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Parikh vectors
6VSX_1 9ENQ_1 5TUN_1 Letter Amino acid
9 31 9 Q Glutamine
10 51 19 K Lycine
11 54 16 S Serine
11 60 5 T Threonine
9 95 7 R Arginine
2 29 19 N Asparagine
5 21 8 C Cysteine
13 121 12 L Leucine
8 79 8 P Proline
0 9 4 W Tryptophan
7 48 13 Y Tyrosine
14 83 15 V Valine
16 138 18 A Alanine
11 101 24 G Glycine
10 54 8 I Isoleucine
7 21 4 M Methionine
6 69 9 D Aspartic acid
15 79 10 E Glutamic acid
18 32 2 H Histidine
6 60 5 F Phenylalanine 

6VSX_1|Chain A|DNA helicase|Bacillus subtilis (1423)
>9ENQ_1|Chain A|DNA polymerase catalytic subunit|Human alphaherpesvirus 1 strain KOS (10306)
>5TUN_1|Chain A|Cathepsin K|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 \(w=f(c)\)
6VSX , Knot 90 188 0.83 38 135 182 
MGSSHHHHHHSSGLVPRGSHMRSTRQIVEFTKAMLQDGADIEPFNRSGEMPLVVKTEGHESLCQKLAQEIGRLKKKGHETIAVICKTAHQCIQAHAHMSEYTDVRLIHKENQPFQKGVCVIPVYLAKGIEFDAVLVYDASEEHYHTEHDRRLLYTACTRAMHMLAVFYTGEASPFVTAVPPHLYQIAE
9ENQ , Knot 434 1235 0.83 40 329 1051 
MFSGGGGPLSPGGKSAARAASGFFAPAGPRGASRGPPPCLRQNFYNPYLAPVGTQQKPTGPTQRHTYYSECDEFRFIAPRVLDEDAPPEKRAGVHDGHLKRAPKVYCGGDERDVLRVGSGGFWPRRSRLWGGVDHAPAGFNPTVTVFHVYDILENVEHAYGMRAAQFHARFMDAITPTGTVITLLGLTPEGHRVAVHVYGTRQYFYMNKEEVDRHLQCRAPRDLCERMAAALRESPGASFRGISADHFEAEVVERTDVYYYETRPALFYRVYVRSGRVLSYLCDNFCPAIKKYEGGVDATTRFILDNPGFVTFGWYRLKPGRNNTLAQPRAPMAFGTSSDVEFNCTADNLAIEGGMSDLPAYKLMCFDIECKAGGEDELAFPVAGHPEDLVIQISCLLYDLSTTALEHVLLFSLGSCDLPESHLNELAARGLPTPVVLEFDSEFEMLLAFMTLVKQYGPEFVTGYNIINFDWPFLLAKLTDIYKVPLDGYGRMNGRGVFRVWDIGQSHFQKRSKIKVNGMVNIDMYGIITDKIKLSSYKLNAVAEAVLKDKKKDLSYRDIPAYYAAGPAQRGVIGEYCIQDSLLVGQLFFKFLPHLELSAVARLAGINITRTIYDGQQIRVFTCLLRLADQKGFILPDTQGRFRGAGGEAPKRPAAAREDEERPEEEGEDEDEREEGGGEREPEGARETAGRHVGYQGARVLDPTSGFHVNPVVVFDFASLYPSIIQAHNLCFSTLSLRADAVAHLEAGKDYLEIEVGGRRLFFVKAHVRESLLSILLRDWLAMRKQIRSRIPQSSPEEAVLLDKQQAAIKVVCNSVYGFTGVQHGLLPCLHVAATVTTIGREMLLATREYVHARWAAFEQLLADFPEAADMRAPGPYSMRIIYGDTDSIFVLCRGLTAAGLTAVGDKMASHISRALFLPPIKLECEKTFTKLLLIAKKKYIGVIYGGKMLIKGVDLVRKNNCAFINRTSRALVDLLFYDDTVSGAAAALAERPAEEWLARPLPEGLQAFGAVLVDAHRRITDPERDIQDFVLTAELSRHPRAYTNKRLAHLTVYYKLMARRAQVPSIKDRIPYVIVAQTREVEETVARLAALRELDAAAPGDEPAPPAALPSPAKRPRETPSPADPPGGASKPRKLLVSELAEDPAYAIAHGVALNTDYYFSHLLGAACVTFKALFGNNAKITESLLKRFIPEVWHPPDDVAARLRTAGFGAVGAGATAEETRRMLHRAFDTLA
5TUN , Knot 99 215 0.82 40 148 209 
APDSVDYRKKGYVTPVKNQGQCGSCWAFSSVGALEGQLKKKTGKLLNLSPQNLVDCVSENDGCGGGYMTNAFQYVQKNRGIDSEDAYPYVGQEESCMYNPTGKAAKCRGYREIPEGNEKALKRAVARVGPVSVAIDASLTSFQFYSKGVYYDESCNSDNLNHAVLAVGYGIQKGNKHWIIKNSWGENWGNKGYILMARNKNNACGIANLASFPKM

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 \(c\).

\(|P_{f(6VSX_1)}(2) \setminus P_{f(9ENQ_1)}(2)|=16\), \(|P_{f(9ENQ_1)}(2) \setminus P_{f(6VSX_1)}(2)|=210\). 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 1:11000000000011110100100000110100111001101011000101111100010001000110011010001000111100010001010101000001011000001100110111101101101011110010000000000001100100011011111001010111011110100110
Pair \(Z_2\) Length of longest common subsequence
6VSX_1,9ENQ_1 226 4
6VSX_1,5TUN_1 183 4
9ENQ_1,5TUN_1 211 4

Newick tree

 
[
	9ENQ_1:11.63,
	[
		6VSX_1:91.5,5TUN_1:91.5
	]:23.13
]

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 \((0.85)(0.8)(\frac{1423 }{\log_{20} 1423}-\frac{188}{\log_{20}188})=326.\)
Status Protein1 Protein2 d d1/2
Query variables 6VSX_1 9ENQ_1 413 237
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]