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Parikh vectors
7RRB_1 6OBH_1 8GAF_1 Letter Amino acid
28 20 15 A Alanine
16 11 19 R Arginine
14 10 17 N Asparagine
30 11 18 K Lycine
10 11 8 M Methionine
20 18 8 P Proline
5 4 1 W Tryptophan
22 7 24 D Aspartic acid
25 18 23 G Glycine
11 6 9 H Histidine
18 15 13 I Isoleucine
19 4 16 F Phenylalanine
24 17 19 E Glutamic acid
12 4 8 Y Tyrosine
26 15 20 V Valine
8 3 3 C Cysteine
18 15 12 Q Glutamine
48 18 20 L Leucine
27 9 12 S Serine
13 16 18 T Threonine 

7RRB_1|Chains A, B|Indoleamine 2,3-dioxygenase 1|Homo sapiens (9606)
>6OBH_1|Chains A, D|CA|Human immunodeficiency virus 1 (11676)
>8GAF_1|Chains A, B, C, D, E, F, L[auth M]|Cas7|Neisseria lactamica (486)
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 \(w=f(c)\)
7RRB , Knot 173 394 0.87 40 227 372 
MISKEYHIDEEVGFALPNPQENLPDFYNDWMFIAKHLPDLIESGQLRERVEKLNMLSIDHLTDHKSQRLARLVLGCITMAYVWGKGHGDVRKVLPRNIAVPYCQLSKKLELPPILVYADCVLANWKKKDPNKPLTYENMDVLFSFRDGDCSKGFFLVSLLVEIAAASAIKVIPTVFKAMQMQERDTLLKALLEIASCLEKALQVFHQIHDHVNPKAFFSVLRIYLSGWKGNPQLSDGLVYEGFWEDPKEFAGGSAGQSSVFQCFDVLLGIQQTAGGGHAAQFLQDMRRYMPPAHRNFLCSLESNPSVREFVLSKGDAGLREAYDACVKALVSLRSYHLQIVTKYILIPASQQPKENKTSEDPSKLEAKGTGGTDLMNFLKTVRSTTEKSLLKEG
6OBH , Knot 107 232 0.83 40 159 224 
MPIVQNLQGQMVHQCISPRTLNAWVKVVEEKAFSPEVIPMFSALSEGATPQDLNTMLNTVGGHQAAMQMLKETINEEAAEWDRLHPVHAGPIAPGQMREPRGSDIAGTTSTLQEQIGWMTHNPPIPVGEIYKRWIILGLNKIVRMYSPTSILDIRQGPKEPFRDYVDRFYKTLRAEQASQEVKNAATETLLVQNANPDCKTILKALGPGATLEEMMTACQGVGGPGHKARVL
8GAF , Knot 127 283 0.84 40 188 273 
TIEKRYDFVFLFDVQDGNPNGDPDAGNLPRIDPQTGEGLVTDVCLKRKVRNFIQMTQNDEHHDIFIREKGILNNLIDEAHEQENVKGKEKGEKTEAARQYMCSRYYDIRTFGAVMTTGKNAGQVRGPVQLTFSRSIDPIMTLEHSITRMAVTNEKDASETGDNRTMGRKFTVPYGLYRCHGFISTHFAKQTGFSENDLELFWQALVNMFDHDHSAARGQMNARGLYVFEHSNNLGDAPADSLFKRIQVVKKDGVEVVRSFDDYLVSVDDKNLEETKLLRKLGG

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 \(c\).

\(|P_{f(7RRB_1)}(2) \setminus P_{f(6OBH_1)}(2)|=131\), \(|P_{f(6OBH_1)}(2) \setminus P_{f(7RRB_1)}(2)|=63\). 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 1:1100000100011111101000110100011111001101100101000100101101001000000011011110101101110101010011100111100010001011111101001110100001001100001011101001000011111011101111011011101101101000001101110110010011011001000101011101101010110101010011100111001001111011000110010111110001111011011001000111100011001000101001110010111001001010111010000101100011111000100000000100101010110011011001000000011001
Pair \(Z_2\) Length of longest common subsequence
7RRB_1,6OBH_1 194 4
7RRB_1,8GAF_1 169 3
6OBH_1,8GAF_1 191 3

Newick tree

 
[
	6OBH_1:99.86,
	[
		7RRB_1:84.5,8GAF_1:84.5
	]:15.36
]

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 \((0.85)(0.8)(\frac{626 }{\log_{20} 626}-\frac{232}{\log_{20}232})=111.\)
Status Protein1 Protein2 d d1/2
Query variables 7RRB_1 6OBH_1 145 114
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]