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Parikh vectors
1IGV_1 1BRK_1 1ONZ_1 Letter Amino acid
2 6 11 N Asparagine
0 2 11 H Histidine
2 7 18 I Isoleucine
0 0 11 M Methionine
6 9 24 S Serine
0 3 6 W Tryptophan
1 7 9 Y Tyrosine
3 4 16 V Valine
2 9 13 A Alanine
0 6 19 R Arginine
4 9 18 D Aspartic acid
13 3 30 E Glutamic acid
5 10 14 G Glycine
2 9 13 T Threonine
4 4 14 Q Glutamine
12 7 29 L Leucine
10 8 23 K Lycine
5 4 14 F Phenylalanine
4 3 22 P Proline
0 0 6 C Cysteine 

1IGV_1|Chain A|VITAMIN D-DEPENDENT CALCIUM-BINDING PROTEIN, INTESTINAL|Bos taurus (9913)
>1BRK_1|Chains A, B, C|BARNASE|Bacillus amyloliquefaciens (1390)
>1ONZ_1|Chain A|Protein-tyrosine phosphatase, non-receptor type 1|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 \(w=f(c)\)
1IGV , Knot 39 75 0.74 30 56 68 
KSPEELKGIFEKYAAKEGDPNQLSKEELKLLLQTEFPSLLKGPSTLDELFEELDKNGDGEVSFEEFQVLVKKISQ
1BRK , Knot 57 110 0.81 36 86 105 
AQVINTFDGVADYLQTYHKLPDNYITKSEAQALGWVASKGNLADVAPGKSIGGDIFSNREGKLPGKSGRTWREADINYTSGFRNSDRILYSSDWLAYKTTDHYQTFTKIR
1ONZ , Knot 141 321 0.84 40 207 309 
MEMEKEFEQIDKSGSWAAIYQDIRHEASDFPCRVAKLPKNKNRNRYRDVSPFDHSRIKLHQEDNDYINASLIKMEEAQRSYILTQGPLPNTCGHFWEMVWEQKSRGVVMLNRVMEKGSLKCAQYWPQKEEKEMIFEDTNLKLTLISEDIKSYYTVRQLELENLTTQETREILHFHYTTWPDFGVPESPASFLNFLFKVRESGSLSPEHGPVVVHCSAGIGRSGTFCLADTCLLLMDKRKDPSSVDIKKVLLEMRKFRMGLIQTADQLRFSYLAVIEGAKFIMGDSSVQDQWKELSHEDLEPPPEHIPPPPRPPKRILEPHN

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 \(c\).

\(|P_{f(1IGV_1)}(2) \setminus P_{f(1BRK_1)}(2)|=44\), \(|P_{f(1BRK_1)}(2) \setminus P_{f(1IGV_1)}(2)|=74\). 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 1:001001011100011001010010000101110001101101100100110010001010101001011100100
Pair \(Z_2\) Length of longest common subsequence
1IGV_1,1BRK_1 118 3
1IGV_1,1ONZ_1 189 3
1BRK_1,1ONZ_1 197 3

Newick tree

 
[
	1ONZ_1:10.11,
	[
		1IGV_1:59,1BRK_1:59
	]:47.11
]

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 \((0.85)(0.8)(\frac{185 }{\log_{20} 185}-\frac{75}{\log_{20}75})=36.8\)
Status Protein1 Protein2 d d1/2
Query variables 1IGV_1 1BRK_1 49 40
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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