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Parikh vectors
1TCW_1 5AQO_1 6FSY_1 Letter Amino acid
2 12 9 Q Glutamine
5 23 8 E Glutamic acid
2 6 1 H Histidine
11 26 7 I Isoleucine
6 28 12 K Lycine
3 18 5 F Phenylalanine
1 1 3 W Tryptophan
4 28 6 D Aspartic acid
8 31 5 V Valine
2 10 7 Y Tyrosine
4 22 3 R Arginine
10 26 13 L Leucine
2 7 5 M Methionine
3 37 5 A Alanine
5 18 12 N Asparagine
13 30 2 G Glycine
6 12 12 P Proline
3 22 2 S Serine
9 27 8 T Threonine
0 2 2 C Cysteine 

1TCW_1|Chains A, B|SIV PROTEASE|Simian immunodeficiency virus (11723)
>5AQO_1|Chains A, C, E|HEAT SHOCK COGNATE 71 KDA PROTEIN|HOMO SAPIENS (9606)
>6FSY_1|Chain A|Bromodomain-containing 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 \(w=f(c)\)
1TCW , Knot 53 99 0.82 38 77 96 
PQFSLWRRPVVTAHIEGQPVEVLLDTGADDSIVTGIELGPHYTPKIVGGIGGFINTKEYKNVKIEVLGKRIKGTIMTGDTPINIFGRNLLTALGMSLNL
5AQO , Knot 164 386 0.84 40 214 369 
GPLGSMSKGPAVGIDLGTTYSCVGVFQHGKVEIIANDQGNRTTPSYVAFTDTERLIGDAAKNQVAMNPTNTVFDAKRLIGRRFDDAVVQSDMKHWPFMVVNDAGRPKVQVEYKGETKSFYPEEVSSMVLTKMKEIAEAYLGKTVTNAVVTVPAYFNDSQRQATKDAGTIAGLNVLRIINEPTAAAIAYGLDKKVGAERNVLIFDLGGGTFDVSILTIEDGIFEVKSTAGDTHLGGEDFDNRMVNHFIAEFKRKHKKDISENKRAVRRLRTACERAKRTLSSSTQASIEIDSLYEGIDFYTSITRARFEELNADLFRGTLDPVEKALRDAKLDKSQIHDIVLVGGSTRIPKIQKLLQDFFNGKELNKSINPDEAVAYGAAVQAAILS
6FSY , Knot 66 127 0.84 40 110 123 
SMNPPPPETSNPNKPKRQTNQLQYLLRVVLKTLWKHQFAWPFQQPVDAVKLNLPDYYKIIKTPMDMGTIKKRLENNYYWNAQECIQDFNTMFTNCYIYNKPGDDIVLMAEALEKLFLQKINELPTEE

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 \(c\).

\(|P_{f(1TCW_1)}(2) \setminus P_{f(5AQO_1)}(2)|=21\), \(|P_{f(5AQO_1)}(2) \setminus P_{f(1TCW_1)}(2)|=158\). 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 1:101011001110101010110111001100011011011100010111111111000000010101110010101101001101110011011110101
Pair \(Z_2\) Length of longest common subsequence
1TCW_1,5AQO_1 179 3
1TCW_1,6FSY_1 137 3
5AQO_1,6FSY_1 196 4

Newick tree

 
[
	5AQO_1:10.88,
	[
		1TCW_1:68.5,6FSY_1:68.5
	]:32.38
]

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 \((0.85)(0.8)(\frac{485 }{\log_{20} 485}-\frac{99}{\log_{20}99})=115.\)
Status Protein1 Protein2 d d1/2
Query variables 1TCW_1 5AQO_1 145 90
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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