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Parikh vectors
6XNE_1 7CWB_1 2CLP_1 Letter Amino acid
0 3 6 W Tryptophan
1 21 11 N Asparagine
5 9 26 E Glutamic acid
0 13 15 P Proline
1 17 12 D Aspartic acid
0 26 29 G Glycine
3 27 17 V Valine
1 10 11 M Methionine
2 11 19 R Arginine
0 11 9 I Isoleucine
6 29 38 L Leucine
1 7 19 H Histidine
5 11 12 K Lycine
1 17 16 F Phenylalanine
1 16 19 S Serine
0 24 19 T Threonine
1 17 33 A Alanine
0 12 7 C Cysteine
1 14 15 Q Glutamine
1 11 14 Y Tyrosine 

6XNE_1|Chain A|GCN4-p1 Peptide with me-F16|Saccharomyces cerevisiae (4932)
>7CWB_1|Chain A|3C-like proteinase|Severe acute respiratory syndrome coronavirus 2 (2697049)
>2CLP_1|Chains A, B, C, D, E, F, G, H, I, J, K|AFLATOXIN B1 ALDEHYDE REDUCTASE MEMBER 3|HOMO SAPIENS (9606)
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 \(w=f(c)\)
6XNE , Knot 20 30 0.75 28 28 28 
RMKQLEDKVEELLSKFYHLENEVARLKKLV
7CWB , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
2CLP , Knot 147 347 0.82 40 209 326 
MHHHHHHSSGVDLGTENLYFQSMARPATVLGAMEMGRRMDAPTSAAVTRAFLERGHTEIDTAFVYSEGQSETILGGLGLRLGGSDCRVKIDTKAIPLFGNSLKPDSLRFQLETSLKRLQCPRVDLFYLHMPDHSTPVEETLRACHQLHQEGKFVELGLSNYAAWEVAEICTLCKSNGWILPTVYQGMYNAITRQVETELFPCLRHFGLRFYAFNPLAGGLLTGKYKYEDKDGKQPVGRFFGNTWAEMYRNRYWKEHHFEGIALVEKALQAAYGASAPSMTSATLRWMYHHSQLQGAHGDAVILGMSSLEQLEQNLAAAEEGPLEPAVVDAFNQAWHLVAHECPNYFR

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 \(c\).

\(|P_{f(6XNE_1)}(2) \setminus P_{f(7CWB_1)}(2)|=13\), \(|P_{f(7CWB_1)}(2) \setminus P_{f(6XNE_1)}(2)|=181\). 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 1:010010001001100100100011010011
Pair \(Z_2\) Length of longest common subsequence
6XNE_1,7CWB_1 194 3
6XNE_1,2CLP_1 193 3
7CWB_1,2CLP_1 173 4

Newick tree

 
[
	6XNE_1:99.93,
	[
		2CLP_1:86.5,7CWB_1:86.5
	]:13.43
]

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 \((0.85)(0.8)(\frac{336 }{\log_{20} 336}-\frac{30}{\log_{20}30})=99.6\)
Status Protein1 Protein2 d d1/2
Query variables 6XNE_1 7CWB_1 127 69.5
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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