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Parikh vectors
7CEM_1 4CQH_1 8PFH_1 Letter Amino acid
4 9 15 F Phenylalanine
8 14 21 S Serine
15 39 16 A Alanine
11 19 28 E Glutamic acid
12 23 13 G Glycine
3 17 7 H Histidine
8 8 28 I Isoleucine
5 7 5 M Methionine
8 24 22 T Threonine
8 20 16 V Valine
8 11 17 D Aspartic acid
1 3 1 C Cysteine
5 18 11 Q Glutamine
12 29 7 P Proline
10 21 21 R Arginine
3 7 18 N Asparagine
19 46 28 L Leucine
11 2 14 K Lycine
4 4 3 W Tryptophan
3 8 9 Y Tyrosine 

7CEM_1|Chains A, B|Ribosomal RNA large subunit methyltransferase H,Ribosomal RNA large subunit methyltransferase H|Escherichia coli K-12 (83333)
>4CQH_1|Chain A|BACTERIOPHYTOCHROME|DEINOCOCCUS RADIODURANS (1299)
>8PFH_1|Chain A|CDC10 isoform 1|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 \(w=f(c)\)
7CEM , Knot 77 158 0.82 40 121 150 
MIPGKPFDTPQLAAELERWKLDGRDVSLLIGGPEGLSPACKAAAEQSWSLSALTLPHPLVRVLVAESLYRAWSITTNHPYHREGSMKLQLVAVGTKMPDWVQTGFTEYLRRFPKDMPFELIEIPAGKRGKNADIKRILDKEGEQMLAAAGKNRIVTLD
4CQH , Knot 136 329 0.79 40 178 306 
MARDPQPFFPPLYLGGPEITTENCEREPIHIPGSIQPHGALLTADGHSGEVLQVSLNAATFLGQEPTVLRGQTLAALLPDQWPALQTALPPGCQDALQYRATLDWPAAGHLSLTVHRVAELLILEFEPTEAWDSIGPHALRNAMFALESAPNLRALAEVATQTVRELSGFDRVMLYKFAPDATGEVIAEARREGMQAYLGHRFPASTTPAQARALYTRHLLRLTADTRAAAVPLDPVLNPQTNAPTPLGGAVLRATSPMHMQYLRNMGVGSSLSVSVVVGGQLWGLIVCHHQTPYVLPPDLRTTLEYLGRLLSLQVQRKEAEFHHHHHH
8PFH , Knot 131 300 0.83 40 186 288 
MADLNWMGFQFNIMVVGQSGLGKSTLINTLFASHLIDSATGDDISALPVTKTTEMKISTHTLVEDRVRLNINVIDTPGFGDFIDNSKAWEPIVKYIKEQHSQYLRKELTAQRERFITDTRVHAILYFLQPNGKELSRLDVEALKRLTEIANVIPVIGKSDTLTLDERTEFRELIQNEFEKYNFKIYPYDSEELTDEELELNRSVRSIIPFAVVGSENEIEINGETFRGRKTRWSAINVEDINQCDFVYLREFLIRTHLQDLIETTSYIHYEGFRARQLIALKENANSRSSAHMSSNAIQR

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 \(c\).

\(|P_{f(7CEM_1)}(2) \setminus P_{f(4CQH_1)}(2)|=39\), \(|P_{f(4CQH_1)}(2) \setminus P_{f(7CEM_1)}(2)|=96\). 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 1:11110110010111010010101001011111101101100111000101011011011101111001001101000010000101010111110011011001100010011001110110111100100101001100010011111100011010
Pair \(Z_2\) Length of longest common subsequence
7CEM_1,4CQH_1 135 4
7CEM_1,8PFH_1 175 3
4CQH_1,8PFH_1 172 4

Newick tree

 
[
	8PFH_1:92.28,
	[
		7CEM_1:67.5,4CQH_1:67.5
	]:24.78
]

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 \((0.85)(0.8)(\frac{487 }{\log_{20} 487}-\frac{158}{\log_{20}158})=96.7\)
Status Protein1 Protein2 d d1/2
Query variables 7CEM_1 4CQH_1 117 84.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]