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Parikh vectors
1CNH_1 2XVH_1 7TGH_1 Letter Amino acid
1 7 5 C Cysteine
17 23 22 V Valine
10 17 11 N Asparagine
19 33 4 D Aspartic acid
10 13 3 Q Glutamine
12 20 2 H Histidine
8 27 18 Y Tyrosine
7 21 7 R Arginine
9 21 34 I Isoleucine
12 25 32 F Phenylalanine
18 25 21 S Serine
7 14 6 W Tryptophan
13 32 16 A Alanine
22 27 18 G Glycine
26 31 42 L Leucine
24 27 6 K Lycine
1 14 8 M Methionine
17 22 8 P Proline
12 28 8 T Threonine
14 37 13 E Glutamic acid 

1CNH_1|Chain A|CARBONIC ANHYDRASE II|Homo sapiens (9606)
>2XVH_1|Chains A, B, C|FLAVIN-CONTAINING MONOOXYGENASE|METHYLOPHAGA AMINISULFIDIVORANS (230105)
>7TGH_1|Chain A[auth 1]|NADH-ubiquinone oxidoreductase chain 1|Tetrahymena thermophila (5911)
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 \(w=f(c)\)
1CNH , Knot 111 259 0.79 40 174 248 
SHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIEFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK
2XVH , Knot 195 464 0.86 40 259 446 
MATRIAILGAGPSGMAQLRAFQSAQEKGAEIPELVCFEKQADWGGQWNYTWRTGLDENGEPVHSSMYRYLWSNGPKECLEFADYTFDEHFGKPIASYPPREVLWDYIKGRVEKAGVRKYIRFNTAVRHVEFNEDSQTFTVTVQDHTTDTIYSEEFDYVVCCTGHFSTPYVPEFEGFEKFGGRILHAHDFRDALEFKDKTVLLVGSSYSAEDIGSQCYKYGAKKLISCYRTAPMGYKWPENWDERPNLVRVDTENAYFADGSSEKVDAIILCTGYIHHFPFLNDDLRLVTNNRLWPLNLYKGVVWEDNPKFFYIGMQDQWYSFNMFDAQAWYARDVIMGRLPLPSKEEMKADSMAWREKELTLVTAEEMYTYQGDYIQNLIDMTDYPSFDIPATNKTFLEWKHHKKENIMTFRDHSYRSLMTGTMAPKHHTPWIDALDDSLEAYLSDKSEIPVAKEALEHHHHHH
7TGH , Knot 123 284 0.81 40 166 265 
MGFWIAPLFVNILSLPLYLVMLVYNIVCMLLITLVIASITLIERKVLSLVQRRVGPHYVGYRGRLQYIADALKLFIKGIVVPEGSNKFWFVAIPSAAGAICYTFWINSMWGPSVSIFDLEYNLVYATILSILFSFCIMLTGYFSKSKYAFMASIRCAILMLNIEIFLGLLVINLIFISESFCFSVFVIYQEIIWLIFIFFGVSGLIFITFLLETNRAPFDLAEAESELVTGYSVEYGGFYFALYYLGEYFHLFFFSMVISIVLFGGWELPNFLYLFLLNDFNIL

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 \(c\).

\(|P_{f(1CNH_1)}(2) \setminus P_{f(2XVH_1)}(2)|=41\), \(|P_{f(2XVH_1)}(2) \setminus P_{f(1CNH_1)}(2)|=126\). 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 1:0001101000110010001111010000110100001000101011010000100101100101101010000001110111101000110101011010101000010000011010110100001011011001011111111101101011100110110010001001010010101111001001001101001111001011110011010000110100101010101001110010110110000101010
Pair \(Z_2\) Length of longest common subsequence
1CNH_1,2XVH_1 167 3
1CNH_1,7TGH_1 184 3
2XVH_1,7TGH_1 193 3

Newick tree

 
[
	7TGH_1:97.60,
	[
		1CNH_1:83.5,2XVH_1:83.5
	]:14.10
]

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 \((0.85)(0.8)(\frac{723 }{\log_{20} 723}-\frac{259}{\log_{20}259})=128.\)
Status Protein1 Protein2 d d1/2
Query variables 1CNH_1 2XVH_1 166 127
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]