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Parikh vectors
8WHZ_1 6ITM_1 3VLD_1 Letter Amino acid
34 15 33 K Lycine
32 14 23 T Threonine
39 9 28 N Asparagine
30 14 9 Q Glutamine
22 15 32 I Isoleucine
20 2 3 W Tryptophan
18 9 23 R Arginine
32 8 16 G Glycine
60 31 68 L Leucine
28 5 22 Y Tyrosine
8 3 6 C Cysteine
47 22 47 E Glutamic acid
21 9 10 M Methionine
27 12 23 F Phenylalanine
27 10 15 P Proline
34 13 37 S Serine
31 9 38 V Valine
38 13 22 A Alanine
33 11 34 D Aspartic acid
16 16 11 H Histidine 

8WHZ_1|Chain A|Processed angiotensin-converting enzyme 2|Homo sapiens (9606)
>6ITM_1|Chains A, C|Bile acid receptor|Homo sapiens (9606)
>3VLD_1|Chains A, B|DNA mismatch repair protein HSM3|Saccharomyces cerevisiae (559292)
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 \(w=f(c)\)
8WHZ , Knot 246 597 0.87 40 289 570 
STIEEQAKTFLDKFNHEAEDLFYQSSLASWNYNTNITEENVQNMNNAGDKWSAFLKEQSTLAQMYPLQEIQNLTVKLQLQALQQNGSSVLSEDKSKRLNTILNTMSTIYSTGKVCNPDNPQECLLLEPGLNEIMANSLDYNERLWAWESWRSEVGKQLRPLYEEYVVLKNEMARANHYEDYGDYWRGDYEVNGVDGYDYSRGQLIEDVEHTFEEIKPLYEHLHAYVRAKLMNAYPSYISPIGCLPAHLLGDMWGRFWTNLYSLTVPFGQKPNIDVTDAMVDQAWDAQRIFKEAEKFFVSVGLPNMTQGFWENSMLTDPGNVQKAVCHPTAWDLGKGDFRILMCTKVTMDDFLTAHHEMGHIQYDMAYAAQPFLLRNGANEGFHEAVGEIMSLSAATPKHLKSIGLLSPDFQEDNETEINFLLKQALTIVGTLPFTYMLEKWRWMVFKGEIPKDQWMKKWWEMKREIVGVVEPVPHDETYCDPASLFHVSNDYSFIRYYTRTLYQFQFQEALCQAAKHEGPLHKCDISNSTEAGQKLFNMLRLGKSEPWTLALENVVGAKNMNVRPLLNYFEPLFTWLKDQNKNSFVGWSTDWSPYAD
6ITM , Knot 106 240 0.80 40 158 225 
MGHHHHHHGSTELTPDQQTLLHFIMDSYNKQRMPQEITNKILKEAFSAEENFLILTEMATNHVQVLVEFTKKLPGFQTLDHEDQIALLKGSAVEAMFLRSAEIFNKKLPSGHSDLLEARIRNSGISDEYITPMFSFYKSIGELKMTQEEYALLTAIVILSPDRQYIKDREAVEKLQEPLLDVLQKLCKIHQPENPQHFACLLGRLTELRTFNHHHAEMLMSWRVNDHKFTPLLCEIWDVQ
3VLD , Knot 196 500 0.81 40 233 450 
MGSSHHHHHHSSGLVPRGSHMSEKETNYVENLLTQLENELNEDNLPEDINTLLRKCSLNLVTVVSLPDMDVKPLLATIKRFLTSNVSYDSLNYDYLLDVVDKLVPMADFDDVLEVYSAEDLVKALRSEIDPLKVAACRVIENSQPKGLFATSNIIDILLDILFDEKVENDKLITAIEKALERLSTDELIRRRLFDNNLPYLVSVKGRMETVSFVRLIDFLTIEFQFISGPEFKDIIFCFTKEEILKSVEDILVFIELVNYYTKFLLEIRNQDKYWALRHVKKILPVFAQLFEDTENYPDVRAFSTNCLLQLFAEVSRIEEDEYSLFKTMDKDSLKIGSEAKLITEWLELINPQYLVKYHKDVVENYFHVSGYSIGMLRNLSADEECFNAIRNKFSAEIVLRLPYLEQMQVVETLTRYEYTSKFLLNEMPKVMGSLIGDGSAGAIIDLETVHYRNSALRNLLDKGEEKLSVWYEPLLREYSKAVNGKNYSTGSETKIADCR

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 \(c\).

\(|P_{f(8WHZ_1)}(2) \setminus P_{f(6ITM_1)}(2)|=168\), \(|P_{f(6ITM_1)}(2) \setminus P_{f(8WHZ_1)}(2)|=37\). 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 1:001000100110010001001100001101000001000010010011001011100000110101100100101010101100010011000000010011001001000101001001000111011100111001000001111001000110010110000111000110100000010010100010110100000101100100010010110001010101011010100101110111011101110110010010111100101010011100110100110010011101111010011100011001101001100101101101010111000101001101000110100011011011110011001100111011010110100100111101010000000101110011011101110011001011110101100011001101000111110111000000011011010000011000000100101001100110001110000100000110011011011000110111001111001010111001011101100000001111000101010
Pair \(Z_2\) Length of longest common subsequence
8WHZ_1,6ITM_1 205 4
8WHZ_1,3VLD_1 160 4
6ITM_1,3VLD_1 179 6

Newick tree

 
[
	6ITM_1:10.04,
	[
		8WHZ_1:80,3VLD_1:80
	]:21.04
]

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 \((0.85)(0.8)(\frac{837 }{\log_{20} 837}-\frac{240}{\log_{20}240})=164.\)
Status Protein1 Protein2 d d1/2
Query variables 8WHZ_1 6ITM_1 211 144
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]