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Parikh vectors
4AOJ_1 7VZR_1 5FKD_1 Letter Amino acid
12 35 0 K Lycine
17 33 0 S Serine
24 48 0 V Valine
15 32 0 D Aspartic acid
7 8 22 C Cysteine
18 33 0 Q Glutamine
21 31 0 E Glutamic acid
12 22 0 H Histidine
14 54 0 I Isoleucine
17 48 0 P Proline
11 49 0 T Threonine
25 60 28 A Alanine
23 37 0 R Arginine
13 61 0 F Phenylalanine
32 79 31 G Glycine
10 26 0 M Methionine
6 40 0 W Tryptophan
11 41 0 Y Tyrosine
5 39 0 N Asparagine
36 89 0 L Leucine 

4AOJ_1|Chains A, B, C|HIGH AFFINITY NERVE GROWTH FACTOR RECEPTOR|HOMO SAPIENS (9606)
>7VZR_1|Chains A, G[auth a]|Photosynthetic reaction center subunit M|Chloracidobacterium thermophilum B (981222)
>5FKD_1|Chain A|SAM-I RIBOSWITCH|THERMOANAEROBACTER TENGCONGENSIS (119072)
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 \(w=f(c)\)
4AOJ , Knot 143 329 0.84 40 209 317 
GAMGSGSSLSPTEGKGSGLQGHIIENPQYFSDACVHHIKRRDIVLKWELGEGAFGKVFLAECHNLLPEQDKMLVAVKALKEASESARQDFQREAELLTMLQHQHIVRFFGVCTEGRPLLMVFEYMRHGDLNRFLRSHGPDAKLLAGGEDVAPGPLGLGQLLAVASQVAAGMVYLAGLHFVHRDLATRNCLVGQGLVVKIGDFGMSRDIYSTDYYRVGGRTMLPIRWMPPESILYRKFTTESDVWSFGVVLWEIFTYGKQPWYQLSNTEAIDCITQGRELERPRACPPEVYAIMRGCWQREPQQRHSIKDVHARLQALAQAPPVYLDVLG
7VZR , Knot 322 865 0.84 40 314 794 
MASFSSYANGVKRWYQKLELPMPPERIFGAHMMLIGGLACLIGTYFFASMTMWNDGYVNLTLRPRLISLGIYDPYDTEQIQRVWLPLIGEFSTSKLPFFGQYPLTMTDFRLFGWGCFHIGLGLWLVYAGAAHYYGARGGATIGEIFWLLPYVPGLKGLCQIKWFTPEGPWYKVGLPWGSFANTPWPILRRTYADALSPHTIYIGLLFFIWGFVLWFVLDKPPVPLQPAQVMTPNGLMPLEQAPFPYGWFDPYLNQVMHPMNTINGETTMCFVWGVLFVALGAYWWYRPPRSINITHLEDTKAVFHVHLTAIGYVSFALAIVGFLALRNHPSYLMLNDMNVIIYGKKIVNPGRMIHNMITFNHVQVGLLYVAAGVFHGGQYLHGLNISGAYKQARSKFITWFQNPDLQTKIVGTTMFVSFVTVVFGYGMICWNTGAELDLNFGIYQFRSFRAIQMDGEAGNIGYRVFRPKNPWDPTAGGDWVKNPDGTAKLVKARNLQVGDRILNEELGIGSSPTYSFTTIEEINYKPEWGQPKLYAVQWGSWTHFLRKVNPLFWVDKGIWYLQNQKTFEATRKADEAYLAAHLKAVSLLNQIDDAQTEEAKQKAQAELDKFRPELEKAHANMLEWNERLASTPAVLYSNLRDQHRDGEINDAIFFWLMIGGWLFGFIPLLRIAFHNYQSPWYRDFEWRKQSPDFPCIGPVKGGTCGVSIQDQLWFCILFSIKPLSAIAWYLDGGWIATMMARGNEAYYLTHNISHTGGVFLYMWNETTWIWTDNHLTAMLLLGHLIWFVSFALWFKDRGSRAEGGDIQSRWVRLMGKRLGIKTLQEVRFPVSNLATAKLWGTVFFYTGTFVLVFLYFADGFFQNR
5FKD , Knot 31 94 0.50 8 16 48 
GGCUUAUCAAGAGAGGGAGAGCGACUGGCGCGAAGAUCCCCGGCAACCAGAAAUGGUGCCAAUUCCUGCAGCGGAAACGUUGAAAGAUGAGCCG

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 \(c\).

\(|P_{f(4AOJ_1)}(2) \setminus P_{f(7VZR_1)}(2)|=38\), \(|P_{f(7VZR_1)}(2) \setminus P_{f(4AOJ_1)}(2)|=143\). 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 1:11110100101001010110101100100100101001000011101011011110111100001110000111110110010001000100010110110000110111100010111111001001010011000110101111100111111111011111001111110111101100011000011101111011011100010000000111001111011110011000100000110111111011001001100100001100100100100101011010111010100010000010010101011101111010111
Pair \(Z_2\) Length of longest common subsequence
4AOJ_1,7VZR_1 181 4
4AOJ_1,5FKD_1 213 3
7VZR_1,5FKD_1 316 4

Newick tree

 
[
	5FKD_1:14.54,
	[
		4AOJ_1:90.5,7VZR_1:90.5
	]:56.04
]

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 \((0.85)(0.8)(\frac{1194 }{\log_{20} 1194}-\frac{329}{\log_{20}329})=227.\)
Status Protein1 Protein2 d d1/2
Query variables 4AOJ_1 7VZR_1 290 198.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]