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Parikh vectors
8PGN_1 6YIR_1 7PKA_1 Letter Amino acid
32 27 24 A Alanine
10 21 8 R Arginine
9 11 8 N Asparagine
1 1 0 C Cysteine
18 29 13 E Glutamic acid
15 13 11 P Proline
23 14 11 S Serine
9 12 8 Y Tyrosine
13 25 5 D Aspartic acid
5 13 9 Q Glutamine
24 26 11 G Glycine
3 0 3 W Tryptophan
30 27 13 V Valine
11 33 14 I Isoleucine
3 14 3 M Methionine
4 18 9 F Phenylalanine
14 15 2 T Threonine
11 14 8 H Histidine
25 29 24 L Leucine
6 31 8 K Lycine 

8PGN_1|Chain A|Beta-lactamase VIM-1|Pseudomonas aeruginosa (287)
>6YIR_1|Chain A|Oligosaccharides import ATP-binding protein MsmX|Bacillus subtilis (strain 168) (224308)
>7PKA_1|Chains A, B|Glutathione S-transferase|Synechocystis sp. PCC 6803 (1148)
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 \(w=f(c)\)
8PGN , Knot 117 266 0.81 40 152 253 
MLKVISSLLVYMTASVMAVASPLAHSGEPSGEYPTVNEIPVGEVRLYQIADGVWSHIATQSFDGAVYPSNGLIVRDGDELLLIDTAWGAKNTAALLAEIEKQIGLPVTRAVSTHFHDDRVGGVDVLRAAGVATYASPSTRRLAEAEGNEIPTHSLEGLSSSGDAVRFGPVELFYPGAAHSTDNLVVYVPSANVLYGGCAVHELSSTSAGNVADADLAEWPTSVERIQKHYPEAEVVIPGHGLPGGLDLLQHTANVVKAHKNRSVAE
6YIR , Knot 158 373 0.83 38 207 351 
MAELRMEHIYKFYDQKEPAVDDFNLHIADKEFIVFVGPSGCGASTTLRMVAGLEEISKGDFYIEGKRVNDVAPKDRDIAMVFQNYALYPHMTVYDNIAFGLKLRKMPKPEIKKRVEEAAKILGLEEYLHRKPKALSGGQRQRVALGRAIVRDAKVFLMDEPLSNLDAKLRVQMRAEIIKLHQRLQTTTIYVTHDQTEALTMATRIVVMKDGKIQQIGTPKDVYEFPENVFVGGFIGSPAMNFFKGKLTDGLIKIGSAALTVPEGKMKVLREKGYIGKEVIFGIRPEDIHDELIVVESYKNSSIKAKINVAELLGSEIMIYSQIDNQDFIARIDARLDIQSGDELTVAFDMNKGHFFDSETEVRIRLEHHHHHH
7PKA , Knot 90 192 0.82 38 126 183 
MIKLYGAPQSRASIIQWYLEELSLPYEFVNVNLKEGEHRQAPYLAINPFGKVPAIADGNFHLWESGAILLYLAEKASTIPADAQARALVNQWILFANSTLANGLFIEAVREKEMPRLLQSLEKILGRSPFILGEKFSVVDVAVGSILAYVPIMLKLNFDDYPAVAAYVQGLVQRPAFQASIGARLEHHHHHH

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 \(c\).

\(|P_{f(8PGN_1)}(2) \setminus P_{f(6YIR_1)}(2)|=55\), \(|P_{f(6YIR_1)}(2) \setminus P_{f(8PGN_1)}(2)|=110\). 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 1:11011001110101011111011100101010010100111101010011011100110001011101001111001001111001111000111110100011111001100010000111101101111100101000011010100110001011000101101111011011110000011101101011011011001000011011010110110010010000101011111011111101100010110100000110
Pair \(Z_2\) Length of longest common subsequence
8PGN_1,6YIR_1 165 4
8PGN_1,7PKA_1 154 4
6YIR_1,7PKA_1 155 9

Newick tree

 
[
	6YIR_1:81.02,
	[
		8PGN_1:77,7PKA_1:77
	]:4.02
]

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 \((0.85)(0.8)(\frac{639 }{\log_{20} 639}-\frac{266}{\log_{20}266})=104.\)
Status Protein1 Protein2 d d1/2
Query variables 8PGN_1 6YIR_1 132 112
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]