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Parikh vectors
5EIK_1 5FTH_1 7ZYK_1 Letter Amino acid
15 29 14 G Glycine
15 7 13 F Phenylalanine
10 18 11 T Threonine
31 20 30 L Leucine
12 28 20 K Lycine
11 13 23 Y Tyrosine
8 12 24 R Arginine
11 14 26 D Aspartic acid
7 4 2 C Cysteine
5 3 9 Q Glutamine
13 9 14 P Proline
20 22 15 S Serine
17 20 15 A Alanine
8 20 22 E Glutamic acid
18 10 16 H Histidine
5 13 9 M Methionine
2 9 15 N Asparagine
16 16 19 I Isoleucine
6 4 5 W Tryptophan
28 20 26 V Valine 

5EIK_1|Chain A|Uncharacterized protein Y57A10A.28|Caenorhabditis elegans (6239)
>5FTH_1|Chains A, B, C|GLUTAMATE RECEPTOR 2|RATTUS NORVEGICUS (10116)
>7ZYK_1|Chain A|Casein kinase II subunit alpha|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 \(w=f(c)\)
5EIK , Knot 114 258 0.81 40 164 244 
MGWVPDEWSIDHDTLIDAGGYVQKLKLYPYFDAAHYVLTCLSVRHDLGPDAISFSRKHPFSCWLSCMLMSFAGSFLSCFLLGEPIISPLKQHADILLGSIVWYLVFYSPFDVVFRLATWFPVKLGLSVLKEVQRTHKIAAGVKHAVRIYPESYLVQILVGVAKGAGSGVVKIVEQLARGTWHPTNHEILRPSFTTKACVIASIVFTLERHSMYVTAPHDLVYLCVVGFFIYFKLASLCLSVHDVLMPIENVLHHHHHH
5FTH , Knot 129 291 0.83 40 184 275 
MHHHHHHHHSSGLVPRGSAMGSGNDTSRGANKTVVVTTILESPYVMMKKNHEMLEGNERYEGYCVDLAAEIAKHCGFKYKLTIVGDGKYGARDADTKIWNGMVGELVYGKADIAIAPLTITLVREEVIDFSKPFMSLGISIMIKKGTPIESAEDLSKQTEIAYGTLDSGSTKEFFRRSKIAVFDKMWTYMRSAEPSVFVRTTAEGVARVRKSKGKYAYLLESTMNEYIEQRKPCDTMKVGGNLDSKGYGIATPMGSSLGKPVNLAVLKLSEQGVLDKLKNKWWYDKGECGA
7ZYK , Knot 147 328 0.86 40 194 315 
SGPVPSRARVYTDVNTHRPSEYWDYESHVVEWGNQDDYQLVRKLGRGKYSEVFEAINITNNEKVVVKILKPVAAAKIKREIKILENLRGGPNIITLADIVKDPVSRTPALVFEHVNNTDFKQLYQTLTDYDIRFYMYEILKALDYCHSMGIMHRDVKPHNVMIDHEHRKLRLIDWGLAEFYHPGQEYNVRVASRYFKGPELLVDYQMYDYSLDMWSLGCMLASMIFRKEPFFHGHDNYDQLVRIAKVLGTEDLYDYIDKYNIELDPRFNDILGRHSRKRWERFVHSENQHLVSPEALDFLDKLLRYDHQSRLTAREAMEHPYFYTVVK

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 \(c\).

\(|P_{f(5EIK_1)}(2) \setminus P_{f(5FTH_1)}(2)|=70\), \(|P_{f(5FTH_1)}(2) \setminus P_{f(5EIK_1)}(2)|=90\). 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 1:111110010100001101110100101010101100110010100011101101000011001100111011101100111101110110001011110111011100110111011011110111011001000001111100110101000110111111011101110110011010101000011010100010111011101000010101100110101111110101101010100111110011000000
Pair \(Z_2\) Length of longest common subsequence
5EIK_1,5FTH_1 160 6
5EIK_1,7ZYK_1 170 4
5FTH_1,7ZYK_1 172 4

Newick tree

 
[
	7ZYK_1:87.25,
	[
		5EIK_1:80,5FTH_1:80
	]:7.25
]

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 \((0.85)(0.8)(\frac{549 }{\log_{20} 549}-\frac{258}{\log_{20}258})=82.6\)
Status Protein1 Protein2 d d1/2
Query variables 5EIK_1 5FTH_1 103 97.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]