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Parikh vectors
2XHU_1 2HQY_1 5AIG_1 Letter Amino acid
55 24 18 A Alanine
26 24 8 D Aspartic acid
18 8 3 Q Glutamine
18 7 3 H Histidine
14 18 8 F Phenylalanine
31 10 4 P Proline
23 15 2 Y Tyrosine
42 19 9 R Arginine
31 16 8 G Glycine
25 22 8 I Isoleucine
50 8 2 S Serine
15 17 5 N Asparagine
20 8 3 C Cysteine
12 11 8 M Methionine
41 16 10 T Threonine
9 4 4 W Tryptophan
32 12 8 V Valine
29 21 10 E Glutamic acid
59 27 3 L Leucine
29 18 1 K Lycine 

2XHU_1|Chains A, B|RNA-directed RNA polymerase|Hepatitis C virus genotype 1b (strain HC-J4) (420174)
>2HQY_1|Chains A, B|conserved hypothetical protein|Bacteroides thetaiotaomicron (226186)
>5AIG_1|Chains A, B|LIMONENE-1,2-EPOXIDE HYDROLASE|UNIDENTIFIED (32644)
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 \(w=f(c)\)
2XHU , Knot 232 579 0.85 40 275 537 
ASMSYTWTGALITPCAAEESKLPINPLSNSLLRHHNMVYATTSRSASLRQKKVTFDRLQVLDDHYRDVLKEMKAKASTVKAKLLSIEEACKLTPPHSAKSKFGYGAKDVRNLSSRAVNHIRSVWEDLLEDTETPIDTTIMAKSEVFCVQPEKGGRKPARLIVFPDLGVRVCEKMALYDVVSTLPQAVMGSSYGFQYSPKQRVEFLVNTWKSKKCPMGFSYDTRCFDSTVTESDIRVEESIYQCCDLAPEARQAIRSLTERLYIGGPLTNSKGQNCGYRRCRASGVLTTSCGNTLTCYLKATAACRAAKLQDCTMLVNGDDLVVICESAGTQEDAAALRAFTEAMTRYSAPPGDPPQPEYDLELITSCSSNVSVAHDASGKRVYYLTRDPTTPLARAAWETARHTPINSWLGNIIMYAPTLWARMILMTHFFSILLAQEQLEKALDCQIYGACYSIEPLDLPQIIERLHGLSAFTLHSYSPGEINRVASCLRKLGVPPLRTWRHRARSVRAKLLSQGGRAATCGRYLFNWAVRTKLKLTPIPAASQLDLSGWFVAGYSGGDIYHSLSRARPRLEHHHHHH
2HQY , Knot 135 305 0.84 40 191 293 
MIPFKDITLADRDTITAFTMKSDRRNCDLSFSNLCSWRFLYDTQFAVIDDFLVFKFWAGEQLAYMMPVGNGDLKAVLRKLIEDADKEKHNFCMLGVCSNMRADLEAILPERFIFTEDRAYADYIYLRSDLATLKGKKFQAKRNHINRFRNTYPDYEYTPITPDRIQECLDLEAEWCKVNNCDQQEGTGNERRALIYALHNFEALGLTGGILHVNGKIVAFTFGMPINHETFGVHVEKADTSIDGAYAMINYEFANRIPEQYIYINREEDLGIEGLRKAKLSYQPVTILEKYMACLKDHPMDMVRW
5AIG , Knot 65 125 0.83 40 101 123 
MTPIETVTAFIAHWNSGDMEAMYDLCAEDVVWHNIPMEPIAGKPAMRAAVEGFMANVSQCDWQVHAIAANGATVLTERTDGFTFTNGRRATIRVMGTFECDAERRIIAWRDYFDMLEFQREFAGA

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 \(c\).

\(|P_{f(2XHU_1)}(2) \setminus P_{f(2HQY_1)}(2)|=138\), \(|P_{f(2HQY_1)}(2) \setminus P_{f(2XHU_1)}(2)|=54\). 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 1:101000101111010110000111011000110000110100000101000010100101100000011001010100101011010010010110010001101100100100011001001100110000011000111000110101001100110111110111010001110011001101111000110001000101110010000011110000001000100001010001000001110100110010001011111000010001000001011100001001000101011001101000011101001111000110000111101100110000111101101000101100000010110010100100100010011101110010001100111011101101110111100110111100010011000101100010110110110010110110100001101001100100111111001000100101011001101100100110111000101011111001010111111001101000100101010000000
Pair \(Z_2\) Length of longest common subsequence
2XHU_1,2HQY_1 192 4
2XHU_1,5AIG_1 224 4
2HQY_1,5AIG_1 176 3

Newick tree

 
[
	2XHU_1:10.20,
	[
		2HQY_1:88,5AIG_1:88
	]:21.20
]

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 \((0.85)(0.8)(\frac{884 }{\log_{20} 884}-\frac{305}{\log_{20}305})=156.\)
Status Protein1 Protein2 d d1/2
Query variables 2XHU_1 2HQY_1 202 152
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]