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Parikh vectors
3HLU_1 1YWX_1 9CFX_1 Letter Amino acid
3 5 23 S Serine
4 3 18 R Arginine
5 7 22 D Aspartic acid
8 13 28 E Glutamic acid
11 9 21 V Valine
7 6 12 N Asparagine
5 3 11 Q Glutamine
7 11 19 K Lycine
6 3 16 M Methionine
6 5 27 T Threonine
8 11 29 A Alanine
1 0 5 C Cysteine
6 9 30 I Isoleucine
3 3 12 F Phenylalanine
1 2 19 P Proline
0 0 4 W Tryptophan
3 2 16 Y Tyrosine
5 2 28 G Glycine
0 0 9 H Histidine
7 8 26 L Leucine 

3HLU_1|Chains A, B|uncharacterized protein DUF2179|Eubacterium ventriosum (411463)
>1YWX_1|Chain A|30S ribosomal protein S24e|Methanococcus maripaludis (39152)
>9CFX_1|Chains A[auth B], B[auth C], C[auth D]|Actin, alpha skeletal muscle|Oryctolagus cuniculus (9986)
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 \(w=f(c)\)
3HLU , Knot 50 96 0.79 36 82 91 
SNANGDQQTMVYIVSAKRKIIADRMLQELDLGVTMLQAVGAYKNNETEVIMCVMRKATLVKVRNLLKEVDPDAFMIVSTANEVFGEGFKNQYETEI
1YWX , Knot 52 102 0.78 34 80 98 
MDISIISDRNNPLLQRREIKFTVSFDAATPSIKDVKMKLVAVLNANKQVLVVDTLDQIFGKLEAEGYAKIYNDEKAMATIETKSVLEKNKIEEEAEAEVAEE
9CFX , Knot 159 375 0.83 40 219 360 
DEDETTALVCDNGSGLVKAGFAGDDAPRAVFPSIVGRPRHQGVMVGMGQKDSYVGDEAQSKRGILTLKYPIEHGIITNWDDMEKIWHHTFYNELRVAPEEHPTLLTEAPLNPKANREKMTQIMFETFNVPAMYVAIQAVLSLYASGRTTGIVLDSGDGVTHNVPIYEGYALPHAIMRLDLAGRDLTDYLMKILTERGYSFVTTAEREIVRDIKEKLCYVALDFENEMATAASSSSLEKSYELPDGQVITIGNERFRCPETLFQPSFIGMESAGIHETTYNSIMKCDIDIRKDLYANNVMSGGTTMYPGIADRMQKEITALAPSTMKIKIIAPPERKYSVWIGGSILASLSTFQQMWITKQEYDEAGPSIVHRKCF

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 \(c\).

\(|P_{f(3HLU_1)}(2) \setminus P_{f(1YWX_1)}(2)|=55\), \(|P_{f(1YWX_1)}(2) \setminus P_{f(3HLU_1)}(2)|=53\). 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 1:001010000110110100011100110010111011011110000000111011001011010011001010111110010011101100000001
Pair \(Z_2\) Length of longest common subsequence
3HLU_1,1YWX_1 108 3
3HLU_1,9CFX_1 199 5
1YWX_1,9CFX_1 193 4

Newick tree

 
[
	9CFX_1:10.79,
	[
		3HLU_1:54,1YWX_1:54
	]:54.79
]

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 \((0.85)(0.8)(\frac{198 }{\log_{20} 198}-\frac{96}{\log_{20}96})=33.4\)
Status Protein1 Protein2 d d1/2
Query variables 3HLU_1 1YWX_1 41 40.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]