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Parikh vectors
1JLL_1 5MUB_1 2BHL_1 Letter Amino acid
30 5 29 I Isoleucine
6 3 13 M Methionine
8 6 26 F Phenylalanine
11 7 28 D Aspartic acid
5 5 7 C Cysteine
18 10 35 E Glutamic acid
41 21 31 G Glycine
6 3 12 H Histidine
0 6 7 W Tryptophan
7 5 20 Q Glutamine
14 18 43 L Leucine
22 11 29 K Lycine
15 13 24 P Proline
11 27 23 S Serine
29 12 29 A Alanine
5 6 32 R Arginine
26 24 21 T Threonine
4 9 25 N Asparagine
6 8 21 Y Tyrosine
24 19 34 V Valine 

1JLL_1|Chains A, C[auth D]|succinyl-CoA synthetase alpha subunit|Escherichia coli (562)
>5MUB_1|Chains A, D[auth C], J[auth I], M[auth L], P[auth O], V[auth U]|ACC1 Fab fragment heavy chain|Mus musculus (10090)
>2BHL_1|Chains A, B|GLUCOSE-6-PHOSPHATE 1-DEHYDROGENASE|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 \(w=f(c)\)
1JLL , Knot 124 288 0.81 38 160 267 
SILIDKNTKVICQGFTGSQGTFHSEQAIAYGTKMVGGVTPGKGGTTHLGLPVFNTVREAVAATGATASVIYVPAPFCKDSILEAIDAGIKLIITITEGIPTLDMLTVKVKLDEAGVRMIGPNCPGVITPGECKIGIQPGHIHKPGKVGIVSRSGTLTYEAVKQTTDYGFGQSTCVGIGGDPIPGSNFIDILEMFEKDPQTEAIVMIGEIGGSAEEEAAAYIKEHVTKPVVGYIAGVTAPKGKRMGHAGAIIAGGKGTADEKFAALEAAGVKTVRSLADIGEALKTVLK
5MUB , Knot 100 218 0.82 40 149 209 
EVKLEESGGGLVQPGGSMKLSCAASGFTFSDAWMDWVRQSPEKGLEWVAEIRNKVNNHATNYAESVKGRFTISRDDSRSVVYLQMNNLKPEDTGIYYCTGLTFDYWGQGTTLTVSSAKTTAPSVYPLAPVCGGTTGSSVTLGCLVKGYFPEPVTLTWNSGSLSSGVHTFPALLLSGLYTLSSSVTVTSNTWPSQTITCNVAHPASSTKVDKKIEPRGP
2BHL , Knot 204 489 0.86 40 255 468 
VQSDTHIFIIMGASGDLAKKKIYPTIWWLFRDGLLPENTFIVGYARSRLTVADIRKQSEPFFKATPEEKLKLEDFFARNSYVAGQYDDAASYQRLNSHMNALHLGSQANRLFYLALPPTVYEAVTKNIHESCMSQIGWNRIIVEKPFGRDLQSSDRLSNHISSLFREDQIYRIDHYLGKEMVQNLMVLRFANRIFGPIWNRDNIACVILTFKEPFGTEGRGGYFDEFGIIRDVMQNHLLQMLCLVAMEKPASTNSDDVRDEKVKVLKCISEVQANNVVLGQYVGNPDGEGEATKGYLDDPTVPRGSTTATFAAVVLYVENERWDGVPFILRCGKALNERKAEVRLQFHDVAGDIFHQQCKRNELVIRVQPNEAVYTKMMTKKPGMFFNPEESELDLTYGNRYKNVKLPDAYERLILDVFCGSQMHFVRSDELREAWRIFTPLLHQIELEKPKPIPYIYGSRGPTEADELMKRVGFQYEGTYKWVNPHKL

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 \(c\).

\(|P_{f(1JLL_1)}(2) \setminus P_{f(5MUB_1)}(2)|=90\), \(|P_{f(5MUB_1)}(2) \setminus P_{f(1JLL_1)}(2)|=79\). 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 1:011100000110011010010100001110100111110110110001111110010011110110101101111100001101101110111010011101011010101001110111100111101100011101101001101111000101000110000001110000111110111100110110110001000111111011101000111010001001111011110110100110111111110101000111101111001001101101100110
Pair \(Z_2\) Length of longest common subsequence
1JLL_1,5MUB_1 169 4
1JLL_1,2BHL_1 175 4
5MUB_1,2BHL_1 198 3

Newick tree

 
[
	2BHL_1:96.21,
	[
		1JLL_1:84.5,5MUB_1:84.5
	]:11.71
]

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 \((0.85)(0.8)(\frac{506 }{\log_{20} 506}-\frac{218}{\log_{20}218})=83.0\)
Status Protein1 Protein2 d d1/2
Query variables 1JLL_1 5MUB_1 102 91.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]