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Parikh vectors
3NPZ_1 6RRU_1 2MTS_1 Letter Amino acid
11 53 14 A Alanine
8 65 0 D Aspartic acid
6 11 0 C Cysteine
9 44 1 H Histidine
13 41 2 I Isoleucine
24 101 13 L Leucine
12 58 2 R Arginine
10 39 2 N Asparagine
18 53 1 E Glutamic acid
10 51 1 K Lycine
7 38 4 Y Tyrosine
7 62 5 G Glycine
5 28 0 M Methionine
17 74 3 S Serine
8 57 0 T Threonine
9 57 0 Q Glutamine
5 44 4 F Phenylalanine
8 63 4 P Proline
2 21 2 W Tryptophan
10 73 5 V Valine 

3NPZ_1|Chain A|Prolactin|Homo sapiens (9606)
>6RRU_1|Chain A|Alpha-mannosidase 2|Drosophila melanogaster (7227)
>2MTS_1|Chain A|HEPATITIS C VIRUS P7 PROTEIN|Hepatitis C virus (420174)
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 \(w=f(c)\)
3NPZ , Knot 96 199 0.85 40 152 194 
LPICPGGAARCQVTLRDLFDRAVVLSHYIHNLSSEMFSEFDKRYTHGRGFITKAINSCHTSSLATPEDKEQAQQMNQKDFLSLIVSILRSWNEPLYHLVTEVRGMQEAPEAILSKAVEIEEQTKRLLEGMELIVSQVHPETKENEIYPVWSGLPSLQMADEESRLSAYYNLLHCLRRDSHKIDNYLKLLKCRIIHNNNC
6RRU , Knot 383 1033 0.85 40 338 949 
DDPIRPPLKVARSPRPGQCQDVVQDVPNVDVQMLELYDRMSFKDIDGGVWKQGWNIKYDPLKYNAHHKLKVFVVPHSHNDPGWIQTFEEYYQHDTKHILSNALRHLHDNPEMKFIWAEISYFARFYHDLGENKKLQMKSIVKNGQLEFVTGGWVMPDEANSHWRNVLLQLTEGQTWLKQFMNVTPTASWAIDPFGHSPTMPYILQKSGFKNMLIQRTHYSVKKELAQQRQLEFLWRQIWDNKGDTALFTHMMPFYSYDIPHTCGPDPKVCCQFDFKRMGSFGLSCPWKVPPRTISDQNVAARSDLLVDQWKKKAELYRTNVLLIPLGDDFRFKQNTEWDVQRVNYERLFEHINSQAHFNVQAQFGTLQEYFDAVHQAERAGQAEFPTLSGDFFTYADRSDNYWSGYYTSRPYHKRMDRVLMHYVRAAEMLSAWHSWDGMARIEERLEQARRELSLFQHHDGITGTAKTHVVVDYEQRMQEALKACQMVMQQSVYRLLTKPSIYSPDFSFSYFTLDDSRWPGSGVEDSRTTIILGEDILPSKHVVMHNTLPHWREQLVDFYVSSPFVSVTDLANNPVEAQVSPVWSWHHDTLTKTIHPQGSTTKYRIIFKARVPPMGLATYVLTISDSKPEHTSYASNLLLRKNPTSLPLGQYPEDVKFGDPREISLRVGNGPTLAFSEQGLLKSIQLTQDSPHVPVHFKFLKYGVRSHGDRSGAYLFLPNGPASPVELGQPVVLVTKGKLESSVSVGLPSVVHQTIMRGGAPEIRNLVDIGSLDNTEIVMRLETHIDSGDIFYTDLNGLQFIKRRRLDKLPLQANYYPIPSGMFIEDANTRLTLLTGQPLGGSSLASGELEIMQDRRLASDDERGLGQGVLDNKPVLHIYRLVLEKVNNCVRPSELHPAGYLTSAAHKASQSLLDPLDKFIFAENEWIGAQGQFGGDHPSAREDLDVSVMRRLTKSSAKTQRVGYVLHRTNLMQCGTPEEHTQKLDVCHLLPNVARCERTTLTFLQNLEHLDGMVAPEVCPMETAAYVSSHSS
2MTS , Knot 32 63 0.70 30 50 57 
ALENLVVLNAASVAGAHGILSFLVFFSAAWYIKGRLAPGAAYAFYGVWPLLLLLLALPPRAYA

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 \(c\).

\(|P_{f(3NPZ_1)}(2) \setminus P_{f(6RRU_1)}(2)|=11\), \(|P_{f(6RRU_1)}(2) \setminus P_{f(3NPZ_1)}(2)|=197\). 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 1:1110111110001010011001111000100100011001000000101110011000000011010000010010000110111011001001100110010110011011100110100000011011011100101000000101110111010110000010100011001000000100010110001100000
Pair \(Z_2\) Length of longest common subsequence
3NPZ_1,6RRU_1 208 4
3NPZ_1,2MTS_1 150 3
6RRU_1,2MTS_1 292 3

Newick tree

 
[
	6RRU_1:13.80,
	[
		3NPZ_1:75,2MTS_1:75
	]:64.80
]

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 \((0.85)(0.8)(\frac{1232 }{\log_{20} 1232}-\frac{199}{\log_{20}199})=276.\)
Status Protein1 Protein2 d d1/2
Query variables 3NPZ_1 6RRU_1 353 208
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]