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Parikh vectors
2DIG_1 5JBN_1 2DSR_1 Letter Amino acid
9 6 8 G Glycine
5 15 9 L Leucine
1 9 1 F Phenylalanine
3 8 2 T Threonine
4 14 2 V Valine
3 8 6 R Arginine
0 0 6 C Cysteine
6 10 5 E Glutamic acid
2 8 3 I Isoleucine
1 10 0 M Methionine
1 1 1 W Tryptophan
3 8 6 P Proline
13 9 4 S Serine
4 2 1 Y Tyrosine
1 19 5 A Alanine
1 6 3 N Asparagine
4 8 5 D Aspartic acid
1 8 5 Q Glutamine
1 15 5 H Histidine
5 6 5 K Lycine 

2DIG_1|Chain A|Lamin-B receptor|Homo sapiens (9606)
>5JBN_1|Chains A, B|Phosphopantetheine adenylyltransferase|Escherichia coli (strain K12) (83333)
>2DSR_1|Chain A[auth G]|Insulin-like growth factor-binding protein 4|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 \(w=f(c)\)
2DIG , Knot 37 68 0.76 38 54 62 
GSSGSSGMPSRKFADGEVVRGRWPGSSLYYEVEILSHDSTSQLYTVKYKDGTELELKENDIKSGPSSG
5JBN , Knot 80 170 0.80 38 126 162 
MDPMQKRAIYPGTFDPITNGHIDIVTRATQMFDHVILAIAASPSKKPMFTLEERVALAQQATAHLGNVEVVGFSDLMANFARNQHATVLIRGLRAVADFEYEMQLAHMNRHLMPELESVFLMPSKEWSFISSSLVKEVARHQGDVTHFLPENVHQALMAKLAVDHHHHHH
2DSR , Knot 45 82 0.80 38 70 80 
GSCQSELHRALERLAASQSRTHEDLYIIPIPNCDRNGNFHPKQCHPALDGQRGKCWCVDRKTGVKLPGGLEPKGELDCHQLA

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 \(c\).

\(|P_{f(2DIG_1)}(2) \setminus P_{f(5JBN_1)}(2)|=32\), \(|P_{f(5JBN_1)}(2) \setminus P_{f(2DIG_1)}(2)|=104\). 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 1:10010011100011010110101110010001011000000010010000100101000010011001
Pair \(Z_2\) Length of longest common subsequence
2DIG_1,5JBN_1 136 3
2DIG_1,2DSR_1 98 2
5JBN_1,2DSR_1 148 3

Newick tree

 
[
	5JBN_1:77.02,
	[
		2DIG_1:49,2DSR_1:49
	]:28.02
]

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 \((0.85)(0.8)(\frac{238 }{\log_{20} 238}-\frac{68}{\log_{20}68})=55.7\)
Status Protein1 Protein2 d d1/2
Query variables 2DIG_1 5JBN_1 70 47
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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