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Parikh vectors
3IBT_1 7FLW_1 2NBT_1 Letter Amino acid
10 5 0 W Tryptophan
5 7 1 Y Tyrosine
21 18 2 A Alanine
17 10 5 P Proline
14 10 3 F Phenylalanine
15 18 6 S Serine
12 21 6 T Threonine
9 14 2 V Valine
15 7 3 G Glycine
26 29 4 L Leucine
14 5 1 H Histidine
12 23 4 I Isoleucine
6 8 0 M Methionine
19 14 3 D Aspartic acid
20 9 5 Q Glutamine
6 1 10 C Cysteine
14 15 1 E Glutamic acid
5 19 2 K Lycine
15 11 4 R Arginine
9 14 4 N Asparagine 

3IBT_1|Chain A|1H-3-hydroxy-4-oxoquinoline 2,4-dioxygenase|Pseudomonas putida (303)
>7FLW_1|Chain A|Pre-mRNA-splicing factor 8|Saccharomyces cerevisiae S288C (559292)
>2NBT_1|Chains A, B|NEURONAL BUNGAROTOXIN|Bungarus multicinctus (8616)
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 \(w=f(c)\)
3IBT , Knot 122 264 0.86 40 181 257 
MQSLNVNGTLMTYSESGDPHAPTLFLLSGWCQDHRLFKNLAPLLARDFHVICPDWRGHDAKQTDSGDFDSQTLAQDLLAFIDAKGIRDFQMVSTSHGCWVNIDVCEQLGAARLPKTIIIDWLLQPHPGFWQQLAEGQHPTEYVAGRQSFFDEWAETTDNADVLNHLRNEMPWFHGEMWQRACREIEANYRTWGSPLDRMDSLPQKPEICHIYSQPLSQDYRQLQLEFAAGHSWFHPRHIPGRTHFPSLENPVAVAQAIREFLQA
7FLW , Knot 117 258 0.84 40 175 251 
GAMNSSNYAELFNNDIKLFVDDTNVYRVTVHKTFEGNVATKAINGCIFTLNPKTGHLFLKIIHTSVWAGQKRLSQLAKWKTAEEVSALVRSLPKEEQPKQIIVTRKAMLDPLEVHMLDFPNIAIRPTELRLPFSAAMSIDKLSDVVMKATEPQMVLFNIYDDWLDRISSYTAFSRLTLLLRALKTNEESAKMILLSDPTITIKSYHLWPSFTDEQWITIESQMRDLILTEYGRKYNVNISALTQTEIKDIILGQNIKA
2NBT , Knot 38 66 0.80 36 60 63 
RTCLISPSSTPQTCPNGQDICFLKAQCDKFCSIRGPVIEQGCVATCPQFRSNYRSLLCCTTDNCNH

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 \(c\).

\(|P_{f(3IBT_1)}(2) \setminus P_{f(7FLW_1)}(2)|=103\), \(|P_{f(7FLW_1)}(2) \setminus P_{f(3IBT_1)}(2)|=97\). 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 1:100101010110000010101101111011000001100111111001011010101001000001010000110011111010110010110000101101010001111011001110111010111100110100100011100011001100000101100100011110101100100010100001101100100110010100100011000000101011110011010011100011010011111011001101
Pair \(Z_2\) Length of longest common subsequence
3IBT_1,7FLW_1 200 3
3IBT_1,2NBT_1 183 4
7FLW_1,2NBT_1 179 3

Newick tree

 
[
	3IBT_1:97.86,
	[
		2NBT_1:89.5,7FLW_1:89.5
	]:8.36
]

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 \((0.85)(0.8)(\frac{522 }{\log_{20} 522}-\frac{258}{\log_{20}258})=75.2\)
Status Protein1 Protein2 d d1/2
Query variables 3IBT_1 7FLW_1 98 96
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]