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Parikh vectors
6EYM_1 6NTR_1 7QKM_1 Letter Amino acid
7 27 29 D Aspartic acid
6 15 27 E Glutamic acid
8 14 3 F Phenylalanine
10 7 43 P Proline
14 7 11 N Asparagine
8 19 49 G Glycine
2 8 6 M Methionine
13 22 27 V Valine
6 13 34 A Alanine
4 2 12 H Histidine
10 8 15 I Isoleucine
8 24 44 K Lycine
6 9 45 S Serine
5 24 35 T Threonine
9 15 14 R Arginine
1 2 2 C Cysteine
4 12 19 Q Glutamine
13 14 21 L Leucine
1 1 0 W Tryptophan
3 9 5 Y Tyrosine 

6EYM_1|Chain A|Galectin-3|Homo sapiens (9606)
>6NTR_1|Chains A, B, C, D|ATP/GTP-binding site-containing protein A|Brucella abortus (235)
>7QKM_1|Chains A, B, C, D, E, F|Microtubule-associated protein tau|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 \(w=f(c)\)
6EYM , Knot 68 138 0.81 40 112 136 
PLIVPYNLPLPGGVVPRMLITILGTVKPNANRIALDFQRGNDVAFHFNPRFNENNRRVIVCNTKLDNNWGREERQSVFPFESGKPFKIQVLVEPDHFKVAVNDAHLLQYNHRVKKLNEISKLGISGDIDLTSASYTMI
6NTR , Knot 112 252 0.82 40 159 237 
SAVRLVPHRAIYDLTLDRADEKSGISGLTGRMVYEFNGSACEGYTTNFRFVTRVDMDEQPQRVTDQQTTTFEDADGKDFRFVNKTFVDKELVKEVRGDAKLEDGKTVVKLSKPKENTLDLKGTQFPTRHMEELIGKAEAGQKFYQTTLFDASEDADRVVATTVVVGKQQAVPDDETKVMGKFSKDQVWPVTIAYFDDKEQQDGMPIYRINFKLYRNGITRDMTMDYGDFSMRGKLVKLDIYDTGKNKTGCSK
7QKM , Knot 167 441 0.76 38 190 383 
MAEPRQEFEVMEDHAGTYGLGDRKDQGGYTMHQDQEGDTDAGLKESPLQTPTEDGSEEPGSETSDAKSTPTAEDVTAPLVDEGAPGKQAAAQPHTEIPEGTTAEEAGIGDTPSLEDEAAGHVTQARMVSKSKDGTGSDDKKAKGADGKTKIATPRGAAPPGQKGQANATRIPAKTPPAPKTPPSSGEPPKSGDRSGYSSPGSPGTPGSRSRTPSLPTPPTREPKKVAVVRTPPKSPSSAKSRLQTAPVPMPDLKNVKSKIGSTENLKHQPGGGKVQIINKKLDLSNVQSKCGSKDNIKHVPGGGSVQIVYKPVDLSKVTSKCGSLGNIHHKPGGGQVEVKSEKLDFKDRVQSKIGSLDNITHVPGGGNKKIETHKLTFRENAKAKTDHGAEIVYKSPVVSGDTSPRHLSNVSSTGSIDMVDSPQLATLADEVSASLAKQGL

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 \(c\).

\(|P_{f(6EYM_1)}(2) \setminus P_{f(6NTR_1)}(2)|=62\), \(|P_{f(6NTR_1)}(2) \setminus P_{f(6EYM_1)}(2)|=109\). 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 1:111110011111111101110111010101001110100100111010101000000111000010001100000011110010110101110100101110010110000010010010011101010100100011
Pair \(Z_2\) Length of longest common subsequence
6EYM_1,6NTR_1 171 4
6EYM_1,7QKM_1 172 4
6NTR_1,7QKM_1 139 4

Newick tree

 
[
	6EYM_1:90.52,
	[
		6NTR_1:69.5,7QKM_1:69.5
	]:21.02
]

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 \((0.85)(0.8)(\frac{390 }{\log_{20} 390}-\frac{138}{\log_{20}138})=76.1\)
Status Protein1 Protein2 d d1/2
Query variables 6EYM_1 6NTR_1 97 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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]