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Parikh vectors
9ICY_1 6NOO_1 8TIO_1 Letter Amino acid
0 30 17 S Serine
0 1 0 W Tryptophan
3 26 26 A Alanine
0 16 36 E Glutamic acid
1 22 23 G Glycine
0 8 10 H Histidine
0 12 15 F Phenylalanine
0 15 30 P Proline
2 11 27 T Threonine
0 14 36 V Valine
0 7 25 R Arginine
0 12 31 D Aspartic acid
1 1 7 C Cysteine
0 5 5 M Methionine
0 6 17 N Asparagine
0 14 9 Q Glutamine
0 20 14 I Isoleucine
0 16 35 K Lycine
0 36 42 L Leucine
0 6 13 Y Tyrosine 

9ICY_1|Chain A[auth T]|DNA (5'-D(*CP*AP*TP*TP*AP*GP*A)-3')|
>6NOO_1|Chains A, B|Maintenance of carboxysome positioning A protein, MCDA|Cyanothece (43988)
>8TIO_1|Chain A|Beta-arrestin-1|Bos taurus (9913)
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 \(w=f(c)\)
9ICY , Knot 6 7 0.55 8 6 5 
CATTAGA
6NOO , Knot 118 278 0.79 40 161 261 
MGSSHHHHHHSSGLVPRGSHMKTLVIASLSGGQGKTTTAFFLGKLLSQSAKVLFIDADPQSNLTFFLGHEVEPSAPTLLELIKDMVEPADAVYSLANSNQFLIPSDDGLSNAQEYLASSGMGAVVLKARLKPLSEYFDYCIIDSPPARTQISIATIGAADQLLIPAEASTKGVNSLIRTLEIVQSLEKLGAFTGSILGVIPFRDKWFGLSQSKDSAGAIAAMKEVAPQLRIFPSILESERYKQALNQGILLSELGYPDLEKPFEGVKEALGIKQLVQI
8TIO , Knot 172 418 0.82 38 212 392 
MGDKGTRVFKKASPNGKLTVYLGKRDFVDHIDLVEPVDGVVLVDPEYLKERRVYVTLTCAFRYGREDLDVLGLTFRKDLFVANVQSFPPAPEDKKPLTRLQERLIKKLGEHAYPFTFEIPPNLPCSVTLQPGPEDTGKACGVDYEVKAFCAENLEEKIHKRNSVRLVIRKVQYAPERPGPQPTAETTRQFLMSDKPLHLEASLDKEIYYHGEPISVNVHVTNNTNKTVKKIKISVRQYADICLFNTAQYKCPVAMEEADDTVAPSSTFCKVYTLTPFLANNREKRGLALDGKLKHEDTNLASSTLLREGANREILGIIVSYKVKVKLVVSRGGLLGDLASSDVAVELPFTLMHPKPKEEPPHREVPEHETPVDTNLIELDTNDDDAAAEDFARQRLKGMKDDKEEEEDGTGSPRLNDR

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 \(c\).

\(|P_{f(9ICY_1)}(2) \setminus P_{f(6NOO_1)}(2)|=1\), \(|P_{f(6NOO_1)}(2) \setminus P_{f(9ICY_1)}(2)|=156\). 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 1:0100111
Pair \(Z_2\) Length of longest common subsequence
9ICY_1,6NOO_1 157 3
9ICY_1,8TIO_1 210 2
6NOO_1,8TIO_1 151 4

Newick tree

 
[
	9ICY_1:97.76,
	[
		6NOO_1:75.5,8TIO_1:75.5
	]:22.26
]

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 \((0.85)(0.8)(\frac{285 }{\log_{20} 285}-\frac{7}{\log_{20}7})=95.3\)
Status Protein1 Protein2 d d1/2
Query variables 9ICY_1 6NOO_1 112 57
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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