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Parikh vectors
9CCI_1 7RNZ_1 5PKJ_1 Letter Amino acid
9 11 15 Q Glutamine
7 13 26 E Glutamic acid
3 12 18 H Histidine
10 12 19 F Phenylalanine
7 7 8 W Tryptophan
11 8 19 Y Tyrosine
19 17 14 V Valine
3 7 18 R Arginine
9 19 14 D Aspartic acid
5 1 7 C Cysteine
38 18 18 S Serine
11 13 25 A Alanine
22 22 27 G Glycine
15 26 24 L Leucine
14 24 23 K Lycine
5 10 20 N Asparagine
3 2 11 M Methionine
13 17 18 P Proline
18 12 20 T Threonine
6 9 20 I Isoleucine 

9CCI_1|Chain A[auth B]|M39 Fab heavy chain|Homo sapiens (9606)
>7RNZ_1|Chain A|Carbonic anhydrase 2|Homo sapiens (9606)
>5PKJ_1|Chain A|Lysine-specific demethylase 4D|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 \(w=f(c)\)
9CCI , Knot 102 228 0.81 40 149 215 
EVQLVQSGAEVKKPGESLKISCQGSGYSFSSFWIGWVRQMPGKGLEWMGIIYGGDSDTRYSPSFQGQVSISADKSLSTAYLQWSSLKPSDTAMYYCARTFGTYYDNTEDWFFDFWGHGTLVTVSSASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKKVEPKSC
7RNZ , Knot 112 260 0.79 40 176 249 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK
5PKJ , Knot 160 364 0.86 40 229 347 
MHHHHHHSSGVDLGTENLYFQSMETMKSKANCAQNPNCNIMIFHPTKEEFNDFDKYIAYMESQGAHRAGLAKIIPPKEWKARETYDNISEILIATPLQQVASGRAGVFTQYHKKKKAMTVGEYRHLANSKKYQTPPHQNFEDLERKYWKNRIYNSPIYGADISGSLFDENTKQWNLGHLGTIQDLLEKECGVVIEGVNTPYLYFGMWKTTFAWHTEDMDLYSINYLHLGEPKTWYVVPPEHGQRLERLARELFPGSSRGCGAFLRHKVALISPTVLKENGIPFNRITQEAGEFMVTFPYGYHAGFNHGFNCAEAINFATPRWIDYGKMASQCSCGEARVTFSMDAFVRILQPERYDLWKRGQDR

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 \(c\).

\(|P_{f(9CCI_1)}(2) \setminus P_{f(7RNZ_1)}(2)|=67\), \(|P_{f(7RNZ_1)}(2) \setminus P_{f(9CCI_1)}(2)|=94\). 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 1:010110011010011001010001010010011111100111011011111011000000010101010101000100101010010100011000100110000000011101110101101001000110111111000000110111101100011011010100111001100111110001100100110110001100001001000100001000101000
Pair \(Z_2\) Length of longest common subsequence
9CCI_1,7RNZ_1 161 4
9CCI_1,5PKJ_1 194 3
7RNZ_1,5PKJ_1 185 4

Newick tree

 
[
	5PKJ_1:99.07,
	[
		9CCI_1:80.5,7RNZ_1:80.5
	]:18.57
]

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 \((0.85)(0.8)(\frac{488 }{\log_{20} 488}-\frac{228}{\log_{20}228})=75.0\)
Status Protein1 Protein2 d d1/2
Query variables 9CCI_1 7RNZ_1 95 88
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]