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Parikh vectors
1WGP_1 6OLP_1 5FSZ_1 Letter Amino acid
6 22 14 P Proline
4 43 7 N Asparagine
3 19 12 C Cysteine
0 10 8 H Histidine
4 40 12 I Isoleucine
6 34 12 K Lycine
2 12 1 M Methionine
7 24 27 A Alanine
12 27 17 E Glutamic acid
7 44 17 T Threonine
2 14 10 Y Tyrosine
1 18 5 Q Glutamine
1 11 3 W Tryptophan
8 38 17 V Valine
12 24 18 R Arginine
8 18 13 D Aspartic acid
13 32 21 G Glycine
17 30 26 L Leucine
8 16 11 F Phenylalanine
16 30 17 S Serine 

1WGP_1|Chain A|Probable cyclic nucleotide-gated ion channel 6|Arabidopsis thaliana (3702)
>6OLP_1|Chains A, C, E|Envelope glycoprotein gp120|Human immunodeficiency virus 1 (11676)
>5FSZ_1|Chain A|MACRODOMAIN|TRYPANOSOMA CRUZI (5693)
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 \(w=f(c)\)
1WGP , Knot 66 137 0.79 38 98 126 
GSSGSSGVRRVPLFENMDERLLDAICERLKPCLFTEKSYLVREGDPVNEMLFIIRGRLESVTTDGGRSGFYNRSLLKEGDFCGDELLTWALDPKSGSNLPSSTRTVKALTEVEAFALIADELKFVASQFRRSGPSSG
6OLP , Knot 206 506 0.84 40 255 473 
MRVKEIRKNYQHLWKWGIMLLGILMICSAAEQLWVTVYYGVPVWKEATTTLFCASDARAYDTEVHNVWATHACVPTDPNPQEVVLENVTENFNMWKNNMVEQMHEDIISLWDQSLKPCVKLTPLCVTLNCTDLRNATNTNATNTTSSSRGTMEGGEIKNCSFNITTSMRDKVQKEYALFYKLDVVPIKNDNTSYRLISCNTSVITQACPKVSFEPIPIHYCAPAGFAILKCNDKKFNGTGPCTNVSTVQCTHGIRPVVSTQLLLNGSLAEEEVVIRSANFTDNAKIIIVQLNKSVEINCTRPNNNTRKSIHIGPGRAFYTTGEIIGDIRQAHCNISGTKWNDTLKQIVVKLKEQFGNKTIVFNHSSGGDPEIVMHSFNCGGEFFYCNSTQLFNSTWNDTTGSNYTGTIVLPCRIKQIVNMWQEVGKAMYAPPIKGQIRCSSNITGLILIRDGGKNRSENTEIFRPGGGDMRDNWRSELYKYKVVKIEPLGIAPTKAKRRVVQREKR
5FSZ , Knot 121 268 0.84 40 179 262 
SMEEVGKGGQSAKFVTYSTSEPAKLFGEAGRRSHFRARVLEATLSPLEKAIFDVPIEEWLTVDRSKFSGWRCAVPHPVTADELKPVDSSHDILRHIALYNGPVTDLQLDAIVNAANKTCLGGKGVDGAIHAAAGPLLVRECATFKGCDTGQCRLTKGYNLPARYVLHTVGPIGERPEELRSCYRSILSLAHRNRLRSIGFCCVSTGVYGYPLIPATRIAVDETIEYLKQHFSAFDLCCFACFKLEEYNAYTDCLRAWLGTAPNASTDQ

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 \(c\).

\(|P_{f(1WGP_1)}(2) \setminus P_{f(6OLP_1)}(2)|=21\), \(|P_{f(6OLP_1)}(2) \setminus P_{f(1WGP_1)}(2)|=178\). 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 1:10010011001111001000110110001010110000011001011001111101010010001100110000110010101001101110100100110000010110010111111001011100100011001
Pair \(Z_2\) Length of longest common subsequence
1WGP_1,6OLP_1 199 4
1WGP_1,5FSZ_1 161 5
6OLP_1,5FSZ_1 180 4

Newick tree

 
[
	6OLP_1:99.19,
	[
		1WGP_1:80.5,5FSZ_1:80.5
	]:18.69
]

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 \((0.85)(0.8)(\frac{643 }{\log_{20} 643}-\frac{137}{\log_{20}137})=145.\)
Status Protein1 Protein2 d d1/2
Query variables 1WGP_1 6OLP_1 184 115
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]