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Parikh vectors
5DLE_1 9IRE_1 5GGK_1 Letter Amino acid
7 4 2 N Asparagine
7 7 8 T Threonine
0 1 3 W Tryptophan
13 7 13 A Alanine
3 9 10 D Aspartic acid
1 4 1 C Cysteine
2 8 3 Q Glutamine
1 1 3 M Methionine
5 6 15 S Serine
3 4 3 Y Tyrosine
3 10 10 R Arginine
5 7 18 G Glycine
12 7 3 I Isoleucine
2 17 16 L Leucine
13 12 9 K Lycine
2 2 9 P Proline
12 21 9 E Glutamic acid
7 8 5 H Histidine
3 4 5 F Phenylalanine
12 12 19 V Valine 

5DLE_1|Chains A, B, C, D|Pts system, fructose-specific iiabc component|Borrelia burgdorferi (strain ATCC 35210 / B31 / CIP 102532 / DSM 4680) (224326)
>9IRE_1|Chains A, B|Ferric uptake regulation protein|Bacillus cereus VD154 (1053238)
>5GGK_1|Chains A, B|Protein O-linked-mannose beta-1,2-N-acetylglucosaminyltransferase 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 \(w=f(c)\)
5DLE , Knot 54 113 0.75 38 86 106 
MAHHHHHHTKAEKIVAVTACPVGVAHTYIAAKKIENEAKKQGYSIRVETQGSIGIENALTEEEIKNASVVILAVDKDIDEKRFEGKRVYKVSTVKAINNTENIIKESFNAPVF
9IRE , Knot 72 151 0.79 40 115 146 
MEERIERIKKQLHAASYKLTPQREATVRVLLENEEDHLSAEDVYLLVKEKSPEIGLATVYRTLELLSELKVVDKINFGDGVSRYDLRQEGAQRFHHHLICTQCGAVQEIQEDLLGEVERKVEHDWSFKVKDHRLTFHGICKNCQENETDEK
5GGK , Knot 73 164 0.75 40 114 155 
GPLGSGSGPRRVLDVEVYSSRSKVYVAVDGTTVLEDEAREQGRGIHVIVLNQATGHVMAKRVFDTYSPHEDEAMVLFLNMVAPGRVLICTVKDEGSFHLKDTAKALLRSLGSQAGPALGWRDTWAFVGRKGGPVFGEKHSKSPALSSWGDPVLLKTDVPLSSAE

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 \(c\).

\(|P_{f(5DLE_1)}(2) \setminus P_{f(9IRE_1)}(2)|=44\), \(|P_{f(9IRE_1)}(2) \setminus P_{f(5DLE_1)}(2)|=73\). 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 1:11000000001001111010111110001110010001000100101000101110011000010010111111000100001010010010010110000011000101111
Pair \(Z_2\) Length of longest common subsequence
5DLE_1,9IRE_1 117 3
5DLE_1,5GGK_1 134 3
9IRE_1,5GGK_1 143 4

Newick tree

 
[
	5GGK_1:72.52,
	[
		5DLE_1:58.5,9IRE_1:58.5
	]:14.02
]

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 \((0.85)(0.8)(\frac{264 }{\log_{20} 264}-\frac{113}{\log_{20}113})=47.7\)
Status Protein1 Protein2 d d1/2
Query variables 5DLE_1 9IRE_1 60 51.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]