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Parikh vectors
1RUW_1 5CEL_1 5LPQ_1 Letter Amino acid
2 9 4 W Tryptophan
4 27 38 A Alanine
1 30 8 N Asparagine
0 20 5 C Cysteine
6 50 33 G Glycine
5 24 27 L Leucine
4 46 18 T Threonine
3 20 8 Y Tyrosine
0 20 17 Q Glutamine
4 18 24 E Glutamic acid
7 13 14 K Lycine
4 21 18 V Valine
3 15 20 F Phenylalanine
7 19 17 P Proline
8 50 30 S Serine
2 7 32 R Arginine
6 25 24 D Aspartic acid
0 4 10 H Histidine
2 10 22 I Isoleucine
1 6 17 M Methionine 

1RUW_1|Chain A|Myosin-3 isoform|Saccharomyces cerevisiae (4932)
>5CEL_1|Chain A|1,4-BETA-D-GLUCAN CELLOBIOHYDROLASE I|Hypocrea jecorina (51453)
>5LPQ_1|Chains A, B|Queuine tRNA-ribosyltransferase|Zymomonas mobilis subsp. mobilis (strain ATCC 31821 / ZM4 / CP4) (264203)
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 \(w=f(c)\)
1RUW , Knot 38 69 0.77 34 60 67 
KDPKFEAAYDFPGSGSSSELPLKKGDIVFISRDEPSGWSLAKLLDGSKEGWVPTAYMTPYKDTRNTVPV
5CEL , Knot 177 434 0.82 40 221 399 
QSACTLQSETHPPLTWQKCSSGGTCTQQTGSVVIDANWRWTHATNSSTNCYDGNTWSSTLCPDNETCAKNCCLDGAAYASTYGVTTSGNSLSIDFVTQSAQKNVGARLYLMASDTTYQEFTLLGNEFSFDVDVSQLPCGLNGALYFVSMDADGGVSKYPTNTAGAKYGTGYCDSQCPRDLKFINGQANVEGWEPSSNNANTGIGGHGSCCSQMDIWEANSISEALTPHPCTTVGQEICEGDGCGGTYSDNRYGGTCDPDGCDWNPYRLGNTSFYGPGSSFTLDTTKKLTVVTQFETSGAINRYYVQNGVTFQQPNAELGSYSGNELNDDYCTAEEAEFGGSSFSDKGGLTQFKKATSGGMVLVMSLWDDYYANMLWLDSTYPTNETSSTPGAVRGSCSTSSGVPAQVESQSPNAKVTFSNIKFGPIGSTGNPSG
5LPQ , Knot 161 386 0.82 40 215 372 
MVEATAQETDRPRFSFSIAAREGKARTGTIEMKRGVIRTPAFMPVGTAATVKALKPETVRATGADIILGNTYHLMLRPGAERIAKLGGLHSFMGWDRPILTDSGGYQVMSLSSLTKQSEEGVTFKSHLDGSRHMLSPERSIEIQHLLGSDIVMAFDECTPYPATPSRAASSMERSMRWAKRSRDAFDSRKEQAENAALFGIQQGSVFENLRQQSADALAEIGFDGYAVGGLAVGEGQDEMFRVLDFSVPMLPDDKPHYLMGVGKPDDIVGAVERGIDMFDCVLPTRSGRNGQAFTWDGPINIRNARFSEDLKPLDSECHCAVCQKWSRAYIHHLIRAGEILGAMLMTEHNIAFYQQLMQKIRDSISEGRFSQFAQDFRARYFARNS

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 \(c\).

\(|P_{f(1RUW_1)}(2) \setminus P_{f(5CEL_1)}(2)|=16\), \(|P_{f(5CEL_1)}(2) \setminus P_{f(1RUW_1)}(2)|=177\). 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 1:001010110011101000011100101111000010110110110100011110101010000000111
Pair \(Z_2\) Length of longest common subsequence
1RUW_1,5CEL_1 193 3
1RUW_1,5LPQ_1 189 4
5CEL_1,5LPQ_1 184 4

Newick tree

 
[
	1RUW_1:96.64,
	[
		5LPQ_1:92,5CEL_1:92
	]:4.64
]

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 \((0.85)(0.8)(\frac{503 }{\log_{20} 503}-\frac{69}{\log_{20}69})=131.\)
Status Protein1 Protein2 d d1/2
Query variables 1RUW_1 5CEL_1 161 93
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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