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Parikh vectors
1WAV_1 3ZNQ_1 3GIZ_1 Letter Amino acid
0 17 16 A Alanine
2 23 11 E Glutamic acid
2 38 18 L Leucine
0 22 12 P Proline
0 22 10 R Arginine
2 15 7 N Asparagine
2 15 14 Q Glutamine
1 31 11 G Glycine
0 13 2 H Histidine
2 12 9 Y Tyrosine
0 10 3 W Tryptophan
0 15 8 D Aspartic acid
4 5 4 C Cysteine
2 22 7 I Isoleucine
0 14 8 F Phenylalanine
2 15 31 S Serine
1 22 17 T Threonine
0 12 10 K Lycine
0 5 0 M Methionine
1 19 13 V Valine 

1WAV_1|Chains A, C, E, G, I, K|INSULIN|Sus scrofa (9823)
>3ZNQ_1|Chains A, B|D-AMINO-ACID OXIDASE|HOMO SAPIENS (9606)
>3GIZ_1|Chain A[auth L]|Fab fragment of anti-CD20 antibody Ofatumumab, light chain|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 \(w=f(c)\)
1WAV , Knot 15 21 0.72 22 20 19 
GIVEQCCTSICSLYQLENYCN
3ZNQ , Knot 148 347 0.83 40 210 335 
MRVVVIGAGVIGLSTALCIHERYHSVLQPLDIKVYADRFTPLTTTDVAAGLWQPYLSDPNNPQEADWSQQTFDYLLSHVHSPNAENLGLFLISGYNLFHEAIPDPSWKDTVLGFRKLTPRELDMFPDYGYGWFHTSLILEGKNYLQWLTERLTERGVKFFQRKVESFEEVAREGADVIVNCTGVWAGALQRDPLLQPGRGQIMKVDAPWMKHFILTHDPERGIYNSPYIIPGTQTVTLGGIFQLGNWSELNNIQDHNTIWEGCCRLEPTLKNARIIGERTGFRPVRPQIRLEREQLRTGPSNTEVIHNYGHGGYGLTIHWGCALEAAKLFGRILEEKKLSRMPPSHL
3GIZ , Knot 94 211 0.79 38 148 199 
EIVLTQSPATLSLSPGERATLSCRASQSVSSYLAWYQQKPGQAPRLLIYDASNRATGIPARFSGSGSGTDFTLTISSLEPEDFAVYYCQQRSNWPITFGQGTRLEIKRTVAAPSVFIFPPSDEQLKSGTASVVCLLNNFYPREAKVQWKVDNALQSGNSQESVTEQDSKDSTYSLSSTLTLSKADYEKHKVYACEVTHQGLSSPVTKSFNR

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 \(c\).

\(|P_{f(1WAV_1)}(2) \setminus P_{f(3ZNQ_1)}(2)|=8\), \(|P_{f(3ZNQ_1)}(2) \setminus P_{f(1WAV_1)}(2)|=198\). 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 1:111000000100100100000
Pair \(Z_2\) Length of longest common subsequence
1WAV_1,3ZNQ_1 206 2
1WAV_1,3GIZ_1 152 2
3ZNQ_1,3GIZ_1 176 4

Newick tree

 
[
	3ZNQ_1:10.53,
	[
		1WAV_1:76,3GIZ_1:76
	]:25.53
]

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 \((0.85)(0.8)(\frac{368 }{\log_{20} 368}-\frac{21}{\log_{20}21})=112.\)
Status Protein1 Protein2 d d1/2
Query variables 1WAV_1 3ZNQ_1 141 74.5
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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