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Parikh vectors
2HXH_1 1AYP_1 8OTN_1 Letter Amino acid
37 8 36 A Alanine
13 4 6 H Histidine
31 6 27 L Leucine
19 13 9 K Lycine
22 10 17 S Serine
30 10 13 T Threonine
15 3 10 Q Glutamine
37 11 29 G Glycine
20 5 7 F Phenylalanine
20 2 13 P Proline
26 2 20 I Isoleucine
10 1 10 M Methionine
4 0 4 W Tryptophan
21 10 13 R Arginine
16 6 8 N Asparagine
27 5 15 D Aspartic acid
12 14 1 C Cysteine
38 3 11 E Glutamic acid
19 8 6 Y Tyrosine
34 3 17 V Valine 

2HXH_1|Chain A|Tubulin alpha chain|Sus scrofa (9823)
>1AYP_1|Chains A, B, C, D, E, F|PHOSPHOLIPASE A2|Homo sapiens (9606)
>8OTN_1|Chains A, B, C, D|Enoyl-[acyl-carrier-protein] reductase [NADH]|Mycobacterium tuberculosis (1773)
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 \(w=f(c)\)
2HXH , Knot 191 451 0.86 40 255 425 
MRECISIHVGQAGVQIGNACWELYCLEHGIQPDGQMPSDKTIGGGDDSFNTFFSETGAGKHVPRAVFVDLEPTVIDEVRTGTYRQLFHPEQLITGKEDAANNYARGHYTIGKEIIDLVLDRIRKLADQCTGLQGFSVFHSFGGGTGSGFTSLLMERLSVDYGKKSKLEFSIYPAPQVSTAVVEPYNSILTTHTTLEHSDCAFMVDNEAIYDICRRNLDIERPTYTNLNRLIGQIVSSITASLRFDGALNVDLTEFQTNLVPYPRAHFPLATYAPVISAEKAYHEQLSVAEITNACFEPANQMVKCDPRHGKYMACCLLYRGDVVPKDVNAAIATIKTKRTIQFVDWCPTGFKVGINYEPPTVVPGGDLAKVQRAVCMLSNTTAIAEAWARLDHKFDLMYAKRAFVHWYVGEGMEEGEFSEAREDMAALEKDYEEVGVDSVEGEGEEEGEEY
1AYP , Knot 61 124 0.79 38 99 120 
NLVNFHRMIKLTTGKEAALSYGFYGCHCGVGGRGSPKDATDRCCVTHDCCYKRLEKRGCGTKFLSYKFSNSGSRITCAKQDSCRSQLCECDKAAATCFARNKTTYNKKYQYYSNKHCRGSTPRC
8OTN , Knot 116 272 0.79 40 169 256 
GSHMTGLLDGKRILVSGIITDSSIAFHIARVAQEQGAQLVLTGFDRLRLIQRITDRLPAKAPLLELDVQNEEHLASLAGRVTEAIGAGNKLDGVVHSIGFMPQTGMGINPFFDAPYADVSKGIHISAYSYASMAKALLPIMNPGGSIVGMDFDPSRAMPAYNWMTVAKSALESVNRFVAREAGKYGVRSNLVAAGPIRTLAMSAIVGGALGEEAGAQIQLLEEGWDQRAPIGWNMKDATPVAKTVCALLSDWLPATTGDIIYADGGAHTQLL

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 \(c\).

\(|P_{f(2HXH_1)}(2) \setminus P_{f(1AYP_1)}(2)|=192\), \(|P_{f(1AYP_1)}(2) \setminus P_{f(2HXH_1)}(2)|=36\). 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 1:1000101011011101101010100100110101011000011110001001100011100110111101010110010010000110100110100011000101000110011011100100110000110110110011110101100111001010010000101010111010011101000110000010000011110001100100001010010000100111011001010101011101010010001110101011110011110100100001011010010101100110001001001100110010111001011110100000101101010110111000110111110110100110110000111011101000101101001110101101100101001000111100000011100101010001000
Pair \(Z_2\) Length of longest common subsequence
2HXH_1,1AYP_1 228 4
2HXH_1,8OTN_1 172 3
1AYP_1,8OTN_1 172 4

Newick tree

 
[
	1AYP_1:10.49,
	[
		2HXH_1:86,8OTN_1:86
	]:19.49
]

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 \((0.85)(0.8)(\frac{575 }{\log_{20} 575}-\frac{124}{\log_{20}124})=131.\)
Status Protein1 Protein2 d d1/2
Query variables 2HXH_1 1AYP_1 176 110
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]