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Parikh vectors
4HFL_1 8DFS_1 3EXL_1 Letter Amino acid
4 4 9 C Cysteine
11 10 9 K Lycine
10 9 12 T Threonine
4 3 1 W Tryptophan
6 24 16 R Arginine
11 6 7 N Asparagine
7 17 6 D Aspartic acid
2 14 14 E Glutamic acid
6 5 7 H Histidine
13 10 15 S Serine
4 7 10 Y Tyrosine
13 20 9 A Alanine
6 5 8 Q Glutamine
11 12 6 I Isoleucine
9 17 15 L Leucine
10 11 6 F Phenylalanine
9 16 14 P Proline
12 17 13 V Valine
20 13 14 G Glycine
8 7 6 M Methionine 

4HFL_1|Chain A|Putative cytoplasmic protein|Enterobacter cloacae (550)
>8DFS_1|Chain A|pre-crRNA processing endonuclease|Desulfovibrio vulgaris (882)
>3EXL_1|Chain A|mouse p53 core domain|Mus musculus (10090)
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 \(w=f(c)\)
4HFL , Knot 84 176 0.82 40 133 171 
HMASMTGGQQMGRMSHMRPAFGAAWNRFKEVNVNVEQVGKLLGGKVQHNIDAGIFKNACPIRMSYVLNYCGIPVPSNSKYATVTGSDKKRYMFRVKDMIAFLPTVLGKADISVSSPTPAQFAGKQGIIIFTGHGWLDATGHVTLWNGNICSDDCHFLGSPGNGSFIPTNATFWSLK
8DFS , Knot 108 227 0.86 40 160 222 
MTHGAVKTYGIRLRVWGDYACFTRPEMKVERVSYDVMPPSAARGILEAIHWKPAIRWIVDRIHVLRPIVFDNVRRNEVSSKIPKPNPATAMRDRKPLYFLVDDGSNRQQRAATLLRNVDYVIEAHFELTDKAGAEDNAGKHLDIFRRRARAGQSFQQPCLGCREFPASFELLEGDVPLSCYAGEKRDLGYMLLDIDFERDMTPLFFKAVMEDGVITPPSRTSPEVRA
3EXL , Knot 95 197 0.85 40 145 191 
SQKTYQGNYGFHLGFLQSGTAKSVMCTYSPPLNKLFCQLAKTCPVQLWVSATPPAGSRVRAMAIYKKSQHMTEVVRRCPHHERCSDGDGLAPPQHLIRVEGNLYPEYLEDRQTFRHSVVVPYEPPEAGSEYTTIHYKYMCNSSCMGGMNRRPILTIITLEDSSGNLLGRDSFEVRVCACPGRDRRTEEENFRKKEVL

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 \(c\).

\(|P_{f(4HFL_1)}(2) \setminus P_{f(8DFS_1)}(2)|=74\), \(|P_{f(8DFS_1)}(2) \setminus P_{f(4HFL_1)}(2)|=101\). 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 1:01101011001101001011111110010010101001101111010001011110010110100110001111100000101010000001101001111110111010101001011011100111110101110101010110101000000111011010111001011010
Pair \(Z_2\) Length of longest common subsequence
4HFL_1,8DFS_1 175 3
4HFL_1,3EXL_1 176 3
8DFS_1,3EXL_1 179 3

Newick tree

 
[
	3EXL_1:89.16,
	[
		4HFL_1:87.5,8DFS_1:87.5
	]:1.66
]

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 \((0.85)(0.8)(\frac{403 }{\log_{20} 403}-\frac{176}{\log_{20}176})=67.5\)
Status Protein1 Protein2 d d1/2
Query variables 4HFL_1 8DFS_1 88 77.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]