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Parikh vectors
6TRK_1 6AAK_1 7WIY_1 Letter Amino acid
4 27 36 S Serine
9 11 35 K Lycine
5 12 28 F Phenylalanine
6 17 25 D Aspartic acid
3 18 28 G Glycine
3 10 17 I Isoleucine
6 39 47 L Leucine
1 6 10 M Methionine
23 18 27 A Alanine
5 4 17 N Asparagine
7 19 23 P Proline
5 12 21 Y Tyrosine
3 23 28 R Arginine
0 8 13 C Cysteine
3 7 10 H Histidine
7 3 27 T Threonine
0 4 4 W Tryptophan
9 16 31 V Valine
2 15 19 Q Glutamine
9 18 44 E Glutamic acid 

6TRK_1|Chains A, B|Pollen allergen Phl p 6|Phleum pratense (15957)
>6AAK_1|Chains A, B, C, D|Tyrosine-protein kinase JAK3|Homo sapiens (9606)
>7WIY_1|Chains A, B, C, D|Tryptophan 5-hydroxylase 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 \(w=f(c)\)
6TRK , Knot 54 110 0.77 36 82 104 
GKATTEEQKLIEDVNASFRAAMATTANVPPADKYKTFEAAFTVSYKRNLADAVSKAPQLVPKLDEVYNAAYNAADHAAPEDKYEAFVLHFSEALRIIAGTPEVHAVKPGA
6AAK , Knot 129 287 0.84 40 177 278 
PTIFEERHLKYISQLGKGNFGSVELCRYDPLGDNTGALVAVKQLQHSGPDQQRDFQREIQILKALHSDFIVKYRGVSYGPGRQSLRLVMEYLPSGCLRDFLQRHRARLDASRLLLYSSQICKGMEYLGSRRCVHRDLAARNILVESEAHVKIADFGLAKLLPLDKDYYVVREPGQSPIFWYAPESLSDNIFSRQSDVWSFGVVLYELFTYCDKSCSPSAEFLRMMGSERDVPALSRLLELLEEGQRLPAPPACPAEVHELMKLCWAPSPQDRPSFSALGPQLDMLWS
7WIY , Knot 201 490 0.84 40 255 458 
MQPAMMMFSSKYWARRGFSLDSAVPEEHQLLGSSTLNKPNSGKNDDKGNKGSSKREAATESGKTAVVFSLKNEVGGLVKALRLFQEKRVNMVHIESRKSRRRSSEVEIFVDCECGKTEFNELIQLLKFQTTIVTLNPPENIWTEEEELEDVPWFPRKISELDKCSHRVLMYGSELDADHPGFKDNVYRQRRKYFVDVAMGYKYGQPIPRVEYTEEETKTWGVVFRELSKLYPTHACREYLKNFPLLTKYCGYREDNVPQLEDVSMFLKERSGFTVRPVAGYLSPRDFLAGLAYRVFHCTQYIRHGSDPLYTPEPDTCHELLGHVPLLADPKFAQFSQEIGLASLGASDEDVQKLATCYFFTIEFGLCKQEGQLRAYGAGLLSSIGELKHALSDKACVKAFDPKTTCLQECLITTFQEAYFVSESFEEAKEKMRDFAKSITRPFSVYFNPYTQSIEILKDTRSIENVVQDLRSDLNTVCDALNKMNQYLGI

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 \(c\).

\(|P_{f(6TRK_1)}(2) \setminus P_{f(6AAK_1)}(2)|=39\), \(|P_{f(6AAK_1)}(2) \setminus P_{f(6TRK_1)}(2)|=134\). 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 1:10100000011001010101111001011110000010111010000011011001101110100100110011001110000011110100110111101010110111
Pair \(Z_2\) Length of longest common subsequence
6TRK_1,6AAK_1 173 3
6TRK_1,7WIY_1 205 3
6AAK_1,7WIY_1 174 4

Newick tree

 
[
	7WIY_1:97.75,
	[
		6TRK_1:86.5,6AAK_1:86.5
	]:11.25
]

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 \((0.85)(0.8)(\frac{397 }{\log_{20} 397}-\frac{110}{\log_{20}110})=87.4\)
Status Protein1 Protein2 d d1/2
Query variables 6TRK_1 6AAK_1 112 75
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]