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Parikh vectors
5XXX_1 3ESS_1 7PCV_1 Letter Amino acid
13 15 3 H Histidine
22 11 6 S Serine
16 11 5 N Asparagine
27 5 6 D Aspartic acid
12 0 5 C Cysteine
33 15 4 V Valine
36 23 6 A Alanine
35 21 7 G Glycine
26 13 5 I Isoleucine
9 4 7 M Methionine
20 12 5 P Proline
30 18 6 T Threonine
15 7 4 Q Glutamine
31 26 9 E Glutamic acid
31 12 8 L Leucine
4 3 2 W Tryptophan
18 11 2 Y Tyrosine
21 13 10 R Arginine
19 6 11 K Lycine
20 4 8 F Phenylalanine 

5XXX_1|Chains A, C, E, G, I, K, M, O, Q|Tubulin alpha-1A chain|Sus scrofa (9823)
>3ESS_1|Chain A|Cholix toxin|Vibrio cholerae (666)
>7PCV_1|Chains A, B|RNA-binding protein 5|Homo sapiens (9606)
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 \(w=f(c)\)
5XXX , Knot 187 438 0.86 40 252 418 
RECISIHVGQAGVQIGNACWELYCLEHGIQPDGQMPSDKTIGGGDDSFNTFFSETGAGKHVPRAVFVDLEPTVIDEVRTGTYRQLFHPEQLITGKEDAANNYARGHYTIGKEIIDLVLDRIRKLADQCTGLQGFSVFHSFGGGTGSGFTSLLMERLSVDYGKKSKLEFSIYPAPQVSTAVVEPYNSILTTHTTLEHSDCAFMVDNEAIYDICRRNLDIERPTYTNLNRLIGQIVSSITASLRFDGALNVDLTEFQTNLVPYPRGHFPLATYAPVISAEKAYHEQLSVAEITNACFEPANQMVKCDPRHGKYMACCLLYRGDVVPKDVNAAIATIKTKRTIQFVDWCPTGFKVGINYEPPTVVPGGDLAKVQRAVCMLSNTTAIAEAWARLDHKFDLMYAKRAFVHWYVGEGMEEGEFSEAREDMAALEKDYEEVGVDS
3ESS , Knot 103 230 0.81 38 151 222 
MGSSHHHHHHSSGENLYFQGSHMAVITPQGVTNWTYQELEATHQALTREGYVFVGYHGTNHVAAQTIVNRIAPVPRGNNTENEEKWGGLYVATHAEVAHGYARIKEGTGEYGLPTRAERDARGVMLRVYIPRASLERFYRTNTPLENAEEHITQVIGHSLPLRNEAFTGPESAGGEDETVIGWDMAIHAVAIPSTIPGNAYEELAIDEEAVAKEQSISTKPPYKERKDEL
7PCV , Knot 63 119 0.84 40 103 117 
MGERESKTIMLRGLPTTITESDIREMMESFEGPQPADVRLMKRKTGVSRGFAFVEFYHLQDATSWMEANQKKLVIQGKHIAMHYSNPRPKFEDWLCNKCCLNNFRKRLKCFRCGADKFD

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 \(c\).

\(|P_{f(5XXX_1)}(2) \setminus P_{f(3ESS_1)}(2)|=140\), \(|P_{f(3ESS_1)}(2) \setminus P_{f(5XXX_1)}(2)|=39\). 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 1:000101011011101101010100100110101011000011110001001100011100110111101010110010010000110100110100011000101000110011011100100110000110110110011110101100111001010010000101010111010011101000110000010000011110001100100001010010000100111011001010101011101010010001110101011110011110100100001011010010101100110001001001100110010111001011110100000101101010110111000110111110110100110110000111011101000101101001110101101100101001000111100000011100
Pair \(Z_2\) Length of longest common subsequence
5XXX_1,3ESS_1 179 3
5XXX_1,7PCV_1 209 3
3ESS_1,7PCV_1 186 4

Newick tree

 
[
	7PCV_1:10.86,
	[
		5XXX_1:89.5,3ESS_1:89.5
	]:12.36
]

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 \((0.85)(0.8)(\frac{668 }{\log_{20} 668}-\frac{230}{\log_{20}230})=123.\)
Status Protein1 Protein2 d d1/2
Query variables 5XXX_1 3ESS_1 163 119.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]