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Parikh vectors
3KTK_1 5PZM_1 2LFB_1 Letter Amino acid
8 16 7 N Asparagine
1 21 2 C Cysteine
21 59 6 L Leucine
20 57 8 A Alanine
21 23 3 I Isoleucine
11 48 5 S Serine
10 38 4 T Threonine
6 23 3 Y Tyrosine
10 42 12 R Arginine
16 27 10 E Glutamic acid
13 30 8 K Lycine
6 13 2 M Methionine
7 35 5 V Valine
7 30 4 P Proline
0 10 2 W Tryptophan
12 26 2 D Aspartic acid
7 18 7 Q Glutamine
14 31 5 G Glycine
5 12 1 H Histidine
4 15 4 F Phenylalanine 

3KTK_1|Chains A, B, C, D, E, F, G, O[auth H], P[auth I], Q[auth J], R[auth K], S[auth L], T[auth M], U[auth N]|ATP-dependent Clp protease proteolytic subunit|Bacillus subtilis (1423)
>5PZM_1|Chains A, B|RNA-directed RNA polymerase|Hepatitis C virus genotype 1b (isolate Con1) (333284)
>2LFB_1|Chain A|LFB1/HNF1 TRANSCRIPTION FACTOR|Rattus rattus (10117)
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 \(w=f(c)\)
3KTK , Knot 93 199 0.82 38 140 193 
NLIPTVIEQTNRGERAYDIYSRLLKDRIIMLGSAIDDNVANSIVSQLLFLAAEDPEKEISLYINSPGGSITAGMAIYDTMQFIKPKVSTICIGMAASMGAFLLAAGEKGKRYALPNSEVMIHQPLGGAQGQATEIEIAAKRILLLRDKLNKVLAERTGQPLEVIERDTDRDNFKSAEEALEYGLIDKILTHTEDKKHHH
5PZM , Knot 230 574 0.84 40 278 537 
MSMSYTWTGALITPCAAEETKLPINALSNSLLRHHNLVYATTSRSASLRQKKVTFDRLQVLDDHYRDVLKEMKAKASTVKAKLLSVEEACKLTPPHSARSKFGYGAKDVRNLSSKAVNHIRSVWKDLLEDTETPIDTTIMAKNEVFCVQPEKGGRKPARLIVFPDLGVRVCEKMALYDVVSTLPQAVMGSSYGFQYSPGQRVEFLVNAWKAKKCPMGFAYDTRCFDSTVTENDIRVEESIYQCCDLAPEARQAIRSLTERLYIGGPLTNSKGQNCGYRRCRASGVLTTSCGNTLTCYLKAAAACRAAKLQDCTMLVCGDDLVVICESAGTQEDEASLRAFTEAMTRYSAPPGDPPKPEYDLELITSCSSNVSVAHDASGKRVYYLTRDPTTPLARAAWETARHTPVNSWLGNIIMYAPTLWARMILMTHFFSILLAQEQLEKALDCQIYGACYSIEPLDLPQIIQRLHGLSAFSLHSYSPGEINRVASCLRKLGVPPLRVWRHRARSVRARLLSQGGRAATCGKYLFNWAVRTKLKLTPIPAASQLDLSSWFVAGYSGGDIYHSLSRARPRWFM
2LFB , Knot 53 100 0.81 40 86 97 
MARIDPTKKGRRNRFKWGPASQQILFQAYERQKNPSKEERETLVEECNRAECIQRGVSPSQAQGLGSNLVTEVRVYNWFANRRKEEAFRHKLAMDTYKLN

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 \(c\).

\(|P_{f(3KTK_1)}(2) \setminus P_{f(5PZM_1)}(2)|=26\), \(|P_{f(5PZM_1)}(2) \setminus P_{f(3KTK_1)}(2)|=164\). 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 1:0111011000001001001000110001111101100011001100111111001000101010011101011111000101101010010111110111111111001000111000111001111101010010111001111000100111000101101100000000100100110011100110000000000
Pair \(Z_2\) Length of longest common subsequence
3KTK_1,5PZM_1 190 3
3KTK_1,2LFB_1 148 3
5PZM_1,2LFB_1 236 3

Newick tree

 
[
	5PZM_1:11.07,
	[
		3KTK_1:74,2LFB_1:74
	]:42.07
]

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 \((0.85)(0.8)(\frac{773 }{\log_{20} 773}-\frac{199}{\log_{20}199})=160.\)
Status Protein1 Protein2 d d1/2
Query variables 3KTK_1 5PZM_1 203 135
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]