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Parikh vectors
1JTJ_1 2DGA_1 5QEV_1 Letter Amino acid
0 25 17 V Valine
0 13 14 Q Glutamine
7 45 14 G Glycine
0 20 11 M Methionine
0 40 25 S Serine
0 30 13 T Threonine
0 14 6 W Tryptophan
6 35 13 A Alanine
0 25 11 N Asparagine
0 37 23 K Lycine
0 32 22 P Proline
0 54 18 D Aspartic acid
0 28 18 I Isoleucine
0 35 29 L Leucine
0 28 14 F Phenylalanine
0 28 9 Y Tyrosine
0 22 19 R Arginine
7 7 4 C Cysteine
0 27 30 E Glutamic acid
0 20 11 H Histidine 

1JTJ_1|Chain A|HIV-1Lai SL1|
>2DGA_1|Chain A|Beta-glucosidase|Triticum aestivum (4565)
>5QEV_1|Chain A|Tyrosine-protein phosphatase non-receptor type 1|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 \(w=f(c)\)
1JTJ , Knot 11 23 0.50 8 11 16 
CUUGCUGAAGCACGCACGGCAAG
2DGA , Knot 228 565 0.85 40 283 529 
MHHHHHHSSGLVPRGSGMKETAAAKFERQHMDSPDLGTDDDDKAMAGTPSKPAEPIGPVFTKLKPWQIPKRDWFDKDFLFGASTSAYQIEGAWNEDGKGPSTWDHFCHTYPERISDMTNGDVAANSYHLYEEDVKALKDMGMKVYRFSISWSRILPDGTGKVNQAGIDYYNKLINSLIDNDIVPYVTIWHWDTPQALEDKYGGFLNRQIVDDYKQFAEVCFKNFGDRVKNWFTFNEPHTYCCFSYGEGIHAPGRCSPGMDCAVPEGDSLREPYTAGHHILLAHAEAVQLFKARYNMHGDSKIGMAFDVMGYEPYQDSFLDDQARERSIDYNMGWFLEPVVRGDYPFSMRSLIGDRLPMFTKEEQEKLASSCDIMGLNYYTSRFSKHVDMSPDFTPTLNTDDAYASSETTGSDGNDIGPITGTYWIYMYPKGLTDLLLIMKEKYGNPPVFITENGIADVEGDESMPDPLDDWKRLDYLQRHISAVKDAIDQGADVRGHFTWGLIDNFEWSLGYSSRFGLVYIDKNDGNKRKLKKSAKWFSKFNSVPKPLLKTTNNNATMTAASVSV
5QEV , Knot 141 321 0.84 40 205 309 
MEMEKEFEQIDKSGSWAAIYQDIRHEASDFPSRVAKLPKNKNRNRYRDVSPFDHSRIKLHQEDNDYINASLIKMEEAQRSYILTQGPLPNTVGHFWEMVWEQKSRGVVMLNRVMEKGSLKCAQYWPQKEEKEMIFEDTNLKLTLISEDIKSYYTVRQLELENLTTQETREILHFHYTTWPDFGVPESPASFLNFLFKVRESGSLSPEHGPVVVHCSAGIGRSGTFCLADTCLLLMDKRKDPSSVDIKKVLLEMRKFRMGLIQTADQLRFSYLAVIEGAKFIMGDSSVQDQWKELSHEDLEPPPEHIPPPPRPPKRILEPHN

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 \(c\).

\(|P_{f(1JTJ_1)}(2) \setminus P_{f(2DGA_1)}(2)|=6\), \(|P_{f(2DGA_1)}(2) \setminus P_{f(1JTJ_1)}(2)|=278\). 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 1:00010011110101010110111
Pair \(Z_2\) Length of longest common subsequence
1JTJ_1,2DGA_1 284 2
1JTJ_1,5QEV_1 208 2
2DGA_1,5QEV_1 182 4

Newick tree

 
[
	1JTJ_1:13.76,
	[
		5QEV_1:91,2DGA_1:91
	]:42.76
]

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 \((0.85)(0.8)(\frac{588 }{\log_{20} 588}-\frac{23}{\log_{20}23})=172.\)
Status Protein1 Protein2 d d1/2
Query variables 1JTJ_1 2DGA_1 224 115.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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