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Parikh vectors
4XRD_1 6RHS_1 6ZRP_1 Letter Amino acid
6 12 13 R Arginine
15 27 15 E Glutamic acid
12 31 19 I Isoleucine
14 22 18 K Lycine
12 13 12 F Phenylalanine
8 15 7 P Proline
19 45 21 A Alanine
5 16 14 Q Glutamine
5 6 7 H Histidine
9 21 16 S Serine
2 2 10 W Tryptophan
13 36 16 G Glycine
11 29 20 L Leucine
5 20 16 N Asparagine
14 20 14 D Aspartic acid
2 0 0 C Cysteine
2 9 9 M Methionine
14 15 12 T Threonine
5 10 6 Y Tyrosine
13 21 20 V Valine 

4XRD_1|Chains A, B, C, D, E|Alkyl hydroperoxide reductase subunit C|Salmonella enterica subsp. enterica serovar Typhimurium (90371)
>6RHS_1|Chain A|Putative L-lactate oxidase|Pediococcus acidilactici DSM 20284 (862514)
>6ZRP_1|Chains A, D|Beta-lactamase|Klebsiella pneumoniae (573)
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 \(w=f(c)\)
4XRD , Knot 88 186 0.82 40 134 177 
SLINTKIKPFKNQAFKNGEFIEVTEKDTEGRWSVFFFYPADFTFVCPTELGDVADHYEELQKLGVDVYSVSTDTHFTHKAWHSSSETIAKIKYAMIGDPTGALTRNFDNMREDEGLADRATFVVDPQGIIQAIEVTAEGIGRDASDLLRKIKAAQYVAAHPGEVCPAKFKEGEATLAPSLDLVGKI
6RHS , Knot 156 370 0.83 38 203 348 
SMTMINGYEQSDREEKIDILNLESLEKQAEEIIPAGGFGYIAGGSEDEWTLKQNRMAFHHRQIAPKALSGIEKPELNTEIFGIPLNTPVMMAPAAAQGLAHSQGEKDTARGLAAVGGLMAQSTYSSVSIAETAAAGGDAPQFFQLYMSKDWNFNESLLDEAKKANVKAIILTVDATVDGYREADIKNKFTFPLPMANLIKFSEGNGQGKGIEEIYASAAQNIRPEDVKRIADYTNLPVIVKGIQTPEDAIRAIDAGAAGIYVSNHGGRQLNGGPASFDVLEDIATAVNKQVPIIFDSGVRRGSDVFKALASGADLVALGRPVIYGLALGGAKGVQSVFEHLNHELEIVMQLAGTKTIEDVKNNSLLNIKY
6ZRP , Knot 121 265 0.85 38 175 261 
MRVLALSAVFLVASIIGMPAVAKEWQENKSWNAHFTEHKSQGVVVLWNENKQQGFTNNLKRANQAFLPASTFKIPNSLIALDLGVVKDEHQVFKWDGQTRDIATWNRDHNLITAMKYSVVPVYQEFARQIGEARMSKMLHAFDYGNEDISGNVDSFWLDGGIRISATEQISFLRKLYHNKLHVSERSQRIVKQAMLTEANGDYIIRAKTGYSTRIEPKIGWWVGWVELDDNVWFFAMNMDMPTSDGLGLRQAITKEVLKQEKIIP

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 \(c\).

\(|P_{f(4XRD_1)}(2) \setminus P_{f(6RHS_1)}(2)|=45\), \(|P_{f(6RHS_1)}(2) \setminus P_{f(4XRD_1)}(2)|=114\). 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 1:011000101100011001011010000001010111101101011010011011000001001110100100000100011000000110100111101011100010010000111001011101011101101010111001001100101100111011010110100101011101011101
Pair \(Z_2\) Length of longest common subsequence
4XRD_1,6RHS_1 159 4
4XRD_1,6ZRP_1 159 4
6RHS_1,6ZRP_1 162 4

Newick tree

 
[
	6ZRP_1:80.50,
	[
		4XRD_1:79.5,6RHS_1:79.5
	]:1.00
]

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 \((0.85)(0.8)(\frac{556 }{\log_{20} 556}-\frac{186}{\log_{20}186})=106.\)
Status Protein1 Protein2 d d1/2
Query variables 4XRD_1 6RHS_1 134 99.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]