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Parikh vectors
2RLK_1 5IKM_1 4CSM_1 Letter Amino acid
4 19 15 R Arginine
4 21 27 E Glutamic acid
0 6 14 F Phenylalanine
3 16 17 S Serine
0 6 1 W Tryptophan
1 20 15 V Valine
1 7 4 H Histidine
0 9 27 I Isoleucine
1 21 23 K Lycine
4 13 16 P Proline
5 8 10 Y Tyrosine
1 19 11 T Threonine
1 13 13 N Asparagine
0 6 2 C Cysteine
1 6 5 Q Glutamine
1 29 8 G Glycine
0 10 3 M Methionine
4 11 11 A Alanine
1 7 14 D Aspartic acid
4 22 20 L Leucine 

2RLK_1|Chain A|Peptide YY|Sus scrofa (9823)
>5IKM_1|Chain A|NS5 Methyl Transferase|Dengue virus 1 (11053)
>4CSM_1|Chains A, B|CHORISMATE MUTASE|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
2RLK , Knot 21 37 0.68 32 33 35 
YPAKPEAPGEDASPEELSRYYASLRHYLNLVTRQRYX
5IKM , Knot 121 269 0.84 40 181 259 
SNAGTGAQGETLGEKWKRQLNQLSKSEFNTYKRSGIMEVDRSEAKEGLKRGETTKHAVSRGTAKLRWFVERNLVKPEGKVIDLGCGRGGWSYYCAGLKKVTEVKGYTKGGPGHEEPIPMATYGWNLVKLHSGKDVFFIPPEKCDTLLCDIGESSPNPTIEEGRTLRVLKMVEPWLRGNQFCIKILNPYMPSVVETLEQMQRKHGGMLVRNPLSRNSTHEMYWVSCGTGNIVSAVNMTSRMLLNRFTMAHRKPTYERDVDLGAGTRHVAV
4CSM , Knot 117 256 0.84 40 167 246 
MDFTKPETVLNLQNIRDELVRMEDSIIFKFIERSHFATCPSVYEANHPGLEIPNFKGSFLDWALSNLEIAHSRIRRFESPDETPFFPDKIQKSFLPSINYPQILAPYAPEVNYNDKIKKVYIEKIIPLISKRDGDDKNNFGSVATRDIECLQSLSRRIHFGKFVAEAKFQSDIPLYTKLIKSKDVEGIMKNITNSAVEEKILERLTKKAEVYGVDPTNESGERRITPEYLVKIYKEIVIPITKEVEVEYLLRRLEE

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 \(c\).

\(|P_{f(2RLK_1)}(2) \setminus P_{f(5IKM_1)}(2)|=13\), \(|P_{f(5IKM_1)}(2) \setminus P_{f(2RLK_1)}(2)|=161\). 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 1:0110101110010100100001010001011000000
Pair \(Z_2\) Length of longest common subsequence
2RLK_1,5IKM_1 174 3
2RLK_1,4CSM_1 156 3
5IKM_1,4CSM_1 184 3

Newick tree

 
[
	5IKM_1:93.06,
	[
		2RLK_1:78,4CSM_1:78
	]:15.06
]

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 \((0.85)(0.8)(\frac{306 }{\log_{20} 306}-\frac{37}{\log_{20}37})=88.0\)
Status Protein1 Protein2 d d1/2
Query variables 2RLK_1 5IKM_1 113 63.5
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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