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Parikh vectors
7YXT_1 8QGW_1 6SXV_1 Letter Amino acid
14 7 26 N Asparagine
18 5 11 Q Glutamine
17 10 34 I Isoleucine
27 14 32 K Lycine
5 0 12 W Tryptophan
32 7 39 A Alanine
10 5 14 M Methionine
20 10 23 P Proline
13 5 20 T Threonine
12 8 25 Y Tyrosine
23 17 38 E Glutamic acid
8 5 7 C Cysteine
17 4 17 H Histidine
48 18 36 L Leucine
19 9 13 F Phenylalanine
31 5 26 S Serine
16 5 24 R Arginine
27 7 29 G Glycine
26 12 42 V Valine
22 7 34 D Aspartic acid 

7YXT_1|Chains A, B, C, D|Indoleamine 2,3-dioxygenase 1|Homo sapiens (9606)
>8QGW_1|Chains A, C|NADH-quinone oxidoreductase subunit E|Aquifex aeolicus VF5 (224324)
>6SXV_1|Chains A, B|GH51 a-l-arabinofuranosidase|Geobacillus stearothermophilus (1422)
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 \(w=f(c)\)
7YXT , Knot 173 405 0.85 40 228 380 
MGSSHHHHHHSSGSAAYHIDEEVGFALPNPQENLPDFYNDWMFIAKHLPDLIESGQLRERVEKLNMLSIDHLTDHKSQRLARLVLGCITMAYVWGKGHGDVRKVLPRNIAVPYCQLSAALELPPILVYADCVLANWKKKDPNKPLTYENMDVLFSFRDGDCSKGFFLVSLLVEIAAASAIKVIPTVFKAMQMQERDTLLKALLEIASCLEKALQVFHQIHDHVNPKAFFSVLRIYLSGWKGNPQLSDGLVYEGFWEDPKEFAGGSAGQSSVFQCFDVLLGIQQTAGGGHAAQFLQDMRRYMPPAHRNFLCSLESNPSVREFVLSKGDAGLREAYDACVKALVSLRSYHLQIVTKYILIPASQQPKENKTSEDPSKLEAKGTGGTDLMNFLKTVRSTTEKSLLKEG
8QGW , Knot 80 160 0.84 38 127 158 
MFKTEFEFPEELKTKLQEHINYFPKKRQAILLCLHEIQNYYGYIPPESLKPLADMLELPLNHVEGVVAFYDMFDREDKAKYRIRVCVSIVCHLMGTNKLLKALENILGIKPGEVTPDGKFKIVPVQCLGACSEAPVFMVNDDEYKFESEVQLNEILSRYT
6SXV , Knot 207 502 0.85 40 264 474 
MATKKATMIIEKDFKIAEIDKRIYGSFIEHLGRAVYGGIYEPGHPQADENGFRQDVIELVKELQVPIIRYPGGNFVSGYNWEDGVGPKEQRPRRLDLAWKSVETNEIGLNEFMDWAKMVGAEVNMAVNLGTRGIDAARNLVEYCNHPSGSYYSDLRIAHGYKEPHKIKTWCLGNEMDGPWQIGHKTAVEYGRIACEAAKVMKWVDPTIELVVCGSSNRNMPTFAEWEATVLDHTYDHVDYISLHQYYGNRDNDTANYLALSLEMDDFIRSVVAIADYVKAKKRSKKTIHLSFDEWNVWYHSNEADKLIEPWTVAPPLLEDIYNFEDALLVGCMLITLMKHADRVKIACLAQLVNVIAPIMTEKNGPAWKQTIYYPFMHASVYGRGVALHPVISSPKYDSKDFTDVPYLESIAVYNEEKEEVTIFAVNRDMEDALLLECDVRSFEDYRVIEHIVLEHDNVKQTNSAQSSPVVPHRNGDAQLSDRKVSATLPKLSWNVIRLGKR

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 \(c\).

\(|P_{f(7YXT_1)}(2) \setminus P_{f(8QGW_1)}(2)|=145\), \(|P_{f(8QGW_1)}(2) \setminus P_{f(7YXT_1)}(2)|=44\). 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 1:110000000000101100100011111101000110100011111001101100101000100101101001000000011011110101101110101010011100111100010111011111101001110100001001100001011101001000011111011101111011011101101101000001101110110010011011001000101011101101010110101010011100111001001111011000110010111110001111011011001000111100011001000101001110010111001001010111010000101100011111000100000000100101010110011011001000000011001
Pair \(Z_2\) Length of longest common subsequence
7YXT_1,8QGW_1 189 5
7YXT_1,6SXV_1 158 4
8QGW_1,6SXV_1 205 3

Newick tree

 
[
	8QGW_1:10.29,
	[
		7YXT_1:79,6SXV_1:79
	]:25.29
]

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 \((0.85)(0.8)(\frac{565 }{\log_{20} 565}-\frac{160}{\log_{20}160})=117.\)
Status Protein1 Protein2 d d1/2
Query variables 7YXT_1 8QGW_1 148 103
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]