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Parikh vectors
3BEC_1 4OMW_1 6NRC_1 Letter Amino acid
12 4 5 M Methionine
23 6 4 S Serine
18 8 7 T Threonine
4 2 0 W Tryptophan
34 5 2 G Glycine
33 21 11 L Leucine
23 16 13 K Lycine
28 16 8 A Alanine
15 3 4 R Arginine
22 6 3 N Asparagine
1 5 0 C Cysteine
16 15 15 E Glutamic acid
26 8 5 D Aspartic acid
17 9 9 Q Glutamine
14 8 0 P Proline
10 4 2 Y Tyrosine
27 10 5 V Valine
4 2 3 H Histidine
20 10 9 I Isoleucine
16 4 2 F Phenylalanine 

3BEC_1|Chain A|Penicillin-binding protein 5|Escherichia coli (562)
>4OMW_1|Chains A, B, C, D|Beta-lactoglobulin|Capra hircus (9925)
>6NRC_1|Chain A[auth 1]|Prefoldin subunit 1|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 \(w=f(c)\)
3BEC , Knot 157 363 0.85 40 207 346 
DDLNIKTMIPGVPQIDAESYILIDYNSGKVLAEQNADVRRDPASLTKMMTSYVIGQAMKAGKFKETDLVTIGNDAWATGNPVFKGSSLMFLKPGMQVPVSQLIRGINLQSGNDACVAMADFAAGSQDAFVGLMNSYVNALGLKNTHFQTVHGLDADGQYSSARDMALIGQALIRDVPNEYSIYKEKEFTFNGIRQLNRNGLLWDNSLNVDGIKTGHTDKAGYNLVASATEGQMRLISAVMGGRTFKGREAESKKLLTWGFRFFETVNPLKVGKEFASEPVWFGDSDRASLGVDKDVYLTIPRGRMKDLKASYVLNSSELHAPLQKNQVVGTINFQLDGKTIEQRPLVVLQEIPEGNFGDPVID
4OMW , Knot 80 162 0.83 40 120 157 
IIVTQTMKGLDIQKVAGTWYSLAMAASDISLLDAQSAPLRVYVEELKPTPEGNLEILLQKWENGECAQKKIIAEKTKIPAVFKIDALNENKVLVLDTDYKKYLLFCMENSAEPEQSLACQCLVRTPEVDKEALEKFDKALKALPMHIRLAFNPTQLEGQCHV
6NRC , Knot 55 107 0.80 34 87 104 
AFTELQAKVIDTQQKVKLADIQIEQLNRTKKHAHLTDTEIMTLVDETNMYEGVGRMFILQSKEAIHSQLLEKQKIAEEKIKELEQKKSYLERSVKEAEDNIREMLMA

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 \(c\).

\(|P_{f(3BEC_1)}(2) \setminus P_{f(4OMW_1)}(2)|=127\), \(|P_{f(4OMW_1)}(2) \setminus P_{f(3BEC_1)}(2)|=40\). 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 1:001010011111101010001110000101110001010001101001100011101101101000011011001110101110100111101110111001101101001001011110111100011111100010111100001001011010100001001111101110011000010000010101100100011110001010110010000110011101001010110111110010100100001101110110010110110011001111100001011100010101101010010100110000101110000111010101010010001111100110101101110
Pair \(Z_2\) Length of longest common subsequence
3BEC_1,4OMW_1 167 4
3BEC_1,6NRC_1 186 4
4OMW_1,6NRC_1 135 3

Newick tree

 
[
	3BEC_1:94.31,
	[
		4OMW_1:67.5,6NRC_1:67.5
	]:26.81
]

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 \((0.85)(0.8)(\frac{525 }{\log_{20} 525}-\frac{162}{\log_{20}162})=105.\)
Status Protein1 Protein2 d d1/2
Query variables 3BEC_1 4OMW_1 131 93.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]