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Parikh vectors
2XXS_1 7MCS_1 7HNJ_1 Letter Amino acid
11 33 9 I Isoleucine
6 24 7 K Lycine
4 21 9 P Proline
8 38 15 S Serine
4 32 14 V Valine
9 25 10 D Aspartic acid
8 37 7 E Glutamic acid
8 14 11 F Phenylalanine
4 15 7 Y Tyrosine
1 4 4 C Cysteine
12 16 10 N Asparagine
1 17 10 H Histidine
10 70 12 L Leucine
0 5 6 W Tryptophan
1 35 8 A Alanine
3 34 10 Q Glutamine
8 30 14 G Glycine
1 12 6 M Methionine
5 24 10 T Threonine
3 37 9 R Arginine 

2XXS_1|Chain A|PROTEIN MXIG|SHIGELLA FLEXNERI (623)
>7MCS_1|Chains A, B, C, D, E, F, G|Transposon Tn7 transposition protein TnsC|Escherichia coli (562)
>7HNJ_1|Chain A[auth B]|E3 ubiquitin-protein ligase TRIM21|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 \(w=f(c)\)
2XXS , Knot 55 107 0.80 38 87 104 
NSNLAPFRLLVKLTNGVGDEFPLYYGNNLIVLGRTIETLEFGNDNFPENIIPVTDSKSDGIIYLTISKDNICQFSDEKGEQIDINSQFNSFEYDGISFHLKNMREDK
7MCS , Knot 209 523 0.83 40 252 487 
MGATRIQAVYRDTGVEAYRDNPFIEALPPLQESVNSAASLKSSLQLTSSDLQKSRVIRAHTICRIPDDYFQPLGTHLLLSERISVMIRGGYVGRNPKTGDLQKHLQNGYERVQTGELETFRFEEARSTAQSLLLIGCSGSGKTTSLHRILATYPQVIYHRELNVEQVVYLKIDCSHNGSLKEICLNFFRALDRALGSNYERRYGLKRHGIETMLALMSQIANAHVLGLLVIDEIQHLSRSRSGGSQEMLNFFVTMVNIIGVPVMLIGTPKAREIFEADLRSARRGAGFGAIFWDPIQQTQRGKPNQEWIAFTDNLWQLQLLQRKDALLSDEVRDVWYELSQGVMDIVVKLFVLAQLRALALGNERITAGLLRQVYQDELKPVHPMLEALRSGIPERIARYSDLVVPEIDKRLIQLQLDIAAIQEQTPEEKALQELDTEDQRHLYLMLKEDYDSSLLIPTIKKAFSQNPTMTRQKLLPLVLQWLMEGETVVSELEKPSKSKKVSPNSSSVDKLAAALEHHHHHH
7HNJ , Knot 91 188 0.84 40 148 179 
MHHHHHHMVHITLDRNTANSWLIISKDRRQVRMGDTHQNVSDNKERFSNYPMVLGAQRFSSGKMYWEVDVTQKEAWDLGVCRDSVQRKGQFSLSPENGFWTIWLWQDSYEAGTSPQTTLHIQVPPCQIGIFVDYEAGVVSFYNITDHGSLIYTFSECVFAGPLRPFFNVGFNYSGGNAAPLKLCPLKM

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 \(c\).

\(|P_{f(2XXS_1)}(2) \setminus P_{f(7MCS_1)}(2)|=22\), \(|P_{f(7MCS_1)}(2) \setminus P_{f(2XXS_1)}(2)|=187\). 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 1:00011110111010011100111001001111100100101100011001111000000111010100001001000010010100010010001101010010000
Pair \(Z_2\) Length of longest common subsequence
2XXS_1,7MCS_1 209 3
2XXS_1,7HNJ_1 159 4
7MCS_1,7HNJ_1 182 6

Newick tree

 
[
	7MCS_1:10.41,
	[
		2XXS_1:79.5,7HNJ_1:79.5
	]:23.91
]

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 \((0.85)(0.8)(\frac{630 }{\log_{20} 630}-\frac{107}{\log_{20}107})=152.\)
Status Protein1 Protein2 d d1/2
Query variables 2XXS_1 7MCS_1 192 114.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]