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Parikh vectors
8AVS_1 1GUL_1 6BXB_1 Letter Amino acid
7 30 37 L Leucine
1 7 6 M Methionine
2 9 16 N Asparagine
0 0 26 C Cysteine
1 12 25 Q Glutamine
4 11 29 G Glycine
0 6 6 H Histidine
5 11 23 A Alanine
0 10 33 D Aspartic acid
2 16 23 I Isoleucine
2 7 41 S Serine
4 15 29 V Valine
0 11 20 R Arginine
2 17 36 E Glutamic acid
10 17 34 K Lycine
1 14 23 P Proline
5 1 3 W Tryptophan
1 12 25 F Phenylalanine
1 8 23 T Threonine
3 8 8 Y Tyrosine 

8AVS_1|Chains A, B|Bacteriocin aureocin A53|Staphylococcus aureus (1280)
>1GUL_1|Chains A, B, C, D, E, F, G, H|Glutathione S-transferase A4|Homo sapiens (9606)
>6BXB_1|Chains A, B|Chimera protein of Integrin beta-3 and Integrin alpha-L|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 \(w=f(c)\)
8AVS , Knot 27 51 0.69 32 40 47 
MSWLNFLKYIAKYGKKAVSAAWKYKGKVLEWLNVGPTLEWVWQKLKKIAGL
1GUL , Knot 103 222 0.83 38 151 216 
MAARPKLHYPNGRGRMESVRWVLAAAGVEFDEEFLETKEQLYKLQDGNHLLFQQVPMVEIDGMKLVQTRSILHYIADKHNLFGKNLKERTLIDMYVEGTLDLLELLIMHPFLKPDDQQKEVVNMAQKAIIRYFPVFEKILRGHGQSFLVGNQLSLADVILLQTILALEEKIPNILSAFPFLQEYTVKLSNIPTIKRFLEPGSKKKPPPDEIYVRTVYNIFRP
6BXB , Knot 196 466 0.86 40 243 448 
GPNICTTRGVSSCQQCLAVSPMCAWCSDEALPPGSPRCDLKENLLKDNCAPESIEFPVSEARVLEDRPLSDKGSGDSSQVTQVSPQRIALRLRPDDSKNFSIQVRQVEDGNVDLVFLFDGSMSLQPDEFQKILDFMKDVMKKLSNTSYQFAAVQFSTSYKTEFDFSDYVKWKDPDALLKHVKHMLLLTNTFGAINYVATEVFREELGARPDATKVLIIITDGEATDSGNIDAAKDIIRYIIGIGKHFQTKESQETLHKFASKPASEFVKILDTFEKLKDLFTELQKKIRSKVELEVRDLPEELSLSFNATCLNNEVIPGLKSCMGLKIGDTVSFSIEAKVRGCPQEKEKSFTIKPVGFKDSLIVQVTFDCDCACQAQAEPNSHRCNNGNGTFECGVCRCGPGWLGSQCECSEEDYRPSQQDECSPREGQPVCSQRGECLCGQCVCHSSDFGKITGKYCETRELYFQ

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 \(c\).

\(|P_{f(8AVS_1)}(2) \setminus P_{f(1GUL_1)}(2)|=20\), \(|P_{f(1GUL_1)}(2) \setminus P_{f(8AVS_1)}(2)|=131\). 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 1:101101100110010011011100010110110111010111001001111
Pair \(Z_2\) Length of longest common subsequence
8AVS_1,1GUL_1 151 3
8AVS_1,6BXB_1 223 3
1GUL_1,6BXB_1 190 4

Newick tree

 
[
	6BXB_1:11.37,
	[
		8AVS_1:75.5,1GUL_1:75.5
	]:35.87
]

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 \((0.85)(0.8)(\frac{273 }{\log_{20} 273}-\frac{51}{\log_{20}51})=72.7\)
Status Protein1 Protein2 d d1/2
Query variables 8AVS_1 1GUL_1 94 56
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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