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Parikh vectors
2DFA_1 7SLI_1 5OHJ_1 Letter Amino acid
40 18 25 A Alanine
25 12 11 G Glycine
3 0 4 C Cysteine
8 19 21 K Lycine
20 4 16 R Arginine
11 7 30 D Aspartic acid
9 5 15 F Phenylalanine
9 6 35 S Serine
4 5 27 T Threonine
8 11 24 H Histidine
27 19 44 L Leucine
22 14 29 E Glutamic acid
6 9 22 I Isoleucine
4 3 21 M Methionine
16 4 11 P Proline
1 2 5 W Tryptophan
7 3 11 Y Tyrosine
4 1 25 N Asparagine
3 5 19 Q Glutamine
23 7 26 V Valine 

2DFA_1|Chain A|Hypothetical UPF0271 protein TTHB195|Thermus thermophilus (300852)
>7SLI_1|Chain A|Myoglobin|Physeter catodon (9755)
>5OHJ_1|Chains A, B|cAMP-specific 3',5'-cyclic phosphodiesterase 4B|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 \(w=f(c)\)
2DFA , Knot 108 250 0.79 40 135 229 
MKVDLNADAGESYGAFAYGHDREIFPLVSSANLACGFHGGSPGRILEAVRLAKAHGVAVGAHPGFPDLVGFGRREMALSPEEVYADVLYQIGALSAFLKAEGLPLHHVKPHGALYLKACRDRETARAIALAVKAFDPGLPLVVLPGTVYEEEARKAGLRVVLEAFPERAYLRSGQLAPRSMPGSWITDPEEAARRALRMVLEGKVEALDGGEVAVRADTLCIHGDNPNAPEVARAVREALEQAGVEVRAF
7SLI , Knot 74 154 0.80 38 111 148 
MVLSEGEWQLVLHVWAKVEADVAGHGQDILIRLFKSHPETLEKIDRLKHLKTEAEMKASEDLKKFGVTGLTALGAILKKKGHHEAELKPLAQSHATKHKIPIKYLEFASEAIIHVLHSRHPGDFGADAQGAMNKALELFRKDIAAKYKELGYQG
5OHJ , Knot 175 421 0.83 40 228 396 
MSVSEMASNKFKRMLNRELTHLSEMSRSGNQVSEYISNTFLDKQNDVEIPKVTAEEAPQPMTQISGVKKLMHSSSLNNTSISRFGVNTENEDHLAKELEDLNKWGLNIFNVAGYSHNRPLTCIMYAIFQERDLLKTFRISSDTFITYMMTLEDHYHSDVAYHNSLHAADVAQSTHVLLSTPALDAVFTDLEILAAIFAAAIHDVDHPGVSNQFLINTNSELALMYNDESVLENHHLAVGFKLLQEEHCDIFMNLTKKQRQTLRKMVIDMVLATDMSKHMSLLADLKTMVETKKVTSSGVLLLDNYTDRIQVLRNMVHCADLSNPTKSLELYRQWTDRIMEEFFQQGDKERERGMEISPMCDKHTASVEKSQVGFIDYIVHPLWETWADLVQPDAQDILDTLEDNRNWYQSMIPQSHHHHHH

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 \(c\).

\(|P_{f(2DFA_1)}(2) \setminus P_{f(7SLI_1)}(2)|=80\), \(|P_{f(7SLI_1)}(2) \setminus P_{f(2DFA_1)}(2)|=56\). 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 1:1010101011000111101000011111001011011011011011011011010111111011110111110001110100101011001111011101011110010101110101000000101111110110111111111101000010011101110111001010010111001110110010011001101110101011011011101001010100101101101100110011101011
Pair \(Z_2\) Length of longest common subsequence
2DFA_1,7SLI_1 136 4
2DFA_1,5OHJ_1 203 3
7SLI_1,5OHJ_1 185 4

Newick tree

 
[
	5OHJ_1:10.02,
	[
		2DFA_1:68,7SLI_1:68
	]:37.02
]

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 \((0.85)(0.8)(\frac{404 }{\log_{20} 404}-\frac{154}{\log_{20}154})=74.8\)
Status Protein1 Protein2 d d1/2
Query variables 2DFA_1 7SLI_1 90 73
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]