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Parikh vectors
8HFQ_1 8GZE_1 7KJB_1 Letter Amino acid
7 14 26 K Lycine
11 10 22 T Threonine
1 7 6 W Tryptophan
8 12 22 R Arginine
2 5 4 C Cysteine
13 16 30 G Glycine
16 33 29 L Leucine
23 7 25 A Alanine
1 5 8 H Histidine
8 15 28 I Isoleucine
1 5 17 M Methionine
10 13 12 N Asparagine
11 13 20 D Aspartic acid
5 16 14 F Phenylalanine
10 11 15 Y Tyrosine
6 21 28 V Valine
9 15 14 Q Glutamine
6 19 27 E Glutamic acid
4 11 20 P Proline
10 18 23 S Serine 

8HFQ_1|Chains A, AA[auth C], CA[auth G], C[auth E], EA[auth K], E[auth I], GA[auth O], G[auth M], IA[auth S], I[auth Q], K[auth U], M[auth W], O[auth Y], Q[auth a], S[auth c], U[auth e], W[auth g], Y[auth i]|C-phycocyanin alpha subunit|Synechocystis sp. (strain PCC 6803 / Kazusa) (1111708)
>8GZE_1|Chains A, B|Protein-L-histidine N-pros-methyltransferase|Homo sapiens (9606)
>7KJB_1|Chain A|Ephrin type-A receptor 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 \(w=f(c)\)
8HFQ , Knot 78 162 0.81 40 120 159 
MKTPLTEAVSTADSQGRFLSSTELQIAFGRLRQANAGLQAAKALTDNAQSLVNGAAQAVYNKFPYTTQTQGNNFAADQRGKDKCARDIGYYLRIVTYCLVAGGTGPLDEYLIAGIDEINRTFDLSPSWYVEALKYIKANHGLSGDARDEANSYLDYAINALS
8GZE , Knot 120 266 0.84 40 188 257 
NHQWYVCNREKLCESLQAVFVQSYLDQGTQIFLNNSIEKSGWLFIQLYHSFVSSVFSLFMSRTSINGLLGRGSMFVFSPDQFQRLLKINPDWKTHRLLDLGAGDGEVTKIMSPHFEEIYATELSETMIWQLQKKKYRVLGINEWQNTGFQYDVISCLNLLDRCDQPLTLLKDIRSVLEPTRGRVILALVLPFHPYVENVGGKWEKPSEILEIKGQNWEEQVNSLPEVFRKAGFVIEAFTRLPYLCEGDMYNDYYVLDDAVFVLKPV
7KJB , Knot 166 390 0.84 40 235 368 
GPGDPHTYEDPNQAVLKFTTEIHPSCVTRQKVIGAGEFGEVYKGMLKTSSGKKEVPVAIKTLKAGYTEKQRVDFLGEAGIMGQFSHHNIIRLEGVISKYKPMMIITEYMENGALDKFLREKDGEFSVLQLVGMLRGIAAGMKYLANMNYVHRDLAARNILVNSNLVCKVSDFGLSRVLEDDPEATYTTSGGKIPIRWTAPEAISYRKFTSASDVWSFGIVMWEVMTYGERPYWELSNHEVMKAINDGFRLPTPMDCPSAIYQLMMQCWQQERARRPKFADIVSILDKLIRAPDSLKTLADFDPRVSIRLPETSGEEGVPFRTVSEWLESIKMQQYTEHFMAAGYTAIEKVVQMTNDDIKRIGVRLPGHQKRIAYSLLGLKDQVNTVGIPI

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 \(c\).

\(|P_{f(8HFQ_1)}(2) \setminus P_{f(8GZE_1)}(2)|=52\), \(|P_{f(8GZE_1)}(2) \setminus P_{f(8HFQ_1)}(2)|=120\). 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 1:100110011001000101100001011110100101110110110001001101110110001100000010011100010000100110010110001111101110001111100100010101010101100101001101010001000100110110
Pair \(Z_2\) Length of longest common subsequence
8HFQ_1,8GZE_1 172 4
8HFQ_1,7KJB_1 201 3
8GZE_1,7KJB_1 179 3

Newick tree

 
[
	7KJB_1:98.02,
	[
		8HFQ_1:86,8GZE_1:86
	]:12.02
]

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 \((0.85)(0.8)(\frac{428 }{\log_{20} 428}-\frac{162}{\log_{20}162})=79.0\)
Status Protein1 Protein2 d d1/2
Query variables 8HFQ_1 8GZE_1 101 80.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]