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Parikh vectors
1RHT_1 4FOM_1 5AOH_1 Letter Amino acid
0 9 11 R Arginine
7 24 30 G Glycine
0 1 5 M Methionine
0 12 9 F Phenylalanine
0 24 19 S Serine
0 24 15 T Threonine
0 6 6 W Tryptophan
6 17 24 A Alanine
0 15 13 N Asparagine
6 6 4 C Cysteine
0 11 12 Q Glutamine
0 20 17 L Leucine
0 27 18 V Valine
0 13 13 E Glutamic acid
0 15 11 H Histidine
0 23 9 I Isoleucine
0 17 10 K Lycine
0 18 15 P Proline
0 11 15 Y Tyrosine
0 15 17 D Aspartic acid 

1RHT_1|Chain A|RNA (5'-R(P*GP*GP*GP*AP*CP*UP*GP*AP*CP*GP*AP*UP*CP*AP*CP*GP*CP*AP*GP*UP*CP*UP*AP*U)-3')|null
>4FOM_1|Chain A|Poliovirus receptor-related protein 3|Homo sapiens (9606)
>5AOH_1|Chains A, B|Spore coat protein CotH|Serratia sp. (strain ATCC 39006) (104623)
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 \(w=f(c)\)
1RHT , Knot 11 24 0.48 8 13 20 
GGGACUGACGAUCACGCAGUCUAU
4FOM , Knot 132 308 0.81 40 188 288 
GPIIVEPHVTAVWGKNVSLKCLIEVNETITQISWEKIHGKSSQTVAVHHPQYGFSVQGEYQGRVLFKNYSLNDATITLHNIGFSDSGKYICKAVTFPLGNAQSSTTVTVLVEPTVSLIKGPDSLIDGGNETVAAICIAATGKPVAHIDWEGDLGEMESTTTSFPNETATIISQYKLFPTRFARGRRITCVVKHPALEKDIRYSFILDIQYAPEVSVTGYDGNWFVGRKGVNLKCNADANPPPFKSVWSRLDGQWPDGLLASDNTLHFVHPLTFNYSGVYICKVTNSLGQRSDQKVIYISDPPHHHHHH
5AOH , Knot 123 273 0.84 40 186 264 
EILPIDEVNVDGGDFYVGLVFGKEDYAAHANTHLTPFSIMRTEVTYHQYQALQAWAETRGYQISGGCNGATFEDCWPSEKDGGRHPVTNVSWWDAVIFANALSAQHNLQPYYVTADGQALKIPPEEGTDRGIRENPQASGYRLPTLAEWQVAARGGNKGLSDGTYGSRYAGKDQPASVANLPVSGTQTFSTLPVASKQPNSLGLYDMSGNVSEWLNENYAVKGGKKMYYFCGGSYMDRVGSLASCDVHTPGFAMSDIGFRLVRPIDDKHHHHH

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 \(c\).

\(|P_{f(1RHT_1)}(2) \setminus P_{f(4FOM_1)}(2)|=12\), \(|P_{f(4FOM_1)}(2) \setminus P_{f(1RHT_1)}(2)|=187\). 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 1:111100110110010101100010
Pair \(Z_2\) Length of longest common subsequence
1RHT_1,4FOM_1 199 2
1RHT_1,5AOH_1 189 2
4FOM_1,5AOH_1 178 5

Newick tree

 
[
	1RHT_1:99.56,
	[
		5AOH_1:89,4FOM_1:89
	]:10.56
]

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 \((0.85)(0.8)(\frac{332 }{\log_{20} 332}-\frac{24}{\log_{20}24})=101.\)
Status Protein1 Protein2 d d1/2
Query variables 1RHT_1 4FOM_1 131 70
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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