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Parikh vectors
2MXT_1 1IEQ_1 1WHU_1 Letter Amino acid
2 33 10 I Isoleucine
1 32 4 P Proline
5 30 11 S Serine
6 52 7 V Valine
9 54 5 A Alanine
7 39 7 D Aspartic acid
11 48 4 L Leucine
0 8 0 W Tryptophan
2 21 4 Y Tyrosine
3 23 3 N Asparagine
8 24 12 E Glutamic acid
5 63 6 G Glycine
7 33 10 K Lycine
1 18 1 M Methionine
1 8 1 C Cysteine
6 19 2 Q Glutamine
2 10 2 H Histidine
4 20 6 F Phenylalanine
2 42 4 T Threonine
3 28 5 R Arginine 

2MXT_1|Chain A|Heterogeneous nuclear ribonucleoprotein Q|Homo sapiens (9606)
>1IEQ_1|Chain A|BETA-D-GLUCAN GLUCOHYDROLASE ISOENZYME EXO1|Hordeum vulgare (4513)
>1WHU_1|Chain A|polynucleotide phosphorylase; 3'-5' RNA exonuclease|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 \(w=f(c)\)
2MXT , Knot 47 85 0.82 38 73 80 
GSENFQTLLDAGLPQKVAEKLDEIYVAGLVAHSDLDERAIEALKEFNEDGALAVLQQFKDSDLSHVQNKSAFLCGVMKTYRQREK
1IEQ , Knot 241 605 0.85 40 273 564 
DYVLYKDATKPVEDRVADLLGRMTLAEKIGQMTQIERLVATPDVLRDNFIGSLLSGGGSVPRKGATAKEWQDMVDGFQKACMSTRLGIPMIYGIDAVHGQNNVYGATIFPHNVGLGATRDPYLVKRIGEATALEVRATGIQYAFAPCIAVCRDPRWGRCYESYSEDRRIVQSMTELIPGLQGDVPKDFTSGMPFVAGKNKVAACAKHFVGDGGTVDGINENNTIINREGLMNIHMPAYKNAMDKGVSTVMISYSSWNGVKMHANQDLVTGYLKDTLKFKGFVISDWEGIDRITTPAGSDYSYSVKASILAGLDMIMVPNKYQQFISILTGHVNGGVIPMSRIDDAVTRILRVKFTMGLFENPYADPAMAEQLGKQEHRDLAREAARKSLVLLKNGKTSTDAPLLPLPKKAPKILVAGSHADNLGYQCGGWTIEWQGDTGRTTVGTTILEAVKAAVDPSTVVVFAENPDAEFVKSGGFSYAIVAVGEHPYTETKGDNLNLTIPEPGLSTVQAVCGGVRCATVLISGRPVVVQPLLAASDALVAAWLPGSEGQGVTDALFGDFGFTGRLPRTWFKSVDQLPMNVGDAHYDPLFRLGYGLTTNATKKY
1WHU , Knot 53 104 0.79 38 81 98 
GSSGSSGPQKIFTPSAEIVKYTKIIAMEKLYAVFTDYEHDKVSRDEAVNKIRLDTEEHLKEKFPEVDQFEIIESFNIVAKEVFRSIILNEYKRCDGRDSGPSSG

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 \(c\).

\(|P_{f(2MXT_1)}(2) \setminus P_{f(1IEQ_1)}(2)|=16\), \(|P_{f(1IEQ_1)}(2) \setminus P_{f(2MXT_1)}(2)|=216\). 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 1:1000100110111100110010010111111000100011011001000111111001000010010000111011100000000
Pair \(Z_2\) Length of longest common subsequence
2MXT_1,1IEQ_1 232 3
2MXT_1,1WHU_1 108 3
1IEQ_1,1WHU_1 226 3

Newick tree

 
[
	1IEQ_1:12.49,
	[
		2MXT_1:54,1WHU_1:54
	]:74.49
]

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 \((0.85)(0.8)(\frac{690 }{\log_{20} 690}-\frac{85}{\log_{20}85})=176.\)
Status Protein1 Protein2 d d1/2
Query variables 2MXT_1 1IEQ_1 222 125.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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