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Parikh vectors
9BNP_1 2YRL_1 4MTY_1 Letter Amino acid
21 0 1 C Cysteine
43 9 12 T Threonine
9 1 7 W Tryptophan
12 2 8 Y Tyrosine
23 6 13 E Glutamic acid
30 11 22 G Glycine
28 6 24 K Lycine
16 1 12 F Phenylalanine
22 4 17 P Proline
33 10 17 V Valine
20 4 13 A Alanine
21 1 7 R Arginine
46 4 10 N Asparagine
28 8 26 L Leucine
8 0 2 M Methionine
31 17 18 S Serine
16 8 19 D Aspartic acid
20 7 11 Q Glutamine
9 0 12 H Histidine
32 3 9 I Isoleucine 

9BNP_1|Chains A, C[auth E], F[auth I]|Envelope glycoprotein Gp120|Human immunodeficiency virus 1 (11676)
>2YRL_1|Chain A|KIAA1837 protein|Homo sapiens (9606)
>4MTY_1|Chain A|Carbonic anhydrase 2|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 \(w=f(c)\)
9BNP , Knot 192 468 0.84 40 232 437 
NLWVTVYYGVPVWKDAETTLFCASDAKAYETEKHNVWATHACVPTDPNPQEIHLENVTEEFNMWKNNMVEQMHTDIISLWDQSLKPCVKLTPLCVTLQCTNVTNNITDGELKNCSFNMTTELRDKKQKVYSLFYRLDVVQINENQGNRSNNSNKEYRLINCNTSACTQACPKVSFEPIPIHYCAPAGFAILKCKDKKFNGTGPCPSVSTVQCTHGIKPVVSTQLLLNGSLAEEEVMIRSENITNNAKNILVQFNTPVQINCTRPNNNTRKSIRIGPGQAFYATGDIIGDIRQAHCNVSKATWNETLGKVVKQLRKHFGNNTIIRFANSSGGDLEVTTHSFNCGGEFFYCNTSGLFNSTWISNTSVQGSNSTGSNDSITLPCRIKQIINMWQRIGQCMYAPPIQGVIRCVSNITGLILTRDGGSTNSTTETFRPGGGDMRDNWRSELYKYKVVKIEPLGVAPTRCKRRV
2YRL , Knot 52 102 0.78 34 78 95 
GSSGSSGQADAGPDKELTLPVDSTTLDGSKSSDDQKIISYLWEKTQGPDGVQLENANSSVATVTGLQVGTYVFTLTVKDERNLQSQSSVNVIVKEESGPSSG
4MTY , Knot 112 260 0.79 40 176 249 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK

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 \(c\).

\(|P_{f(9BNP_1)}(2) \setminus P_{f(2YRL_1)}(2)|=175\), \(|P_{f(2YRL_1)}(2) \setminus P_{f(9BNP_1)}(2)|=21\). 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 1:011101001111100100011010010100000001110010110010100101001000101100011001000110110001010101011010100001000100101000010100010000001001100101101000010000000000011000001000101010101111000111111110000001010110101001000011011100011101011000111000010001001110100110100001000000010111101101010111010010001001010001101100100011000110110001101010000100110110000011100011000010100001000010110010011011001100101111011100100101111000110000000010111101000100010000110101111110000001
Pair \(Z_2\) Length of longest common subsequence
9BNP_1,2YRL_1 196 4
9BNP_1,4MTY_1 176 4
2YRL_1,4MTY_1 158 3

Newick tree

 
[
	9BNP_1:97.39,
	[
		4MTY_1:79,2YRL_1:79
	]:18.39
]

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 \((0.85)(0.8)(\frac{570 }{\log_{20} 570}-\frac{102}{\log_{20}102})=138.\)
Status Protein1 Protein2 d d1/2
Query variables 9BNP_1 2YRL_1 173 104
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]