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Parikh vectors
2KLJ_1 4REU_1 7VJI_1 Letter Amino acid
4 4 26 P Proline
4 10 34 A Alanine
10 19 45 E Glutamic acid
17 21 45 L Leucine
6 9 23 F Phenylalanine
4 2 12 W Tryptophan
4 6 11 M Methionine
21 4 25 R Arginine
7 8 19 N Asparagine
12 7 23 D Aspartic acid
6 1 5 C Cysteine
11 12 9 Q Glutamine
6 6 17 H Histidine
6 7 23 I Isoleucine
14 9 19 Y Tyrosine
14 5 32 G Glycine
1 9 35 K Lycine
17 12 39 S Serine
4 11 13 T Threonine
6 2 27 V Valine 

2KLJ_1|Chain A|Gamma-crystallin D|Homo sapiens (9606)
>4REU_1|Chains A, B, C, D, E, F|Ferritin|Escherichia coli str. K-12 substr. MDS42 (1110693)
>7VJI_1|Chain A|DNA photolyase|Methanosarcina mazei Go1 (192952)
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 \(w=f(c)\)
2KLJ , Knot 83 174 0.82 40 121 164 
QGKITLYEDRGFQGRHYECSSDHTNLQPYLSRCNSARVDSGCWMLYEQPNYSGLQYFLRRGDYADHQQWMGLSDSVRSCRLIPHSGSHRIRLYEREDYRGQMIEFTEDCSCLQDRFRFNEIHSLNVLEGSWVLYELSNYRGRQYLLMPGDYRRYQDWGATNARVGSLRRVIDFS
4REU , Knot 78 164 0.80 40 122 158 
LKPEMIEKLNEQMNLELYSSLLYQQMSAWCSYHTFEGAAAFLRRHAQEEMTHMQRLFDYLTDTGNLPRINTVESPFAEYSSLDELFQETYKHEQLITQKINELAHAAMTNQDYPTFNFLQWYVSEQHEEEKLFKSIIDKLSLAGKSGEGLYFIDKELSTLDTQN
7VJI , Knot 196 482 0.83 40 243 441 
MGSSHHHHHHSSGLVPRGSHMNPKRIRALKSGKQGDGPVVYWMSRDQRAEDNWALLFSRAIAKEANVPVVVVFCLTDEFLEAGIRQYEFMLKGLQELEVSLSRKKIPSFFLRGDPGEKISRFVKDYNAGTLVTDFSPLRIKNQWIEKVISGISIPFFEVDAHNVVPCWEASQKHEYAAHTFRPKLYALLPEFLEEFPELEPNSVTPELSAGAGMVETLSDVLETGVKALLPERALLKNKDPLFEPWHFEPGEKAAKKVMESFIADRLDSYGALRNDPTKNMLSNLSPYLHFGQISSQRVVLEVEKAESNPGSKKAFLDEILIWKEISDNFCYYNPGYDGFESFPSWAKESLNAHRNDVRSHIYTLEEFEAGKTHDPLWNASQMELLSTGKMHGYTRMYWAKKILEWSESPEKALEIAICLNDRYELDGRDPNGYAGIAWSIGGVHDRAWGEREVTGKIRYMSYEGCKRKFDVKLYIEKYSAL

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 \(c\).

\(|P_{f(2KLJ_1)}(2) \setminus P_{f(4REU_1)}(2)|=76\), \(|P_{f(4REU_1)}(2) \setminus P_{f(2KLJ_1)}(2)|=77\). 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 1:010101000011010000000000010101000001010010111000100011001100100100001111000100001110010001010000000101101000000100010100100101101011100100001000111110000000111001011010011010
Pair \(Z_2\) Length of longest common subsequence
2KLJ_1,4REU_1 153 4
2KLJ_1,7VJI_1 198 3
4REU_1,7VJI_1 207 3

Newick tree

 
[
	7VJI_1:10.28,
	[
		2KLJ_1:76.5,4REU_1:76.5
	]:31.78
]

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 \((0.85)(0.8)(\frac{338 }{\log_{20} 338}-\frac{164}{\log_{20}164})=52.7\)
Status Protein1 Protein2 d d1/2
Query variables 2KLJ_1 4REU_1 66 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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]