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Parikh vectors
1CYV_1 4DQH_1 5VSE_1 Letter Amino acid
1 3 21 R Arginine
8 13 16 G Glycine
12 7 12 K Lycine
9 6 14 V Valine
5 5 12 A Alanine
6 4 22 D Aspartic acid
9 3 19 E Glutamic acid
4 15 22 I Isoleucine
1 2 3 M Methionine
2 2 12 F Phenylalanine
7 8 11 T Threonine
5 4 14 N Asparagine
1 1 5 H Histidine
2 0 17 S Serine
6 5 11 Q Glutamine
9 10 18 L Leucine
5 6 10 P Proline
0 2 4 W Tryptophan
6 1 12 Y Tyrosine
0 2 20 C Cysteine 

1CYV_1|Chain A|CYSTATIN A|Homo sapiens (9606)
>4DQH_1|Chains A, B|Wild-type HIV-1 protease dimer|Human immunodeficiency virus 1 (11676)
>5VSE_1|Chains A, B|Histone-lysine N-methyltransferase EHMT2|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 \(w=f(c)\)
1CYV , Knot 50 98 0.78 36 80 94 
MIPGGLSEAKPATPEIQEIVDKVKPQLEEKTNETYGKLEAVQYKTQVVAGTNYYIKVRAGDNKYLHLKVFKSLPGQNEDLVLTGYQVDKNKDDELTGF
4DQH , Knot 51 99 0.79 38 77 94 
PQITLWKRPLVTICIGGQLKEALLDTGADDTVIEEMNLPGKWKPKMIGGIGGFIKVRQYDQIIICIAGHKAIGTVLVGPTPVNIIGRNLLTQIGATLNF
5VSE , Knot 127 275 0.86 40 196 270 
TRTEKIICRDVARGYENVPIPCVNGVDGEPCPEDYKYISENCETSTMNIDRNITHLQHCTCVDDCSSSNCLCGQLSIRCWYDKDGRLLQEFNKIEPPLIFECNQACSCWRNCKNRVVQSGIKVRLQLYRTAKMGWGVRALQTIPQGTFICEYVGELISDAEADVREDDSYLFDLDNKDGEVYCIDARYYGNISRFINHLCDPNIIPVRVFMLHQDLRFPRIAFFSSRDIRTGEELGFDYGDRFWDIKSKYFTCQCGSEKCKHSAEAIALEQSRLA

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 \(c\).

\(|P_{f(1CYV_1)}(2) \setminus P_{f(4DQH_1)}(2)|=53\), \(|P_{f(4DQH_1)}(2) \setminus P_{f(1CYV_1)}(2)|=50\). 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 1:11111100101101010011001010100000000101011000001111000010101100001010110011100001110100100000001011
Pair \(Z_2\) Length of longest common subsequence
1CYV_1,4DQH_1 103 4
1CYV_1,5VSE_1 184 4
4DQH_1,5VSE_1 201 4

Newick tree

 
[
	5VSE_1:10.20,
	[
		1CYV_1:51.5,4DQH_1:51.5
	]:55.70
]

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 \((0.85)(0.8)(\frac{197 }{\log_{20} 197}-\frac{98}{\log_{20}98})=32.4\)
Status Protein1 Protein2 d d1/2
Query variables 1CYV_1 4DQH_1 39 38.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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