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Parikh vectors
6CBP_1 2MVX_1 7EJW_1 Letter Amino acid
2 2 9 E Glutamic acid
1 3 7 H Histidine
0 0 1 W Tryptophan
0 1 4 Y Tyrosine
2 3 31 A Alanine
4 1 17 R Arginine
1 1 12 Q Glutamine
2 0 14 P Proline
1 2 17 S Serine
0 6 36 V Valine
5 2 9 I Isoleucine
0 2 35 L Leucine
0 2 12 K Lycine
5 1 8 N Asparagine
1 3 18 D Aspartic acid
2 0 2 C Cysteine
2 6 24 G Glycine
0 1 8 M Methionine
0 3 9 F Phenylalanine
2 0 12 T Threonine 

6CBP_1|Chain A[auth P]|Man9-V3 glycopeptide|Human immunodeficiency virus 1 (11676)
>2MVX_1|Chains A, B, C, D, E, F, G, H, I, J|Amyloid beta A4 protein|Homo sapiens (9606)
>7EJW_1|Chains A, B|Transcriptional antiactivator FleN|Pseudomonas aeruginosa PAO1 (208964)
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 \(w=f(c)\)
6CBP , Knot 19 30 0.71 26 25 27 
EINCTRPNNNTRPGEIIGDIRQAHCNISRA
2MVX , Knot 25 39 0.78 32 37 37 
DAEFRHDSGYEVHHQKLVFFADVGSNKGAIIGLMVGGVV
7EJW , Knot 122 285 0.80 40 176 265 
GPLGSMKQMGSMHPVQVIAVTGGKGGVGKTNVSVNLALALADLGRRVMLLDADLGLANVDVLLGLTPKRTLADVIEGRCELRDVLLLGPGGVRIVPAASGTQSMVHLSPMQHAGLIQAFSDISDNLDVLVVDTAAGIGDSVVSFVRAAQEVLLVVCDEPTSITDAYALIKLLNRDHGMTRFRVLANMAHSPQEGRNLFAKLTKVTDRFLDVALQYVGVIPYDESVRKAVQKQRAVYEAFPRSKASLAFKAVAQKVDSWPLPANPRGHLEFFVERLVQHPATGSAV

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 \(c\).

\(|P_{f(6CBP_1)}(2) \setminus P_{f(2MVX_1)}(2)|=23\), \(|P_{f(2MVX_1)}(2) \setminus P_{f(6CBP_1)}(2)|=35\). 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 1:010000100000110111010010001001
Pair \(Z_2\) Length of longest common subsequence
6CBP_1,2MVX_1 58 3
6CBP_1,7EJW_1 183 3
2MVX_1,7EJW_1 165 3

Newick tree

 
[
	7EJW_1:99.19,
	[
		6CBP_1:29,2MVX_1:29
	]:70.19
]

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 \((0.85)(0.8)(\frac{69 }{\log_{20} 69}-\frac{30}{\log_{20}30})=15.2\)
Status Protein1 Protein2 d d1/2
Query variables 6CBP_1 2MVX_1 19 17
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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