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Parikh vectors
6ECD_1 3NME_1 7AYN_1 Letter Amino acid
5 8 14 N Asparagine
8 9 5 Q Glutamine
9 21 4 K Lycine
21 12 10 P Proline
11 15 14 S Serine
2 5 2 C Cysteine
17 21 2 E Glutamic acid
18 7 1 H Histidine
8 7 0 M Methionine
19 17 20 V Valine
28 15 13 A Alanine
18 18 3 R Arginine
24 20 7 D Aspartic acid
34 24 13 G Glycine
34 28 10 L Leucine
12 12 6 F Phenylalanine
14 20 8 I Isoleucine
11 19 14 T Threonine
1 4 2 W Tryptophan
9 12 10 Y Tyrosine 

6ECD_1|Chain A|Vlm2|Streptomyces tsusimaensis (285482)
>3NME_1|Chains A, B|SEX4 glucan phosphatase|Arabidopsis thaliana (3702)
>7AYN_1|Chains A, B, C, D|Type 1 fimbrin D-mannose specific adhesin|Escherichia coli K12 (83333)
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 \(w=f(c)\)
6ECD , Knot 126 303 0.79 40 173 277 
MHHHHHHHHENLYFQGGSSTAGDPTAKLVRLNPRGGDGPGIVFAPPAGGTVLGYIELARHLKGFGEIHGVEAPGLGAGETPVYPSFEEMVQFCSDSAAGVAGDGVYIGGHALGGHIAFYLATMLLDRGIRPKGLIILDTPPRLGDIPVADADLTEEETKVFILAMGIGGMLDQDRDALKDLPYEEAKQLLLDRAKNDPRVSAFLSEDYLDRFLRLQMHQLMYSRDVVLPQRKLDIPIHVFRTKNHAPEVARLFSAWENYAAGEVTFVDIPGDHATMLRAPHVSEVAQLLDRHCGLPSDDGPRG
3NME , Knot 131 294 0.84 40 189 285 
GSHMYRHELGMNYNFIRPDLIVGSCLQTPEDVDKLRKIGVKTIFCLQQDPDLEYFGVDISSIQAYAKKYSDIQHIRCEIRDFDAFDLRMRLPAVVGTLYKAVKRNGGVTYVHSTAGMGRAPAVALTYMFWVQGYKLMEAHKLLMSKRSCFPKLDAIRNATIDILTGLKRKTVTLTLKDKGFSRVEISGLDIGWGQRIPLTLGKGTGFWILKRELPEGQFEYKYIIDGEWTHNEAEPFIGPNKDGHTNNYAKVVDDPTSVDGTTRERLSSEDPELLEEERSKLIQFLETCSEAEV
7AYN , Knot 73 158 0.78 38 114 153 
FACKTANGTAIPIGGGSANVYVNLAPVVNVGQNLVVDLSTQIFCHNDYPETITDYVTLQRGSAYGGVLSNFSGTVKYSGSSYPFPTTSETPRVVYNSATDKPWPVALYLTPVSSAGGVAIKAGSLIAVLILRQTNNYNSDDFQFVWNIYANNDVVVPT

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 \(c\).

\(|P_{f(6ECD_1)}(2) \setminus P_{f(3NME_1)}(2)|=75\), \(|P_{f(3NME_1)}(2) \setminus P_{f(6ECD_1)}(2)|=91\). 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 1:100000000001010110001101010110101011011111111111101110101100101110101101111111001101010011010000111111011011101111011101101110011010111110011011011110101000000111111111111000001100110001001110010001010111000010011010100110000111100010111011000001101101101100011101011011100101101101001101100001110001101
Pair \(Z_2\) Length of longest common subsequence
6ECD_1,3NME_1 166 4
6ECD_1,7AYN_1 165 3
3NME_1,7AYN_1 165 4

Newick tree

 
[
	3NME_1:82.83,
	[
		6ECD_1:82.5,7AYN_1:82.5
	]:0.33
]

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 \((0.85)(0.8)(\frac{597 }{\log_{20} 597}-\frac{294}{\log_{20}294})=84.8\)
Status Protein1 Protein2 d d1/2
Query variables 6ECD_1 3NME_1 105 103
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]