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Parikh vectors
6MEL_1 3DBM_1 8AMW_1 Letter Amino acid
1 20 13 R Arginine
11 19 5 D Aspartic acid
11 35 21 P Proline
34 29 29 A Alanine
8 19 12 N Asparagine
36 32 35 G Glycine
8 8 15 M Methionine
12 37 19 F Phenylalanine
14 30 24 S Serine
25 30 32 V Valine
4 2 2 C Cysteine
6 5 11 H Histidine
30 20 21 I Isoleucine
29 32 9 K Lycine
16 29 19 T Threonine
0 5 8 W Tryptophan
9 14 8 Q Glutamine
19 37 13 E Glutamic acid
15 50 35 L Leucine
7 20 11 Y Tyrosine 

6MEL_1|Chain A|Succinate--CoA ligase subunit alpha|Campylobacter jejuni (197)
>3DBM_1|Chain A|Cytochrome P450 74A2|Parthenium argentatum (35935)
>8AMW_1|Chains A, B, C, D, E, F, G, H|Aquaporin-7|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 \(w=f(c)\)
6MEL , Knot 131 295 0.84 38 171 275 
MSILVNKNTKVIVQGFTGKEATFHAEQCMAYGTNIVGGITPHKGGQTHLGKPVFDTVADAVKATKADVSLIFVPAFAVGDSVIEAADAGIKLAVVITEHTPVKDMMFAKQYANKKGMKIIGPNCPGIITSEECKLGIMPGFIFKKGCVGLISKSGTLTYEAANQVVQGGYGISTAVGIGGDPIIGLAYKELLSEFQKDDETKAIVMIGEIGGSLEVEAAKFIKENISKPVVAFIAGATAPKGKRMGHAGAIVGSADESAAAKKEALKSYGIHVVDSPALIGEEIQKILGENLYFQ
3DBM , Knot 187 473 0.81 40 227 444 
MDPSSKPLREIPGSYGIPFFQPIKDRLEYFYGTGGRDEYFRSRMQKYQSTVFRANMPPGPFVSSNPKVIVLLDAKSFPILFDVSKVEKKDLFTGTYMPSTKLTGGYRVLSYLDPSEPRHAQLKNLLFFMLKNSSNRVIPQFETTYTELFEGLEAELAKNGKAAFNDVGEQAAFRFLGRAYFNSNPEETKLGTSAPTLISSWVLFNLAPTLDLGLPWFLQEPLLHTFRLPAFLIKSTYNKLYDYFQSVATPVMEQAEKLGVPKDEAVHNILFAVCFNTFGGVKILFPNTLKWIGLAGENLHTQLAEEIRGAIKSYGDGNVTLEAIEQMPLTKSVVYESLRIEPPVPPQYGKAKSNFTIESHDATFEVKKGEMLFGYQPFATKDPKVFDRPEEYVPDRFVGDGEALLKYVWWSNGPETESPTVENKQCAGKDFVVLITRLFVIELFRRYDSFEIELGESPLGAAVTLTFLKRASI
8AMW , Knot 145 342 0.82 40 191 333 
MVQASGHRRSTRGSKMVSWSVIAKIQEILQRKMVREFLAEFMSTYVMMVFGLGSVAHMVLNKKYGSYLGVNLGFGFGVTMGVHVAGRISGAHMNAAVTFANCALGRVPWRKFPVYVLGQFLGSFLAAATIYSLFYTAILHFSGGQLMVTGPVATAGIFATYLPDHMTLWRGFLNEAWLTGMLQLCLFAITDQENNPALPGTEALVIGILVVIIGVSLGMNTGYAINPSRDLPPRIFTFIAGWGKQVFSNGENWWWVPVVAPLLGAYLGGIIYLVFIGSTIPREPLKLEDSVAYEDHGITVLPKMGSHEPTISPLTPVSVSPANRSSVHPAPPLHESMALEHF

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 \(c\).

\(|P_{f(6MEL_1)}(2) \setminus P_{f(3DBM_1)}(2)|=53\), \(|P_{f(3DBM_1)}(2) \setminus P_{f(6MEL_1)}(2)|=109\). 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 1:1011100000111011010010101000110100111110100110001101110011011010010101111111111001101101110111110000110011110001000110111100111100000011111111100101111000101000110011011011001111110111111000110010000000111111011101010110110001001111111110110100110111111010001110001100011011001111100100111001010
Pair \(Z_2\) Length of longest common subsequence
6MEL_1,3DBM_1 162 4
6MEL_1,8AMW_1 170 4
3DBM_1,8AMW_1 162 4

Newick tree

 
[
	8AMW_1:83.68,
	[
		6MEL_1:81,3DBM_1:81
	]:2.68
]

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 \((0.85)(0.8)(\frac{768 }{\log_{20} 768}-\frac{295}{\log_{20}295})=129.\)
Status Protein1 Protein2 d d1/2
Query variables 6MEL_1 3DBM_1 164 130.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]