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Parikh vectors
5PNO_1 2CNN_1 8TUI_1 Letter Amino acid
25 3 18 A Alanine
27 12 32 G Glycine
18 3 4 H Histidine
23 11 15 K Lycine
18 3 16 P Proline
20 11 6 I Isoleucine
14 7 24 V Valine
8 0 6 W Tryptophan
18 11 5 R Arginine
20 10 5 N Asparagine
14 11 8 D Aspartic acid
24 12 13 L Leucine
19 7 10 F Phenylalanine
18 12 43 S Serine
7 5 5 C Cysteine
15 1 13 Q Glutamine
26 11 8 E Glutamic acid
11 4 3 M Methionine
20 10 19 T Threonine
19 3 12 Y Tyrosine 

5PNO_1|Chain A|Lysine-specific demethylase 4D|Homo sapiens (9606)
>2CNN_1|Chain A|Caspase-3|Homo sapiens (9606)
>8TUI_1|Chain A[auth H]|Lirilumab Fab Heavy Chain|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 \(w=f(c)\)
5PNO , Knot 160 364 0.86 40 229 347 
MHHHHHHSSGVDLGTENLYFQSMETMKSKANCAQNPNCNIMIFHPTKEEFNDFDKYIAYMESQGAHRAGLAKIIPPKEWKARETYDNISEILIATPLQQVASGRAGVFTQYHKKKKAMTVGEYRHLANSKKYQTPPHQNFEDLERKYWKNRIYNSPIYGADISGSLFDENTKQWNLGHLGTIQDLLEKECGVVIEGVNTPYLYFGMWKTTFAWHTEDMDLYSINYLHLGEPKTWYVVPPEHGQRLERLARELFPGSSRGCGAFLRHKVALISPTVLKENGIPFNRITQEAGEFMVTFPYGYHAGFNHGFNCAEAINFATPRWIDYGKMASQCSCGEARVTFSMDAFVRILQPERYDLWKRGQDR
2CNN , Knot 72 147 0.81 38 113 144 
SGISLDNSYKMDYPEMGLCIIINNKNFHKSTGMTSRSGTDVDAANLRETFRNLKYEVRNKNDLTREEIVELMRDVSKEDHSKRSSFVCVLLSHGEEGIIFGTNGPVDLKKITNFFRGDRCRSLTGKPKLFIIQACRGTELDCGIETD
8TUI , Knot 111 265 0.78 40 156 236 
QSQVQLVQSGAEVKKPGSSVKVSCKASGGTFSFYAISWVRQAPGQGLEWMGGFIPIFGAANYAQKFQGRVTITADESTSTAYMELSSLRSDDTAVYYCARIPSGSYYYDYDMDVWGQGTTVTVSSASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKKVEPKSCASLVPRGSGWSHPQFEKGGGSGGGSGGGSWSHPQFEK

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 \(c\).

\(|P_{f(5PNO_1)}(2) \setminus P_{f(2CNN_1)}(2)|=164\), \(|P_{f(2CNN_1)}(2) \setminus P_{f(5PNO_1)}(2)|=48\). 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 1:1000000001101100010100100100010010010001111010000100100011010001100111101111001010000001001111011001101011110000000011011000011000000011000100100001000100011011010101100000010110110100110000111101100101011110001110000101001001011010010111100100100110011110001011110001111010110001111001000110111011010011100110010110110101100101100000101010101011101101000011001000
Pair \(Z_2\) Length of longest common subsequence
5PNO_1,2CNN_1 212 3
5PNO_1,8TUI_1 191 4
2CNN_1,8TUI_1 179 3

Newick tree

 
[
	5PNO_1:10.40,
	[
		8TUI_1:89.5,2CNN_1:89.5
	]:14.90
]

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 \((0.85)(0.8)(\frac{511 }{\log_{20} 511}-\frac{147}{\log_{20}147})=106.\)
Status Protein1 Protein2 d d1/2
Query variables 5PNO_1 2CNN_1 139 96.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]