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Parikh vectors
5NXG_1 5UQA_1 2LIT_1 Letter Amino acid
19 0 4 D Aspartic acid
13 2 7 E Glutamic acid
22 1 12 G Glycine
24 0 15 K Lycine
12 0 4 F Phenylalanine
10 2 7 N Asparagine
17 2 4 S Serine
7 0 1 W Tryptophan
9 2 4 I Isoleucine
1 4 2 C Cysteine
26 2 8 L Leucine
1 0 2 M Methionine
17 0 3 P Proline
17 1 3 V Valine
13 0 8 A Alanine
11 2 2 Q Glutamine
11 0 5 H Histidine
12 1 9 T Threonine
8 2 5 Y Tyrosine
7 0 3 R Arginine 

5NXG_1|Chain A|Carbonic anhydrase 2|Homo sapiens (9606)
>5UQA_1|Chains A, C, E, G, I, K|Insulin, chain A|Homo sapiens (9606)
>2LIT_1|Chain A|Cytochrome c iso-1|Saccharomyces cerevisiae (559292)
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 \(w=f(c)\)
5NXG , Knot 110 257 0.79 40 173 246 
HWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK
5UQA , Knot 15 21 0.72 22 20 19 
GIVEQCCTSICSLYQLENYCN
2LIT , Knot 59 108 0.85 40 92 105 
TEFKAGSAKKGATLFKTRCLQCHTVEKGGPHKVGPNLHGIFGRHSGQAEGYSYTDANIKKNVLWDENNMSEYLTNHAKYIPGTKMAFGGLKKEKDRNDLITYLKKATE

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 \(c\).

\(|P_{f(5NXG_1)}(2) \setminus P_{f(5UQA_1)}(2)|=165\), \(|P_{f(5UQA_1)}(2) \setminus P_{f(5NXG_1)}(2)|=12\). 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 1:01101000110010001111010000110100001000101011010000100101100101101010000001110111101000110101011010101000010000011010110100001011011001011111111101101011100110110010001001010010101111001001001101001111001011110011010000110100101010101001110010110110000101010
Pair \(Z_2\) Length of longest common subsequence
5NXG_1,5UQA_1 177 2
5NXG_1,2LIT_1 167 4
5UQA_1,2LIT_1 104 2

Newick tree

 
[
	5NXG_1:94.70,
	[
		2LIT_1:52,5UQA_1:52
	]:42.70
]

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 \((0.85)(0.8)(\frac{278 }{\log_{20} 278}-\frac{21}{\log_{20}21})=86.5\)
Status Protein1 Protein2 d d1/2
Query variables 5NXG_1 5UQA_1 107 58
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]