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Parikh vectors
2HIU_1 1OOC_1 8CHW_1 Letter Amino acid
0 29 2 D Aspartic acid
2 32 1 I Isoleucine
2 16 2 L Leucine
1 31 5 T Threonine
2 26 3 N Asparagine
2 11 0 Q Glutamine
0 9 1 H Histidine
0 4 3 M Methionine
0 4 1 W Tryptophan
2 14 4 Y Tyrosine
1 23 7 V Valine
0 28 2 A Alanine
2 35 5 S Serine
0 19 8 K Lycine
4 2 1 C Cysteine
2 11 5 E Glutamic acid
1 36 5 G Glycine
0 14 6 F Phenylalanine
0 9 4 P Proline
0 8 3 R Arginine 

2HIU_1|Chain A|INSULIN|Homo sapiens (9606)
>1OOC_1|Chains A, B|Pectate lyase A|Erwinia chrysanthemi (556)
>8CHW_1|Chains A, B|Transcriptional activator protein Pur-alpha|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 \(w=f(c)\)
2HIU , Knot 15 21 0.72 22 20 19 
GIVEQCCTSICSLYQLENYCN
1OOC , Knot 149 361 0.81 40 193 336 
AELVSDKALESAPTVGWASQNGFTTGGAAATSDNIYIVTNISEFTSALSAGAEAKIIQIKGTIDISGGTPYTDFADQKARSQINIPANTTVIGLGTDAKFINGSLIIDGTDGTNNVIIRNVYIQTPIDVEPHYEKGDGWNAEWDAMNITNGAHHVWIDHVTISDGNFTDDMYTTKDGETYVQHDGALDIKRGSDYVTISNSLIDQHDKTMLIGHSDSNGSQDKGKLHVTLFNNVFNRVTERAPRVRYGSIHSFNNVFKGDAKDPVYRYQYSFGIGTSGSVLSEGNSFTIANLSASKACKVVKKFNGSIFSDNGSVLNGSAVDLSGCGFSAYTSKIPYIYDVQPMTTELAQSITDNAGSGKL
8CHW , Knot 41 68 0.84 38 62 66 
GPMAELPEGTSLTVDNKRFFFDVGSNKYGVFMRVSEVKPTYRNSITVPYKVWAKFGHTFCKYSEEMKK

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 \(c\).

\(|P_{f(2HIU_1)}(2) \setminus P_{f(1OOC_1)}(2)|=12\), \(|P_{f(1OOC_1)}(2) \setminus P_{f(2HIU_1)}(2)|=185\). 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
2HIU_1,1OOC_1 197 2
2HIU_1,8CHW_1 76 2
1OOC_1,8CHW_1 177 3

Newick tree

 
[
	1OOC_1:10.86,
	[
		2HIU_1:38,8CHW_1:38
	]:67.86
]

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 \((0.85)(0.8)(\frac{382 }{\log_{20} 382}-\frac{21}{\log_{20}21})=116.\)
Status Protein1 Protein2 d d1/2
Query variables 2HIU_1 1OOC_1 141 75
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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