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Parikh vectors
2AOC_1 8UJG_1 3OQT_1 Letter Amino acid
5 21 4 Q Glutamine
13 52 7 G Glycine
2 28 3 F Phenylalanine
7 31 8 V Valine
5 34 8 A Alanine
4 35 5 D Aspartic acid
1 38 4 S Serine
2 14 1 W Tryptophan
1 16 1 Y Tyrosine
4 43 5 R Arginine
4 34 3 E Glutamic acid
14 24 4 I Isoleucine
2 16 3 M Methionine
6 39 1 P Proline
0 12 0 C Cysteine
1 14 3 H Histidine
10 52 4 L Leucine
7 22 1 K Lycine
8 24 4 T Threonine
3 14 1 N Asparagine 

2AOC_1|Chains A, B|HIV-1 PROTEASE|Human immunodeficiency virus type 1 (BH5 ISOLATE) (11682)
>8UJG_1|Chains A, B|Glycosyl transferase|Toxoplasma gondii ME49 (508771)
>3OQT_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P|Rv1498A PROTEIN|Mycobacterium tuberculosis (1773)
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 \(w=f(c)\)
2AOC , Knot 53 99 0.82 38 77 95 
PQITLWKRPLVTIKIGGQLKEALLDTGADDTVIEEMSLPGRWKPKMIGGIGGFIKVRQYDQIIIEIAGHKAIGTVLVGPTPVNVIGRNLLTQIGATLNF
8UJG , Knot 227 563 0.85 40 276 532 
GARDVQEGVSSFEESGGAQLPPRVHALEVLEGAGPGRLHGRLGIKPDGQPGYTRAPSPPTDLSMPQALARGGGFNLYLSDHLELDRTAPDARHASCRQLHYDLSTLPKASVIIVFYNEPFSTLMRSVHSVLNGTPPQILEELILVDDGSTLPYIREDGNQQLVEYLKLLPAKVRLIRNEVRKGIVGARMKGIRASRAPIFAILDSHIEVSPQWLEPLLLRIKEDSRRVVMPQIDGIDAETFKHIAGGIGCKLGFLWKLMEHSYEGHQTARLPPEERQPSPTDFQTSPAMAGGLFAANKAFFFDVGAYDEDFQFWGTENLELSFRLWQCGGVLECAPCSRVYHIFRKGGSGYSSPGDSITINKMRTMLWMDEYADLAWRVIGKPRVNYRPESLEKRREWRKRKGCKSFRWFMENVFPEGDVVTLDDVPYLGPLRNDKIGMCLDNMGWASPGHAVGLEYCHGGDTQTFMFFRKVGHVMPVNDDEACLQPSGRLDWCRGTAQFWWDFTSSGQLMFRETKQCLSAFGRKLRMVECDDTDPYQIWSWTAYNPPDTFTFPSVSRSIRSG
3OQT , Knot 41 70 0.83 38 63 68 
MSNHTYRVIEIVGTSPDGVDAAIQGGLARAAQTMRALDWFEVQSIRGHLVDGAVAHFQVTMKVGFRLEDS

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 \(c\).

\(|P_{f(2AOC_1)}(2) \setminus P_{f(8UJG_1)}(2)|=13\), \(|P_{f(8UJG_1)}(2) \setminus P_{f(2AOC_1)}(2)|=212\). 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 1:101011001110101110100111001100011001011101010111111111010000011101110011101111101101110011001110101
Pair \(Z_2\) Length of longest common subsequence
2AOC_1,8UJG_1 225 4
2AOC_1,3OQT_1 108 3
8UJG_1,3OQT_1 239 3

Newick tree

 
[
	8UJG_1:13.32,
	[
		2AOC_1:54,3OQT_1:54
	]:76.32
]

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 \((0.85)(0.8)(\frac{662 }{\log_{20} 662}-\frac{99}{\log_{20}99})=163.\)
Status Protein1 Protein2 d d1/2
Query variables 2AOC_1 8UJG_1 210 121
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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