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Parikh vectors
5PIK_1 8EYO_1 5HGI_1 Letter Amino acid
20 30 17 T Threonine
8 3 3 W Tryptophan
14 30 26 D Aspartic acid
15 23 18 Q Glutamine
18 16 20 H Histidine
18 23 28 S Serine
25 44 24 A Alanine
26 34 34 E Glutamic acid
24 61 42 L Leucine
20 36 23 I Isoleucine
19 26 22 F Phenylalanine
18 26 27 P Proline
14 40 27 V Valine
18 30 28 R Arginine
7 9 11 C Cysteine
27 42 25 G Glycine
19 24 13 Y Tyrosine
20 24 9 N Asparagine
23 32 25 K Lycine
11 15 9 M Methionine 

5PIK_1|Chain A|Lysine-specific demethylase 4D|Homo sapiens (9606)
>8EYO_1|Chains A, B|NADP-dependent malic enzyme, mitochondrial|Homo sapiens (9606)
>5HGI_1|Chain A|Serine/threonine-protein kinase/endoribonuclease IRE1|Homo sapiens (9606)
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 \(w=f(c)\)
5PIK , Knot 160 364 0.86 40 229 347 
MHHHHHHSSGVDLGTENLYFQSMETMKSKANCAQNPNCNIMIFHPTKEEFNDFDKYIAYMESQGAHRAGLAKIIPPKEWKARETYDNISEILIATPLQQVASGRAGVFTQYHKKKKAMTVGEYRHLANSKKYQTPPHQNFEDLERKYWKNRIYNSPIYGADISGSLFDENTKQWNLGHLGTIQDLLEKECGVVIEGVNTPYLYFGMWKTTFAWHTEDMDLYSINYLHLGEPKTWYVVPPEHGQRLERLARELFPGSSRGCGAFLRHKVALISPTVLKENGIPFNRITQEAGEFMVTFPYGYHAGFNHGFNCAEAINFATPRWIDYGKMASQCSCGEARVTFSMDAFVRILQPERYDLWKRGQDR
8EYO , Knot 227 568 0.84 40 267 536 
MLKKRGYDVTRNPHLNKGMAFTLEERLQLGIHGLIPPCFLSQDVQLLRIMRYYERQQSDLDKYIILMTLQDRNEKLFYRVLTSDVEKFMPIVYTPTVGLACQHYGLTFRRPRGLFITIHDKGHLATMLNSWPEDNIKAVVVTDGERILGLGDLGCYGMGIPVGKLALYTACGGVNPQQCLPVLLDVGTNNEELLRDPLYIGLKHQRVHGKAYDDLLDEFMQAVTDKFGINCLIQFEDFANANAFRLLNKYRNKYCMFNDDIQGTASVAVAGILAALRITKNKLSNHVFVFQGAGEAAMGIAHLLVMALEKEGVPKAEATRKIWMVDSKGLIVKGRSHLNHEKEMFAQDHPEVNSLEEVVRLVKPTAIIGVAAIAGAFTEQILRDMASFHERPIIFALSNPTSKAECTAEKCYRVTEGRGIFASGSPFKSVTLEDGKTFIPGQGNNAYVFPGVALGVIAGGIRHIPDEIFLLTAEQIAQEVSEQHLSQGRLYPPLSTIRDVSLRIAIKVLDYAYKHNLASYYPEPKDKEAFVRSLVYTPDYDSFTLDSYTWPKEAMNVQTVTRENLYFQ
5HGI , Knot 180 431 0.84 40 234 415 
GSSPSLEQDDGDEETSVVIVGKISFCPKDVLGHGAEGTIVYRGMFDNRDVAVKRILPECFSFADREVQLLRESDEHPNVIRYFCTEKDRQFQYIAIELCAATLQEYVEQKDFAHLGLEPITLLQQTTSGLAHLHSLNIVHRDLKPHNILISMPNAHGKIKAMISDFGLCKKLAVGRHSFSRRSGVPGTEGWIAPEMLSEDCKENPTYTVDIFSAGCVFYYVISEGSHPFGKSLQRQANILLGACSLDCLHPEKHEDVIARELIEKMIAMDPQKRPSAKHVLKHPFFWSLEKQLQFFQDVSDRIEKESLDGPIVKQLERGGRAVVKMDWRENITVPLQTDLRKFRTYKGGSVRDLLRAMRNKKHHYRELPAEVRETLGSLPDDFVCYFTSRFPHLLAHTYRAMELCSHERLFQPYYFHEPPEPQPPVTPDAL

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 \(c\).

\(|P_{f(5PIK_1)}(2) \setminus P_{f(8EYO_1)}(2)|=70\), \(|P_{f(8EYO_1)}(2) \setminus P_{f(5PIK_1)}(2)|=108\). 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 1:1000000001101100010100100100010010010001111010000100100011010001100111101111001010000001001111011001101011110000000011011000011000000011000100100001000100011011010101100000010110110100110000111101100101011110001110000101001001011010010111100100100110011110001011110001111010110001111001000110111011010011100110010110110101100101100000101010101011101101000011001000
Pair \(Z_2\) Length of longest common subsequence
5PIK_1,8EYO_1 178 6
5PIK_1,5HGI_1 177 4
8EYO_1,5HGI_1 147 5

Newick tree

 
[
	5PIK_1:93.28,
	[
		5HGI_1:73.5,8EYO_1:73.5
	]:19.78
]

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 \((0.85)(0.8)(\frac{932 }{\log_{20} 932}-\frac{364}{\log_{20}364})=151.\)
Status Protein1 Protein2 d d1/2
Query variables 5PIK_1 8EYO_1 194 157.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]