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Parikh vectors
1FML_1 8QLU_1 2PQN_1 Letter Amino acid
12 15 7 I Isoleucine
22 11 4 P Proline
23 8 9 Y Tyrosine
20 8 6 A Alanine
19 3 4 N Asparagine
30 13 12 E Glutamic acid
42 13 13 L Leucine
19 7 3 F Phenylalanine
5 0 2 W Tryptophan
24 16 7 D Aspartic acid
3 2 2 C Cysteine
9 10 7 Q Glutamine
26 5 11 K Lycine
12 13 6 S Serine
14 12 9 T Threonine
14 25 8 V Valine
19 23 8 R Arginine
15 18 7 G Glycine
8 13 1 H Histidine
15 3 3 M Methionine 

1FML_1|Chains A, B|RETINOL DEHYDRATASE|Spodoptera frugiperda (7108)
>8QLU_1|Chain A|Woronin body major protein hexA|Aspergillus fumigatus (746128)
>2PQN_1|Chain A|Mitochondria fission 1 protein|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
1FML , Knot 152 351 0.84 40 198 334 
MEKQQDLPFPYEFRELNPEEDKLVKANLGAFPTTYVKLGPKGYMVYRPYLKDAANIYNMPLRPTDVFVASYQRSGTTMTQELVWLIENDLNFEAAKTYMSLRYIYLDGFMIYDPEKQEEYNDILPNPENLDMERYLGLLEYSSRPGSSLLAAVPPTEKRFVKTHLPLSLMPPNMLDTVKMVYLARDPRDVAVSSFHHARLLYLLNKQSNFKDFWEMFHRGLYTLTPYFEHVKEAWAKRHDPNMLFLFYEDYLKDLPGCIARIADFLGKKLSEEQIQRLCEHLNFEKFKNNGAVNMEDYREIGILADGEHFIRKGKAGCWRDYFDEEMTKQAEKWIKDNLKDTDLRYPNMEL
8QLU , Knot 97 218 0.79 38 139 210 
MGYYDDDGNYHSFRRGVERAVDRITHPFHHHHHHHDREEVVIADERGPVRYRDGVREDVRIVEPRASKTTAESVPIPCHFIRIGDILILQGRPCQVIRISVSPQTGQHRYLGVDLFTRQLQEESSFVSNPSPSVVVQTMLGPVYKTYRILDLHEDGTITAMTETGDVKQALPVVTQGQLFRKIRDAFSEGRGSVRALVINDGGRELVVDYKVIHGSRL
2PQN , Knot 68 129 0.85 40 106 127 
MTKVDFWPTLKDAYEPLYPQQLEILRQQVVSEGGPTATIQSRFNYAWGLIKSTDVNDERLGVKILTDIYKEAESRRRECLYYLTIGCYKLGEYSMAKRYVDTLFEHERNNKQVGALKSMVEDKIQKETL

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 \(c\).

\(|P_{f(1FML_1)}(2) \setminus P_{f(8QLU_1)}(2)|=124\), \(|P_{f(8QLU_1)}(2) \setminus P_{f(1FML_1)}(2)|=65\). 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 1:100000111100100101000011010111110001011101011001010011010011101001111000001001000111110001010110001010010101111001000000001110100101000111100000110011111110000110001110111101100101101100100111001001011011000001001101100110010101001001110000101111100001001110110110111001000010010001010010001110100000111110100110010110100010001000100110001000010010101
Pair \(Z_2\) Length of longest common subsequence
1FML_1,8QLU_1 189 4
1FML_1,2PQN_1 170 4
8QLU_1,2PQN_1 151 3

Newick tree

 
[
	1FML_1:94.18,
	[
		2PQN_1:75.5,8QLU_1:75.5
	]:18.68
]

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 \((0.85)(0.8)(\frac{569 }{\log_{20} 569}-\frac{218}{\log_{20}218})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 1FML_1 8QLU_1 129 103.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]