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Parikh vectors
8BIT_1 9ARU_1 1AHS_1 Letter Amino acid
26 67 8 P Proline
37 105 5 S Serine
32 94 6 T Threonine
1 32 2 C Cysteine
40 94 15 G Glycine
30 71 6 I Isoleucine
13 11 2 M Methionine
29 61 5 D Aspartic acid
45 58 0 K Lycine
30 78 4 F Phenylalanine
50 51 4 E Glutamic acid
52 94 16 V Valine
16 18 0 H Histidine
49 107 6 L Leucine
9 12 3 W Tryptophan
22 53 6 Y Tyrosine
28 78 16 A Alanine
34 42 9 R Arginine
18 88 7 N Asparagine
9 66 6 Q Glutamine 

8BIT_1|Chains A, B|4-hydroxybutyrate--CoA ligase 1|Metallosphaera sedula DSM 5348 (399549)
>9ARU_1|Chains A, D[auth B], G[auth C]|Spike glycoprotein|Severe acute respiratory syndrome coronavirus 2 (2697049)
>1AHS_1|Chains A, B, C|AFRICAN HORSE SICKNESS VIRUS (SEROTYPE 4) VP7|African horsesickness virus (40050)
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 \(w=f(c)\)
8BIT , Knot 229 570 0.85 40 255 533 
MHHHHHHVTVQDFFRKFIEFQNSPNEKSLQEIVKLVGQLDLRRFNWVRDVFEDIHVKERGSKTALIWRDINTGEEAKLSYHELSLMSNRVLSTLRKHGLKKGDVVYLMTKVHPMHWAVFLAVIKGGFVMVPSATNLTVAEMKYRFSDLKPSAIISDSLRASVMEEALGSLKVEKFLIDGKRETWNSLEDESSNAEPEDTRGEDVIINYFTSGTTGMPKRVIHTAVSYPVGSITTASIVGVRESDLHLNLSATGWAKFAWSSFFSPLLVGATVVGINYEGKLDTRRYLGEVENLGVTSFCAPPTAWRQFITLDLDQFRFERLRSVVSAGEPLNPEVIKIWKDKFNLTIRDFYGQTETTAMVGNFPFLKVKPGSMGKPHPLYDIRLLDDEGKEITKPYEVGHITVKLNPRPIGLFLGYSDEKKNMESFREGYYYTGDKAYFDEEGYFYFVGRGDDVIKTSDYRVGPFEVESALLEHPAVAEAAVVGVPDTVRWQLVKAYIVLKKGYMPSKELAEEIREKMKTLLSPYKVPRIIEFVDELPKTISGKIRRVELRKREEEKRKKGEVGQNEYVF
9ARU , Knot 451 1280 0.84 40 328 1097 
MFVFLVLLPLVSSQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHVSGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLGVYYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPINLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQDVNCTEVPVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSPGSASSVASQSIIAYTMSLGAENSCAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSPIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTICSGWTFGAGPALQIPFPMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTPSALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGRSLEVLFQGPGSAWSHPQFEKGGGSGGGGSGGSAWSHPQFEK
1AHS , Knot 62 126 0.79 36 99 120 
TGPYAGAVEVQQSGRYYVPQGRTRGGYINSNIAEVCMDAGAAGQVNALLAPRRGDAVMIYFVWRPLRIFCDPQGASLESAPGTFVTVDGVNVAAGDVVAWNTIAPVNVGNPGARRSILQFEVLWYT

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 \(c\).

\(|P_{f(8BIT_1)}(2) \setminus P_{f(9ARU_1)}(2)|=25\), \(|P_{f(9ARU_1)}(2) \setminus P_{f(8BIT_1)}(2)|=98\). 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 1:100000010100110011010001000010011011101010010110011001010001000111100100100101000010110001100100011001011011001011011111111011111110100101101000100101011100010101100111010100111010000100100000010100001001110010010011100110011001110100101111000010101010111011100110111111011110001010000011010011100101110110011010100101001001101101101011011000101010010100000111101111010110110101100101100010010010011010101010111111100000001001001000010010100010101110100110000001111010011100111101111111001010110101110010110001100100010011010011011011001100101010010100000000001011000011
Pair \(Z_2\) Length of longest common subsequence
8BIT_1,9ARU_1 123 4
8BIT_1,1AHS_1 208 4
9ARU_1,1AHS_1 245 4

Newick tree

 
[
	1AHS_1:12.30,
	[
		8BIT_1:61.5,9ARU_1:61.5
	]:64.80
]

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 \((0.85)(0.8)(\frac{1850 }{\log_{20} 1850}-\frac{570}{\log_{20}570})=317.\)
Status Protein1 Protein2 d d1/2
Query variables 8BIT_1 9ARU_1 407 293
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]