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Parikh vectors
5CSR_1 9EVI_1 5XKT_1 Letter Amino acid
13 14 14 D Aspartic acid
19 4 13 E Glutamic acid
13 31 10 S Serine
18 26 22 G Glycine
18 11 12 I Isoleucine
16 10 11 K Lycine
11 10 6 F Phenylalanine
2 18 4 Q Glutamine
19 32 21 L Leucine
6 4 5 M Methionine
9 31 9 P Proline
17 37 18 A Alanine
11 9 10 R Arginine
6 14 6 N Asparagine
2 6 2 C Cysteine
14 28 16 T Threonine
15 24 14 V Valine
9 1 2 H Histidine
0 5 2 W Tryptophan
8 9 3 Y Tyrosine 

5CSR_1|Chains A, B, C, D|Triosephosphate isomerase|Thermoplasma acidophilum (strain ATCC 25905 / DSM 1728 / JCM 9062 / NBRC 15155 / AMRC-C165) (273075)
>9EVI_1|Chains A, B|Lipase B|Moesziomyces antarcticus (84753)
>5XKT_1|Chains A, B|Urease accessory protein UreG|Klebsiella pneumoniae subsp. rhinoscleromatis SB3432 (861365)
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 \(w=f(c)\)
5CSR , Knot 107 226 0.85 38 150 216 
MDMYTAIVNLKTYREATGANFTRFMEKFEPVQGKFELIFSPSLLDLEKAAKCGKFRFFAQHVDAEPYGAYTGHVPMDMMIDLGITGSILNHSERRLPRDTIINTLKKASKLDFTIVLCVENAEEAKYFREYEPDFIAYEPRDLIGGDVSVSTAKPEIIEDIVKIYEGTGTSVLVGAGIKTGEDVRRSIGLGARGILVASGVVKSADPTKSLNSLIELKLEHHHHHH
9EVI , Knot 142 324 0.84 40 184 305 
ATPLVKRLPSGSDPAFSQPKSVLDAGLTCQGASPSSVSKPILLVPGTGTTGPQSFDSNWIPLSTQLGYTPCWISPPPFMLNDTQVNTEYMVNAITALYAGSGNNKLPVLTWSQGGLVAQWGLTFFPSIRSKVDRLMAFAPDYKGTVLAGPLDALAVSAPSVWQQTTGSALTTALRNAGGLTQIVPTTNLYSATDEIVQPQVSNSPLDSSYLFNGKNVQAQAVCGPLFVIDHAGSLTSQFSYVVGRSALRSTTGQARSADYGITDCNPLPANDLTPEQKVAAAALLAPAAAAIVAGPKQNCEPDLMPYARPFAVGKRTCSGIVTP
5XKT , Knot 96 200 0.84 40 144 194 
KHPLRVGVGGPVGSGKTALLEALCKAMRDTWQLAVVTNDIYTKEDQRILTEAGTLAPERIVGVETGGCPHTAIREDASMNLAAVEALSEKFGNLDLIFVESGGDNLSATFSPELADLTIYVIDVAEGEKIPRKGGPGITRSDFLVINKTDLAPYVGASLKVMASDTQRMRGDRPWTFTNLKQGDGLSTIIAFLEDKGMLG

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 \(c\).

\(|P_{f(5CSR_1)}(2) \setminus P_{f(9EVI_1)}(2)|=73\), \(|P_{f(9EVI_1)}(2) \setminus P_{f(5CSR_1)}(2)|=107\). 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 1:1010011101000001011010011001011010101110101101001100101011100101010110010111011101110101100000011000110010010010101110100100100100001011100100111101010010101100110100101001111111001001000111110111110111001010001001101010000000
Pair \(Z_2\) Length of longest common subsequence
5CSR_1,9EVI_1 180 4
5CSR_1,5XKT_1 136 3
9EVI_1,5XKT_1 172 3

Newick tree

 
[
	9EVI_1:93.75,
	[
		5CSR_1:68,5XKT_1:68
	]:25.75
]

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 \((0.85)(0.8)(\frac{550 }{\log_{20} 550}-\frac{226}{\log_{20}226})=92.6\)
Status Protein1 Protein2 d d1/2
Query variables 5CSR_1 9EVI_1 116 98
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]