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Parikh vectors
2UUW_1 8EBU_1 1ZEH_1 Letter Amino acid
21 48 0 R Arginine
30 57 2 S Serine
35 32 0 P Proline
38 54 1 V Valine
38 40 1 G Glycine
15 18 0 H Histidine
24 39 2 I Isoleucine
8 8 0 W Tryptophan
52 54 0 A Alanine
21 48 0 D Aspartic acid
46 42 1 T Threonine
25 63 2 E Glutamic acid
39 60 2 L Leucine
23 61 0 K Lycine
11 22 0 M Methionine
17 33 0 F Phenylalanine
20 22 2 N Asparagine
7 14 4 C Cysteine
24 37 2 Q Glutamine
21 30 2 Y Tyrosine 

2UUW_1|Chain A|RNA-DIRECTED RNA POLYMERASE|SAPPORO VIRUS (95342)
>8EBU_1|Chain A|TFIIH basal transcription factor complex helicase XPB subunit|Homo sapiens (9606)
>1ZEH_1|Chains A, C|INSULIN|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 \(w=f(c)\)
2UUW , Knot 209 515 0.84 40 253 482 
DEFQWKGLPVVKSGLDVGGMPTGTRYHRSPAWPEEQPGETHAPAPFGSGDKRYTFSQTEMLVNGLKPYTEPTAGVPPQLLSRAVTHVRSYIETIIGTHRSPVLTYHQACELLERTTSCGPFVQGLKGDYWDEEQQQYTGVLANHLEQAWDKANKGIAPRNAYKLALKDELRPIEKNKAGKRRLLWGCDAATTLIATAAFKAVATRLQVVTPMTPVAVGINMDSVQMQVMNDSLKGGVLYCLDYSKWDSTQNPAVTAASLAILERFAEPHPIVSCAIEALSSPAEGYVNDIKFVTRGGLPSGMPFTSVVNSINHMIYVAAAILQAYESHNVPYTGNVFQVETIHTYGGGCMYSVCPATASIFHTVLANLTSYGLKPTAADKSDAIKPTNTPVFLKRTFTQTPHGIRALLDITSITRQFYWLKANRTSDPSSPPAFDRQARSAQLENALAYASQHGPVMFDTVRQIAIKTAQGEGLVLVNTNYDQALATYNAWFIGGTVPDPVGHTEGTHKIVFEME
8EBU , Knot 302 782 0.85 40 313 716 
MGKRDRADRDKKKSRKRHYEDEEDDEEDAPGNDPQEAVPSAAGKQVDESGTKVDEYGAKDYRLQMPLKDDHTSRPLWVAPDGHIFLEAFSPVYKYAQDFLVAIAEPVCRPTHVHEYKLTAYSLYAAVSVGLQTSDITEYLRKLSKTGVPDGIMQFIKLCTVSYGKVKLVLKHNRYFVESCHPDVIQHLLQDPVIRECRLRNSEGEATELITETFTSKSAISKTAESSGGPSTSRVTDPQGKSDIPMDLFDFYEQMDKDEEEEEETQTVSFEVKQEMIEELQKRCIHLEYPLLAEYDFRNDSVNPDINIDLKPTAVLRPYQEKSLRKMFGNGRARSGVIVLPCGAGKSLVGVTAACTVRKRCLVLGNSAVSVEQWKAQFKMWSTIDDSQICRFTSDAKDKPIGCSVAISTYSMLGHTTKRSWEAERVMEWLKTQEWGLMILDEVHTIPAKMFRRVLTIVQAHCKLGLTATLVREDDKIVDLNFLIGPKLYEANWMELQNNGYIAKVQCAEVWCPMSPEFYREYVAIKTKKRILLYTMNPNKFRACQFLIKFHERRNDKIIVFADNVFALKEYAIRLNKPYIYGPTSQGERMQILQNFKHNPKINTIFISKVGDTSFDLPEANVLIQISSHGGSRRQEAQRLGRVLRAKKGMVAEEYNAFFYSLVSQDTQEMAYSTKRQRFLVDQGYSFKVITKLAGMEEEDLAFSTKEEQQQLLQKVLAATDLDAEEEVVAGEFGSRSSQASRRFGTMSSMSGADDTVYMEYHSSRSKAPSKHVHPLFKRFRK
1ZEH , Knot 15 21 0.72 22 20 19 
GIVEQCCTSICSLYQLENYCN

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 \(c\).

\(|P_{f(2UUW_1)}(2) \setminus P_{f(8EBU_1)}(2)|=36\), \(|P_{f(8EBU_1)}(2) \setminus P_{f(2UUW_1)}(2)|=96\). 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 1:00101011111001101111101000000111100011000111111010000010000111011010001011111011001100100010011100001110000100110000001111011010010000000011110010011001001111001001110001011000011000111100110011101110111001011011011111101001010110001011110010000100000111011011110011010111001101100110101001011001111011110011001001101111110100000110010110100100011101001011010110011101000110101100001101000111100010001011011101001000101101000001001111000100101001110100011111001001110010101111100000011100011111101101110001000111010
Pair \(Z_2\) Length of longest common subsequence
2UUW_1,8EBU_1 132 5
2UUW_1,1ZEH_1 247 4
8EBU_1,1ZEH_1 305 3

Newick tree

 
[
	1ZEH_1:15.62,
	[
		2UUW_1:66,8EBU_1:66
	]:89.62
]

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 \((0.85)(0.8)(\frac{1297 }{\log_{20} 1297}-\frac{515}{\log_{20}515})=200.\)
Status Protein1 Protein2 d d1/2
Query variables 2UUW_1 8EBU_1 258 210
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]