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Parikh vectors
1RXI_1 8JXS_1 8XFI_1 Letter Amino acid
5 15 11 P Proline
5 24 19 T Threonine
12 14 34 K Lycine
4 26 8 F Phenylalanine
8 15 8 N Asparagine
9 13 16 G Glycine
3 3 4 H Histidine
1 15 0 C Cysteine
11 10 19 E Glutamic acid
12 31 18 I Isoleucine
3 6 3 M Methionine
4 10 4 W Tryptophan
8 32 23 A Alanine
4 17 6 R Arginine
12 10 17 D Aspartic acid
7 30 23 V Valine
2 8 17 Y Tyrosine
3 6 10 Q Glutamine
8 31 25 L Leucine
10 36 18 S Serine 

1RXI_1|Chain A|Arsenate reductase|Staphylococcus aureus (1280)
>8JXS_1|Chain A|D(1A) dopamine receptor|Homo sapiens (9606)
>8XFI_1|Chains A, B|Iron-hydroxamate ABC transporter substrate-binding protein|Streptococcus pyogenes SSI-1 (193567)
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 \(w=f(c)\)
1RXI , Knot 64 131 0.79 40 104 125 
MDKKTIYFISTGNSARSQMAEGWGKEILGEGWNVYSAGIETHGVNPKAIEAMKEVDIDISNHTSDLIDNDILKQSDLVVTLSSDADNNCPILPPNVKKEHWGFDDPAGKEWSEFQRVRDEIKLAIEKFKLR
8JXS , Knot 150 352 0.83 40 201 328 
GTGLVVERDFSVRILTACFLSLLILSTLLGNTLVCAAVIRFRHLRSKVTNFFVISLAVSDLLVAVLVMPWKAVAEIAGFWPFGSFCNIWVAFDIMCSTASIWNLCVISVDRYWAISSPFRYERKMTPKAAFILISVAWTLSVLISFIPVQLSWHKAKPTSPSDGNATSLAETIDNCDSSLSRTYAISSSVISFYIPVAIMIVTYTRIYRIAQKQIRRIAALERAAVHAKNCQTTTGNGKPVECSQPESSFKMSFKRETKVLKTLSVIMGVFVCCWLPFFILNCILPFCGSGETQPFCIDSNTFDVFVWFGWANSALNPIIYAFNADFRKAFSTLLGCYRLCPATNNAIETVS
8XFI , Knot 123 283 0.81 38 177 271 
SNTASKSLSPMPQIAGVTYYGDIPKQPKRVVSLASTYTGYLKKLDMNLVGVTSYDKKNPILAKTVKKAKQVAATDLEAITTLKPDLIVVGSTEENIKQLAEIAPVISIEYRKRDYLQVLSDFGRIFNKEGKAKKWLKDWKTKTAAYEKEVKAVTGDKATFTIMGLYEKDVYLFGKDWGRGGEIIHQAFHYDAPEKVKTEVFKQGYLSLSQEVLPDYIGDYVVIAAEDDKTGSALYESKLWQSIPAVKKHHVIKVNANVFYFTDPLSLEYQLETLREAILSSEN

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 \(c\).

\(|P_{f(1RXI_1)}(2) \setminus P_{f(8JXS_1)}(2)|=45\), \(|P_{f(8JXS_1)}(2) \setminus P_{f(1RXI_1)}(2)|=142\). 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 1:10000101100100100011011100111011010011100011010110110010101000000110001100001110100010000111110100001110011100100100100010111001010
Pair \(Z_2\) Length of longest common subsequence
1RXI_1,8JXS_1 187 3
1RXI_1,8XFI_1 157 3
8JXS_1,8XFI_1 174 4

Newick tree

 
[
	8JXS_1:93.91,
	[
		1RXI_1:78.5,8XFI_1:78.5
	]:15.41
]

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 \((0.85)(0.8)(\frac{483 }{\log_{20} 483}-\frac{131}{\log_{20}131})=104.\)
Status Protein1 Protein2 d d1/2
Query variables 1RXI_1 8JXS_1 132 90
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]