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Parikh vectors
6AQY_1 6OMR_1 9NJR_1 Letter Amino acid
8 3 6 C Cysteine
14 8 6 H Histidine
24 12 9 F Phenylalanine
21 43 21 A Alanine
18 30 18 D Aspartic acid
14 24 8 P Proline
21 20 20 T Threonine
28 5 20 K Lycine
11 11 8 M Methionine
16 12 27 S Serine
9 29 19 R Arginine
15 9 17 Q Glutamine
20 20 27 E Glutamic acid
22 24 31 G Glycine
14 9 22 I Isoleucine
22 36 24 V Valine
21 8 11 N Asparagine
25 36 31 L Leucine
5 9 0 W Tryptophan
13 8 9 Y Tyrosine 

6AQY_1|Chains A, B, C, D, E, F|gdp-l-fucose synthetase|Naegleria fowleri (5763)
>6OMR_1|Chains A, B|PtmU3|Streptomyces platensis (58346)
>9NJR_1|Chains A, B, C, D, E, F|Meiotic recombination protein DMC1|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
6AQY , Knot 149 341 0.85 40 220 324 
MAHHHHHHMAQREVSDQPITLTQDDVILVTGGTGLFGKAVEHIVKKEQIKGKWVFLGSKDGDLRDADACKQPFEKYRPTYVIHLAAFVGGLFKNMNFKVSFWLDNVNMNNNILTCCYDFGVKKTISCLSTCVFPDKIEYPITEEKLHEGPPHFSNNAYAYAKRMLDMLGRWYNEKAVNEGKSCLFTSVIPTNLFGPHDNFNVEAGHVLPGLMHKCYKAQQNGTDFVVFGSGKPLRQFLYSHDAARMLLWTMFNYQSEEPIMLCVSEEDEKSIGQVAQTIKDAFNFTGNMVFDTSKADGQYKKTSSNAKFLRLNPTFQYTPFEQAIKETVQWFLENYETARK
6OMR , Knot 151 356 0.83 40 194 339 
MEKLWLNSADSHVLEPDDLWERALPAALRDRAPRCVRDNGRETVYVDGQVVRRDPLDFADAMRPPGALDHHIRLKDLDDQGIWGEVVFPSRGLWTAVMTDPVLARECIKVYNDWLKSDFLSLSPRLVGAAMVSMLDTDDAVAELRRAADLGYQTVFLAATPPPGREFNMDVWEPLWAAAEEAGMTVSIHIGTGADTVVARGPGGAVINYVETLFPAQRAVAQLVASGALDRHPGLRVLIAEAGCAWVPALADRMDEAYRQHGMFVRPKLSMLPGELVRRQVYASFQHDETAIGAVTAMNYTNVLWGSDYPHLEGTFPRTQEVVTELFAGVDPEVRDLITRRNFTDLFTVPALPATV
9NJR , Knot 142 334 0.82 38 197 319 
MSVTGTEIDSDTAKNILSVDELQNYGINASDLQKLKSGGIYTVNTVLSTTRRHLCKIKGLSEVKVEKIKEAAGKIIQVGFIPATVQLDIRQRVYSLSTGSKQLDSILGGGIMTMSITEVFGEFRCGKTQMSHTLCVTTQLPREMGGGEGKVAYIDTEGTFRPERIKQIAEGYELDPESCLANVSYARALNSEHQMELVEQLGEELSSGDYRLIVVDSIMANFRVDYCGRGELSERQQKLNQHLFKLNRLAEEFNVAVFLTNQVQSDPGASALFASADGRKPIGGHVLAHASATRILLRKGRGDERVAKLQDSPDMPEKECVYVIGEKGITDSSD

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 \(c\).

\(|P_{f(6AQY_1)}(2) \setminus P_{f(6OMR_1)}(2)|=99\), \(|P_{f(6OMR_1)}(2) \setminus P_{f(6AQY_1)}(2)|=73\). 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 1:11000000110001000110100001111011011110110011000010101111100010100101000110000100110111111111001010101110010100011000001110001001000111001001100001001110100010101001101110100001100100011001110011110001010110111111000001000100111110101100110000110111101100000011110100000001101100100110101011100001010000000010110101010001100110001011100000100
Pair \(Z_2\) Length of longest common subsequence
6AQY_1,6OMR_1 172 3
6AQY_1,9NJR_1 167 4
6OMR_1,9NJR_1 155 4

Newick tree

 
[
	6AQY_1:87.04,
	[
		9NJR_1:77.5,6OMR_1:77.5
	]:9.54
]

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 \((0.85)(0.8)(\frac{697 }{\log_{20} 697}-\frac{341}{\log_{20}341})=97.7\)
Status Protein1 Protein2 d d1/2
Query variables 6AQY_1 6OMR_1 124 123
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]