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Parikh vectors
2VPB_1 1IAO_1 8YKH_1 Letter Amino acid
2 7 7 H Histidine
1 12 8 F Phenylalanine
2 9 6 N Asparagine
5 16 8 D Aspartic acid
4 14 7 E Glutamic acid
2 0 2 M Methionine
3 14 13 V Valine
1 5 11 R Arginine
7 2 4 C Cysteine
3 18 19 L Leucine
5 11 37 S Serine
2 4 6 W Tryptophan
5 12 33 G Glycine
1 10 7 K Lycine
2 14 11 P Proline
6 13 19 T Threonine
2 7 12 Y Tyrosine
7 7 15 A Alanine
2 7 15 Q Glutamine
3 12 11 I Isoleucine 

2VPB_1|Chain A|PYGOPUS HOMOLOG 1|HOMO SAPIENS (9606)
>1IAO_1|Chain A|MHC CLASS II I-AD|Mus musculus (10090)
>8YKH_1|Chain A|NT-108 scFv|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 \(w=f(c)\)
2VPB , Knot 37 65 0.79 40 58 61 
GHSSSDPVYPCGICTNEVNDDQDAILCEASCQKWFHRICTGMTETAYGLLTAEASAVWGCDTCMA
1IAO , Knot 95 194 0.86 38 151 190 
EDDIEADHVGFYGTTVYQSPGDIGQYTHEFDGDELFYVDLDKKKTVWRLPEFGQLILFEPQGGLQNIAAEKHNLGILTKRSNFTPATNEAPQATVFPKSPVLLGQPNTLICFVDNIFPPVINITWLRNSKSVTDGVYETSFLVNRDHSFHKLSYLTFIPSDDDIYDCKVEHWGLEEPVLKHWEPEISSADLVPR
8YKH , Knot 105 251 0.77 40 154 222 
EIVLTQSPGTLSLSPGERATLSCRASQSVSSTFLAWYQQKPGQAPRLLIYGASYMATGIPDRFSGSGSGTDFTLTISRLEPEDFAVYYCQQYGSSLTFGGGTKLEIKGGGGSGGGGSGGGGSQVQLQQSGPGLVKPSQTLSLTCSISGDSVSSNSAAWNWIRQSPSRGLEWLGRTYYRSKWYNDYAGTVKSRIAINPDTSKNQFSLHLNSVTPEDTAVYFCARVISVAGYAFDIWGQGTMVTVSSHHHHHH

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 \(c\).

\(|P_{f(2VPB_1)}(2) \setminus P_{f(1IAO_1)}(2)|=35\), \(|P_{f(1IAO_1)}(2) \setminus P_{f(2VPB_1)}(2)|=128\). 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 1:10000011010110000100000111001000011001001100010111010101111000011
Pair \(Z_2\) Length of longest common subsequence
2VPB_1,1IAO_1 163 3
2VPB_1,8YKH_1 166 2
1IAO_1,8YKH_1 173 4

Newick tree

 
[
	8YKH_1:85.82,
	[
		2VPB_1:81.5,1IAO_1:81.5
	]:4.32
]

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 \((0.85)(0.8)(\frac{259 }{\log_{20} 259}-\frac{65}{\log_{20}65})=63.2\)
Status Protein1 Protein2 d d1/2
Query variables 2VPB_1 1IAO_1 82 53.5
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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