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Parikh vectors
4APY_1 7NDB_1 4QIC_1 Letter Amino acid
25 42 16 R Arginine
3 30 2 C Cysteine
12 11 8 M Methionine
18 79 15 F Phenylalanine
16 88 9 N Asparagine
33 51 18 E Glutamic acid
33 94 10 G Glycine
31 107 28 L Leucine
22 105 20 S Serine
32 61 15 D Aspartic acid
15 66 17 Q Glutamine
18 58 11 K Lycine
22 95 13 T Threonine
8 12 2 W Tryptophan
43 81 28 A Alanine
15 26 5 H Histidine
23 71 23 I Isoleucine
27 63 17 P Proline
7 53 5 Y Tyrosine
30 95 14 V Valine 

4APY_1|Chain A|P450 HEME-THIOLATE PROTEIN|Mycobacterium smegmatis str. MC2 155 (246196)
>7NDB_1|Chains A, B, C|Spike glycoprotein|Severe acute respiratory syndrome coronavirus 2 (2697049)
>4QIC_1|Chains A, C|Sensory transduction regulatory protein, Anti-anti-sigma factor PhyR|Bartonella quintana (283165)
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 \(w=f(c)\)
4APY , Knot 180 433 0.84 40 230 406 
MPTPNIPSDFDFLDATLNLERLPVEELAELRKSEPIHWVDVPGGTGGFGDKGYWLVTKHADVKEVSRRSDVFGSSPDGAIPVWPQDMTREAVDLQRAVLLNMDAPQHTRLRKIISRGFTPRAIGRLEDELRSRAQKIAQTAAAQGAGDFVEQVSCELPLQAIAELLGVPQDDRDKLFRWSNEMTAGEDPEYADVDPAMSSFELISYAMKMAEERAVNPTEDIVTKLIEADIDGEKLSDDEFGFFVVMLAVAGNETTRNSITHGMIAFAQNPDQWELYKKERPETAADEIVRWATPVSAFQRTALEDVELGGVQIKKGQRVVMSYRSANFDEEVFEDPHTFNILRSPNPHVGFGGTGAHYCIGANLARMTINLIFNAIADNMPDLKPIGAPERLKSGWLNGIKHWQVDYTGAGKASVSGAPGTCPVAHHHHHHH
7NDB , Knot 451 1288 0.83 40 329 1098 
MFVFLVLLPLVSSQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHVSGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLGVYYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPINLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQDVNCTEVPVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSPGSASSVASQSIIAYTMSLGAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSFIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGAALQIPFAMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTASALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGRSLEVLFQGPGHHHHHHHHGSAWSHPQFEKGGGSGGGSGGSAWSHPQFEK
4QIC , Knot 123 276 0.83 40 177 262 
MSLSTRIAPHLPYLRRFARSVTGSQSSGDAYVSAMLEALVADISIFPRASCDRIGTYWLFCHLFDQTTPNIPEPLPQFGLEQKTSAKLSYLTPRARQAFLLIAVEGFNEQEASEIMNLDARDFRKLLNQASIDISQQIATQVMIIEDEPLIAMDIEQMVESLGHQVVGIARTRKEAVVMYHQKKPRLILADIQLADNSSGIDAVNDILQNDRIPVIFITAFPERLLTGERPEPTFLVTKPFNPDMVKALISQALFFKENASKAALEAGWSHPQFEK

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 \(c\).

\(|P_{f(4APY_1)}(2) \setminus P_{f(7NDB_1)}(2)|=27\), \(|P_{f(7NDB_1)}(2) \setminus P_{f(4APY_1)}(2)|=126\). 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 1:1101011001011010101001110011010000110110111101111001011100010100100000111001011111110010001101001111010110000100110011010111010001000100110011101110110010001110111011111000000110100010110010010101110010110011011000110100011001101010100100001111111111110000000100111111001001010000010011001101101101100011001011110100100111000010100011001001011001010111110110001110110101011101110011010111110010011101100101000111010101111001110000000
Pair \(Z_2\) Length of longest common subsequence
4APY_1,7NDB_1 153 7
4APY_1,4QIC_1 149 5
7NDB_1,4QIC_1 178 8

Newick tree

 
[
	7NDB_1:85.62,
	[
		4APY_1:74.5,4QIC_1:74.5
	]:11.12
]

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 \((0.85)(0.8)(\frac{1721 }{\log_{20} 1721}-\frac{433}{\log_{20}433})=325.\)
Status Protein1 Protein2 d d1/2
Query variables 4APY_1 7NDB_1 413 275
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]