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Parikh vectors
5HKL_1 9JSI_1 6TOR_1 Letter Amino acid
10 61 29 S Serine
4 41 17 Y Tyrosine
13 9 21 H Histidine
4 6 14 M Methionine
1 22 20 F Phenylalanine
4 25 19 P Proline
12 60 27 T Threonine
1 5 1 W Tryptophan
12 49 29 D Aspartic acid
0 1 11 C Cysteine
3 26 16 Q Glutamine
20 40 37 G Glycine
19 44 39 L Leucine
23 31 34 V Valine
25 36 38 A Alanine
18 14 23 R Arginine
1 59 19 N Asparagine
12 44 33 E Glutamic acid
6 54 29 I Isoleucine
2 56 43 K Lycine 

5HKL_1|Chains A, B|Orotate phosphoribosyltransferase|Mycobacterium tuberculosis (strain ATCC 25618 / H37Rv) (83332)
>9JSI_1|Chains A, B, C, D, E, F, G|Iota toxin component Ib|Clostridium perfringens (1502)
>6TOR_1|Chains A, B|Ethanolamine-phosphate phospho-lyase|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 \(w=f(c)\)
5HKL , Knot 80 190 0.73 38 108 174 
HHHHHHIEGRHMAGPDRAELAELVRRLSVVHGRVTLSSGREADYYVDLRRATLHHRASALIGRLMRELTADWDYSVVGGLTLGADPVATAIMHAPGRPIDAFVVRKSAKAHGMQRLIEGSEVTGQRVLVVEDTSTTGNSALTAVHAVQDVGGEVVGVATVVDRATGAAEAIEAEGLRYRSVLGLADLGLD
9JSI , Knot 257 683 0.81 40 249 614 
NNFFDVRFFSAAWEDEDLDTDNDNIPDAYEKNGYTIKDSIAVKWNDSFAEQGYKKYVSSYLESNTAGDPYTDYQKASGSIDKAIKLEARDPLVAAYPVVGVGMENLIISTNEHASSDQGKTVSRATTNSKTDANTVGVSISAGYQNGFTGNITTSYSHTTDNSTAVQDSNGESWNTGLSINKGESAYINANVRYYNTGTAPMYKVTPTTNLVLDGETLATIKAQDNQIGNNLSPNETYPKKGLSPLALNTMDQFNARLIPINYDQLKKLDSGKQIKLETTQVSGNYGTKNSQGQIITEGNSWSNYISQIDSVSASIILDTGSQTFERRVAAKEQGNPEDKTPEITIGEAIKKAFSATKNGELLYFNGIPIDESCVELIFDDNTSEIIKEQLKYLDDKKIYNVKLERGMNILIKVPSYFTNFDEYNNFPASWSNIDTKNQDGLQSVANKLSGETKIIIPMSKLKPYKRYVFSGYSKDPSTSNSITVNIKSKEQKTDYLVPEKDYTKFSYEFETTGKDSSDIEITLTSSGVIFLDNLSITELNSTPEILKEPEIKVPSDQEILDAHNKYYADIKLDTNTGNTYIDGIYFEPTQTNKEALDYIQKYRVEATLQYSGFKDIGTKDKEIRNYLGDQNQPKTNYINFRSYFTSGENVMTYKKLRIYAVTPDNRELLVLSVNLEHHHHHH
6TOR , Knot 205 499 0.85 40 259 476 
MCELYSKRDTLGLRKKHIGPSCKVFFASDPIKIVRAQRQYMFDENGEQYLDCINNVAHVGHCHPGVVKAALKQMELLNTNSRFLHDNIVEYAKRLSATLPEKLSVCYFTNSGSEANDLALRLARQFRGHQDVITLDHAYHGHLSSLIEISPYKFQKGKDVKKEFVHVAPTPDTYRGKYREDHADSASAYADEVKKIIEDAHNSGRKIAAFIAESMQSCGGQIIPPAGYFQKVAEYVHGAGGVFIADEVQVGFGRVGKHFWSFQMYGEDFVPDIVTMGKPMGNGHPVACVVTTKEIAEAFSSSGMEYFNTYGGNPVSCAVGLAVLDIIENEDLQGNAKRVGNYLTELLKKQKAKHTLIGDIRGIGLFIGIDLVKDHLKRTPATAEAQHIIYKMKEKRVLLSADGPHRNVLKIKPPMCFTEEDAKFMVDQLDRILTVLEEAMGTKTESVTSENTPCKTKMLKEAHIELLRDSTTDSKENPSRKRNGMCTDTHSLLSKRLKT

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 \(c\).

\(|P_{f(5HKL_1)}(2) \setminus P_{f(9JSI_1)}(2)|=27\), \(|P_{f(9JSI_1)}(2) \setminus P_{f(5HKL_1)}(2)|=168\). 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 1:0000001010011110010110110010110101010010010001010010100010111101100101010001111101110111011101110110111100010101100110100101001111000000100110110110011101111101100101110110101100001111101110
Pair \(Z_2\) Length of longest common subsequence
5HKL_1,9JSI_1 195 6
5HKL_1,6TOR_1 191 3
9JSI_1,6TOR_1 130 4

Newick tree

 
[
	5HKL_1:10.92,
	[
		6TOR_1:65,9JSI_1:65
	]:39.92
]

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 \((0.85)(0.8)(\frac{873 }{\log_{20} 873}-\frac{190}{\log_{20}190})=188.\)
Status Protein1 Protein2 d d1/2
Query variables 5HKL_1 9JSI_1 234 147
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]