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Parikh vectors
3JYL_1 7QLY_1 5JHF_1 Letter Amino acid
16 22 5 D Aspartic acid
18 8 6 E Glutamic acid
9 22 5 F Phenylalanine
3 6 3 W Tryptophan
33 31 5 V Valine
40 37 6 A Alanine
8 12 4 N Asparagine
13 12 1 Q Glutamine
4 2 2 H Histidine
34 27 9 L Leucine
16 22 2 P Proline
20 58 7 S Serine
13 17 3 Y Tyrosine
11 3 10 R Arginine
18 19 9 K Lycine
4 2 1 M Methionine
2 2 0 C Cysteine
34 46 2 G Glycine
12 20 4 I Isoleucine
17 51 3 T Threonine 

3JYL_1|Chain A|Quinone oxidoreductase|Pseudomonas syringae pv. tomato (323)
>7QLY_1|Chain A|Endothiapepsin|Cryphonectria parasitica (5116)
>5JHF_1|Chains A, D|KLTH0D11660p|Lachancea thermotolerans (strain ATCC 56472 / CBS 6340 / NRRL Y-8284) (559295)
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 \(w=f(c)\)
3JYL , Knot 140 325 0.83 40 183 304 
MAKRIQFSTVGGPEVLEYVDFEPEAPGPQAVVVRNKAIGLNFIDTYYRSGLYPAPFLPSGLGAEGAGVVEAVGDEVTRFKVGDRVAYGTGPLGAYSEVHVLPEANLVKLADSVSFEQAAALMLKGLTVQYLLRQTYQVKPGEIILFHAAAGGVGSLACQWAKALGAKLIGTVSSPEKAAHAKALGAWETIDYSHEDVAKRVLELTDGKKCPVVYDGVGQDTWLTSLDSVAPRGLVVSFGNASGPVSGVNLGILAQKDSVYVTRPTLGSYANNAQNLQTMADELFDMLASGKLKVDGIEQYALKDAAKAQIELSARRTTGSTILIP
7QLY , Knot 170 419 0.81 40 190 380 
MSSPLKNALVTAMLAGGALSSPTKQHVGIPVNASPEVGPGKYSFKQVRNPNYKFNGPLSVKKTYLKYGVPIPAWLEDAVQNSTSGLAERSTGSATTTPIDSLDDAYITPVQIGTPAQTLNLDFDTGSSDLWVFSSETTASEVDGQTIYTPSKSTTAKLLSGATWSISYGDGSSSSGDVYTDTVSVGGLTVTGQAVESAKKVSSSFTEDSTIDGLLGLAFSTLNTVSPTQQKTFFDNAKASLDSPVFTADLGYHAPGTYNFGFIDTTAYTGSITYTAVSTKQGFWEWTSTGYAVGSGTFKSTSIDGIADTGTTLLYLPATVVSAYWAQVSGAKSSSSVGGYVFPCSATLPSFTFGVGSARIVIPGDYIDFGPISTGSSSCFGGIQSSAGIGINIFGDVALKAAFVVFNGATTPTLGFASK
5JHF , Knot 48 87 0.82 38 78 84 
MNSENTIVYVRVAGRARNGFVDPLKFYWDLERDRSLWSSVSKLDNTKKTIDWKRLSREFKAPEHFIRKRSYALFAKHLKLLERQIEA

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 \(c\).

\(|P_{f(3JYL_1)}(2) \setminus P_{f(7QLY_1)}(2)|=63\), \(|P_{f(7QLY_1)}(2) \setminus P_{f(3JYL_1)}(2)|=70\). 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 1:1100101001111011001010101111011110001111011000000110111111011110111110111001001011001101011111000101110101101100101001111110110100110000010110111101111111011001101111011101001001101011111001000000110011010010001110011100011001001110111101101011101101111100001010010110010010010011001101110101010110001100110101010100001001111
Pair \(Z_2\) Length of longest common subsequence
3JYL_1,7QLY_1 133 4
3JYL_1,5JHF_1 179 3
7QLY_1,5JHF_1 166 3

Newick tree

 
[
	5JHF_1:91.97,
	[
		3JYL_1:66.5,7QLY_1:66.5
	]:25.47
]

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 \((0.85)(0.8)(\frac{744 }{\log_{20} 744}-\frac{325}{\log_{20}325})=114.\)
Status Protein1 Protein2 d d1/2
Query variables 3JYL_1 7QLY_1 138 122.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]