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Parikh vectors
5URU_1 1VZQ_1 7UJG_1 Letter Amino acid
0 17 6 R Arginine
2 10 13 N Asparagine
2 20 14 L Leucine
0 18 9 K Lycine
0 7 7 M Methionine
0 14 10 P Proline
2 14 11 S Serine
2 10 8 Y Tyrosine
0 12 9 A Alanine
0 16 9 D Aspartic acid
2 8 9 Q Glutamine
0 5 6 H Histidine
2 16 6 I Isoleucine
0 8 1 W Tryptophan
4 7 10 C Cysteine
2 14 5 E Glutamic acid
1 20 18 G Glycine
0 9 10 F Phenylalanine
1 9 12 T Threonine
1 16 18 V Valine 

5URU_1|Chains A, C, E, G|Insulin Chain A|Homo sapiens (9606)
>1VZQ_1|Chain A[auth H]|THROMBIN HEAVY|HOMO SAPIENS (9606)
>7UJG_1|Chains A, B|3C-like proteinase nsp5|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
5URU , Knot 15 21 0.72 22 20 19 
GIVEQCCTSICSLYQLENYCN
1VZQ , Knot 116 250 0.85 40 181 236 
IVEGSDAEIGMSPWQVMLFRKSPQELLCGASLISDRWVLTAAHCLLYPPWDKNFTENDLLVRIGKHSRTAYERNIEKISMLEKIYIHPRYNWRENLDRDIALMKLKKPVAFSDYIHPVCLPDRETAASLLQAGYKGRVTGWGNLKETGQPSVLQVVNLPIVERPVCKDSTRIRITDNMFCAGYKPDEGKRGDACEGDSGGPFVMKSPFNNRWYQMGIVSWGEGCDRDGKYGFYTHVFRLKKWIQKVIDQF
7UJG , Knot 92 191 0.84 40 149 186 
KMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAG

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 \(c\).

\(|P_{f(5URU_1)}(2) \setminus P_{f(1VZQ_1)}(2)|=12\), \(|P_{f(1VZQ_1)}(2) \setminus P_{f(5URU_1)}(2)|=173\). 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 1:111000000100100100000
Pair \(Z_2\) Length of longest common subsequence
5URU_1,1VZQ_1 185 3
5URU_1,7UJG_1 149 3
1VZQ_1,7UJG_1 182 4

Newick tree

 
[
	1VZQ_1:96.82,
	[
		5URU_1:74.5,7UJG_1:74.5
	]:22.32
]

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 \((0.85)(0.8)(\frac{271 }{\log_{20} 271}-\frac{21}{\log_{20}21})=84.4\)
Status Protein1 Protein2 d d1/2
Query variables 5URU_1 1VZQ_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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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