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Parikh vectors
3MJI_1 3LQJ_1 3LZV_1 Letter Amino acid
24 11 7 V Valine
18 7 3 A Alanine
11 10 4 R Arginine
22 9 4 N Asparagine
2 8 2 C Cysteine
9 10 5 Q Glutamine
16 14 4 E Glutamic acid
3 4 2 W Tryptophan
20 12 4 D Aspartic acid
28 16 11 L Leucine
12 6 2 M Methionine
19 15 7 K Lycine
18 6 1 Y Tyrosine
26 6 13 G Glycine
4 3 1 H Histidine
24 8 12 I Isoleucine
12 7 2 F Phenylalanine
16 7 7 P Proline
24 18 0 S Serine
30 6 8 T Threonine 

3MJI_1|Chains A, B, C, D|Penicillin acylase|Bacillus sphaericus (1421)
>3LQJ_1|Chains A, B|MLL1 PHD3-Bromo|Homo sapiens (9606)
>3LZV_1|Chains A, B|HIV-1 Protease|HIV-1 M:B_ARV2/SF2 (11685)
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 \(w=f(c)\)
3MJI , Knot 144 338 0.82 40 199 328 
MLGSSSLSIRTTDDKSLFARTMDFTMEPDSKVIIVPRNYGIRLLEKENVVINNSYAFVGMGSTDITSPVLYDGVNEKGLMGAMLYYATFATYADEPKKGTRGINPVYVISQVLGNCVTVDDVIEKLTSYTLLNEANIILGFAPPLHYTFTDASGESIVIEPDKTGITIHRKTIGVMTNSPGYEWHQTNLRAYIGVTPNPPQDIMMGDLDLTPFGQGAGGLGLPGDFTPSARFLRVAYWKKYTEKAKNETEGVTNLFHILSSVNIPKGVVLTNEGKTDYTIYTSAMCAQSKNYYFKLYDNSRISAVSLMAENLNSQDLITFEWDRKQDIKQLNQVNVMS
3LQJ , Knot 92 183 0.87 40 142 180 
SGNFCPLCDKCYDDDDYESKMMQCGKCDRWVHSKCENLSDEMYEILSNLPESVAYTCVNCTERHPAEWRLALEKELQISLKQVLTALLNSRTTSHLLRYRQQQPLDLEGVKRKMDQGNYTSVLEFSDDIVKIIQAAINSDGGQPEIKKANSMVKSFFIRQMERVFPWFSVKKSRFWEPNKVSS
3LZV , Knot 54 99 0.83 38 82 94 
PQITLWKRPLVTIRIGGQLKEALLDTGADNTVLEEMNLPGKWKPKMIGGIGGFIKVRQYDQIPVEICGHKAIGTVLVGPTPVNIIGRDLLTQIGCTLNF

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 \(c\).

\(|P_{f(3MJI_1)}(2) \setminus P_{f(3LQJ_1)}(2)|=123\), \(|P_{f(3LQJ_1)}(2) \setminus P_{f(3MJI_1)}(2)|=66\). 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 1:11100010100000001110010101010001111100011011000011100001111110001001110011000111111100101100100100100110110110011100101001100100001100101111111110001001010011101000110100001111000110010000101011101011001111010101110111111111010101011011010000001000001100110110010110111100010000010001101000000101000001011011100100001101010000010010010110
Pair \(Z_2\) Length of longest common subsequence
3MJI_1,3LQJ_1 189 4
3MJI_1,3LZV_1 189 3
3LQJ_1,3LZV_1 154 3

Newick tree

 
[
	3MJI_1:99.65,
	[
		3LQJ_1:77,3LZV_1:77
	]:22.65
]

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 \((0.85)(0.8)(\frac{521 }{\log_{20} 521}-\frac{183}{\log_{20}183})=98.0\)
Status Protein1 Protein2 d d1/2
Query variables 3MJI_1 3LQJ_1 125 95
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]