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Parikh vectors
8DOG_1 7ODT_1 4ZYX_1 Letter Amino acid
1 7 3 C Cysteine
6 20 14 E Glutamic acid
6 3 3 M Methionine
2 21 7 P Proline
12 25 14 S Serine
1 4 2 W Tryptophan
12 37 21 A Alanine
6 13 9 N Asparagine
2 14 11 H Histidine
2 13 16 I Isoleucine
10 14 7 F Phenylalanine
4 11 1 Y Tyrosine
8 32 21 V Valine
6 30 8 R Arginine
9 49 15 G Glycine
13 50 19 L Leucine
12 19 12 K Lycine
7 10 10 T Threonine
11 15 9 D Aspartic acid
7 19 8 Q Glutamine 

8DOG_1|Chains A, B|Dehaloperoxidase B|Amphitrite ornata (129555)
>7ODT_1|Chain A[auth t]|Mitochondrial ribosome-associated GTPase 2|Homo sapiens (9606)
>4ZYX_1|Chain A|Trifunctional purine biosynthetic protein adenosine-3|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 \(w=f(c)\)
8DOG , Knot 68 137 0.81 40 105 134 
GFKQDIATLRGDLRTYAQDIFLAFLNKYPDEKRNFKNYVGKSDQELKSMAKFGDHTEKVFNLMMEVADRATDCVPLASDASTLVQMKQHSGLTTGNFEKLFVALVEYMRASGQSFDSQSWDRFGKNLVSALSSAGMK
7ODT , Knot 163 406 0.80 40 206 379 
MAPARCFSARLRTVFQGVGHWALSTWAGLKPSRLLPQRASPRLLSVGRADLAKHQELPGKKLLSEKKLKRYFVDYRRVLVCGGNGGAGASCFHSEPRKEFGGPDGGDGGNGGHVILRVDQQVKSLSSVLSRYQGFSGEDGGSKNCFGRSGAVLYIRVPVGTLVKEGGRVVADLSCVGDEYIAALGGAGGKGNRFFLANNNRAPVTCTPGQPGQQRVLHLELKTVAHAGMVGFPNAGKSSLLRAISNARPAVASYPFTTLKPHVGIVHYEGHLQIAVADIPGIIRGAHQNRGLGSAFLRHIERCRFLLFVVDLSQPEPWTQVDDLKYELEMYEKGLSARPHAIVANKIDLPEAQANLSQLRDHLGQEVIVLSALTGENLEQLLLHLKVLYDAYAEAELGQGRQPLRW
4ZYX , Knot 93 210 0.79 40 139 199 
MARVAVLISGTGSNLQALIDSTREPNSSAQIDIVISNKAAVAGLDKAERAGIPTRVINHKLYKNRVEFDSAIDLVLEEFSIDIVCLAGFMRILSGPFVQKWNGKMLNIHPSLLPSFKGSNAHEQALETGVTVTGCTVHFVAEDVDAGQIILQEAVPVKRGDTVATLSERVKLAEHKIFPAALQLVASGTVQLGENGKICWVKEEHHHHHH

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 \(c\).

\(|P_{f(8DOG_1)}(2) \setminus P_{f(7ODT_1)}(2)|=38\), \(|P_{f(7ODT_1)}(2) \setminus P_{f(8DOG_1)}(2)|=139\). 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 1:11000110101010001001111110001000001000110000010011011000001101110110010001111001001101000011001010011111100101010010000100110011011001110
Pair \(Z_2\) Length of longest common subsequence
8DOG_1,7ODT_1 177 3
8DOG_1,4ZYX_1 158 3
7ODT_1,4ZYX_1 159 3

Newick tree

 
[
	7ODT_1:85.75,
	[
		8DOG_1:79,4ZYX_1:79
	]:6.75
]

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 \((0.85)(0.8)(\frac{543 }{\log_{20} 543}-\frac{137}{\log_{20}137})=118.\)
Status Protein1 Protein2 d d1/2
Query variables 8DOG_1 7ODT_1 148 98.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]