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Parikh vectors
2OCE_1 7CJK_1 6VTZ_1 Letter Amino acid
58 41 29 G Glycine
53 25 29 K Lycine
12 9 8 M Methionine
25 29 27 F Phenylalanine
38 21 9 P Proline
39 41 26 S Serine
51 26 43 D Aspartic acid
63 33 50 E Glutamic acid
12 12 7 H Histidine
28 21 46 I Isoleucine
4 2 2 W Tryptophan
13 12 27 Y Tyrosine
53 21 16 R Arginine
3 5 5 C Cysteine
88 36 44 L Leucine
40 21 22 T Threonine
92 33 25 A Alanine
20 15 34 N Asparagine
26 13 15 Q Glutamine
62 29 27 V Valine 

2OCE_1|Chain A|Hypothetical protein PA5201|Pseudomonas aeruginosa PAO1 (208964)
>7CJK_1|Chain A|4-hydroxyphenylpyruvate dioxygenase|Arabidopsis thaliana (3702)
>6VTZ_1|Chain A|Non-ribosomal peptide synthetase|Methanobrevibacter ruminantium (strain ATCC 35063 / DSM 1093 / JCM 13430 / OCM 146 / M1) (634498)
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 \(w=f(c)\)
2OCE , Knot 286 780 0.81 40 264 692 
SLDSINTRIAEELSALPSGRVQPQQVAAAVALLDEGSTVPFIARYRKEVTGSLDDTQLRMLEERLRYLRELEERRGAILASIEEQGKLTPELARDIKLADTKTRLEDLYLPYKQKRRTKGQIALEAGLGALADALFDDPTLVPESEAARFVDAEKGFADVKAVLEGAKYILMERFAEDATLLDKLRVFMKNEATLTARVVPGKEQEGAKFSDYFEHDEPLKSAPSHRALAIFRGRNEGVLSASLKVGEEAPGTLHPCEVMIAERFGLSNQGRAADKWLAEVVRWTWKVKLYTHLETDLFGELRDGAEDEAISVFARNLHDLLLAAPAGPRATLGLDPGLRTGVKVAVVDATGKLLDTATVYPHAPKNQWDQTLAVLAALCAKHQVELIAIGNGTASRETDKLAGELIKKYPGMKLTKIMVSEAGASVYSASELAAKEFPELDVSLRGAVSIARRLQDPLAELVKIEPKSIGVGQYQHDVSQLKLARSLDAVVEDCVNAVGVDVNTASAALLARISGLNSTLAQNIVAHRDANGAFRTRDELKKVSRLGEKTFEQAAGFLRVMNGDNPLDASAVHPETYPLVQRIAADTERDIRSLIGDSAFLKRLDPKKFTDETFGLPTVTDILKELDKPGRDPRPEFKTAEFQEGVESLKDLKPGMVLEGVVTNVTNFGAFVDIGVHQDGLVHISALSEKFVKDPYEVVKAGDIVKVKVMEVDIPRNRVGLSMRMSDTPGEKVEGQRGGRPTGSGQPRQERGAPRGQSAPPANNAMAALFANAKQLKKK
7CJK , Knot 187 445 0.85 40 234 425 
MGHQNAAVSENQNHDDGAASSPGFKLVGFSKFVRKNPKSDKFKVKRFHHIEFWCGDATNVARRFSWGLGMRFSAKSDLSTGNMVHASYLLTSGDLRFLFTAPYSPSLSAGEIKPTTTASIPSFDHGSCRSFFSSHGLGVRAVAIEVEDAESAFSISVANGAIPSSPPIVLNEAVTIAEVKLYGDVVLRYVSYKAEDTEKSEFLPGFERVEDASSFPLDYGIRRLDHAVGNVPELGPALTYVAGFTGFHQFAEFTADDVGTAESGLNSAVLASNDEMVLLPINEPVHGTKRKSQIQTYLEHNEGAGLQHLALMSEDIFRTLREMRKRSSIGGFDFMPSPPPTYYQNLKKRVGDVLSDDQIKECEELGILVDRDDQGTLLQIFTKPLGDRPTIFIEIIQRVGCMMKDEEGKAYQSGGCGGFGKGNFSELFKSIEEYEKTLEAKQLVG
6VTZ , Knot 199 491 0.83 40 231 463 
GSGAVELNYVAPKTRLEHEICAIFSSILNIETVGVEDNFFEIGGTSLIASKLIIELLKQGYSVRYDDIFRNKTPRALAKLLSGEDIGEEDLDLTDDIIKNYNYGEINELLQENTWENFFDGENLELGNVLLTGATGFLGIHILYEFIKSEEGKIYCMLRKGKFDSCQERLIDVMNDYFDEDFTDLVGSRIIPIEGDITEIDDFKQLEDEPIDTVINSAALVKHYTADDYIFRVNVDGVINGLKFAQTRNNIKYVQISTISVLSSYSLNEEAYPNQEYDERTLYYEQDLENKYVCSKFLAERAVLQAATKGLPVKIIRVGNLMSRYSDGVFQKNYDTNAFLNNIKTIKKLGAMNPAMASEKVDMSQIDYVAKGILALSKTPEKSRVFHCMNNHYISHRDIVDALNTYGYGIEEVDFEEFKQIYEQNMNENIQGIITADFSIDDFDEEDDFEENVEIEQTVDILHSLGFDWPEADEEYLKRLFDYLNKFDYFE

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 \(c\).

\(|P_{f(2OCE_1)}(2) \setminus P_{f(7CJK_1)}(2)|=77\), \(|P_{f(7CJK_1)}(2) \setminus P_{f(2OCE_1)}(2)|=47\). 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 1:010010001100101110101010011111111001001111100000101010000101100010010010000111110100010101011001011000001001011000000001011101111111011100101110001101101001110101110110011100110010110010111000101010111100001101000100001100110001111101000111010101100111010100111100111000101100111011010101010001000111010011000110111001001111111110101110111001101111010101100101010110001000111111101000101111101010000001110110001110100111001110100100111001101010101110110010011101101010011110000010010110010111000101111010010111110101100011001110001011100000100100110001001111101101001101011010001110011100000100111001110010100100001111010011001001100101010010100110010010111110111001001111101110001110101100011001001101101101011010110001110101000110010100110101010100001110100111100111111101001000
Pair \(Z_2\) Length of longest common subsequence
2OCE_1,7CJK_1 124 5
2OCE_1,6VTZ_1 131 5
7CJK_1,6VTZ_1 133 4

Newick tree

 
[
	6VTZ_1:67.28,
	[
		2OCE_1:62,7CJK_1:62
	]:5.28
]

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 \((0.85)(0.8)(\frac{1225 }{\log_{20} 1225}-\frac{445}{\log_{20}445})=202.\)
Status Protein1 Protein2 d d1/2
Query variables 2OCE_1 7CJK_1 253 199.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]