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Parikh vectors
1ERC_1 6OWW_1 7HSG_1 Letter Amino acid
6 1 4 C Cysteine
5 49 21 E Glutamic acid
2 14 21 G Glycine
0 6 13 H Histidine
0 5 7 W Tryptophan
2 26 22 V Valine
4 22 11 A Alanine
1 10 19 R Arginine
1 11 12 N Asparagine
1 36 23 L Leucine
0 25 6 F Phenylalanine
2 17 9 P Proline
2 12 28 T Threonine
2 53 19 D Aspartic acid
2 10 15 Q Glutamine
1 6 7 M Methionine
3 17 21 S Serine
4 13 8 Y Tyrosine
2 23 15 I Isoleucine
0 31 14 K Lycine 

1ERC_1|Chain A|PHEROMONE ER-1|Euplotes raikovi (5938)
>6OWW_1|Chains A, B, C, D, E, F, G, H|Calsequestrin-2|Homo sapiens (9606)
>7HSG_1|Chains A, B|Oleoyl-acyl carrier protein thioesterase 1, chloroplastic|Arabidopsis thaliana (3702)
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 \(w=f(c)\)
1ERC , Knot 25 40 0.76 32 36 37 
DACEQAAIQCVESACESLCTEGEDRTGCYMYIYSNCPPYV
6OWW , Knot 155 387 0.79 40 195 348 
GHMASRAEEGLNFPTYDGKDRVVSLSEKNFKQVLKKYDLLCLYYHEPVSSDKVTQKQFQLKEIVLELVAQVLEHKAIGFVMVDAKKEAKLAKKLGFDEEGSLYILKGDRTIEFDGEFAADVLVEFLLDLIEDPVEIISSKLEVQAFERIEDYIKLIGFFKSEDSEYYKAFEEAAEHFQPYIKFFATFDKGVAKKLSLKMNEVDFYEPFMDEPIAIPNKPYTEEELVEFVKEHQRPTLRRLRPEEMFETWEDDLNGIHIVAFAEKSDPDGYEFLEILKQVARDNTDNPDLSILWIDPDDFPLLVAYWEKTFKIDLFRPQIGVVNVTDADSVWMEIPDDDDLPTAEELEDWIEDVLSGKINTEDDDEDDDDDDNSDEEDNDDSDDDDDE
7HSG , Knot 131 295 0.84 40 189 276 
MGSLTEDGLSYKEKFVVRSYEVGSNKTATVETIANLLQEVGCNHAQSVGFSTDGFATTTTMRKLHLIWVTARMHIEIYKYPAWGDVVEIETWCQSEGRIGTRRDWILKDSVTGEVTGRATSKWVMMNQDTRRLQKVSDDVRDEYLVFCPQEPRLAFPEENNRSLKKIPKLEDPAQYSMIGLKPRRADLDMNQHVNNVTYIGWVLESIPQEIVDTHELQVITLDYRRECQQDDVVDSLTTTTSEIGGTNGSATSGTQGHNDSQFLHLLRLSGDGQEINRGTTLWRKKPSSHHHHHH

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 \(c\).

\(|P_{f(1ERC_1)}(2) \setminus P_{f(6OWW_1)}(2)|=21\), \(|P_{f(6OWW_1)}(2) \setminus P_{f(1ERC_1)}(2)|=180\). 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 1:0100011100100100010001000010010100001101
Pair \(Z_2\) Length of longest common subsequence
1ERC_1,6OWW_1 201 3
1ERC_1,7HSG_1 195 2
6OWW_1,7HSG_1 172 3

Newick tree

 
[
	1ERC_1:10.98,
	[
		7HSG_1:86,6OWW_1:86
	]:16.98
]

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 \((0.85)(0.8)(\frac{427 }{\log_{20} 427}-\frac{40}{\log_{20}40})=121.\)
Status Protein1 Protein2 d d1/2
Query variables 1ERC_1 6OWW_1 146 80.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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