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Parikh vectors
3RQW_1 3VEO_1 2NPH_1 Letter Amino acid
4 11 3 M Methionine
21 29 1 S Serine
21 71 3 A Alanine
20 48 4 R Arginine
7 11 6 K Lycine
16 35 6 P Proline
13 17 1 Y Tyrosine
19 12 3 N Asparagine
14 34 4 E Glutamic acid
17 47 13 G Glycine
16 31 8 T Threonine
14 11 6 Q Glutamine
4 22 1 H Histidine
32 13 12 I Isoleucine
21 21 2 F Phenylalanine
8 11 2 W Tryptophan
23 46 7 V Valine
19 45 4 D Aspartic acid
2 3 1 C Cysteine
31 58 12 L Leucine 

3RQW_1|Chains A, B, C, D, E, F, G, H, I, J|ELIC Pentameric Ligand Gated Ion Channel from Erwinia Chrysanthemi|Dickeya dadantii (198628)
>3VEO_1|Chain A|O-carbamoyltransferase TobZ|Streptoalloteichus tenebrarius (1933)
>2NPH_1|Chains A, B|PROTEASE RETROPEPSIN|Human immunodeficiency virus 1 (11676)
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 \(w=f(c)\)
3RQW , Knot 142 322 0.85 40 201 307 
APADNAADARPVDVSVSIFINKIYGVNTLEQTYKVDGYIVAQWTGKPRKTPGDKPLIVENTQIERWINNGLWVPALEFINVVGSPDTGNKRLMLFPDGRVIYNARFLGSFSNDMDFRLFPFDRQQFVLELEPFSYNNQQLRFSDIQVYTENIDNEEIDEWWIRGKASTHISDIRYDHLSSVQPNQNEFSRITVRIDAVRNPSYYLWSFILPLGLIIAASWSVFWLESFSERLQTSFTLMLTVVAYAFYTSNILPRLPYTTVIDQMIIAGYGSIFAAILLIIFAHHRQANGVEDDLLIQRCRLAFPLGFLAIGCVLVIRGITL
3VEO , Knot 221 576 0.81 40 234 517 
HHHHHHMRVLGLNGWPRDFHDASAALLVDGRIAAFAEEERLTRKKHGYNTAPVQAAAFCLAQAGLTVDDLDAVAFGWDLPAMYRERLGGWPHSDSEALDILLPRDVFPRRTDPPLHFVQHHLAHAASAYYFSGEDRGAVLIVDGQGEEECVTLAHAEGGKITVLDTVPGAWSLGFFYEHVSEYTGLGGDNPGKLMGLAAHGTTVDETLSAFAFDSDGYRLNLIDPQARDPEDWDEYSVTERAWFAHLERIYRLPPNEFVRRYDPAKGRVVRDTRRDPYEYRDLAATAQAALERAVFGLADSVLARTGERTLFVAGGVGLNATMNGKLLTRSTVDKMFVPPVASDIGVSLGAAAAVAVELGDRIAPMGDTAAWGPEFSPDQVRAALDRTGLAYREPANLEREVAALIASGKVVGWAQGRGEVGPRALGQRSLLGSAHSPTMRDHINLRVKDREWWRPFAPSMLRSVSDQVLEVDADFPYMIMTTKVRAAYAERLPSVVHEDWSTRPQTVTEASNPRYHRMLTELGDLVGDPVCLNTSFNDRGEPIVSSPADALLTFSRLPIDALAVGPYLVTKDLRH
2NPH , Knot 51 99 0.79 40 81 94 
PQVTLWQRPLVTIKIGGQLKEALLDTGADDTVLEEMSLPGRWKPKMIGGIGGFIKVRQYDQILIEICGHKAIGTVLVGPTPVNIIGRNLLTQIGMTLNF

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 \(c\).

\(|P_{f(3RQW_1)}(2) \setminus P_{f(3VEO_1)}(2)|=63\), \(|P_{f(3VEO_1)}(2) \setminus P_{f(3RQW_1)}(2)|=96\). 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 1:1110011010110101011100101100100000101011101010100011001111000010011001111111011011101001000111110101100101110100010101111000011101011000000101001010000100001001110101000100100001001010000100101010110010001101111111111101011110010001000101110111011000011101100011001111101011111111111000010110001110000111111111110111101101
Pair \(Z_2\) Length of longest common subsequence
3RQW_1,3VEO_1 159 3
3RQW_1,2NPH_1 182 3
3VEO_1,2NPH_1 221 3

Newick tree

 
[
	2NPH_1:10.48,
	[
		3RQW_1:79.5,3VEO_1:79.5
	]:27.98
]

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 \((0.85)(0.8)(\frac{898 }{\log_{20} 898}-\frac{322}{\log_{20}322})=155.\)
Status Protein1 Protein2 d d1/2
Query variables 3RQW_1 3VEO_1 192 151
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]