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Parikh vectors
3UAS_1 4UUI_1 7YZJ_1 Letter Amino acid
33 27 16 G Glycine
38 11 8 F Phenylalanine
34 15 29 S Serine
27 14 22 T Threonine
30 24 13 V Valine
36 12 10 R Arginine
23 16 12 K Lycine
30 11 11 P Proline
24 16 12 D Aspartic acid
15 8 10 N Asparagine
27 17 10 E Glutamic acid
15 11 3 H Histidine
1 1 3 W Tryptophan
13 12 9 Y Tyrosine
22 26 7 A Alanine
16 21 10 Q Glutamine
27 17 9 I Isoleucine
53 34 18 L Leucine
8 7 2 M Methionine
4 4 5 C Cysteine 

3UAS_1|Chain A|Cytochrome P450 2B4|Oryctolagus cuniculus (9986)
>4UUI_1|Chains A, B, C, D|N-ACETYLNEURAMINATE LYASE|ESCHERICHIA COLI (562)
>7YZJ_1|Chain A[auth L]|Light chain of FAB fragment|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 \(w=f(c)\)
3UAS , Knot 193 476 0.83 40 238 445 
MAKKTSSKGKLPPGPSPLPVLGNLLQMDRKGLLRSFLRLREKYGDVFTVYLGSRPVVVLCGTDAIREALVDQAEAFSGRGKIAVVDPIFQGYGVIFANGERWRALRRFSLATMRDFGMGKRSVEERIQEEARCLVEELRKSKGALLDNTLLFHSITSNIICSIVFGKRFDYKDPVFLRLLDLFFQSFSLISSFSSQVFELFSGFLKYFPGTHRQIYRNLQEINTFIGQSVEKHRATLDPSNPRDFIDVYLLRMEKDKSDPSSEFHHQNLILTVLSLFFAGTETTSTTLRYGFLLMLKYPHVTERVQKEIEQVIGSHRPPALDDRAKMPYTDAVIHEIQRLGDLIPFGVPHTVTKDTQFRGYVIPKNTEVFPVLSSALHDPRYFETPNTFNPGHFLDANGALKRNEGFMPFSLGKRICLGEGIARTELFLFFTTILQNFSIASPVPPEDIDLTPRESGVGNVPPSYQIRFLARHHHH
4UUI , Knot 134 304 0.84 40 186 290 
MEHHHHHHATNLRGVMAALLTPFDQQQALDKASLRRLVQFNIQQGIDGLYVGGSTGEAFVQSLSEREQVLEIVAEEAKGKIKLIAHVGCVSTAESQQLAASAKRYGFDAVSAVTPFYYPFSFEEHCDHYRAIIDSADGLPMVVFNIPALSGVKLTLDQINTLVTLPGVGALKQTSGDLYQMEQIRREHPDLVLYNGYDNIFASGLLAGADGGIGSTYNIMGWRYQGIVKALKEGDIQTAQKLQTECNKVIDLLIKTGVFRGLKTVLHYMDVVSVPLCRKPFGPVDEKYLPELKALAQQLMQERG
7YZJ , Knot 102 219 0.83 40 159 214 
DVQMTQTPLTLSVTIGQPASISCESSQSLLYSNGKTYLNWLLQRPGQSPKRLIYLVSKLDSGVPDRFTGSGSGTDFTLRISRVEAEDLGVYYCVQGTHFPRTFGGGTKLEIKRADAAPTVSIFPPSSEQLTSGGASVVCFLNNFYPKDINVKWKIDGSERQNGVLNSWTDQDSKDSTYSMSSTLTLTKDEYERHNSYTCEATHKTSTSPIVKSFNRNEC

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 \(c\).

\(|P_{f(3UAS_1)}(2) \setminus P_{f(4UUI_1)}(2)|=103\), \(|P_{f(4UUI_1)}(2) \setminus P_{f(3UAS_1)}(2)|=51\). 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 1:11000000101111101111110110100011100110100001011010110011111010011001110010110101011110111010111110100101100101101001111000100010001001100100001111000111001000110011110010000111101101110010110010001101101110011100001000100100111001000010101001001101011010000001000100001110110111110000000100111111001010001000100111000111100010110001110010011011111110010000010101110000111110011001001001001011011010111000011111011001011011100011111001100101101111001010100011101110001011100000
Pair \(Z_2\) Length of longest common subsequence
3UAS_1,4UUI_1 154 4
3UAS_1,7YZJ_1 175 4
4UUI_1,7YZJ_1 179 3

Newick tree

 
[
	7YZJ_1:92.02,
	[
		3UAS_1:77,4UUI_1:77
	]:15.02
]

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 \((0.85)(0.8)(\frac{780 }{\log_{20} 780}-\frac{304}{\log_{20}304})=130.\)
Status Protein1 Protein2 d d1/2
Query variables 3UAS_1 4UUI_1 159 130.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]