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Parikh vectors
3EFA_1 4IDN_1 7DCN_1 Letter Amino acid
8 20 8 Q Glutamine
5 16 10 H Histidine
7 15 1 Y Tyrosine
10 24 14 V Valine
18 28 20 A Alanine
8 25 9 D Aspartic acid
3 4 4 C Cysteine
9 19 7 T Threonine
11 41 12 E Glutamic acid
6 25 8 I Isoleucine
4 37 5 K Lycine
2 22 5 N Asparagine
12 46 26 L Leucine
2 7 3 W Tryptophan
6 25 2 F Phenylalanine
6 15 12 P Proline
4 32 13 S Serine
12 19 15 R Arginine
11 27 9 G Glycine
3 10 3 M Methionine 

3EFA_1|Chain A|Putative Acetyltransferase|Lactobacillus plantarum WCFS1 (220668)
>4IDN_1|Chains A, B|Atlastin-1|Homo sapiens (9606)
>7DCN_1|Chain A|Probable apo-citrate lyase phosphoribosyl-dephospho-CoA transferase|Escherichia coli (562)
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 \(w=f(c)\)
3EFA , Knot 73 147 0.82 40 117 142 
SNAMKIIFSASPANRAAAYALRQAVFVEERGISADVEFDVKDTDQCEYAVLYLQPDLPITTLRLEPQADHVMRFGRVCTRKAYRGHGWGRQLLTAAEEWATQRGFTHGEIHGELTAQRFYELCGYRVTAGPYDEDGAPVVIMHKQLL
4IDN , Knot 186 457 0.83 40 234 420 
MAKNRRDRNSWGGFSEKTYEWSSEEEEPVKKAGPVQVLIVKDDHSFELDETALNRILLSEAVRDKEVVAVSVAGAFRKGKSFLMDFMLRYMYNQESVDWVGDYNEPLTGFSWRGGSERETTGIQIWSEIFLINKPDGKKVAVLLMDTQGTFDSQSTLRDSATVFALSTMISSIQVYNLSQNVQEDDLQHLQLFTEYGRLAMEETFLKPFQSLIFLVRDWSFPYEFSYGADGGAKFLEKRLKVSGNQHEELQNVRKHIHSCFTNISCFLLPHPGLKVATNPNFDGKLKEIDDEFIKNLKILIPWLLSPESLDIKEINGNKITCRGLVEYFKAYIKIYQGEELPHPKSMLQATAEANNLAAVATAKDTYNKKMEEICGGDKPFLAPNDLQTKHLQLKEESVKLFRGVKKMGGEEFSRRYLQQLESEIDELYIQYIKHNDSKNIFHAARAAALEHHHHHH
7DCN , Knot 85 186 0.79 40 134 179 
GSHMHLLPELASHHAVSIPELLVSRDERQARQHVWLKRHPVPLVSFTVVAPGPIKDSEVTRRIFNHGVTALRALAAKQGWQIQEQAALVSASGPEGMLSIAAPARDLKLATIELEHSHPLGRLWDIDVLTPEGEILSRRDYSLPPRRCLLCEQSAAVCARGKTHQLTDLLNRMEALLNDVDACNVN

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 \(c\).

\(|P_{f(3EFA_1)}(2) \setminus P_{f(4IDN_1)}(2)|=44\), \(|P_{f(4IDN_1)}(2) \setminus P_{f(3EFA_1)}(2)|=161\). 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 1:001101110101100111011001111000110101010100000000111010101110010101010011011010000100101110011011001100011001010101010010010100101110000111111100011
Pair \(Z_2\) Length of longest common subsequence
3EFA_1,4IDN_1 205 4
3EFA_1,7DCN_1 153 3
4IDN_1,7DCN_1 194 3

Newick tree

 
[
	4IDN_1:10.42,
	[
		3EFA_1:76.5,7DCN_1:76.5
	]:29.92
]

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 \((0.85)(0.8)(\frac{604 }{\log_{20} 604}-\frac{147}{\log_{20}147})=132.\)
Status Protein1 Protein2 d d1/2
Query variables 3EFA_1 4IDN_1 167 110.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]