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Parikh vectors
2DNE_1 1HTN_1 4HLT_1 Letter Amino acid
2 6 2 C Cysteine
12 14 29 G Glycine
0 2 11 H Histidine
9 7 15 I Isoleucine
6 15 22 L Leucine
7 16 13 T Threonine
2 4 6 Y Tyrosine
10 15 14 A Alanine
6 10 19 V Valine
8 16 21 K Lycine
6 6 21 P Proline
6 9 14 D Aspartic acid
2 11 11 Q Glutamine
11 15 23 E Glutamic acid
2 6 12 F Phenylalanine
12 8 22 S Serine
1 4 4 W Tryptophan
2 6 14 R Arginine
2 4 6 M Methionine
2 8 9 N Asparagine 

2DNE_1|Chain A|Dihydrolipoyllysine-residue acetyltransferase component of pyruvate dehydrogenase complex|Homo sapiens (9606)
>1HTN_1|Chain A|TETRANECTIN|Homo sapiens (9606)
>4HLT_1|Chain A|Pirin|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 \(w=f(c)\)
2DNE , Knot 55 108 0.79 38 87 101 
GSSGSSGQKVPLPSLSPTMQAGTIARWEKKEGDKINEGDLIAEVETDKATVGFESLEECYMAKILVAEGTRDVPIGAIICITVGKPEDIEAFKNYTLDSSAASGPSSG
1HTN , Knot 91 182 0.86 40 149 179 
GEPPTQKPKKIVNAKKDVVNTKMFEELKSRLDTLSQEVALLKEQQALQTVCLKGTKVHMKCFLAFTQTKTFHEASEDCISRGGTLSTPQTGSENDALYEYLRQSVGNEAEIWLGLNDMAAEGTWVDMTGARIAYKNWETEITAQPDGGKTENCAVLSGAANGKWFDKRCRDQLPYICQFGIV
4HLT , Knot 131 288 0.85 40 190 277 
SSKKVTLSVLSREQSEGVGARVRRSIGRPVLKNLDPFLLFDEFKGGRPGGFPDHPHRGFETVSYLLEGGSMAHEDFCGHTGKMNPGDLQWMTAGRGILHAEMPCSEEPAHGLQLWVNLRSSEKMVEPQYQELKSEEIPKPSKDGVTVAVISGEALGIKSKVYTRTPTLYLDFKLDPGAKHSQPIPKGWTSFIYTISGDVYIGPDDAQQKIEPHHTAVLGEGDSVQVENKDPKRSHFVLIAGEPLREPVIQHGPFVMNTNEEISQAILDFRNAKNGFERAKTWKSKIGN

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 \(c\).

\(|P_{f(2DNE_1)}(2) \setminus P_{f(1HTN_1)}(2)|=43\), \(|P_{f(1HTN_1)}(2) \setminus P_{f(2DNE_1)}(2)|=105\). 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 1:100100100111101010101101101000010010010111010000101110010000110111101000111111101011010010110000100011011001
Pair \(Z_2\) Length of longest common subsequence
2DNE_1,1HTN_1 148 4
2DNE_1,4HLT_1 171 3
1HTN_1,4HLT_1 163 4

Newick tree

 
[
	4HLT_1:86.46,
	[
		2DNE_1:74,1HTN_1:74
	]:12.46
]

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 \((0.85)(0.8)(\frac{290 }{\log_{20} 290}-\frac{108}{\log_{20}108})=57.2\)
Status Protein1 Protein2 d d1/2
Query variables 2DNE_1 1HTN_1 75 57.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]