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Parikh vectors
8HET_1 3HGO_1 9DWD_1 Letter Amino acid
25 38 16 G Glycine
5 3 5 W Tryptophan
16 36 22 A Alanine
18 20 18 E Glutamic acid
13 21 4 Y Tyrosine
6 26 11 R Arginine
19 25 10 S Serine
6 25 15 I Isoleucine
13 15 11 K Lycine
8 12 4 F Phenylalanine
9 21 17 P Proline
16 28 16 V Valine
7 4 2 C Cysteine
3 16 5 H Histidine
8 17 14 Q Glutamine
10 32 16 L Leucine
6 8 12 M Methionine
8 19 16 T Threonine
11 18 11 N Asparagine
13 18 7 D Aspartic acid 

8HET_1|Chain A|Procathepsin L|Homo sapiens (9606)
>3HGO_1|Chains A, B|12-oxophytodienoate reductase 3|Solanum lycopersicum (4081)
>9DWD_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R|Gag|Human immunodeficiency virus type 1 (NEW YORK-5 ISOLATE) (11698)
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 \(w=f(c)\)
8HET , Knot 99 220 0.81 40 157 210 
APRSVDWREKGYVTPVKNQGQCGSCWAFSATGALEGQMFRKTGRLISLSEQNLVDCSGPQGNEGCNGGLMDYAFQYVQDNGGLDSEESYPYEATEESCKYNPKYSVANDTGFVDIPKQEKALMKAVATVGPISVAIDAGHESFLFYKEGIYFEPDCSSEDMDHGVLVVGYGFESTESDNNKYWLVKNSWGEEWGMGGYVKMAKDRRNHCGIASAASYPTV
3HGO , Knot 168 402 0.83 40 220 377 
HHHHHHMASSAQDGNNPLFSPYKMGKFNLSHRVVLAPMTRCRALNNIPQAALGEYYEQRATAGGFLITEGTMISPTSAGYPHVPGIFTKEQVREWKKIVDVVHAKGAVIFCQLWHVGRASHEVYQPAGAAPISSTEKPISNRWRILMPDGTHGIYPKPRAIGTYEISQVVEDYRRSALNAIEAGFDGIEIHGAHGYLIDQFLKDGINDRTDEYGGSLANRCKFITQVVQAVVSAIGADRVGVRVSPAIDYLDAMDSNPLSLGLAVVERLNKIQLHSGSKLAYLHVTQPRYVAYGQTEAGRLGSEEEEARLMRTLRNAYQGTFICSGGYTRELGIEAVAQGDADLVSYGRLFISNPDLVMRIKLNAPLNKYNRKTFYTQDPVVGYTDYPFLQGNGSNGPLSRL
9DWD , Knot 108 232 0.84 40 159 226 
QMVHQAISPRTLNAWVKVVEEKAFSPEVIPMFSALSEGATPQDLNTMLNTVGGHQAAMQMLKETINEEAAEWDRVHPVHAGPIAPGQMREPRGSDIAGTTSTLQEQIGWMTNNPPIPVGEIYKRWIILGLNKIVRMYSPTSILDIRQGPKEPFRDYVDRFYKTLRAEQASQEVKNWMTETLLVQNANPDCKTILKALGPAATLEEMMTACQGVGGPGHKARVLAEAMSQVIN

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 \(c\).

\(|P_{f(8HET_1)}(2) \setminus P_{f(3HGO_1)}(2)|=53\), \(|P_{f(3HGO_1)}(2) \setminus P_{f(8HET_1)}(2)|=116\). 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 1:1100101000101011000100100111010111010110001011010000110001101001001111001100100011100000010010000000010001100011101100001110111011110111011000111000110101000000100111111011000000000011100011001111101011000000011101100101
Pair \(Z_2\) Length of longest common subsequence
8HET_1,3HGO_1 169 4
8HET_1,9DWD_1 158 3
3HGO_1,9DWD_1 177 4

Newick tree

 
[
	3HGO_1:88.88,
	[
		8HET_1:79,9DWD_1:79
	]:9.88
]

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 \((0.85)(0.8)(\frac{622 }{\log_{20} 622}-\frac{220}{\log_{20}220})=113.\)
Status Protein1 Protein2 d d1/2
Query variables 8HET_1 3HGO_1 143 109
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]