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Parikh vectors
8QNA_1 9CVH_1 1WXJ_1 Letter Amino acid
25 107 12 S Serine
4 30 0 C Cysteine
17 18 1 H Histidine
25 71 8 I Isoleucine
32 105 38 L Leucine
13 78 10 F Phenylalanine
37 82 34 A Alanine
25 42 20 R Arginine
18 88 2 N Asparagine
19 95 15 T Threonine
3 12 1 W Tryptophan
18 60 9 D Aspartic acid
28 95 26 V Valine
15 58 10 K Lycine
8 11 4 M Methionine
21 63 20 P Proline
19 53 5 Y Tyrosine
17 65 9 Q Glutamine
20 50 22 E Glutamic acid
38 92 25 G Glycine 

8QNA_1|Chain A|12-oxophytodienoate reductase 3|Solanum lycopersicum (4081)
>9CVH_1|Chains A, B, C|Spike glycoprotein|Severe acute respiratory syndrome coronavirus 2 (2697049)
>1WXJ_1|Chain A|tryptophan synthase alpha chain|Thermus thermophilus (300852)
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 \(w=f(c)\)
8QNA , Knot 168 402 0.83 40 222 376 
HHHHHHMASSAQDGNNPLFSPYKMGKFNLSHRVVLAPMTRCRALNNIPQAALGEYYEQRATAGGFLITEGTMISPTSAGFPHVPGIFTKEQVREWKKIVDVVHAKGAVIFCQLWHVGRASHEVYQPAGAAPISSTEKPISNRWRILMPDGTHGIYPKPRAIGTYEISQVVEDYRRSALNAIEAGFDGIEIHGAHGYLIDQFLKDGINDRTDEYGGSLANRCKFITQVVQAVVSAIGADRVGVRVSPAIDHLDAMDSNPLSLGLAVVERLNKIQLHSGSKLAYLHVTQPRYVAYGQTEAGRLGSEEEEARLMRTLRNAYQGTFICSGGYTRELGIEAVAQGDADLVSYGRLFISNPDLVMRIKLNAPLNKYNAKTFYTQDPVVGYTDYPFLQGNGSNGPLSRL
9CVH , Knot 446 1275 0.83 40 327 1091 
MFVFLVLLPLVSSQCVNLSAWSHPQFEKGGGSGGGSGGSSAWSHPQFEKSALVPRGSTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHVSGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLGVYYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPINLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQGVNCTEVPVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSPGSASSVASQSIIAYTMSLGAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSFIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGAALQIPFAMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTASALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTF
1WXJ , Knot 113 271 0.77 38 140 249 
MTTLEAFAKARSEGRAALIPYLTAGFPSREGFLQAVEEVLPYADLLEIGLPYSDPLGDGPVIQRASELALRKGMSVQGALELVREVRALTEKPLFLMTYLNPVLAWGPERFFGLFKQAGATGVILPDLPPDEDPGLVRLAQEIGLETVFLLAPTSTDARIATVVRHATGFVYAVSVTGVTGMRERLPEEVKDLVRRIKARTALPVAVGFGVSGKATAAQAAVADGVVVGSALVRALEEGRSLAPLLQEIRQGLQRLEANPGLKESSKKPLP

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 \(c\).

\(|P_{f(8QNA_1)}(2) \setminus P_{f(9CVH_1)}(2)|=20\), \(|P_{f(9CVH_1)}(2) \setminus P_{f(8QNA_1)}(2)|=125\). 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 1:000000110010010011101001101010001111110000110011011110000001011111100101101001111011111000010010011011010111110011011010001001111111000001100010111101001101010111000100110000001101101110110101101011001100110000000110110000110011011101111001110101110010110001101111110010010100100110101001001101000110110000010110010010010110011000011101110101011001011100101110101011100001001000011110000111010100111001
Pair \(Z_2\) Length of longest common subsequence
8QNA_1,9CVH_1 145 4
8QNA_1,1WXJ_1 156 4
9CVH_1,1WXJ_1 201 4

Newick tree

 
[
	1WXJ_1:95.06,
	[
		8QNA_1:72.5,9CVH_1:72.5
	]:22.56
]

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 \((0.85)(0.8)(\frac{1677 }{\log_{20} 1677}-\frac{402}{\log_{20}402})=323.\)
Status Protein1 Protein2 d d1/2
Query variables 8QNA_1 9CVH_1 407 265
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]