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Parikh vectors
2WKS_1 3NXF_1 5RFI_1 Letter Amino acid
12 27 9 E Glutamic acid
15 11 26 G Glycine
15 13 21 N Asparagine
10 4 14 Q Glutamine
4 6 7 H Histidine
8 14 11 K Lycine
6 11 17 D Aspartic acid
18 31 11 I Isoleucine
10 7 10 M Methionine
8 5 17 F Phenylalanine
5 22 16 S Serine
6 8 24 T Threonine
0 3 3 W Tryptophan
13 15 17 A Alanine
1 2 12 C Cysteine
12 25 29 L Leucine
7 10 13 P Proline
4 9 11 Y Tyrosine
9 13 27 V Valine
4 21 11 R Arginine 

2WKS_1|Chains A, B, C, D, E, F|3-DEHYDROQUINATE DEHYDRATASE|HELICOBACTER PYLORI (210)
>3NXF_1|Chain A|Retro-Aldolase|artificial gene (32630)
>5RFI_1|Chain A|3C-like proteinase|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
2WKS , Knot 84 167 0.85 38 125 164 
MKILVIQGPNLNMLGHRDPRLYGMVTLDQIHEIMQTFVKQGNLDVELEFFQTNFEGEIIDKIQESVGSDYEGIIINPGAFSHTSIAIADAIMLAGKPVIEVHLTNIQAREEFRKNSYTGAACGGVIMGFGPLGYNMALMAMVNILAEMKAFQEAQKNNPNNPINNQK
3NXF , Knot 113 258 0.81 42 157 246 
MPRYLKGWLEDVVQLSLRRPSVRASRQRPIISLNERILEFNKRNITAIIATYMRKSPWGLDVERDPIEYAKFMERYAVGLSICTEEKYANGSYETLRKIASSVSIPILMADFIVKESQIDDAYNLGADTVPLIVKILTERELESLLEYARSYGMEPIIXINDENDLDIALRIGARFIGICSRDWETLEINKENQRKLISMIPSNVVKVASTGISERNEIEELRKLGVNAFSIISSLMRNPEKIKELIEGSLEHHHHHH
5RFI , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ

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 \(c\).

\(|P_{f(2WKS_1)}(2) \setminus P_{f(3NXF_1)}(2)|=68\), \(|P_{f(3NXF_1)}(2) \setminus P_{f(2WKS_1)}(2)|=100\). 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 1:10111101101011100010101110100100110011001010101011000101011001000110000111101111000011110111111011101010010100010000001110111111111110011111110111010110010000100110000
Pair \(Z_2\) Length of longest common subsequence
2WKS_1,3NXF_1 168 4
2WKS_1,5RFI_1 183 3
3NXF_1,5RFI_1 165 3

Newick tree

 
[
	2WKS_1:89.53,
	[
		3NXF_1:82.5,5RFI_1:82.5
	]:7.03
]

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 \((0.85)(0.8)(\frac{425 }{\log_{20} 425}-\frac{167}{\log_{20}167})=76.5\)
Status Protein1 Protein2 d d1/2
Query variables 2WKS_1 3NXF_1 96 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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]