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Parikh vectors
7WOB_1 8JBM_1 2DBG_1 Letter Amino acid
11 25 5 M Methionine
63 45 2 P Proline
105 55 11 S Serine
80 82 4 A Alanine
41 47 2 R Arginine
88 43 5 N Asparagine
30 23 1 C Cysteine
106 96 15 L Leucine
95 61 4 T Threonine
12 12 0 W Tryptophan
53 24 4 Y Tyrosine
95 76 4 V Valine
97 75 8 G Glycine
26 16 1 H Histidine
58 55 14 K Lycine
66 35 2 Q Glutamine
60 56 6 D Aspartic acid
51 68 7 E Glutamic acid
71 82 5 I Isoleucine
80 45 3 F Phenylalanine 

7WOB_1|Chains A, B[auth C], C[auth E]|Spike glycoprotein|Severe acute respiratory syndrome coronavirus 2 (2697049)
>8JBM_1|Chains A, D[auth C]|Sodium/potassium-transporting ATPase subunit alpha|Sus scrofa (9823)
>2DBG_1|Chain A|Myeloid cell nuclear differentiation antigen|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 \(w=f(c)\)
7WOB , Knot 452 1288 0.83 40 328 1099 
MFVFLVLLPLVSSQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHVSGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLGVYYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPINLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQGVNCTEVPVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSPGSASSVASQSIIAYTMSLGAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSFIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGAALQIPFAMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTASALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVFLSTFLSGLEVLFQGPGGWSHPQFEKGGGSGGGSGGSAWSHPQFEKGGSHHHHHHHH
8JBM , Knot 382 1021 0.86 40 316 915 
MGKGVGRDKYEPAAVSEHGDKKKAKKERDMDELKKEVSMDDHKLSLDELHRKYGTDLSRGLTPARAAEILARDGPNALTPPPTTPEWVKFCRQLFGGFSMLLWIGAILCFLAYGIQAATEEEPQNDNLYLGVVLSAVVIITGCFSYYQEAKSSKIMESFKNMVPQQALVIRNGEKMSINAEEVVVGDLVEVKGGDRIPADLRIISANGCKVDNSSLTGESEPQTRSPDFTNENPLETRNIAFFSTNCVEGTARGIVVYTGDRTVMGRIATLASGLEGGQTPIAAEIEHFIHIITGVAVFLGVSFFILSLILEYTWLEAVIFLIGIIVANVPEGLLATVTVCLTLTAKRMARKNCLVKNLEAVETLGSTSTICSDKTGTLTQNRMTVAHMWFDNQIHEADTTENQSGVSFDKTSATWLALSRIAGLCNRAVFQANQENLPILKRAVAGDASESALLKCIELCCGSVKEMRERYTKIVEIPFNSTNKYQLSIHKNPNTAEPRHLLVMKGAPERILDRCSSILIHGKEQPLDEELKDAFQNAYLELGGLGERVLGFCHLFLPDEQFPEGFQFDTDDVNFPLDNLCFVGLISMIDPPRAAVPDAVGKCRSAGIKVIMVTGDHPITAKAIAKGVGIISEGNETVEDIAARLNIPVSQVNPRDAKACVVHGSDLKDMTSEQLDDILKYHTEIVFARTSPQQKLIIVEGCQRQGAIVAVTGDGVNDSPALKKADIGVAMGIAGSDVSKQAADMILLDDNFASIVTGVEEGRLIFDNLKKSIAYTLTSNIPEITPFLIFIIANIPLPLGTVTILCIDLGTDMVPAISLAYEQAESDIMKRQPRNPKTDKLVNERLISMAYGQIGMIQALGGFFTYFVILAENGFLPIHLLGLRVNWDDRWINDVEDSYGQQWTYEQRKIVEFTCHTAFFVSIVVVQWADLVICKTRRNSVFQQGMKNKILIFGLFEETALAAFLSYCPGMGVALRMYPLKPTWWFCAFPYSLLIFVYDEVRKLIIRRRPGGWVEKETYY
2DBG , Knot 51 103 0.76 38 78 98 
GSSGSSGMVNEYKKILLLKGFELMDDYHFTSIKSLLAYDLGLTTKMQEEYNRIKITDLMEKKFQGVACLDKLIELAKDMPSLKNLVNNLRKEKSKVASGPSSG

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 \(c\).

\(|P_{f(7WOB_1)}(2) \setminus P_{f(8JBM_1)}(2)|=52\), \(|P_{f(8JBM_1)}(2) \setminus P_{f(7WOB_1)}(2)|=40\). 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 1:1111111111100001010000011110000100110010011000110000011111100101101101010010001001111100110110000001101111100100000011110010011101001010001111100000000110001010001000010010011110101001010010011100101010100000110110011011011011101111101001001111000010110000110111110011010100111000001010011001101100000010010100110000010101000110110100101110110100110101100001000110001100010100100011010010010100101001110100100111100101100000110010101111000010001110000100110000101100010001001100100110110001110001101001110010011110101101110101100000110000101010110101110000001111001100110000110010010110101001111011011000000111100110000111110100101010100010011000110111100100000001111111010000000011010011000111001011100011000001111001010100011110100001000101010000000111001010001001101111000000001110100100011100111101001110100100001100111001011011110001001101110011010010110111111000111000011111010011011111110111110110010111100011000001110010011101000100010111010011000101100110010001111001100110010110101010011010100100010001101101010101110010001110000101010100110110011011111010011100001001111000101011001111001001110000100101100000110100011111100010011010100100010001000001010110101101011010001001001100100011010011000010101101100101010001011110011011011101111100101001110111011011001010011000000000
Pair \(Z_2\) Length of longest common subsequence
7WOB_1,8JBM_1 92 5
7WOB_1,2DBG_1 260 3
8JBM_1,2DBG_1 254 3

Newick tree

 
[
	2DBG_1:14.99,
	[
		7WOB_1:46,8JBM_1:46
	]:99.99
]

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 \((0.85)(0.8)(\frac{2309 }{\log_{20} 2309}-\frac{1021}{\log_{20}1021})=307.\)
Status Protein1 Protein2 d d1/2
Query variables 7WOB_1 8JBM_1 390 350.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]