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Parikh vectors
7XRY_1 8RTS_1 8ZBN_1 Letter Amino acid
16 42 12 N Asparagine
9 51 26 E Glutamic acid
11 64 19 K Lycine
12 36 19 P Proline
16 34 11 Q Glutamine
15 21 15 M Methionine
26 34 26 T Threonine
11 27 16 Y Tyrosine
30 55 21 V Valine
5 28 18 R Arginine
10 17 5 C Cysteine
25 23 28 G Glycine
8 22 9 H Histidine
24 115 27 L Leucine
5 9 4 W Tryptophan
23 31 29 A Alanine
12 39 20 D Aspartic acid
9 49 30 I Isoleucine
14 42 11 F Phenylalanine
20 72 24 S Serine 

7XRY_1|Chains A, B|ORF1a|Middle East respiratory syndrome-related coronavirus (1335626)
>8RTS_1|Chains A, B, C, D, E, F, G|Volume-regulated anion channel subunit LRRC8C|Homo sapiens (9606)
>8ZBN_1|Chains A[auth B], B[auth D], C[auth I], D[auth J], E[auth K], F[auth M]|Actin, alpha cardiac muscle 1|Mus musculus (10090)
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 \(w=f(c)\)
7XRY , Knot 130 301 0.82 40 188 288 
SGLVKMSHPSGDVEACMVQVTCGSMTLNGLWLDNTVWCPRHVMCPADQLSDPNYDALLISMTNHSFSVQKHIGAPANLRVVGHAMQGTLLKLTVDVANPSTPAYTFTTVKPGAAFSVLACYNGRPTGTFTVVMRPNYTIKGSFLCGSCGSVGYTKEGSVINFCYMHQMELANGTHTGSAFDGTMYGAFMDKQVHQVQLTDKYCSVNVVAWLYAAILNGCAWFVKPNRTSVVSFNEWALANQFTEFVGTQSVDMLAVKTGVAIEQLLYAIQQLYTGFQGKQILGSTMLEDEFTPEDVNMQIM
8RTS , Knot 308 811 0.84 40 297 717 
MSIPVTEFRQFSEQQPAFRVLKPWWDVFTDYLSVAMLMIGVFGCTLQVMQDKIICLPKRVQPAQNHSSLSNVSQAVASTTPLPPPKPSPANPITVEMKGLKTDLDLQQYSFINQMCYERALHWYAKYFPYLVLIHTLVFMLCSNFWFKFPGSSSKIEHFISILGKCFDSPWTTRALSEVSGEDSEEKDNRKNNMNRSNTIQSGPEGSLVNSQSLKSIPEKFVVDKSTAGALDKKEGEQAKALFEKVKKFRLHVEEGDILYAMYVRQTVLKVIKFLIIIAYNSALVSKVQFTVDCNVDIQDMTGYKNFSCNHTMAHLFSKLSFCYLCFVSIYGLTCLYTLYWLFYRSLREYSFEYVRQETGIDDIPDVKNDFAFMLHMIDQYDPLYSKRFAVFLSEVSENKLKQLNLNNEWTPDKLRQKLQTNAHNRLELPLIMLSGLPDTVFEITELQSLKLEIIKNVMIPATIAQLDNLQELSLHQCSVKIHSAALSFLKENLKVLSVKFDDMRELPPWMYGLRNLEELYLVGSLSHDISRNVTLESLRDLKSLKILSIKSNVSKIPQAVVDVSSHLQKMCIHNDGTKLVMLNNLKKMTNLTELELVHCDLERIPHAVFSLLSLQELDLKENNLKSIEEIVSFQHLRKLTVLKLWHNSITYIPEHIKKLTSLERLSFSHNKIEVLPSHLFLCNKIRYLDLSYNDIRFIPPEIGVLQSLQYFSITCNKVESLPDELYFCKKLKTLKIGKNSLSVLSPKIGNLLFLSYLDVKGNHFEILPPELGDCRALKRARLVVEDALFETLPSDVREQMKADALEVLFQ
8ZBN , Knot 157 370 0.83 40 216 355 
TTALVCDNGSGLVKAGFAGDDAPRAVFPSIVGRPRHQGVMVGMGQKDSYVGDEAQSKRGILTLKYPIEHGIITNWDDMEKIWHHTFYNELRVAPEEHPTLLTEAPLNPKANREKMTQIMFETFNVPAMYVAIQAVLSLYASGRTTGIVLDSGDGVTHNVPIYEGYALPHAIMRLDLAGRDLTDYLMKILTERGYSFVTTAEREIVRDIKEKLCYVALDFENEMATAASSSSLEKSYELPDGQVITIGNERFRCPETLFQPSFIGMESAGIHETTYNSIMKCDIDIRKDLYANNVLSGGTTMYPGIADRMQKEITALAPSTMKIKIIAPPERKYSVWIGGSILASLSTFQQMWISKQEYDEAGPSIVHRKC

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 \(c\).

\(|P_{f(7XRY_1)}(2) \setminus P_{f(8RTS_1)}(2)|=39\), \(|P_{f(8RTS_1)}(2) \setminus P_{f(7XRY_1)}(2)|=148\). 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 1:0111010010101010110100101010111100011010011011001001000111101000010100011111010111011010110101011010011001001011111011100010101010111010001010110100101100001011010010010110100010110101011110001001010000001011111011110101111010000110100111100100111000101111001111001101100100110100111001100010100101011
Pair \(Z_2\) Length of longest common subsequence
7XRY_1,8RTS_1 187 4
7XRY_1,8ZBN_1 178 5
8RTS_1,8ZBN_1 169 4

Newick tree

 
[
	7XRY_1:93.42,
	[
		8ZBN_1:84.5,8RTS_1:84.5
	]:8.92
]

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 \((0.85)(0.8)(\frac{1112 }{\log_{20} 1112}-\frac{301}{\log_{20}301})=215.\)
Status Protein1 Protein2 d d1/2
Query variables 7XRY_1 8RTS_1 273 186
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]