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Parikh vectors
4ENA_1 6LSM_1 3EYB_1 Letter Amino acid
0 16 5 N Asparagine
0 27 16 D Aspartic acid
17 36 11 G Glycine
0 23 10 S Serine
0 18 3 Y Tyrosine
0 16 9 Q Glutamine
0 37 16 E Glutamic acid
0 19 14 K Lycine
0 20 9 F Phenylalanine
0 20 12 P Proline
0 10 8 M Methionine
0 29 8 T Threonine
0 34 15 V Valine
11 36 17 A Alanine
0 21 11 R Arginine
16 12 4 C Cysteine
0 13 6 H Histidine
0 27 13 I Isoleucine
0 32 30 L Leucine
0 4 2 W Tryptophan 

4ENA_1|Chain A|Fluoride riboswitch|Thermotoga petrophila (93929)
>6LSM_1|Chains A, C|Tubulin alpha-1B chain|Sus scrofa (9823)
>3EYB_1|Chains A, B, C, D|Nuclear hormone receptor RXR|Branchiostoma floridae (7739)
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 \(w=f(c)\)
4ENA , Knot 18 52 0.45 8 14 29 
GGGCGAUGAGGCCCGCCCAAACUGCCCUGAAAAGGGCUGAUGGCCUCUACUG
6LSM , Knot 192 450 0.87 40 257 425 
MRECISIHVGQAGVQIGNACWELYCLEHGIQPDGQMPSDKTIGGGDDSFNTFFSETGAGKHVPRAVFVDLEPTVIDEVRTGTYRQLFHPEQLITGKEDAANNYARGHYTIGKEIIDLVLDRIRKLADQCTGLQGFLVFHSFGGGTGSGFTSLLMERLSVDYGKKSKLEFSIYPAPQVSTAVVEPYNSILTTHTTLEHSDCAFMVDNEAIYDICRRNLDIERPTYTNLNRLISQIVSSITASLRFDGALNVDLTEFQTNLVPYPRIHFPLATYAPVISAEKAYHEQLSVAEITNACFEPANQMVKCDPRHGKYMACCLLYRGDVVPKDVNAAIATIKTKRSIQFVDWCPTGFKVGINYQPPTVVPGGDLAKVQRAVCMLSNTTAIAEAWARLDHKFDLMYAKRAFVHWYVGEGMEEGEFSEAREDMAALEKDYEEVGVDSVEGEGEEEGEE
3EYB , Knot 103 219 0.84 40 155 207 
DMPVEKIQEAEMAVEPKDGNMVEQPNDPVTNICQAADKQLVTLVEWAKRIPHFSDLPIDDQVILLRAGWNELLIAAFSHRSIDVKDGILLASGLHVHRSSAHQAGVGTIFDRVLTELVAKMRDMKMDKTELGCLRAIVLFNPDAKGLTDPSLVESLREKVYASLEEYCKQQYPEQPGRFAKLLLRLPALRSIGLKCLEHLFFFKLIGDTPIDTFLMEML

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 \(c\).

\(|P_{f(4ENA_1)}(2) \setminus P_{f(6LSM_1)}(2)|=7\), \(|P_{f(6LSM_1)}(2) \setminus P_{f(4ENA_1)}(2)|=250\). 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 1:1110110111100010001110010000111111110011011000001001
Pair \(Z_2\) Length of longest common subsequence
4ENA_1,6LSM_1 257 3
4ENA_1,3EYB_1 163 2
6LSM_1,3EYB_1 192 4

Newick tree

 
[
	6LSM_1:12.22,
	[
		4ENA_1:81.5,3EYB_1:81.5
	]:40.72
]

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 \((0.85)(0.8)(\frac{502 }{\log_{20} 502}-\frac{52}{\log_{20}52})=137.\)
Status Protein1 Protein2 d d1/2
Query variables 4ENA_1 6LSM_1 185 100
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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