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Parikh vectors
4HOO_1 8SEJ_1 7XVM_1 Letter Amino acid
13 6 6 V Valine
17 0 18 R Arginine
13 1 4 D Aspartic acid
14 1 8 Q Glutamine
14 1 6 S Serine
18 0 10 T Threonine
18 1 3 Y Tyrosine
8 0 0 W Tryptophan
6 0 2 C Cysteine
25 6 8 G Glycine
12 2 2 H Histidine
20 3 7 I Isoleucine
22 2 12 L Leucine
8 1 3 M Methionine
24 2 7 E Glutamic acid
19 2 13 K Lycine
18 2 4 F Phenylalanine
25 3 18 A Alanine
18 1 1 N Asparagine
18 0 6 P Proline 

4HOO_1|Chains A, B|Lysine-specific demethylase 4D|Homo sapiens (9606)
>8SEJ_1|Chains A, B, C, D, E, F, G, H, I, J|Amyloid-beta protein 42|Homo sapiens (9606)
>7XVM_1|Chains A, E, K, O|Histone H3.1|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 \(w=f(c)\)
4HOO , Knot 148 330 0.86 40 219 320 
QNPNCNIMIFHPTKEEFNDFDKYIAYMESQGAHRAGLAKIIPPKEWKARETYDNISEILIATPLQQVASGRAGVFTQYHKKAAAMTVGEYRHLANSKKYQTPPHQNFEDLERKYWKNRIYNSPIYGADISGSLFDENTKQWNLGHLGTIQDLLEKECGVVIEGVNTPYLYFGMWKTTFAWHTEDMDLYSINYLHLGEPKTWYVVPPEHGQRLERLARELFPGSSRGCGAFLRHKVALISPTVLKENGIPFNRITQEAGEFMVTFPYGYHAGFNHGFNCAEAINFATPRWIDYGKMASQCSCGEARVTFSMDAFVRILQPERYDLWKRGQD
8SEJ , Knot 21 34 0.72 30 32 32 
GYEVHHQKLVFFAEDVGSNKGAIIGLMVGGVVIA
7XVM , Knot 67 138 0.79 38 106 130 
GSMARTKQTARKSTGGKAPRKQLATKAARKSAPATGGVKKPHRYRPGTVALREIRRYQKSTELLIRKLPFQRLVREIAQDFKTDLRFQSSAVMALQEACEAYLVGLFEDTNLCAIHAKRVTIMPKDIQLARRIRGERA

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 \(c\).

\(|P_{f(4HOO_1)}(2) \setminus P_{f(8SEJ_1)}(2)|=202\), \(|P_{f(8SEJ_1)}(2) \setminus P_{f(4HOO_1)}(2)|=15\). 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 1:001000111101000010010001101000110011110111100101000000100111101100110101111000000111101100001100000001100010010000100010001101101010110000001011011010011000011110110010101111000111000010100100101101001011110010010011001111000101111000111101011000111100100011011101101001110011001011011010110010110000010101010101110110100001100100
Pair \(Z_2\) Length of longest common subsequence
4HOO_1,8SEJ_1 217 4
4HOO_1,7XVM_1 183 3
8SEJ_1,7XVM_1 116 3

Newick tree

 
[
	4HOO_1:11.94,
	[
		7XVM_1:58,8SEJ_1:58
	]:52.94
]

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 \((0.85)(0.8)(\frac{364 }{\log_{20} 364}-\frac{34}{\log_{20}34})=106.\)
Status Protein1 Protein2 d d1/2
Query variables 4HOO_1 8SEJ_1 139 76
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]