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Parikh vectors
6IPI_1 7EQG_1 4INP_1 Letter Amino acid
27 28 6 R Arginine
3 12 12 N Asparagine
13 15 22 D Aspartic acid
7 3 0 C Cysteine
27 24 19 G Glycine
19 15 11 T Threonine
22 17 26 V Valine
28 33 26 A Alanine
45 46 30 L Leucine
4 3 5 M Methionine
21 26 18 P Proline
22 19 18 S Serine
5 4 7 Y Tyrosine
30 13 7 Q Glutamine
12 5 42 K Lycine
27 24 23 E Glutamic acid
6 12 27 I Isoleucine
18 11 18 F Phenylalanine
5 7 4 W Tryptophan
15 10 13 H Histidine 

6IPI_1|Chain A|DNA-directed DNA/RNA polymerase mu|Homo sapiens (9606)
>7EQG_1|Chain A[auth C]|CRISPR type I-F/YPEST-associated protein Csy2|Pseudomonas aeruginosa (287)
>4INP_1|Chains A, B|Iron (III) ABC transporter, periplasmic iron-binding protein|Helicobacter pylori (563041)
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 \(w=f(c)\)
6IPI , Knot 147 356 0.80 40 198 344 
GSAAAPLSPAWMPAYACQRPTPLTHHNTGLSEALEILAEAAGFEGSEGRLLTFCRAASVLKALPSPVTTLSQLQGLPHFGEHSSRVVQELLEHGVCEEVERVRRSERYQTMKLFTQIFGVGVKTADRWYREGLRTLDDLREQPQKLTQQQKAGLQHHQDLSTPVLRSDVDALQQVVEEAVGQALPGATVTLTGGFRRGKLQGHDVDFLITHPKEGQEAGLLPRVMCRLQDQGLILYHQHQHSCCESPTRLAQQSHMDAFERSFCIFRLPQPGSWKAVRVDLVVAPVSQFPFALLGWTGSKLFQRELRRFSRKEKGLWLNSHGLFDPEQKTFFQAASEEDIFRHLGLEYLPPEQRNA
7EQG , Knot 139 327 0.82 40 182 303 
MSVTDPEALLLLPRLSIQNANAISSPLTWGFPSPGAFTGFVHALQRRVGISLDIELDGVGIVCHRFEAQISQPAGKRTKVFNLTRNPLNRDGSTAAIVEEGRAHLEVSLLLGVHGDGLDDHPAQEIARQVQEQAGAMRLAGGSILPWCNERFPAPNAELLMLGGSDEQRRKNQRRLTRRLLPGFALVSREALLQQHLETLRTTLPEATTLDALLDLCRINFEPPATSSEEEASPPDAAWQVRDKPGWLVPIPAGYNALSPLYLPGEVRNARDRETPLRFVENLFGLGEWLSPHRVAALSDLLWYHHAEPDKGLYRWSTPRFVEHAIA
4INP , Knot 141 334 0.81 38 180 307 
QEVKVKDYFGEQTIKLPVSKIIYLGSFAEVPAMFHTWDRVVGISDYAFKSDIVKATLKDPERIKPMSSDHAAALNVELLKKLSPDLVVTFVGNPKAVEHAKKFGISFLSFQEKTIAEVMEDIDTQAKALEVDASKKLAKMQETLDFIAERLKGVKKKKGVELFHKANKISGHQALDSDILEKGGIDNFGLKYVKFGRADISVEKIVKENPEIIFIWWISPLSPEDVLNNPKFATIKAIKNKQVYKLPTMDIGGPRAPLISLFIALKAHPEAFKGVDINAIVKDYYKVVFDLNDAEVEPFLWHKGELNSKLEGKPIPNPLLGLDSTRTGHHHHHH

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 \(c\).

\(|P_{f(6IPI_1)}(2) \setminus P_{f(7EQG_1)}(2)|=82\), \(|P_{f(7EQG_1)}(2) \setminus P_{f(6IPI_1)}(2)|=66\). 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 1:10111110111111010001011000001100110111011110100101101001101101110110010010111011000001100110011000100100000000101100111111001001000110010010001001000001110000010011100010110011001110111110101011100101010010111001001001111101100100011110000000000010011000010110001011011011010110101111110011111111010011000100100000111100011101000011011000011001110011100001
Pair \(Z_2\) Length of longest common subsequence
6IPI_1,7EQG_1 148 4
6IPI_1,4INP_1 168 4
7EQG_1,4INP_1 164 4

Newick tree

 
[
	4INP_1:85.79,
	[
		6IPI_1:74,7EQG_1:74
	]:11.79
]

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 \((0.85)(0.8)(\frac{683 }{\log_{20} 683}-\frac{327}{\log_{20}327})=98.1\)
Status Protein1 Protein2 d d1/2
Query variables 6IPI_1 7EQG_1 124 118
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]