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Parikh vectors
1MQX_1 2VTJ_1 8PRY_1 Letter Amino acid
0 19 52 A Alanine
0 16 25 D Aspartic acid
3 3 10 C Cysteine
0 9 22 Q Glutamine
1 17 29 E Glutamic acid
1 16 21 F Phenylalanine
0 10 9 Y Tyrosine
0 8 15 N Asparagine
4 16 38 G Glycine
1 16 24 I Isoleucine
0 17 34 T Threonine
2 39 55 L Leucine
0 21 32 K Lycine
0 5 20 M Methionine
1 14 31 S Serine
0 16 28 R Arginine
0 10 14 H Histidine
1 19 16 P Proline
0 4 13 W Tryptophan
1 23 35 V Valine 

1MQX_1|Chain A|LANTIBIOTIC MERSACIDIN|BACILLUS SP. HIL-Y85/54728 (69002)
>2VTJ_1|Chain A|CELL DIVISION PROTEIN KINASE 2|HOMO SAPIENS (9606)
>8PRY_1|Chains A, B, C, D|glycerol kinase|Trypanosoma cruzi (5693)
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 \(w=f(c)\)
1MQX , Knot 12 20 0.60 20 15 17 
CXFXLPGGGGVCXLXSECIX
2VTJ , Knot 134 298 0.85 40 191 284 
MENFQKVEKIGEGTYGVVYKARNKLTGEVVALKKIRLDTETEGVPSTAIREISLLKELNHPNIVKLLDVIHTENKLYLVFEFLHQDLKKFMDASALTGIPLPLIKSYLFQLLQGLAFCHSHRVLHRDLKPQNLLINTEGAIKLADFGLARAFGVPVRTYTHEVVTLWYRAPEILLGCKYYSTAVDIWSLGCIFAEMVTRRALFPGDSEIDQLFRIFRTLGTPDEVVWPGVTSMPDYKPSFPKWARQDFSKVVPPLDEDGRSLLSQMLHYDPNKRISAKAALAHPFFQDVTKPVPHLRL
8PRY , Knot 212 523 0.84 40 258 490 
MAHHHHHHMGAMKYVGALDQGTISTRFIIFDEKQCLVAQHQMPHRQLTPQAGWLEHDPMELYRASVACIVAAVEDLRRRVPSFEKLETIGIANQRETTVAWDRVTKETLYNAIVWSDLRSYEVTANVKKELGGGDDLFFAKMNGLRISTYFSAFKMRWMLENVPKVEEARKRGTLCFGTIDAWLLWKLSGGKVFVTDVTNASRTFLMDIRTCKWSPELCDKLGIPMACLPEIRSNSELFTYVLSDEGGLSTALRHPTPIMGSIADQQGALLGNMCFKEGESKNTYGTGCFLLMTVGERIRFSDHGLLSTVAYQLGNKSPFIYALEGSIAGAGATIEWLKNNMKLIDDVSDCEKFAQTVSDTQGVVFVPAFSGFLAPSWDPSARGSIFGMTLKTTRAHVLRAALLAIALQVVDVLEAMEKDAEVKVQFLRVDGGLTKNHLLMKMQSDLLGINLHRPTMAETTALGAALCAGLAAGVWKSLEEVKTISQQANVWRVIRPAASKDSQQMLRAQWKRAKQQAKWAKL

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 \(c\).

\(|P_{f(1MQX_1)}(2) \setminus P_{f(2VTJ_1)}(2)|=10\), \(|P_{f(2VTJ_1)}(2) \setminus P_{f(1MQX_1)}(2)|=186\). 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 1:00101111111001000010
Pair \(Z_2\) Length of longest common subsequence
1MQX_1,2VTJ_1 196 2
1MQX_1,8PRY_1 263 3
2VTJ_1,8PRY_1 173 3

Newick tree

 
[
	1MQX_1:12.24,
	[
		2VTJ_1:86.5,8PRY_1:86.5
	]:37.74
]

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 \((0.85)(0.8)(\frac{318 }{\log_{20} 318}-\frac{20}{\log_{20}20})=98.8\)
Status Protein1 Protein2 d d1/2
Query variables 1MQX_1 2VTJ_1 127 67.5
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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