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Parikh vectors
4GKU_1 1RAY_1 6EGW_1 Letter Amino acid
11 24 35 K Lycine
5 12 25 F Phenylalanine
20 12 15 T Threonine
4 7 6 W Tryptophan
25 13 22 A Alanine
8 10 16 N Asparagine
15 9 21 I Isoleucine
21 22 23 G Glycine
12 17 14 P Proline
4 8 14 Y Tyrosine
2 1 2 C Cysteine
8 11 14 Q Glutamine
20 13 27 E Glutamic acid
31 26 32 L Leucine
9 1 7 M Methionine
15 18 17 S Serine
13 17 20 V Valine
18 7 15 R Arginine
16 19 18 D Aspartic acid
6 12 10 H Histidine 

4GKU_1|Chain A|Beta-lactamase TEM|Escherichia coli (562)
>1RAY_1|Chain A|CARBONIC ANHYDRASE II|Homo sapiens (9606)
>6EGW_1|Chain A|cAMP-dependent protein kinase catalytic subunit alpha|Cricetulus griseus (10029)
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 \(w=f(c)\)
4GKU , Knot 117 263 0.82 40 176 256 
HPETLVKVKDAEDQLGARVGYIELDLNSGKILESFRPEERFPMMSTFKVLLCGAVLSRVDAGQEQLGRRIHYSQNDLVEYSPVTEKHLTDGMTVRELCSAAITMSDNTAANLLLTTIGGPKELTAFLHNMGDHVTRLDRWEPELNEAIPNDERDTTMPAAMATTLRKLLTGELLTLASRQQLIDWMEADKVAGPLLRSALPAGWFIADKSGAGERGSRGIIAALGPDGKPSRIVVIYTTGSQATMDERNRQIAEIGASLIKHW
1RAY , Knot 111 259 0.79 40 175 248 
SHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK
6EGW , Knot 153 353 0.84 40 212 339 
GHMGNAAAAKKGSEQESVKEFLAKAKEEFLKKWESPSQNTAQLDHFDRIKTLGTGSFGRVMLVKHKETGNHYAMKILDKQKVVKLKQIEHTLNEKRILQAVNFPFLVKLEFSFKDNSNLYMVMEYVPGGEMFSHLRRIGRFSEPHARFYAAQIVLTFEYLHSLDLIYRDLKPENLLIDQQGYIQVTDFGFAKRVKGRTWTLCGTPEYLAPEIILSKGYNKAVDWWALGVLIYEMAAGYPPFFADQPIQIYEKIVSGKVRFPSHFSSDLKDLLRNLLQVDLTKRFGNLKNGVNDIKNHKWFATTDWIAIYQRKVEAPFIPKFKGPGDTSNFDDYEEEEIRVSINEKCGKEFTEF

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 \(c\).

\(|P_{f(4GKU_1)}(2) \setminus P_{f(1RAY_1)}(2)|=86\), \(|P_{f(1RAY_1)}(2) \setminus P_{f(4GKU_1)}(2)|=85\). 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 1:01001101001000111011010101001011001010001111001011101111001011000110010000001100011000010011010010011101000011011100111100101110011001001001010100111000000011111100100110101101100001101101001111110011111111100011100100111111110101001111000100101000000110111011001
Pair \(Z_2\) Length of longest common subsequence
4GKU_1,1RAY_1 171 4
4GKU_1,6EGW_1 152 3
1RAY_1,6EGW_1 155 4

Newick tree

 
[
	1RAY_1:83.38,
	[
		4GKU_1:76,6EGW_1:76
	]:7.38
]

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 \((0.85)(0.8)(\frac{522 }{\log_{20} 522}-\frac{259}{\log_{20}259})=74.9\)
Status Protein1 Protein2 d d1/2
Query variables 4GKU_1 1RAY_1 95 94
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]