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Parikh vectors
7PPA_1 2KMN_1 7AZL_1 Letter Amino acid
7 8 28 G Glycine
8 14 15 I Isoleucine
8 2 7 Y Tyrosine
6 13 32 V Valine
3 11 19 D Aspartic acid
2 6 16 Q Glutamine
11 7 12 K Lycine
2 3 14 F Phenylalanine
7 6 21 P Proline
5 5 28 S Serine
2 0 1 W Tryptophan
9 10 28 A Alanine
4 3 14 N Asparagine
7 19 29 E Glutamic acid
1 4 16 M Methionine
5 3 18 T Threonine
3 10 26 R Arginine
7 2 4 C Cysteine
3 4 14 H Histidine
8 17 44 L Leucine 

7PPA_1|Chains A, B|Bone morphogenetic protein 10|Homo sapiens (9606)
>2KMN_1|Chain A|Peptide deformylase|Escherichia coli K-12 (83333)
>7AZL_1|Chains A, B, C, D|Beta sliding clamp|Escherichia coli 2-427-07_S4_C3 (1444266)
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 \(w=f(c)\)
7PPA , Knot 58 108 0.83 40 91 105 
NAKGNYCKRTPLYIDFKEIGWDSWIIAPPGYEAYECRGVCNYPLAEHLTPTKHAIIQALVHLKNSQKASKACCVPTKLEPISILYLDKGVVTYKFKYEGMAVSECGCR
2KMN , Knot 70 147 0.79 38 106 140 
SVLQVLHIPDERLRKVAKPVEEVNAEIQRIVDDMFETMYAEEGIGLAATQVDIHQRIIVIDVSENRDERLVLINPELLEKSGETGIEEGCLSIPEQRALVPRAEKVKIRALDRDGKPFELEADGLLAICIQHEMDHLVGKLFMDYLS
7AZL , Knot 165 386 0.84 40 216 371 
MGSSHHHHHHSSGLVPRGSHMKFTVEREHLLKPLQQVSGPLGGRPTLPILGNLLLQVADGTLSLTGTDLEMEMVARVALVQPHEPGATTVPARKFFDICRGLPEGAEIAVQLEGERMLVRSGRSRFSLSTLPAADFPNLDDWQSEVEFTLPQATMKRLIEATQFSMAHQDVRYYLNGMLFETEGEELRTVATDGHRLAVCSMPIGQSLPSHSVIVPRKGVIELMRMLDGGDNPLRVQIGSNNIRAHVGDFIFTSKLVDGRFPDYRRVLPKNPDKHLEAGCDLLKQAFARAAILSNEKFRGVRLYVSENQLKITANNPEQEEAEEILDVTYSGAEMEIGFNVSYVLDVLNALKCENVRMMLTDSVSSVQIEDAASQSAAYVVMPMRL

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 \(c\).

\(|P_{f(7PPA_1)}(2) \setminus P_{f(2KMN_1)}(2)|=67\), \(|P_{f(2KMN_1)}(2) \setminus P_{f(7PPA_1)}(2)|=82\). 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 1:010100000011010100111001111111001000011000111001010001110111010000010010011001011011010011100010001111000100
Pair \(Z_2\) Length of longest common subsequence
7PPA_1,2KMN_1 149 3
7PPA_1,7AZL_1 197 3
2KMN_1,7AZL_1 178 4

Newick tree

 
[
	7AZL_1:99.49,
	[
		7PPA_1:74.5,2KMN_1:74.5
	]:24.99
]

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 \((0.85)(0.8)(\frac{255 }{\log_{20} 255}-\frac{108}{\log_{20}108})=46.7\)
Status Protein1 Protein2 d d1/2
Query variables 7PPA_1 2KMN_1 60 52.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]