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Parikh vectors
6AMG_1 1SBN_1 2ZRD_1 Letter Amino acid
19 37 37 A Alanine
4 10 13 Q Glutamine
2 0 0 C Cysteine
16 11 22 K Lycine
10 13 17 T Threonine
4 3 1 W Tryptophan
13 30 28 V Valine
5 2 18 R Arginine
8 18 11 N Asparagine
12 5 27 E Glutamic acid
17 33 37 G Glycine
9 6 4 H Histidine
9 13 25 I Isoleucine
12 15 30 L Leucine
12 14 13 P Proline
7 10 5 Y Tyrosine
8 10 21 D Aspartic acid
3 5 9 M Methionine
13 3 11 F Phenylalanine
13 37 20 S Serine 

6AMG_1|Chains A, B|Cytochrome P460|Nitrosomonas sp. AL212 (153948)
>1SBN_1|Chain A[auth E]|SUBTILISIN NOVO BPN'|Bacillus subtilis (1423)
>2ZRD_1|Chain A|Protein recA|Mycobacterium smegmatis str. MC2 155 (246196)
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 \(w=f(c)\)
6AMG , Knot 92 196 0.82 40 140 188 
MLQFKKTLLSSIAPVLLSIVLANPVIASDAHHAHKGLNYGSFTKEHVLLTPKGYREWVFIGASVTPNELNDDKAAFPEFHNVYIDPTSWGHWKKTGEFRDGTVIVKELAGVGSKASPSGNGYFPGEFNGIAAMVKDSKRYPERPGNWAFFGFESYEAKQGIIQTDETCAACHKEHAAHDMVFTQFYPVLRAGKPSK
1SBN , Knot 114 275 0.77 38 148 254 
AQSVPYGVSQIKAPALHSQGYTGSNVKVAVIDSGIDSSHPDLKVAGGASMVPSETNPFQDNNSHGTHVAGTVAALNNSIGVLGVAPSASLYAVKVLGADGSGQYSWIINGIEWAIANNMDVINMSLGGPSGSAALKAAVDKAVASGVVVVAAAGNEGTSGSSSTVGYPGKYPSVIAVGAVDSSNQRASFSSVGPELDVMAPGVSIQSTLPGNKYGAYNGTSMASPHVAGAAALILSKHPNWTNTQVRSSLENTTTKLGDSFYYGKGLINVQAAAQ
2ZRD , Knot 150 349 0.84 38 187 329 
MAQQAPDREKALELAMAQIDKNFGKGSVMRLGEEVRQPISVIPTGSISLDVALGIGGLPRGRVIEIYGPESSGKTTVALHAVANAQAAGGIAAFIDAEHALDPEYAKKLGVDTDSLLVSQPDTGEQALEIADMLVRSGALDIIVIDSVAALVPRAEIEGEMGDSHVGLQARLMSQALRKMTGALNNSGTTAIFINNLREKIGVMFGSPETTTGGKALKFYASVRLDVRRIETLKDGTDAVGNRTRVKVVKNKVSPPFKQAEFDILYGQGISREGSLIDMGVEHGFIRKSGSWFTYEGEQLGQGKENARKFLLENTDVANEIEKKIKEKLGIGAVVTAEADDVLPAPVDF

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 \(c\).

\(|P_{f(6AMG_1)}(2) \setminus P_{f(1SBN_1)}(2)|=76\), \(|P_{f(1SBN_1)}(2) \setminus P_{f(6AMG_1)}(2)|=84\). 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 1:1101000110011111101111011110010010011001010000111010100011111101010010000111101001010100110100010100101110011111001010101011101011111100000010011011111100001001110000001100000110011100101110110100
Pair \(Z_2\) Length of longest common subsequence
6AMG_1,1SBN_1 160 3
6AMG_1,2ZRD_1 167 4
1SBN_1,2ZRD_1 143 5

Newick tree

 
[
	6AMG_1:84.91,
	[
		1SBN_1:71.5,2ZRD_1:71.5
	]:13.41
]

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 \((0.85)(0.8)(\frac{471 }{\log_{20} 471}-\frac{196}{\log_{20}196})=80.2\)
Status Protein1 Protein2 d d1/2
Query variables 6AMG_1 1SBN_1 98 85.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]