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Parikh vectors
5ODZ_1 2YOH_1 1ARI_1 Letter Amino acid
10 5 4 W Tryptophan
12 9 22 T Threonine
18 22 18 K Lycine
12 8 20 F Phenylalanine
13 13 21 S Serine
11 6 22 R Arginine
0 2 5 C Cysteine
15 5 16 Q Glutamine
15 19 27 E Glutamic acid
16 10 30 G Glycine
13 6 7 H Histidine
18 16 38 L Leucine
5 6 15 P Proline
17 17 23 N Asparagine
7 13 11 Y Tyrosine
15 11 20 D Aspartic acid
16 16 17 I Isoleucine
7 5 8 M Methionine
16 14 26 V Valine
16 7 46 A Alanine 

5ODZ_1|Chains A[auth B], B[auth D]|Beta-lactamase|Enterobacter cloacae (550)
>2YOH_1|Chains A, B|THYMIDYLATE KINASE|PLASMODIUM FALCIPARUM (36329)
>1ARI_1|Chains A, B|ASPARTATE AMINOTRANSFERASE|Escherichia coli (562)
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 \(w=f(c)\)
5ODZ , Knot 114 252 0.83 38 170 245 
HHHHHHENLYFQGKEWQENKSWNAHFTEHKSQGVVVLWNENKQQGFTNNLKRANQAFLPASTFKIPNSLIALDLGVVKDEHQVFKWDGQTRDIATWNRDHNLITAMKYSVVPVYQEFARQIGEARMSKMLHAFDYGNEDISGNVDSFWLDGGIRISATEQISFLRKLYHNKLHVSERSQRIVKQAMLTEANGDYIIRAKTGYDTKIGWWVGWVELDDNVWFFAMNMDMPTSDGLGLRQAITKEVLKQEKIIP
2YOH , Knot 97 210 0.82 40 150 204 
MTDDKKKGKFIVFEGLDRSGKSTQSKLLVEYLKNNNVEVKHLYFPNRETGIGQIISKYLKMENSMSNETIHLLFSANRWEHMNEIKSLLLKGIWVVCDRYAYSGVAYSSGALNLNKTWCMNPDQGLIKPDVVFYLNVPPNYAQNRSDYGEEIYEKVETQKKIYETYKHFAHEDYWINIDATRKIEDIHNDIVKEVTKIKVEPEEFNFLWS
1ARI , Knot 168 396 0.84 40 219 371 
MFENITAAPADPILGLADLFRADERPGKINLGIGVYKDETGKTPVLTSVKKAEQYLLENETTKNYLGIDGIPEFGRCTQELLFGKGSALINDKRARTAQTPGGTGALRVAADFLAKNTSVKRVWVSNPSHPNHKSVFNSAGLEVREYAYYDAENHTLDFDALINSLNEAQAGDVVLFHGCCHNPTGIDPTLEQWQTLAQLSVEKGWLPLFDFAYQGFARGLEEDAEGLRAFAAMHKELIVASSYSKNFGLYNERVGACTLVAADSETVDRAFSQMKAAIRANYSNPPAHGASVVATILSNDALRAIWEQELTDMRQRIQRMRQLFVNTLQEKGANRDFSFIIKQNGMFSFSGLTKEQVLRLREEFGVYAVASGRVNVAGMTPDNMAPLCEAIVAVL

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 \(c\).

\(|P_{f(5ODZ_1)}(2) \setminus P_{f(2YOH_1)}(2)|=97\), \(|P_{f(2YOH_1)}(2) \setminus P_{f(5ODZ_1)}(2)|=77\). 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 1:000000001010100100000101010000001111110000001100010010011111001011001111011110000011010100001101000001101100011110001100110101001101100100010101001110111010100010110010000101000000110011100101001101001000011111111010001111110101100011110011000110000111
Pair \(Z_2\) Length of longest common subsequence
5ODZ_1,2YOH_1 174 3
5ODZ_1,1ARI_1 171 3
2YOH_1,1ARI_1 185 3

Newick tree

 
[
	2YOH_1:91.17,
	[
		5ODZ_1:85.5,1ARI_1:85.5
	]:5.67
]

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 \((0.85)(0.8)(\frac{462 }{\log_{20} 462}-\frac{210}{\log_{20}210})=73.3\)
Status Protein1 Protein2 d d1/2
Query variables 5ODZ_1 2YOH_1 94 85.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]