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Parikh vectors
1VJN_1 6LRP_1 8ZIP_1 Letter Amino acid
1 6 4 W Tryptophan
3 22 11 Q Glutamine
19 36 28 G Glycine
17 14 9 H Histidine
5 18 24 S Serine
2 4 5 C Cysteine
18 29 26 E Glutamic acid
12 18 11 F Phenylalanine
16 32 27 L Leucine
18 12 19 K Lycine
11 22 26 T Threonine
26 19 21 V Valine
6 29 18 R Arginine
8 18 12 N Asparagine
13 20 20 D Aspartic acid
11 16 19 P Proline
6 21 16 Y Tyrosine
9 65 29 A Alanine
14 23 30 I Isoleucine
5 12 15 M Methionine 

1VJN_1|Chains A, B|Zn-dependent hydrolase of metallo-beta-lactamase superfamily TM0207|Thermotoga maritima (2336)
>6LRP_1|Chain A|Isocitrate lyase|Chloroflexus aurantiacus (strain ATCC 29366 / DSM 635 / J-10-fl) (324602)
>8ZIP_1|Chain A[auth J]|Actin, alpha cardiac muscle 1|Mus musculus (10090)
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 \(w=f(c)\)
1VJN , Knot 103 220 0.84 40 153 214 
MGSDKIHHHHHHMKITWFGHACFALEMEGKTIVTDPFDESVGYPIPNVTADVVTESHQHFDHNAHHLVKGNFRVIDRPGAYTVNGVKIKGVETFHDPSHGRERGKNIVFVFEGEGIKVCHLGDLGHVLTPAQVEEIGEIDVLLVPVGGTYTIGPKEAKEVADLLNAKVIIPMHYKTKYLKFNLLPVDDFLKLFDSYERVGNILELFEKPKERKVVVMEVQ
6LRP , Knot 179 436 0.83 40 225 407 
MRGSHHHHHHGSMDRAAQIKQIADSWNTPRFAGIVRPYTPEDVYRLRGSVQIEYTLARMGAERLWNLLHTEPYINALGALTGNQAMQQVKAGLKAIYLSGWQVAADANLAGQMYPDQSLYPANSGPQLVRNINNALRRADQIYHSEGRNDIYWFAPIVADAEAGFGGPLNVFEIMKAYIEAGAAGVHFEDQLASEKKCGHMGGKVLIPTQAAIRNLVAARLAADVMGVPTIIVARTDANAATLLTSDIDERDRPFCTGERTSEGFYRVRAGLDQAIARGLAYAPYADMIWCETSEPNLEEARRFAEAIHAQFPGKLLAYNCSPSFNWKKKLDDATIAAFQRELGAMGYKFQFVTLAGFHALNYSMFELARNYRDRGMAAYSELQQAEFAAEAYGYTATRHQREVGTGYFDEVAQVIAGGEISTTALTGSTEEEQFH
8ZIP , Knot 157 370 0.83 40 216 355 
TTALVCDNGSGLVKAGFAGDDAPRAVFPSIVGRPRHQGVMVGMGQKDSYVGDEAQSKRGILTLKYPIEHGIITNWDDMEKIWHHTFYNELRVAPEEHPTLLTEAPLNPKANREKMTQIMFETFNVPAMYVAIQAVLSLYASGRTTGIVLDSGDGVTHNVPIYEGYALPHAIMRLDLAGRDLTDYLMKILTERGYSFVTTAEREIVRDIKEKLCYVALDFENEMATAASSSSLEKSYELPDGQVITIGNERFRCPETLFQPSFIGMESAGIHETTYNSIMKCDIDIRKDLYANNVLSGGTTMYPGIADRMQKEITALAPSTMKIKIIAPPERKYSVWIGGSILASLSTFQQMWISKQEYDEAGPSIVHRKC

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 \(c\).

\(|P_{f(1VJN_1)}(2) \setminus P_{f(6LRP_1)}(2)|=53\), \(|P_{f(6LRP_1)}(2) \setminus P_{f(1VJN_1)}(2)|=125\). 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 1:1100010000001010111010111010100110011000110111010101100000010001001101010110011100101101011001001001000100111110101101001101101101101001101011111111000111001001101101011111000000101011110011011000001101101100100001111010
Pair \(Z_2\) Length of longest common subsequence
1VJN_1,6LRP_1 178 6
1VJN_1,8ZIP_1 177 3
6LRP_1,8ZIP_1 149 4

Newick tree

 
[
	1VJN_1:93.01,
	[
		8ZIP_1:74.5,6LRP_1:74.5
	]:18.51
]

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 \((0.85)(0.8)(\frac{656 }{\log_{20} 656}-\frac{220}{\log_{20}220})=122.\)
Status Protein1 Protein2 d d1/2
Query variables 1VJN_1 6LRP_1 155 115
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]