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Parikh vectors
7XNC_1 8ZIG_1 2NBG_1 Letter Amino acid
19 16 7 D Aspartic acid
6 5 0 C Cysteine
28 23 10 E Glutamic acid
24 19 5 I Isoleucine
12 8 6 M Methionine
29 19 13 G Glycine
14 14 2 F Phenylalanine
24 24 4 S Serine
7 22 6 T Threonine
23 18 14 V Valine
25 11 7 K Lycine
2 4 0 W Tryptophan
8 12 1 Y Tyrosine
18 25 11 A Alanine
18 18 11 R Arginine
11 14 3 N Asparagine
15 7 5 Q Glutamine
35 29 11 L Leucine
8 3 3 H Histidine
22 19 6 P Proline 

7XNC_1|Chain A|Dual specificity mitogen-activated protein kinase kinase 1|Homo sapiens (9606)
>8ZIG_1|Chain A|Psilocybin synthase|Psilocybe cubensis (181762)
>2NBG_1|Chain A|Translation initiation factor IF-2|Geobacillus stearothermophilus (1422)
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 \(w=f(c)\)
7XNC , Knot 150 348 0.84 40 209 336 
GLEELELDEQQRKRLEAFLTQKQKVGELKDDDFEKISELGAGNGGVVFKVSHKPSGLVMARKLIHLEIKPAIRNQIIRELQVLHECNSPYIVGFYGAFYSDGEISICMEHMDGGSLDQVLKKAGRIPEQILGKVSIAVIKGLTYLREKHKIMHRDVKPSNILVNSRGEIKLCDFGVSGQLIDSMANSFVGTRSYMSPERLQGTHYSVQSDIWSMGLSLVEMAVGRYPIPPPDAKELELMFGCQVEGDAAETPPRPRTPGRPLNKKGMDSRPPMAIFELLDYIVNEPPPKLPSGVFSLEFQDFVNKCLIKNPAERADLKQLMVHAFIKRSDAEEVDFAGWLCSTIGLNQ
8ZIG , Knot 138 310 0.85 40 196 298 
SMHIRNPYRTPIDYQALSEAFPPLKPFVSVNADGTSSVDLTIPEAQRAFTAALLHRDFGLTMTIPEDRLCPTVPNRLNYVLWIEDIFNYTNKTLGLSDDRPIKGVDIGTGASAIYPMLACARFKAWSMVGTEVERKCIDTARLNVVANNLQDRLSILETSIDGPILVPIFEATEEYEYEFTMCNPPFYDGAADMQTSDAAKGFGFGVGAPHSGTVIEMSTEGGESAFVAQMVRESLKLRTRCRWYTSNLGKLKSLKEIVGLLKELEISNYAINEYVQGSTRRYAVAWSFTDIQLPEELSRPSNPELSSLF
2NBG , Knot 62 125 0.79 36 95 120 
GSHMLDHLLEMILLVSEMEELKANPNRRAVGTVIEAKLDKGRGPVATLLVQAGTLKVGDPIVVGTTYGRVRAMVNDSGRRVKEAGPSMPVEITGLHDVPQAGDRFMVFEDEKKARQIGEARAQRQ

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 \(c\).

\(|P_{f(7XNC_1)}(2) \setminus P_{f(8ZIG_1)}(2)|=89\), \(|P_{f(8ZIG_1)}(2) \setminus P_{f(7XNC_1)}(2)|=76\). 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 1:110010100000001011100000110100001001001111011111010001011111001101010111000110010110000010111101110001010101001011010011001101100111010111101100100000110001010011100010101001110101100110011100001010010100001000110111011011110011111010010111100101011001101001101100011000111111011001100111011011101010011000110011001010011101110000100101111100011100
Pair \(Z_2\) Length of longest common subsequence
7XNC_1,8ZIG_1 165 4
7XNC_1,2NBG_1 180 3
8ZIG_1,2NBG_1 171 5

Newick tree

 
[
	2NBG_1:89.46,
	[
		7XNC_1:82.5,8ZIG_1:82.5
	]:6.96
]

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 \((0.85)(0.8)(\frac{658 }{\log_{20} 658}-\frac{310}{\log_{20}310})=96.4\)
Status Protein1 Protein2 d d1/2
Query variables 7XNC_1 8ZIG_1 123 115.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]