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Parikh vectors
1HGV_1 1CEJ_1 6ZPK_1 Letter Amino acid
10 2 8 A Alanine
0 12 2 C Cysteine
2 5 7 Q Glutamine
4 4 17 L Leucine
3 9 8 K Lycine
2 3 7 F Phenylalanine
1 2 14 R Arginine
2 10 5 N Asparagine
1 9 11 E Glutamic acid
3 8 19 S Serine
1 0 2 W Tryptophan
1 7 9 D Aspartic acid
4 7 18 G Glycine
3 3 5 I Isoleucine
1 0 4 M Methionine
1 5 12 P Proline
2 2 3 Y Tyrosine
4 2 24 V Valine
0 2 1 H Histidine
1 4 9 T Threonine 

1HGV_1|Chain A|PH75 INOVIRUS MAJOR COAT PROTEIN|BACTERIOPHAGE PH75 (144736)
>1CEJ_1|Chain A|PROTEIN (MEROZOITE SURFACE PROTEIN 1)|Plasmodium falciparum (5833)
>6ZPK_1|Chain A|Trypanosoma brucei KKT4 463-645|Trypanosoma brucei brucei (strain 927/4 GUTat10.1) (185431)
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 \(w=f(c)\)
1HGV , Knot 27 46 0.75 36 40 44 
MDFNPSEVASQVTNYIQAIAAAGVGVLALAIGLSAAWKYAKRFLKG
1CEJ , Knot 49 96 0.77 36 76 94 
NISQHQCVKKQCPQNSGCFRHLDEREECKCLLNYKQEGDKCVENPNPTCNENNGGCDADAKCTEEDSGSNGKKITCECTKPDSYPLFDGIFCSSSN
6ZPK , Knot 83 185 0.78 40 124 177 
SMSGASSAVGGSTRSPSPVDPKRGAVQPRYFITTSLTEKERNSVMEAIQKLGQRAVLVDNKVDEILPLNTTHIVLRGPPRSVKALCGVVSSKWLVQPSYVFDSLGAGFWLDEEVEGGLRYFPPPLRCQRFLLTMPEGVVKTMLQRVVEFGGGEVVGTKRNGSSNDQDVVVVSSGDELLRFAISRD

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 \(c\).

\(|P_{f(1HGV_1)}(2) \setminus P_{f(1CEJ_1)}(2)|=36\), \(|P_{f(1CEJ_1)}(2) \setminus P_{f(1HGV_1)}(2)|=72\). 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 1:1010100110010001011111111111111110111001001101
Pair \(Z_2\) Length of longest common subsequence
1HGV_1,1CEJ_1 108 2
1HGV_1,6ZPK_1 134 3
1CEJ_1,6ZPK_1 150 3

Newick tree

 
[
	6ZPK_1:75.96,
	[
		1HGV_1:54,1CEJ_1:54
	]:21.96
]

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 \((0.85)(0.8)(\frac{142 }{\log_{20} 142}-\frac{46}{\log_{20}46})=33.8\)
Status Protein1 Protein2 d d1/2
Query variables 1HGV_1 1CEJ_1 44 32
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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