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Parikh vectors
1SLU_1 8XUO_1 6TYR_1 Letter Amino acid
7 33 20 T Threonine
2 14 6 W Tryptophan
8 17 9 Y Tyrosine
7 36 35 A Alanine
6 28 13 Q Glutamine
13 90 66 L Leucine
6 57 14 S Serine
5 37 10 N Asparagine
5 52 8 I Isoleucine
15 43 35 V Valine
9 40 18 D Aspartic acid
12 53 30 E Glutamic acid
9 34 37 G Glycine
4 14 13 M Methionine
2 33 16 F Phenylalanine
8 36 44 P Proline
5 39 32 R Arginine
2 15 1 C Cysteine
2 17 17 H Histidine
15 64 15 K Lycine 

1SLU_1|Chain A|ECOTIN|Escherichia coli (562)
>8XUO_1|Chains A, B[auth G]|NRC2|Solanum lycopersicum (4081)
>6TYR_1|Chain A|Laccase|Thermus thermophilus (strain HB27 / ATCC BAA-163 / DSM 7039) (262724)
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 \(w=f(c)\)
1SLU , Knot 72 142 0.83 40 115 138 
AESVQPLEKIAPYPQAEKGMKRQVIQLTPQEDESTLKVELLIGQTLEVDCNLHRLGGKLENKTLEGWGYDYYVFDKVSSPVSTMMHCPDGKKEKKFVTAYLGDAGMLRYNSKLPIVVYTPDNVDVKYRVWKAEEKIDNAVVR
8XUO , Knot 287 752 0.84 40 287 683 
DDDLSARGSEERKPPVVEEDDVVGFDEEADIVINRLLGESNHLEVVPVVGMPGLGKTTLANKIYKHPKIGYEFFTRIWVYVSQSYRRRELFLNIISKFTRNTKQYHGMCEEDLADEIQEFLGKGGKYLVVLDDVWSDEAWERIKIAFPNNNKPNRVLLTTRDSKVAKQCNPIPHDLKFLTEDESWILLEKKVFHKDKCPPELVLSGKSIAKKCKGLPLAIVVIAGALIGKGKTPREWKQVDDSVSEHLINRDHPENCNKLVQMSYDRLPYDLKACFLYCSAFPGGFQIPAWKLIRLWIAEGFIQYKGHLSLECKGEDNLNDLINRNLVMVMERTSDGQIKTCRLHDMLHEFCRQEAMKEENLFQEIKLGSEQYFPGKRELSTYRRLCIHSSVLDFFSTKPSAEHVRSFLSFSSKKIEMPSADIPTIPKGFPLLRVLDVESINFSRFSREFYQLYHLRYVAFSSDSIKILPKLMGELWNIQTIIINTQQRTLDIQANIWNMERLRHLHTNSSAKLPVPVAPKNSKVTLVNQSLQTLSTIAPESCTEEVFARTPNLKKLGIRGKISVLLDNKSAASLKNVKRLEYLENLKLINDSSIQTSKLRLPPAYIFPTKLRKLTLLDTWLEWKDMSILGQLEHLEVLKMKENGFSGESWESTGGFCSLLVLWIERTNLVSWKASADDFPRLKHLVLICCDNLKEVPIALADIRSFQVMMLQNSTKTAAISARQIQAKKDNQTQQGTKNIAFKLSIFPPDL
6TYR , Knot 177 439 0.81 40 201 397 
GPSFPEPKVVRSQGGLLSLKLSATPTPLAIAGQRATLLTYGGSFPGPTLRVRPRDTVRLTLENRLPEPTNLHWHGLPISPKVDDPFLEIPPGESWTYEFTVPKELAGTFWYHPHLHGRVAPQLFAGLLGALVVESSLDAIPELREAEEHLLVLKDLALQGGRPAPHTPMDWMNGKEGDLVLVNGALRPTLVAQKATLRLRLLNASNARYYRLALQDHPLYLIAADGGFLEEPLEVSELLLAPGERAEVLVRLRKEGRFLLQALPYDRGAMGMMDMGGMAHAMPQGPSRPETLLYLIAPKNPKPLPLPKALSPFPTLPAPVVTRRLVLTEDMMAARFFINGQVFDHRRVDLKGQAQTVEVWEVENQGDMDHPFHLHVHPFQVLSVGGRPFPYRAWKDVVNLKAGEVARLLVPLREKGRTVFHCHIVEHEDRGMMGVLEVG

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 \(c\).

\(|P_{f(1SLU_1)}(2) \setminus P_{f(8XUO_1)}(2)|=21\), \(|P_{f(8XUO_1)}(2) \setminus P_{f(1SLU_1)}(2)|=193\). 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 1:1001011001110101001100011010100000010101111001010001001110100001011100001100100110011001010000011010110111100000111110010010100011010001001110
Pair \(Z_2\) Length of longest common subsequence
1SLU_1,8XUO_1 214 4
1SLU_1,6TYR_1 174 3
8XUO_1,6TYR_1 166 4

Newick tree

 
[
	1SLU_1:10.89,
	[
		6TYR_1:83,8XUO_1:83
	]:18.89
]

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 \((0.85)(0.8)(\frac{894 }{\log_{20} 894}-\frac{142}{\log_{20}142})=209.\)
Status Protein1 Protein2 d d1/2
Query variables 1SLU_1 8XUO_1 266 156
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]