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Parikh vectors
6TWF_1 2XYF_1 1PXA_1 Letter Amino acid
5 2 6 W Tryptophan
21 4 38 R Arginine
8 2 5 C Cysteine
46 13 34 G Glycine
24 7 17 P Proline
21 2 12 F Phenylalanine
36 4 20 D Aspartic acid
16 6 15 Q Glutamine
35 6 12 K Lycine
12 2 6 M Methionine
50 11 46 L Leucine
23 9 19 T Threonine
19 1 16 Y Tyrosine
27 3 35 A Alanine
27 3 3 N Asparagine
30 4 34 E Glutamic acid
34 12 18 I Isoleucine
19 1 9 H Histidine
34 1 21 S Serine
45 6 28 V Valine 

6TWF_1|Chain A|5'-nucleotidase|Homo sapiens (9606)
>2XYF_1|Chains A, B|PROTEASE|HUMAN IMMUNODEFICIENCY VIRUS 1 (Z2/CDC-Z34 ISOLATE) (11683)
>1PXA_1|Chain A|P-HYDROXYBENZOATE HYDROXYLASE|Pseudomonas aeruginosa (287)
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 \(w=f(c)\)
6TWF , Knot 219 532 0.86 40 262 498 
MWELTILHTNDVHSRLEQTSEDSSKCVDASRCMGGVARLFTKVQQIRRAEPNVLLLDAGDQYQGTIWFTVYKGAEVAHFMNALRYDAMALGNHEFDNGVEGLIEPLLKEAKFPILSANIKAKGPLASQISGLYLPYKVLPVGDEVVGIVGYTSKETPFLSNPGTNLVFEDEITALQPEVDKLKTLNVNKIIALGHSGFEMDKLIAQKVRGVDVVVGGHSNTFLYTGNPPSKEVPAGKYPFIVTSDDGRKVPVVQAYAFGKYLGYLKIEFDERGNVISSHGNPILLDSSIPEDPSIKADINKWRIKLDDYSTQELGKTIVYLDGSSQSCRFRECNMGNLICDAMINNNLRHADEMFWNHVSMCILNGGGIRSPIDERNDGTITWENLAAVLPFGGTFDLVQLKGSTLKKAFEHSVHRYGQSTGEFLQVGGIHVVYDLSRKPGDRVVKLDVLCTKCRVPSYDPLKMDEVYKVILPNFLANGGDGFQMIKDELLRHDSGDQDINVVSTYISKMKVIYPAVEGRIKFSLEHHHHHH
2XYF , Knot 52 99 0.80 40 80 94 
PQITLWQRPLVTIKIGGQLKEALLDTGADDTVLEEMSLPGRWKPKMIGGIGGFIKVRQYDQIPIEICGHKAIGTVLVGPTPTNVIGRNLLTQIGCTLNF
1PXA , Knot 163 394 0.82 40 214 373 
MKTQVAIIGAGPSGLLLGQLLHKAGIDNVILERQTPDYVLGRIRAGVLEQGMVDLLREAGVDRRMARDGLVHEGVEIAFAGQRRRIDLKRLSGGKTVTVYGQTEVTRDLMEAREACGATTVYQAAEVRLHDLQGERPYVTFERDGERLRLDCDYIAGCDGFHGISRQSIPAERLKVFERVYPFGWLGLLADTPPVSHELIYANHPRGFALCSQRSATRSRYYVQVPLSEKVEDWSDERFWTELKARLPSEVAEKLVTGPSLEKSIAPLRSFVVEPMQHGRLFLAGDAAHIVPPTGAKGLDLAASDVSTLYRLLLKAYREGRGELLERYSAICLRRIWKAERFSWWMTSVLHRFPDTDAFSQRIQQTELEYYLGSEAGLATIAENYVGLPYEEIE

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 \(c\).

\(|P_{f(6TWF_1)}(2) \setminus P_{f(2XYF_1)}(2)|=200\), \(|P_{f(2XYF_1)}(2) \setminus P_{f(6TWF_1)}(2)|=18\). 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 1:1101011000010001000000000010100011111011001001001010111101100001011101001101101101100011111000100110111011100101111010101011110010110110011111001111110000001110011001110001011010100100101001111100110100111001011011111000011001011000111100111100001001111010111001101010100010110001011110001100101010100101010000000110011010100000010000110110011100010010011100101011011110011000001010100111111111010110101001001100010001000101101111011001000110011010110000011000110100100111101110110110110001100001000101100010010110111010101010000000
Pair \(Z_2\) Length of longest common subsequence
6TWF_1,2XYF_1 218 4
6TWF_1,1PXA_1 164 3
2XYF_1,1PXA_1 196 3

Newick tree

 
[
	2XYF_1:10.91,
	[
		6TWF_1:82,1PXA_1:82
	]:27.91
]

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 \((0.85)(0.8)(\frac{631 }{\log_{20} 631}-\frac{99}{\log_{20}99})=155.\)
Status Protein1 Protein2 d d1/2
Query variables 6TWF_1 2XYF_1 202 118
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]