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Parikh vectors
2ZYN_1 1INZ_1 7FPK_1 Letter Amino acid
31 11 29 L Leucine
15 6 7 Y Tyrosine
33 6 7 G Glycine
19 6 21 T Threonine
40 8 18 A Alanine
23 8 14 N Asparagine
20 9 14 D Aspartic acid
2 3 5 H Histidine
26 2 10 P Proline
12 15 18 S Serine
4 13 11 R Arginine
1 1 1 C Cysteine
24 12 15 E Glutamic acid
9 8 8 M Methionine
15 2 10 F Phenylalanine
7 3 5 W Tryptophan
31 11 14 V Valine
24 7 9 Q Glutamine
16 6 23 I Isoleucine
45 11 19 K Lycine 

2ZYN_1|Chain A|Solute-binding protein|Thermoactinomyces vulgaris (2026)
>1INZ_1|Chain A|EPS15-INTERACTING PROTEIN(EPSIN)|Homo sapiens (9606)
>7FPK_1|Chain A|Pre-mRNA-splicing factor 8|Saccharomyces cerevisiae S288C (559292)
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 \(w=f(c)\)
2ZYN , Knot 165 397 0.83 40 206 377 
CGPKRDPYAKAGKSEGKPDKLVVWENADDGVQLNNTKKWAGEFTKKTGIQVEVVPVALLKQQEKLTLDGPAGKGADLVTWPHDRLGEAVTKGLLQPIQVDNSVKNQFDDVAMKALTYGGKLYGLPKAIESVALIYNKKLMGQVPATYDELFQYAKANNKPDEQKYGVLFEANNFYYTYFLFAAKGAAVFKEQDGTLDPNEIGLNSPEAVQGMNEVQKWFTEARLPQSLKADTVNGLFKSGKVAAVINGPWAIKDYQAAGINVGVAPLPKIDGKDAQTFIGVKGWYLSAYSKYPKYATELMQFLTSKEALASRFKETGEIPPQKELLNDPMIKNNPVVNGFAKQASKGVPMPSIPEMGVVWEPINNAHTFVAQGKQTPEQALNDAVKIMKEKIQTMKQ
1INZ , Knot 73 148 0.82 40 114 143 
GSSRMSTSSLRRQMKNIVHNYSEAEIKVREATSNDPWGPSSSLMSEIADLTYNVVAFSEIMSMIWKRLNDHGKNWRHVYKAMTLMEYLIKTGSERVSQQCKENMYAVQTLKDFQYVDRDGKDQGVNVREKAKQLVALLRDEDRLREER
7FPK , Knot 117 258 0.84 40 175 251 
GAMNSSNYAELFNNDIKLFVDDTNVYRVTVHKTFEGNVATKAINGCIFTLNPKTGHLFLKIIHTSVWAGQKRLSQLAKWKTAEEVSALVRSLPKEEQPKQIIVTRKAMLDPLEVHMLDFPNIAIRPTELRLPFSAAMSIDKLSDVVMKATEPQMVLFNIYDDWLDRISSYTAFSRLTLLLRALKTNEESAKMILLSDPTITIKSYHLWPSFTDEQWITIESQMRDLILTEYGRKYNVNISALTQTEIKDIILGQNIKA

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 \(c\).

\(|P_{f(2ZYN_1)}(2) \setminus P_{f(1INZ_1)}(2)|=141\), \(|P_{f(1INZ_1)}(2) \setminus P_{f(2ZYN_1)}(2)|=49\). 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 1:0110001010110001010011110010011010000011101000011010111111100000101011110110110110001101100111011010001000100111011001101011101100111100001110111000011001010001000001111010010000111110111110000101010011100101101100100110010110010100101110010111110111110000111101111111010100100111101101010000100100110110000111001000101110001100111000111011100100111110110111110110010011101000100110011011000100100
Pair \(Z_2\) Length of longest common subsequence
2ZYN_1,1INZ_1 190 4
2ZYN_1,7FPK_1 173 3
1INZ_1,7FPK_1 161 3

Newick tree

 
[
	2ZYN_1:94.04,
	[
		7FPK_1:80.5,1INZ_1:80.5
	]:13.54
]

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 \((0.85)(0.8)(\frac{545 }{\log_{20} 545}-\frac{148}{\log_{20}148})=115.\)
Status Protein1 Protein2 d d1/2
Query variables 2ZYN_1 1INZ_1 145 98.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]