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Parikh vectors
1KXQ_1 2EPI_1 4CVM_1 Letter Amino acid
14 0 16 Q Glutamine
22 10 24 E Glutamic acid
22 4 24 T Threonine
40 12 34 V Valine
30 7 83 A Alanine
28 4 29 R Arginine
31 7 26 D Aspartic acid
53 5 47 G Glycine
25 9 60 L Leucine
12 0 2 C Cysteine
20 15 13 K Lycine
21 1 17 P Proline
35 6 28 S Serine
19 0 2 W Tryptophan
19 2 5 Y Tyrosine
37 3 11 N Asparagine
9 1 16 H Histidine
26 8 18 I Isoleucine
9 3 8 M Methionine
24 3 12 F Phenylalanine 

1KXQ_1|Chains A, B, C, D|alpha-amylase, pancreatic|Sus scrofa (9823)
>2EPI_1|Chains A, B, C, D|UPF0045 protein MJ1052|Methanocaldococcus jannaschii (243232)
>4CVM_1|Chain A|UDP-N-ACETYLMURAMOYL-TRIPEPTIDE--D-ALANYL-D- ALANINE LIGASE|PSEUDOMONAS AERUGINOSA PAO1 (208964)
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 \(w=f(c)\)
1KXQ , Knot 211 496 0.88 40 266 481 
QYAPQTQSGRTSIVHLFEWRWVDIALECERYLGPKGFGGVQVSPPNENIVVTNPSRPWWERYQPVSYKLCTRSGNENEFRDMVTRCNNVGVRIYVDAVINHMCGSGAAAGTGTTCGSYCNPGSREFPAVPYSAWDFNDGKCKTASGGIESYNDPYQVRDCQLVGLLDLALEKDYVRSMIADYLNKLIDIGVAGFRIDASKHMWPGDIKAVLDKLHNLNTNWFPAGSRPFIFQEVIDLGGEAIKSSEYFGNGRVTEFKYGAKLGTVVRKWSGEKMSYLKNWGEGWGFMPSDRALVFVDNHDNQRGHGAGGSSILTFWDARLYKIAVGFMLAHPYGFTRVMSSYRWARNFVNGEDVNDWIGPPNNNGVIKEVTINADTTCGNDWVCEHRWREIRNMVWFRNVVDGQPFANWWDNGSNQVAFGRGNRGFIVFNNDDWQLSSTLQTGLPGGTYCDVISGDKVGNSCTGIKVYVSSDGTAQFSISNSAEDPFIAIHAESKL
2EPI , Knot 50 100 0.76 34 76 94 
MIFMRKVVAEVSIIPLGKGASVSKYVKKAIEVFKKYDLKVETNAMGTVLEGDLDEILKAFKEAHSTVLNDVDRVVSSLKIDERKDKENTIERKLKAIGEL
4CVM , Knot 186 475 0.80 40 202 421 
MLEPLRLSQLTVALDARLIGEDAVFSAVSTDSRAIGPGELFIALSGPRFDGHDYLAEVAAKGAVAALVEREVADAPLPQLLVRDTRAALGRLGALNRRKFTGPLAAMTGSSGKTTVKEMLASILRTQAGDAESVLATRGNLNNDLGVPLTLLQLAPQHRSAVIELGASRIGEIAYTVELTRPHVAIITNAGTAHVGEFGGPEKIVEAKGEILEGLAADGTAVLNLDDKAFDTWKARASGRPLLTFSLDRPQADFRAADLQRDARGCMGFRLQGVAGEAQVQLNLLGRHNVANALAAAAAAHALGVPLDGIVAGLQALQPVKGRAVAQLTASGLRVIDDSYNANPASMLAAIDILSGFSGRTVLVLGDMGELGSWAEQAHREVGAYAAGKVSALYAVGPLMAHAVQAFGATGRHFADQASLIGALATEDPTTTILIKGSRSAAMDKVVAALCGSSEESHSSVDKLAAALEHHHHHH

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 \(c\).

\(|P_{f(1KXQ_1)}(2) \setminus P_{f(2EPI_1)}(2)|=206\), \(|P_{f(2EPI_1)}(2) \setminus P_{f(1KXQ_1)}(2)|=16\). 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 1:0011000010001101101011011100000111011111010110001110010011100001100010000100001001100000111010101110010101111101000100001100011111001101001000010111000001001000011111011100001001110010011011111101010001111010111001001000111110011110011011101100000110101001001101101100101001001001101111110001111100000001011110011011010100111111110101100110000110011010010011111000111001010100001001100001001001111001101011101100100011110100111110000101000100111110000110100110000110101000101010100010011111010001
Pair \(Z_2\) Length of longest common subsequence
1KXQ_1,2EPI_1 222 3
1KXQ_1,4CVM_1 160 4
2EPI_1,4CVM_1 162 4

Newick tree

 
[
	2EPI_1:10.24,
	[
		1KXQ_1:80,4CVM_1:80
	]:22.24
]

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 \((0.85)(0.8)(\frac{596 }{\log_{20} 596}-\frac{100}{\log_{20}100})=145.\)
Status Protein1 Protein2 d d1/2
Query variables 1KXQ_1 2EPI_1 192 113
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]