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Parikh vectors
4ENR_1 5YQS_1 6DOH_1 Letter Amino acid
37 44 10 T Threonine
19 37 5 Y Tyrosine
64 65 6 A Alanine
44 26 5 R Arginine
48 51 16 E Glutamic acid
60 58 10 L Leucine
59 35 5 P Proline
40 47 7 S Serine
53 69 9 G Glycine
40 74 9 V Valine
58 43 7 D Aspartic acid
2 4 0 C Cysteine
37 37 13 K Lycine
40 25 2 F Phenylalanine
9 14 6 W Tryptophan
33 38 6 N Asparagine
26 24 3 Q Glutamine
28 20 2 H Histidine
44 40 11 I Isoleucine
12 14 1 M Methionine 

4ENR_1|Chains A, B, C, D|Catalase HPII|Escherichia coli (83333)
>5YQS_1|Chains A, B|Isoprimeverose-producing enzyme|Aspergillus oryzae RIB40 (510516)
>6DOH_1|Chain A|Ribonuclease H|Bacillus halodurans (strain ATCC BAA-125 / DSM 18197 / FERM 7344 / JCM 9153 / C-125) (272558)
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 \(w=f(c)\)
4ENR , Knot 285 753 0.83 40 291 692 
MSQHNEKNPHQHQSPLHDSSEAKPGMDSLAPEDGSHRPAAEPTPPGAQPTAPGSLKAPDTRNEKLNSLEDVRKGSENYALTTNQGVRIADDQNSLRAGSRGPTLLEDFILREKITHFDHERIPERIVHARGSAAHGYFQPYKSLSDITKADFLSDPNKITPVFVRFSTVQGGAGSADTVRDIRGFATKFYTEEGIFDLVGNNTPIFFIQDAHKFPDFVHAVKPEPHWAIPQGQSAHDTFWDYVSLQPETLHNVMWAMSDRGIPRSYRTMEGFGIHTFRLINAEGKATFVRFHWKPLAGKASLVWDEAQKLTGRDPDFHRRELWEAIEAGDFPEYELGFQLIPEEDEFKFDFDLLDPTKLIPEELVPVQRVGKMVLNRNPDNFFAENEQAAFHPGHIVPGLDFTNDPLLQGRLFSYTDTQISRLGGPNFHEIPINRPTCPYHNFQRDGMHRMGIDTNPANYEPNSINDNWPRETPPGPKRGGFESYQERVEGNKVRERSPSFGEYYSHPRLFWLSQTPFEQRHIVDGFSFILSKVVRPYIRERVVDQLAHIDLTLAQAVAKNLGIELTDDQLNITPPPDVNGLKKDPSLSLYAIPDGDVKGRVVAILLNDEVRSADLLAILKALKAKGVHAKLLYSRMGEVTADDGTVLPIAATFAGAPSLTVDAVIVPCGNIADIADNGDANYYLMEAYKHLKPIALAGDARKFKATIKIADQGEEGIVEADSADGSFMDELLTLMAAHRVWSRIPKIDKIPA
5YQS , Knot 292 765 0.84 40 290 701 
QDEKPRYKDPSVPVEERVTDLLGRMTLEEKMSQLIQGDITNWMNETTGEFNLTGLEWSTKMRGGMFYVGYPVPWDYIADNVKKAQDYILQNTTLGIPAIVQTESLHGFLIGNATIYNSPIGFACSFNPELIEKMARLIGQEASALGVNHVMGPVVDLARELRFGRVEETYGEDPFLAGEIGYHYTKGIQSHNISANVKHFVGFSQPEQGLNTAPVHGGERYLRTTWLPSFKRAIMDAGAWSIMSAYHSYDGIPAVADYHTLTEILREEWGYKYWVTSDAGASDRVCTAFKLCRADPIDKEAVTLAILPAGNDVEMGGGSYNFETIIDLVNAGKLDIEIVNTAVSRVLRAKFEMGLFENPYNAAPASEWNKLIHTQEAVDLARELDRESIVLLENHDNALPLKKSGSIAVIGPMAHGFMNYGDYVVYESQYRGVTPLDGIKAAVGDKATINYAQGCERWSNDQSGFAEAVEAAKKSDVAVVVVGTWSRDQKELWAGLNATTGEHVDVNSLSLVGAQAPLIKAIIDTGVPTVVVLSSGKPITEPWLSNNTAALVQQFYPSEQGGNALADVLFGDYNPSGKLSVSFPHSVGDLPIYYDYLNSAREIGDAGYIYSNGTLEFGHQYALGNPKAWYPFGYGKSYSSFEYGAVKLDKTNVTEADTVTVSVDVKNTDATREGTEVVQVYVVDEVASVVVPNRLLKGFKKVVIPAGQTKTVEIPLKVQDLGLWNVRMKYVVEPGAFGVLVGSSSEDIRGNATFYVQVDHHHHHH
6DOH , Knot 64 133 0.78 38 106 131 
EEIIWESLSVDVGSQGNPGIVEYKGVDTKTGEVLFEREPIPIGTNNMGEFLAIVHGLRYLKERNSRKPIYSDSQTAIKWVKDKKAKSTLVRNEETALIWKLVDEAEEWLNTHTYETPILKWQTDKWGEIKADY

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 \(c\).

\(|P_{f(4ENR_1)}(2) \setminus P_{f(5YQS_1)}(2)|=56\), \(|P_{f(5YQS_1)}(2) \setminus P_{f(4ENR_1)}(2)|=55\). 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 1:100000001000001100000101110011100100011101011110101110101100000010010010010000110000110110000010110011011001110001001000011001101010110101010001001001011001001011110100101111010010010111001000011101110001111100100110110110101011110100100011001010100100111110001110000010111100101101010101101010111101011100100101001010000110110110110001110111000010101011010011100111100110111000100111000011101101111101000111010110000001001111010011100100100010001100111000110001001000110001111001110000001010010000101100000101111000110000110110111001101010001100110101011011100111010000101011101011000101010111010101011111100010010111110110101101011000110101001011111101111101010111110101101100101000110100010111111010010101011001001110100101011001101111001100110100111
Pair \(Z_2\) Length of longest common subsequence
4ENR_1,5YQS_1 111 4
4ENR_1,6DOH_1 215 4
5YQS_1,6DOH_1 204 4

Newick tree

 
[
	6DOH_1:11.67,
	[
		4ENR_1:55.5,5YQS_1:55.5
	]:61.17
]

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 \((0.85)(0.8)(\frac{1518 }{\log_{20} 1518}-\frac{753}{\log_{20}753})=190.\)
Status Protein1 Protein2 d d1/2
Query variables 4ENR_1 5YQS_1 244 241.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]