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Parikh vectors
7LSJ_1 4JJU_1 4NIZ_1 Letter Amino acid
10 58 6 L Leucine
3 12 1 M Methionine
2 14 0 F Phenylalanine
1 23 1 Y Tyrosine
3 20 0 C Cysteine
3 31 1 G Glycine
7 18 1 H Histidine
2 25 0 I Isoleucine
11 26 1 D Aspartic acid
8 28 6 E Glutamic acid
11 29 5 K Lycine
3 41 0 T Threonine
4 32 2 V Valine
17 54 2 A Alanine
3 42 3 R Arginine
4 31 0 P Proline
2 50 1 S Serine
6 15 2 N Asparagine
5 18 1 Q Glutamine
1 9 0 W Tryptophan 

7LSJ_1|Chains A, B[auth C]|Soluble cytochrome b562|Escherichia coli (562)
>4JJU_1|Chains A, B|Genome polyprotein|Hepatitis C virus (420174)
>4NIZ_1|Chains A, B|General control protein GCN4|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 \(w=f(c)\)
7LSJ , Knot 54 106 0.79 40 80 101 
ADLEDNMETLNDNLKVIEKADNAAQVKDALTKMRAAALDAQKATPPKLEDKSPDSPEMWHFRAGFDHLVGHIDDALKLANEGKVKEAQAAAEQLKCHCNHCHQHYR
4JJU , Knot 231 576 0.85 40 274 533 
SMSYTWTGALITPCAAEESKLPINPLSNSLLRHHNMVYATTSRSASLRQKKVTFDRLQVLDDHYRDVLKEMKAKASTVKAKLLSIEEACKLTPPHSAKSKFGYGAKDVRNLSSRAVNHIRSVWEDLLEDTETPIDTTIMAKSEVFCVQPEKGGRKPARLIVFPDLGVRVCEKMALYDVVSTLPQAVMGSSYGFQYSPKQRVEFLVNTWKSKKCPMGFSYDTRCFDSTVTESDIRVEESIYQCCDLAPEARQAIRSLTERLYIGGPLTNSKGQNCGYRRCRASGVLTTSCGNTLTCYLKATAACRAAKLQDCTMLVNGDDLVVICESAGTQEDAAALRAFTEAMTRYSAPPGDPPQPEYDLELITSCSSNVSVAHDASGKRVYYLTRDPTTPLARAAWETARHTPINSWLGNIIMYAPTLWARMILMTHFFSILLAQEQLEKALDCQIYGACYSIEPLDLPQIIERLHGLSAFTLHSYSPGEINRVASCLRKLGVPPLRTWRHRARSVRAKLLSQGGRAATCGRYLFNWAVRTKLKLTPIPAASQLDLSGWFVAGYSGGDIYHSLSRARPRHHHHHH
4NIZ , Knot 23 35 0.77 30 33 33 
XRMKQLEDKAEELLSKNYHLENEVARLKKLVGERX

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 \(c\).

\(|P_{f(7LSJ_1)}(2) \setminus P_{f(4JJU_1)}(2)|=22\), \(|P_{f(4JJU_1)}(2) \setminus P_{f(7LSJ_1)}(2)|=216\). 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 1:1010001001000101100100110100110010111101001011010000100101101011100111010011011001010010111001000000000000
Pair \(Z_2\) Length of longest common subsequence
7LSJ_1,4JJU_1 238 4
7LSJ_1,4NIZ_1 89 4
4JJU_1,4NIZ_1 259 3

Newick tree

 
[
	4JJU_1:14.28,
	[
		7LSJ_1:44.5,4NIZ_1:44.5
	]:96.78
]

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 \((0.85)(0.8)(\frac{682 }{\log_{20} 682}-\frac{106}{\log_{20}106})=166.\)
Status Protein1 Protein2 d d1/2
Query variables 7LSJ_1 4JJU_1 214 125
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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