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Parikh vectors
1DII_1 5NLW_1 1CNR_1 Letter Amino acid
22 5 2 R Arginine
30 2 5 P Proline
9 2 0 W Tryptophan
47 13 2 V Valine
30 6 0 K Lycine
22 4 0 M Methionine
42 10 5 A Alanine
28 6 3 N Asparagine
6 2 6 C Cysteine
25 8 0 Q Glutamine
9 6 0 H Histidine
26 2 4 I Isoleucine
28 8 6 T Threonine
21 6 2 Y Tyrosine
27 5 1 D Aspartic acid
31 3 1 E Glutamic acid
46 13 4 G Glycine
31 9 2 L Leucine
21 2 1 F Phenylalanine
20 13 2 S Serine 

1DII_1|Chains A, C[auth B]|P-CRESOL METHYLHYDROXYLASE|Pseudomonas putida (303)
>5NLW_1|Chain A|nanobody Nb36|Lama glama (9844)
>1CNR_1|Chain A|CRAMBIN|Crambe hispanica subsp. abyssinica (3721)
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 \(w=f(c)\)
1DII , Knot 212 521 0.84 40 268 493 
MSEQNNAVLPKGVTQGEFNKAVQKFRALLGDDNVLVESDQLVPYNKIMMPVENAAHAPSAAVTATTVEQVQGVVKICNEHKIPIWTISTGRNFGYGSAAPVQRGQVILDLKKMNKIIKIDPEMCYALVEPGVTFGQMYDYIQENNLPVMLSFSAPSAIAGPVGNTMDRGVGYTPYGEHFMMQCGMEVVLANGDVYRTGMGGVPGSNTWQIFKWGYGPTLDGMFTQANYGICTKMGFWLMPKPPVFKPFEVIFEDEADIVEIVDALRPLRMSNTIPNSVVIASTLWEAGSAHLTRAQYTTEPGHTPDSVIKQMQKDTGMGAWNLYAALYGTQEQVDVNWKIVTDVFKKLGKGRIVTQEEAGDTQPFKYRAQLMSGVPNLQEFGLYNWRGGGGSMWFAPVSEARGSECKKQAAMAKRVLHKYGLDYVAEFIVAPRDMHHVIDVLYDRTNPEETKRADACFNELLDEFEKEGYAVYRVNTRFQDRVAQSYGPVKRKLEHAIKRAVDPNNILAPGRSGIDLNNDF
5NLW , Knot 61 125 0.78 40 101 120 
MQVQLVESGGGLVQAGGSLRLSCVVSGSAVSDYAMGWYRQAPGKQRELVAAIYNSGRTNYVDSVKGRFTISKDNAKKTVYLQMNSLKPEDTADYFCNLLGATTMSNAVWGQGTQVTVSSHHHHHH
1CNR , Knot 27 46 0.75 30 39 44 
TTCCPSIVARSNFNVCRLPGTPEALCATYTGCIIIPGATCPGDYAN

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 \(c\).

\(|P_{f(1DII_1)}(2) \setminus P_{f(5NLW_1)}(2)|=184\), \(|P_{f(5NLW_1)}(2) \setminus P_{f(1DII_1)}(2)|=17\). 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 1:10000011110110010100110010111100011100001110001111100110110111010010010111010000011110100100110101111001011101001001101010100111011101101000100001111101011011111110010011100101001110011011110101000111111100010110110110101110010011000111111101111011011100010110110110110100011001111001101101010010000011001001100100001111101011101000010101011001100110101100001100011000101101110100111001011110111111001010000001111001100011001101111100100110110000010000010101001100100010110010001000110001110001001100110100111110011010001
Pair \(Z_2\) Length of longest common subsequence
1DII_1,5NLW_1 201 4
1DII_1,1CNR_1 253 3
5NLW_1,1CNR_1 118 3

Newick tree

 
[
	1DII_1:12.44,
	[
		5NLW_1:59,1CNR_1:59
	]:68.44
]

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 \((0.85)(0.8)(\frac{646 }{\log_{20} 646}-\frac{125}{\log_{20}125})=150.\)
Status Protein1 Protein2 d d1/2
Query variables 1DII_1 5NLW_1 185 113
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]