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Parikh vectors
5XOD_1 3GRI_1 3PSF_1 Letter Amino acid
12 13 44 R Arginine
8 7 6 C Cysteine
17 34 91 L Leucine
11 35 97 E Glutamic acid
4 9 15 M Methionine
10 14 41 F Phenylalanine
8 12 48 Y Tyrosine
14 14 65 S Serine
13 26 48 T Threonine
5 1 8 W Tryptophan
12 36 65 A Alanine
12 16 66 N Asparagine
15 36 36 G Glycine
4 15 28 H Histidine
8 31 85 I Isoleucine
5 29 76 D Aspartic acid
12 13 44 Q Glutamine
4 28 81 K Lycine
11 23 37 P Proline
14 32 49 V Valine 

5XOD_1|Chain A|Mothers against decapentaplegic homolog 2|Homo sapiens (9606)
>3GRI_1|Chains A, B|Dihydroorotase|Staphylococcus aureus subsp. aureus MW2 (196620)
>3PSF_1|Chain A|Transcription elongation factor SPT6|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
5XOD , Knot 94 199 0.83 40 159 195 
GPDLQPVTYSEPAFWCSIAYYELNQRVGETFHASQPSLTVDGFTDPSNSERFCLGLLSNVNRNATVEMTRRHIGRGVRLYYIGGEVFAECLSDSAIFVQSPNCNQRYGWHPATVCKIPPGCNLKIFNNQEFAALLAQSVNQGFEAVYQLTRMCTIRMSFVKGWGAEYRRQTVTSTPCWIELHLNGPLQWLDKVLTQMGS
3GRI , Knot 174 424 0.82 40 218 401 
MKLIKNGKVLQNGELQQADILIDGKVIKQIAPAIEPSNGVDIIDAKGHFVSPGFVDVHVHLREPGGEYKETIETGTKAAARGGFTTVCPMPNTRPVPDSVEHFEALQKLIDDNAQVRVLPYASITTRQLGKELVDFPALVKEGAFAFTDDGVGVQTASMMYEGMIEAAKVNKAIVAHCEDNSLIYGGAMHEGKRSKELGIPGIPNICESVQIARDVLLAEAAGCHYHVCHVSTKESVRVIRDAKRAGIHVTAEVTPHHLLLTEDDIPGNNAIYKMNPPLRSTEDREALLEGLLDGTIDCIATDHAPHARDEKAQPMEKAPFGIVGSETAFPLLYTHFVKNGDWTLQQLVDYLTIKPCETFNLEYGTLKENGYADLTIIDLDSEQEIKGEDFLSKADNTPFIGYKVYGNPILTMVEGEVKFEGDK
3PSF , Knot 376 1030 0.84 40 311 910 
GIDPFTGLSSDKIDEMYDIFGDGHDYDWALEIENEELENGNDNNEAEEEEIDEETGAIKSTKKKISLQDIYDLEDLKKNLMTEGDMKIRKTDIPERYQELRAGITDYGNMSSEDQELERNWIAEKISVDKNFDANYDLTEFKEAIGNAIKFITKENLEVPFIYAYRRNYISSREKDGFLLTEDDLWDIVSLDIEFHSLVNKKDYVQRFYAELHIDDPIVTEYFKNQNTASIAELNSLQDIYDYLEFKYANEINEMFINHTGKTGKKHLKNSSYEKFKASPLYQAVSDIGISAEDVGENISSQHQIHPPVDHPSSKPVEVIESILNANSGDLQVFTSNTKLAIDTVQKYYSLELSKNTKIREKVRSDFSKYYLADVVLTAKGKKEIQKGSLYEDIKYAINRTPMHFRRDPDVFLKMVEAESLNLLSVKLHMSSQAQYIEHLFQIALETTNTSDIAIEWNNFRKLAFNQAMDKIFQDISQEVKDNLTKNCQKLVAKTVRHKFMTKLDQAPFIPNVRDPKIPKILSLTCGQGRFGADAIIAVYVNRKGDFIRDYKIVDNPFDKTNPEKFEDTLDNIIQSCQPNAIGINGPNPKTQKFYKRLQEVLHKKQIVDSRGHTIPIIYVEDEVAIRYQNSERAAQEFPNKPPLVKYCIALARYMHSPLLEYANLTSEEVRSLSIHPHQNLLSSEQLSWALETAFVDIVNLVSVEVNKATDNNYYASALKYISGFGKRKAIDFLQSLQRLNEPLLARQQLITHNILHKTIFMNSAGFLYISWNEKRQKYEDLEHDQLDSTRIHPEDYHLATKVAADALEYDPDTIAEKEEQGTMSEFIELLREDPDRRAKLESLNLESYAEELEKNTGLRKLNNLNTIVLELLDGFEELRNDFHPLQGDEIFQSLTGESEKTFFKGSIIPVRVERFWHNDIICTTNSEVECVVNAQRHAGAQLRRPANEIYEIGKTYPAKVIYIDYANITAEVSLLDHDVKQQYVPISYSKDPSIWDLKQELEDAEEERKLMMAEARAKRTHRVINHPYY

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 \(c\).

\(|P_{f(5XOD_1)}(2) \setminus P_{f(3GRI_1)}(2)|=63\), \(|P_{f(3GRI_1)}(2) \setminus P_{f(5XOD_1)}(2)|=122\). 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 1:1101011000011110011000100011001010010101011001000001011110010001010100001101101001110111001000111100100000011011010011110010110000111111001001101100100100101011011110000001000101101010111011001100110
Pair \(Z_2\) Length of longest common subsequence
5XOD_1,3GRI_1 185 3
5XOD_1,3PSF_1 202 4
3GRI_1,3PSF_1 151 6

Newick tree

 
[
	5XOD_1:10.97,
	[
		3GRI_1:75.5,3PSF_1:75.5
	]:27.47
]

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 \((0.85)(0.8)(\frac{623 }{\log_{20} 623}-\frac{199}{\log_{20}199})=120.\)
Status Protein1 Protein2 d d1/2
Query variables 5XOD_1 3GRI_1 153 112
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]