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Parikh vectors
6MRU_1 4ZLW_1 3WYI_1 Letter Amino acid
24 16 22 E Glutamic acid
33 20 21 L Leucine
4 7 9 P Proline
2 4 4 W Tryptophan
13 2 10 Y Tyrosine
4 7 11 R Arginine
13 9 23 N Asparagine
32 3 24 K Lycine
5 3 7 M Methionine
12 10 12 F Phenylalanine
24 15 15 S Serine
2 3 1 C Cysteine
19 6 29 I Isoleucine
7 3 5 H Histidine
23 11 7 V Valine
17 9 8 Q Glutamine
15 7 13 G Glycine
22 6 14 T Threonine
30 18 9 A Alanine
23 9 12 D Aspartic acid 

6MRU_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M|Hemolysin E, chromosomal|Escherichia coli (strain K12) (83333)
>4ZLW_1|Chains A, B, C, D, E, F, G, H|Ferritin|Pseudo-nitzschia multiseries (37319)
>3WYI_1|Chain A|Isoprenyl transferase|Staphylococcus aureus (1280)
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 \(w=f(c)\)
6MRU , Knot 134 324 0.79 40 171 297 
MGSSHHHHHHSQDLDEVDAGSMTEIVADKTVEVVKNAIETADGALDLYNKYLDQVIPWQTFDETIKELSRFKQEYSQAASVLVGDIKTLLMDSQDKYFEATQTVYEWCGVATQLLAAYILLFDEYNEKKASAQKDILIKVLDDGITKLNEAQKSLLVSSQSFNNASGKLLALDSQLTNDFSEKSSYFQSQVDKIRKEAYAGAAAGVVAGPFGLIISYSIAAGVVEGKLIPELKNKLKSVQNFFTTLSNTVKQANKDIDAAKLKLTTEIVAIGEIKTETETTRFYVDYDDLMLSLLKEAAKKMINTCNEYQKRHGKKTLFEVPEV
4ZLW , Knot 80 168 0.81 40 124 162 
GSEELLDLFNRQVTQEFTASQVYLSASIWFDQNDWEGMAAYMLAESAEEREHGLGFVDFANKRNIPIELQAVPAPVSCAEWSSPEDVWQSILELEQANTRSLLNLAEAASTCHDFAVMAFLNPFHLQQVNAEDKIGSILAKVTDENRTPGLLRSLDVVSFLGPCLFRS
3WYI , Knot 115 256 0.83 40 174 244 
MFKKLINKKNTINNYNEELDSSNIPEHIAIIMDGNGRWAKKRKMPRIKGHYEGMQTIKKITRIASDIGVKYLTLYAFSTENWSRPESEVNYIMNLPVNFLKTFLPELIEKNVKVETIGFTDKLPKSTIEAINNAKEKTANNTGLKLIFAINYGGRAELVHSIKNMFDELHQQGLNSDIIDETYINNHLMTKDYPDPELLIRTSGEQRISNFLIWQVSYSEFIFNQKLWPDFDEDELIKCIKIYQSRQRRFGGLSEE

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 \(c\).

\(|P_{f(6MRU_1)}(2) \setminus P_{f(4ZLW_1)}(2)|=107\), \(|P_{f(4ZLW_1)}(2) \setminus P_{f(6MRU_1)}(2)|=60\). 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 1:110000000000010010110100111000101100110010111010000100111100100010010010000001101111010011100000010100010010111001111011110000000101000111011001100100100011100001001010111100010001000000100010010001011111111111111110001111110101110100010010011001000100100010110101000111110100000000101000011101100110011000000000010001101101
Pair \(Z_2\) Length of longest common subsequence
6MRU_1,4ZLW_1 167 3
6MRU_1,3WYI_1 145 3
4ZLW_1,3WYI_1 174 3

Newick tree

 
[
	4ZLW_1:89.11,
	[
		6MRU_1:72.5,3WYI_1:72.5
	]:16.61
]

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 \((0.85)(0.8)(\frac{492 }{\log_{20} 492}-\frac{168}{\log_{20}168})=94.9\)
Status Protein1 Protein2 d d1/2
Query variables 6MRU_1 4ZLW_1 116 89
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]