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Parikh vectors
4RRD_1 6RJS_1 7XYZ_1 Letter Amino acid
15 26 45 A Alanine
8 10 35 R Arginine
7 15 16 F Phenylalanine
8 26 36 V Valine
7 8 14 H Histidine
4 38 9 I Isoleucine
11 10 32 P Proline
8 22 14 T Threonine
9 33 18 K Lycine
3 6 9 M Methionine
1 4 5 W Tryptophan
6 14 6 Y Tyrosine
1 22 4 N Asparagine
1 17 30 Q Glutamine
8 24 27 G Glycine
14 24 61 L Leucine
5 28 22 D Aspartic acid
1 6 13 C Cysteine
8 20 37 E Glutamic acid
11 24 33 S Serine 

4RRD_1|Chain A|Probable threonine--tRNA ligase 2|Aeropyrum pernix K1 (272557)
>6RJS_1|Chains A, B, C, D|Methionine adenosyltransferase|Ureaplasma urealyticum serovar 7 str. ATCC 27819 (519849)
>7XYZ_1|Chains A, B, C, D|Tripartite motif-containing protein 72|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 \(w=f(c)\)
4RRD , Knot 64 136 0.77 40 100 130 
MRLLYLHADRFEYKTVKPALKNPPDPPGEASFGEALVVFTTVEDGDGPQTVMYAASDIASHSSRLKVTTVILYPYAHLSSRLAKPMAAHKRLIELEGALRTKFPGHVHRAPFGWYKSFSIACKGHPLAELSRSFTE
6RJS , Knot 156 377 0.81 40 210 361 
MQYKKIITSESVGAGHPDKICDQISDAILDECLSQDQNSRVACEVLACNRLIVIAGEITTHAYVDVVKTAWEIIKPLGYDENDFTIISNVNKQSVDIAQSVDKTNKNLIGAGDQGIVFGYACDETPQYMPLTSVLAHELLKEIERQRRSKEFIKIQADMKSQVSIDYSNSTPLIETMLVSIQHDEDYDVEYFNKKVSAIMEQIAKKYNLNTNFKKIINSSGRFVIGGPIGDTGLTGRKIIVDTYGGVGHHGGGAFSGKDPTKVDRSASYFARWIAKNVVAAKLAKQCEIQLAFAIGQPQPVAMYVNTFNTNLIDETKIFEAIKKSFNFDIKTFINDLNLWTTKYLPVATYGHFGRDDLDLSWEKLNKVEDLIKNSKH
7XYZ , Knot 194 466 0.85 40 221 433 
GSLLRQELSCPLCLQLFDAPVTAECGHSFCRACLIRVAGEPAADGTVACPSCQAPTRPQALSTNLQLSRLVEGLAQVPQGHCEEHLDPLSIYCEQDRTLVCGVCASLGSHRGHRLLPAAEAQARLKTQLPQQKMQLQEASMRKEKTVAVLEHQLVEVEETVRQFRGAVGEQLGKMRMFLAALESSLDREAERVRGDAGVALRRELSSLNSYLEQLRQMEKVLEEVADKPQTEFLMKFSLVTSRLQKILSESPPPARLDIQLPVISDDFKFQVWKHMFRHLMPALEELTFDPSSAHPSLVVSSSGRRVECSDQKAPPAGEDTRQFDKAVAVVAQQLLSQGEHYWEVEVGDKPRWALGVMAADASRRGRLHAVPSQGLWLLGLRDGKILEAHVEAKEPRALRTPERPPARIGLYLSFADGVLAFYDASNPDVLTPIFSFHERLPGPVYPIFDVCWHDKGKNAQPLLLV

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 \(c\).

\(|P_{f(4RRD_1)}(2) \setminus P_{f(6RJS_1)}(2)|=40\), \(|P_{f(6RJS_1)}(2) \setminus P_{f(4RRD_1)}(2)|=150\). 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 1:1011010100100001011100110111010110111110010010110011011001100000101001110101010001101111000110101110001110100111110001011001011101000100
Pair \(Z_2\) Length of longest common subsequence
4RRD_1,6RJS_1 190 3
4RRD_1,7XYZ_1 173 4
6RJS_1,7XYZ_1 179 4

Newick tree

 
[
	6RJS_1:94.14,
	[
		4RRD_1:86.5,7XYZ_1:86.5
	]:7.64
]

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 \((0.85)(0.8)(\frac{513 }{\log_{20} 513}-\frac{136}{\log_{20}136})=111.\)
Status Protein1 Protein2 d d1/2
Query variables 4RRD_1 6RJS_1 140 94
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]