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Parikh vectors
2NYF_1 5SUS_1 6YLL_1 Letter Amino acid
48 1 19 G Glycine
43 0 19 S Serine
20 0 11 Y Tyrosine
22 1 23 E Glutamic acid
25 0 15 R Arginine
32 0 12 N Asparagine
32 1 21 D Aspartic acid
7 2 3 C Cysteine
31 0 10 Q Glutamine
14 0 13 H Histidine
60 1 44 L Leucine
42 4 17 A Alanine
20 0 17 P Proline
16 0 12 M Methionine
19 2 11 F Phenylalanine
4 0 2 W Tryptophan
45 2 19 V Valine
18 0 15 K Lycine
31 0 13 T Threonine
40 2 23 I Isoleucine 

2NYF_1|Chain A|Nostoc punctiforme phenylalanine ammonia lyase|Nostoc punctiforme (63737)
>5SUS_1|Chains A, B, C, D|16mer A-beta peptide: ORN-CYS-VAL-PHE-MEA-CYS-GLU-ASP-ORN-ALA-ILE-ILE-GLY-LEU-ORN-VAL|Homo sapiens (9606)
>6YLL_1|Chains A, B|Mitogen-activated protein kinase 6|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 \(w=f(c)\)
2NYF , Knot 228 569 0.84 40 259 532 
MNITSLQQNITRSWQIPFTNSSDSIVTVGDRNLTIDEVVNVARHGTQVRLTDNADVIRGVQASCDYINNAVETAQPIYGVTSGFGGMADVVISREQAAELQTNLIWFLKSGAGNKLSLADVRAAMLLRANSHLYGASGIRLELIQRIETFLNAGVTPHVYEFGSIGASGDLVPLSYITGALIGLDPSFTVDFDGKEMDAVTALSRLGLPKLQLQPKEGLAMMNGTSVMTGIAANCVYDAKVLLALTMGVHALAIQGLYGTNQSFHPFIHQCKPHPGQLWTADQMFSLLKDSSLVREELDGKHEYRGKDLIQDRYSLRCLAQFIGPIVDGVSEITKQIEVEMNSVTDNPLIDVENQVSYHGGNFLGQYVGVTMDRLRYYIGLLAKHIDVQIALLVSPEFSNGLPPSLVGNSDRKVNMGLKGLQISGNSIMPLLSFYGNSLADRFPTHAEQFNQNINSQGYISANLTRRSVDIFQNYMAIALMFGVQAVDLRTYKMKGHYDARTCLSPNTVQLYTAVCEVVGKPLTSVRPYIWNDNEQCLDEHIARISADIAGGGLIVQAVEHIFSSLKST
5SUS , Knot 12 16 0.69 18 15 14 
ACVFFCEDAAIIGLAV
6YLL , Knot 138 319 0.83 40 194 302 
MNIHGFDLGSRYMDLKPLGCGGNGLVFSAVDNDCDKRVAIKKIVLTDPQSVKHALREIKIIRRLDHDNIVKVFEILGPSGSQLTDDVGSLTELNSVYIVQEYMETDLANVLEQGPLLEEHARLFMYQLLRGLKYIHSANVLHRDLKPANLFINTEDLVLKIGDFGLARIMDPHYSHKGHLSEGLVTKWYRSPRLLLSPNNYTKAIDMWAAGCIFAEMLTGKTLFAGAHELEQMQLILESIPVVHEEDRQELLSVIPVYIRNDMTEPHKPLTQLLPGISREAVDFLEQILTFSPMDRLTAEEALSHPYMSIYSFPMDEPI

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 \(c\).

\(|P_{f(2NYF_1)}(2) \setminus P_{f(5SUS_1)}(2)|=250\), \(|P_{f(5SUS_1)}(2) \setminus P_{f(2NYF_1)}(2)|=6\). 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 1:10100100010001011100000011011000101001101100100101000101101101000010011001011011001111110111000011010001111100111001011010111110100010110110101100100110111010100110111010111100101111110101010101001011011001111010101001111101001101111001001011111011101111011010000101110000101101101001101100001100010100000100110000010011011111101100100010101001000111010001000110111001110100100011111001010111110101001111011100000101110110101001111101010011001100100100010001010101000010110001111111110110100001010001000101001010011001110110010101100000010001101010111111110110011001000
Pair \(Z_2\) Length of longest common subsequence
2NYF_1,5SUS_1 256 3
2NYF_1,6YLL_1 171 4
5SUS_1,6YLL_1 191 3

Newick tree

 
[
	5SUS_1:12.69,
	[
		2NYF_1:85.5,6YLL_1:85.5
	]:35.19
]

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 \((0.85)(0.8)(\frac{585 }{\log_{20} 585}-\frac{16}{\log_{20}16})=175.\)
Status Protein1 Protein2 d d1/2
Query variables 2NYF_1 5SUS_1 223 115
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]