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Parikh vectors
3SIZ_1 7SWK_1 2CMM_1 Letter Amino acid
10 15 2 M Methionine
9 16 6 F Phenylalanine
9 32 6 S Serine
17 34 8 V Valine
20 29 4 R Arginine
16 36 14 E Glutamic acid
21 46 11 G Glycine
17 39 18 L Leucine
0 6 2 W Tryptophan
4 14 3 Y Tyrosine
26 58 17 A Alanine
3 16 5 Q Glutamine
22 32 9 I Isoleucine
7 28 5 T Threonine
3 4 0 C Cysteine
2 12 12 H Histidine
10 22 4 P Proline
7 17 1 N Asparagine
18 24 7 D Aspartic acid
7 18 19 K Lycine 

3SIZ_1|Chains A, B|Orotidine 5'-phosphate decarboxylase|Methanothermobacter thermautotrophicus (187420)
>7SWK_1|Chains A, B|Betaine aldehyde dehydrogenase|Klebsiella pneumoniae subsp. pneumoniae (strain HS11286) (1125630)
>2CMM_1|Chain A|MYOGLOBIN|Physeter catodon (9755)
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 \(w=f(c)\)
3SIZ , Knot 98 228 0.77 38 145 215 
MRSRRVDVMDVMNRLILAMDLMNRDDALRVTGEVREYIDTVKIGYPLVLSEGMDIIAEFRKRFGCRIIADFKVADIPETNEKICRATFKAGADAIIVHGFPGADSVRACLNVAEEMGREVFLLTEMAHPGAEMFIQGAADEIARMGVDLGVKNYVGPSTRPERLSRLREIIGQDSFLISPGVGAQGGDPGETLRFADAIIVGRSIYLADNPAAAAAGIIESIKDLLNP
7SWK , Knot 201 498 0.83 40 241 466 
MAHHHHHHMSRMAEQQLYINGGYTSATSGRTFETINPATGEVLATVQAAGREDVDRAVESAQRGQKIWAAMTAMERSRILRRAVDLLRQRNDELARLETLDTGKPLSETAAVDIVTGADVLEYYAGLIPALEGSQIPLRDSSFVYTRREPLGVVAGIGAWNYPIQIALWKSAPALAAGNAMIFKPSEVTPLTALKLAEIYSEAGLPDGVFNVLPGIGAETGQYLTEHPDIAKISFTGGVASGKKVMANSAASSLKEVTMELGGKSPLIIADDADLDLAADIAMMANFYSSGQVCTNGTRVFVPAKQKAEFEHKILERVARIRPGDLFADDTNFGPLVSFPHRDNVLRYIESGKREGARLLCGGEALKGDGFDNGSWVAPTVFTDCSDEMTIVREEIFGPVMSILSYADEAEVIRRANATEYGLAAGVVTPNLNRAHRIIHQLEAGICWINSWGESPAEMPVGGYKHSGIGRENGVMTLQSYTQVKSIQVEMGKFQSIF
2CMM , Knot 73 153 0.80 38 111 148 
VLSEGEWQLVLHVWAKVEADVAGHGQDILIRLFKSHPETLEKFDRFKHLKTEAEMKASEDLKKHGVTVLTALGAILKKKGHHEAELKPLAQSHATKHKIPIKYLEFISEAIIHVLHSRHPGDFGADAQGAMNKALELFRKDIAAKYKELGYQG

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 \(c\).

\(|P_{f(3SIZ_1)}(2) \setminus P_{f(7SWK_1)}(2)|=32\), \(|P_{f(7SWK_1)}(2) \setminus P_{f(3SIZ_1)}(2)|=128\). 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 1:100001011011001111101100001101010100010010110111100110111010001100111010110110000010010101110111101111100101010110011001111001101110111011100110111011100011100010010010011100011101111101101100101101111100101100111111111001001101
Pair \(Z_2\) Length of longest common subsequence
3SIZ_1,7SWK_1 160 4
3SIZ_1,2CMM_1 126 5
7SWK_1,2CMM_1 192 4

Newick tree

 
[
	7SWK_1:95.32,
	[
		3SIZ_1:63,2CMM_1:63
	]:32.32
]

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 \((0.85)(0.8)(\frac{726 }{\log_{20} 726}-\frac{228}{\log_{20}228})=138.\)
Status Protein1 Protein2 d d1/2
Query variables 3SIZ_1 7SWK_1 171 123.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]