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Parikh vectors
4WYM_1 4JLE_1 7GTB_1 Letter Amino acid
15 5 14 Q Glutamine
18 5 14 G Glycine
6 4 11 H Histidine
10 7 11 M Methionine
9 11 25 S Serine
16 2 13 T Threonine
15 6 17 V Valine
20 5 13 A Alanine
16 25 30 E Glutamic acid
4 14 14 F Phenylalanine
18 2 22 P Proline
4 2 6 W Tryptophan
11 8 19 R Arginine
11 24 23 K Lycine
4 7 9 Y Tyrosine
15 11 18 I Isoleucine
7 5 18 D Aspartic acid
4 2 4 C Cysteine
18 13 29 L Leucine
10 10 11 N Asparagine 

4WYM_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|Capsid protein p24|Human immunodeficiency virus type 1 group M subtype B (11698)
>4JLE_1|Chains A, B|PHIST|Plasmodium falciparum (36329)
>7GTB_1|Chain A|Tyrosine-protein phosphatase non-receptor type 1|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 \(w=f(c)\)
4WYM , Knot 106 231 0.83 40 158 223 
PIVQNLQGQMVHQCISPRTLNAWVKVVEEKAFSPEVIPMFSALSCGATPQDLNTMLNTVGGHQAAMQMLKETINEEAAEWDRLHPVHAGPIAPGQMREPRGSDIAGTTSTLQEQIGWMTHNPPIPVGEIYKRWIILGLNKIVRMYSPTSILDIRQGPKEPFRDYVDRFYKTLRAEQASQEVKNAATETLLVQNANPDCKTILKALGPGATLEEMMTACQGVGGPGHKARVL
4JLE , Knot 78 168 0.79 40 113 158 
GPLGSSLSDEINKCDMKKYTAEEINEMINSSNEFINRNDMNIIFSYVHESEREKFKKVEENIFKFIQSIVETYKIPDEYKMRKFKFAHFEMQGYALKQEKFLLEYAFLSLNGKLCERKKFKEVLEYVKREWIEFRKSMFDVWKEKLASEFREHGEMLNQKRKLKQHEL
7GTB , Knot 141 321 0.84 40 205 309 
MEMEKEFEQIDKSGSWAAIYQDIRHEASDFPSRVAKLPKNKNRNRYRDVSPFDHSRIKLHQEDNDYINASLIKMEEAQRSYILTQGPLPNTVGHFWEMVWEQKSRGVVMLNRVMEKGSLKCAQYWPQKEEKEMIFEDTNLKLTLISEDIKSYYTVRQLELENLTTQETREILHFHYTTWPDFGVPESPASFLNFLFKVRESGSLSPEHGPVVVHCSAGIGRSGTFCLADTCLLLMDKRKDPSSVDIKKVLLEMRKFRMGLIQTADQLRFSYLAVIEGAKFIMGDSSVQDQWKELSHEDLEPPPEHIPPPPRPPKRILEPHN

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 \(c\).

\(|P_{f(4WYM_1)}(2) \setminus P_{f(4JLE_1)}(2)|=108\), \(|P_{f(4JLE_1)}(2) \setminus P_{f(4WYM_1)}(2)|=63\). 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 1:111001010110001010010111011000110101111101100110100100110011100111011000100011010010110111111101001010011100001000111100011111101000111111001101001001101001100110001001000101001000100110001110010100001101111110100110100111111001011
Pair \(Z_2\) Length of longest common subsequence
4WYM_1,4JLE_1 171 3
4WYM_1,7GTB_1 195 4
4JLE_1,7GTB_1 180 4

Newick tree

 
[
	7GTB_1:96.44,
	[
		4WYM_1:85.5,4JLE_1:85.5
	]:10.94
]

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 \((0.85)(0.8)(\frac{399 }{\log_{20} 399}-\frac{168}{\log_{20}168})=68.9\)
Status Protein1 Protein2 d d1/2
Query variables 4WYM_1 4JLE_1 90 76.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]