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Parikh vectors
5PDF_1 4YJS_1 5INZ_1 Letter Amino acid
2 5 6 C Cysteine
6 17 2 G Glycine
4 9 0 I Isoleucine
6 11 0 P Proline
4 19 0 A Alanine
6 16 0 N Asparagine
8 5 0 H Histidine
11 12 0 S Serine
9 24 0 E Glutamic acid
8 7 0 F Phenylalanine
8 7 0 T Threonine
4 19 0 Y Tyrosine
8 23 4 V Valine
5 13 6 R Arginine
12 13 0 D Aspartic acid
14 25 0 K Lycine
6 14 0 M Methionine
2 6 0 W Tryptophan
2 9 0 Q Glutamine
13 27 0 L Leucine 

5PDF_1|Chain A|Bromodomain adjacent to zinc finger domain protein 2B|Homo sapiens (9606)
>4YJS_1|Chain A|Tyrosine-protein kinase SYK|Homo sapiens (9606)
>5INZ_1|Chains A, B, C|Theta defensin-2, L-peptide|Papio anubis (9555)
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 \(w=f(c)\)
5PDF , Knot 66 138 0.78 40 109 129 
MHHHHHHSSGVDLGTENLYFQSMSVKKPKRDDSKDLALCSMILTEMETHEDAWPFLLPVNLKLVPGYKKVIKKPMDFSTIREKLSSGQYPNLETFALDVRLVFDNCETFNEDDSDIGRAGHNMRKYFEKKWTDTFKVS
4YJS , Knot 127 281 0.85 40 188 270 
PEEIRPKEVYLDRKLLTLEDKELGSGNFGTVKKGYYQMKKVVKTVAVKILKNEANDPALKDELLAEANVMQQLDNPYIVRMIGICEAESWMLVMEMAELGPLNKYLQQNRHVKDKNIIELVHQVSMGMKYLEESNFVHRDLAARNVLLVTQHYAKISDFGLSKALRADENYYKAQTHGKWPVKWYAPECINYYKFSSKSDVWSFGVLMWEAFSYGQKPYRGMKGSEVTAMLEKGERMGCPAGCPREMYDLMNLCWTYDVENRPGFAAVELRLRNYYYDVVN
5INZ , Knot 7 18 0.37 8 7 8 
GVCRCVCRRGVCRCVCRR

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 \(c\).

\(|P_{f(5PDF_1)}(2) \setminus P_{f(4YJS_1)}(2)|=47\), \(|P_{f(4YJS_1)}(2) \setminus P_{f(5PDF_1)}(2)|=126\). 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 1:100000000110110001010010100100000001110011100100000111111110101111000110011010010001001001010011101011100000100000011011001000100010001010
Pair \(Z_2\) Length of longest common subsequence
5PDF_1,4YJS_1 173 3
5PDF_1,5INZ_1 114 2
4YJS_1,5INZ_1 191 2

Newick tree

 
[
	4YJS_1:99.92,
	[
		5PDF_1:57,5INZ_1:57
	]:42.92
]

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 \((0.85)(0.8)(\frac{419 }{\log_{20} 419}-\frac{138}{\log_{20}138})=84.3\)
Status Protein1 Protein2 d d1/2
Query variables 5PDF_1 4YJS_1 109 78.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]