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Parikh vectors
5DAR_1 7YXA_1 4HRD_1 Letter Amino acid
19 68 18 A Alanine
0 14 8 N Asparagine
0 10 6 Q Glutamine
0 83 17 L Leucine
0 31 16 V Valine
0 14 12 E Glutamic acid
0 23 2 P Proline
0 17 13 T Threonine
0 13 11 Y Tyrosine
0 28 9 R Arginine
0 23 11 D Aspartic acid
23 26 25 G Glycine
0 15 8 F Phenylalanine
0 19 15 S Serine
0 5 6 W Tryptophan
19 11 4 C Cysteine
0 15 5 H Histidine
0 16 12 I Isoleucine
0 19 11 K Lycine
0 8 3 M Methionine 

5DAR_1|Chains A, D|74 nt fragment of 23S rRNA|Methanocaldococcus jannaschii (2190)
>7YXA_1|Chains A, B|Sphingosine 1-phosphate receptor 5,Soluble cytochrome b562|Homo sapiens (9606)
>4HRD_11|Chains K, Y|Proteasome component PRE2|Saccharomyces cerevisiae (559292)
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 \(w=f(c)\)
5DAR , Knot 26 74 0.50 8 16 48 
GCCUAAGACAGCGGGGAGGUUGGCUUAGAAGCAGCCAUCCUUUAAAGAGUGCGUAACAGCUCACCCGUCGAGGC
7YXA , Knot 176 458 0.78 40 207 398 
MKTIIALSYIFCLVFADYKDDDDAGRAMESGLLRPAPVSEVIVLHYNYTGKLRGARYQPGAGLRADAVVCLAVCAFIVLENLAVLLVLGRHPRFHAPMFLLLGSLTLSDLLAGAAYAANILLSGPLTLKLSPALWFAREGGVFVALTASVLSLLAIALERSLTMARRGPAPVSSRGRTLAMAAAAWGVSLLLGLLPALGWNCLGRLDACSTVLPLYAKAYVLFCVLAFVGILAAICALYARIYCQVRANAADLEDNWETLNDNLKVIEKADNAAQVKDALTKMRAAALDAQKATPPKLEDKSPDSPEMKDFRHGFDILVGQIDDALKLANEGKVKEAQAAAEQLKTTRNAYIQKYLRRKPRSLALLRTLSVVLLAFVACWGPLFLLLLLDVACPARTCPVLLQADPFLGLAMANSLLNPIIYTLTNRDLRHALLRLVGRPLEVLFQGPHHHHHHHHHH
4HRD , Knot 99 212 0.83 40 154 207 
TTTLAFRFQGGIIVAVDSRATAGNWVASQTVKKVIEINPFLLGTMAGGAADCQFWETWLGSQCRLHELREKERISVAAASKILSNLVYQYKGAGLSMGTMICGYTRKEGPTIYYVDSDGTRLKGDIFCVGSGQTFAYGVLDSNYKWDLSVEDALYLGKRSILAAAHRDAYSGGSVNLYHVTEDGWIYHGNHDVGELFWKVKEEEGSFNNVIG

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 \(c\).

\(|P_{f(5DAR_1)}(2) \setminus P_{f(7YXA_1)}(2)|=10\), \(|P_{f(7YXA_1)}(2) \setminus P_{f(5DAR_1)}(2)|=201\). 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 1:10001111011011111110011000111110110010000001111110101011011000100010011110
Pair \(Z_2\) Length of longest common subsequence
5DAR_1,7YXA_1 211 4
5DAR_1,4HRD_1 160 3
7YXA_1,4HRD_1 173 4

Newick tree

 
[
	7YXA_1:10.36,
	[
		5DAR_1:80,4HRD_1:80
	]:21.36
]

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 \((0.85)(0.8)(\frac{532 }{\log_{20} 532}-\frac{74}{\log_{20}74})=137.\)
Status Protein1 Protein2 d d1/2
Query variables 5DAR_1 7YXA_1 174 98.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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