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Parikh vectors
4XAJ_1 1ROW_1 6YHM_1 Letter Amino acid
59 3 23 L Leucine
29 9 7 P Proline
11 0 1 W Tryptophan
18 1 15 Y Tyrosine
69 20 16 A Alanine
35 7 20 G Glycine
39 3 18 E Glutamic acid
5 2 3 H Histidine
35 5 20 I Isoleucine
11 2 5 M Methionine
26 5 13 F Phenylalanine
31 9 41 S Serine
17 2 13 R Arginine
32 6 13 D Aspartic acid
32 13 25 V Valine
19 1 3 Q Glutamine
44 7 13 K Lycine
33 9 22 T Threonine
28 4 27 N Asparagine
3 1 1 C Cysteine 

4XAJ_1|Chains A, B, D[auth C], E[auth D]|Maltose-binding periplasmic protein,Nuclear receptor subfamily 2 group E member 1|Escherichia coli O157:H7 (83334)
>1ROW_1|Chains A, B|MSP-domain protein like family member|Caenorhabditis elegans (6239)
>6YHM_1|Chain A|Cytotoxic necrotizing factor|Yersinia pseudotuberculosis (633)
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 \(w=f(c)\)
4XAJ , Knot 226 576 0.83 40 256 526 
MAKIEEGKLVIWINGDKGYNGLAEVGKKFEKDTGIKVTVEHPDKLEEKFPQVAATGDGPDIIFWAHDRFGGYAQSGLLAEITPDKAFQDKLYPFTWDAVRYNGKLIAYPIAVEALSLIYNKDLLPNPPKTWEEIPALDKELKAKGKSALMFNLQEPYFTWPLIAADGGYAFKYENGKYDIKDVGVDNAGAKAGLTFLVDLIKNKHMNADTDYSIAEAAFNKGETAMTINGPWAWSNIDTSKVNYGVTVLPTFKGQPSKPFVGVLSAGINAASPNKELAKEFLENYLLTDEGLEAVNKDKPLGAVALKSYEEELAKDPRIAATMENAQKGEIMPNIPQMSAFWYAVRTAVINAASGRQTVDEALKDAQTNAAAEFTESVCESAARLLFMSIKWAKSVPAFSTLSLQDQLMLLEDAWRELFVLGIAQWAIPVDANTLLAVSGMNGDNTDSQRLTLIISEIQALQEVVARFRQLRLDATEFACLKCIVTFKAVPTHSGSELRSFRNAAAIAALQDEAQLTLNSYIHTRYPTQPVRFGKLLLLLPALRSISPSTIEEVFFKKTIGNVPITRLLSDMYKSS
1ROW , Knot 55 109 0.79 38 80 105 
MSLTADPPACTVPAAGVSSTHKLVNGGAEKIVFKIKSSNNNEYRIAPVFGFVDPSGSKDVVITRTAGAPKEDKLVVHFASAPADATDAQAAFVAVAPAGTVTIPMSATA
6YHM , Knot 130 299 0.82 40 171 283 
GSHMNVAEISSINFRRLNSGNINVLKGRGVFSSRRLREIYLRFDAANADELRPGDVYVKKTKFDSMGYDSHFYNEGIGINGAPTLNTYTGEYVADSSSQGATYWLKYNLTNETSIIKVSNSARGANGIKIALEEIEENKPVVITSGTLTGCTVVFARKGEYFYAVHTGNSESLIGFTSTSGVAKAIEVLSSLSELEVPALPDVINNNTLVEYLSDNFDSALISYSSSSLKPNSMINISRENVSTFSYYTDDIQLPSFGTSVTILVRTNDNTVVRSLSESYTMNSNSSKMVVFNVLQKDF

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 \(c\).

\(|P_{f(4XAJ_1)}(2) \setminus P_{f(1ROW_1)}(2)|=188\), \(|P_{f(1ROW_1)}(2) \setminus P_{f(4XAJ_1)}(2)|=12\). 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 1:110100101111101001001110110010000110101001001000110111010110111110001110100111101010011000101101011000101110111101101100001110110010011110001010100111101001010111111011011000010001001110011101110111011000010100000110111001001101011111001000010011011101010100111111011101101000110011000110001101100001111111000000110010111010010010111011010111011001110110100010011001000111010001000110111101011001111001010001111001100111111101111101001111011010000000101110010110011101001010100110100110101110001001001001111111000101010001000010011011011111111001010010011100011011100110010000
Pair \(Z_2\) Length of longest common subsequence
4XAJ_1,1ROW_1 200 3
4XAJ_1,6YHM_1 163 3
1ROW_1,6YHM_1 151 4

Newick tree

 
[
	4XAJ_1:95.88,
	[
		6YHM_1:75.5,1ROW_1:75.5
	]:20.38
]

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 \((0.85)(0.8)(\frac{685 }{\log_{20} 685}-\frac{109}{\log_{20}109})=166.\)
Status Protein1 Protein2 d d1/2
Query variables 4XAJ_1 1ROW_1 207 122
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]