKFGLPGVLKLFACGGWNYDAFMDTETHYRIQGGTIGLINAMLTDSGAEVRMSVPVTAVEQVNGGVKIKTDDDEIITAGVVVMTVPLNTYKHIGFTPALSKGKQRFIKEGQLSKGAKLYVHVKQNLGRVFAFADEQQPLNWVQTHDYSDELGTILSITIARKETIDVNDRDAVTREVQKMFPGVEVLGTAAYDWTADPFSLGAWAAYGVGQLSRLKDLQAAEGRILFAGAETSNGWHAVIDGAVESGLRAGREVKQLLSLEHHHHHH
3AAU , Knot 101 223 0.81 40 140 216 
IVGGYTCGANTVPYQVSLNSGYHFCGGSLINSQWVVSAAHCYKSGIQVRLGEDNINVVEGNEQFISASKSIVHPSYNSNTLNNDIMLIKLKSAASLNSRVASISLPTSCASAGTQCLISGWGNTKSSGTSYPDVLKCLKAPILSDSSCKSAYPGQITSNMFCAGYLEGGKDSCQGDSGGPVVCSGKLQGIVSWGSGCAQKNKPGVYTKVCNYVSWIKQTIASN
8AXZ , Knot 171 398 0.85 40 225 388 
GGSMNGQLNGFHEAFIEEGTFLFTSESVGEGHPDKICDQISDAVLDAHLQQDPDAKVACETVAKTGMILLAGEITSRAAVDYQKVVREAVKHIGYDDSSKGFDYKTCNVLVALEQQSPDIAQGVHLDRNEEDIGAGDQGLMFGYATDETEECMPLTIVLAHKLNAKLAELRRNGTLPWLRPDSKTQVTVQYMQDRGAVLPIRVHTIVISVQHDEEVCLDEMRDALKEKVIKAVVPAKYLDEDTIYHLQPSGRFVIGGPQGDAGLTGRKIIVDTYGGWGAHGGGAFSGKDYTKVDRSAAYAARWVAKSLVKGGLCRRVLVQVSYAIGVSHPLSISIFHYGTSQKSERELLEIVKKNFDLRPGVIVRDLDLKKPIYQRTAAYGHFGRDSFPWEVPKKLKY

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 \(c\).

\(|P_{f(7KHO_1)}(2) \setminus P_{f(3AAU_1)}(2)|=145\), \(|P_{f(3AAU_1)}(2) \setminus P_{f(7KHO_1)}(2)|=51\). 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 1:1000000001100001110111101111111110110100000010010111000111111111110110001101000111010001110010001110010111111011010111010001111100110010001110001010010100110010111001000110111010011100010010000110010011100100101000111011000001111111011101110001110000000101101111011100011010101110110010111010000001101111110111000001110111001000110010100110101010001101111100001101100000000110110101100001010000110001001111101110110010101101111110111010010010110101111110000110111011100110110010011010000000
Pair \(Z_2\) Length of longest common subsequence
7KHO_1,3AAU_1 196 3
7KHO_1,8AXZ_1 143 4
3AAU_1,8AXZ_1 183 4

Newick tree

 
[
	3AAU_1:10.39,
	[
		7KHO_1:71.5,8AXZ_1:71.5
	]:29.89
]

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 \((0.85)(0.8)(\frac{713 }{\log_{20} 713}-\frac{223}{\log_{20}223})=137.\)
Status Protein1 Protein2 d d1/2
Query variables 7KHO_1 3AAU_1 174 125.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]