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Parikh vectors
3WWX_1 6CNY_1 2TPT_1 Letter Amino acid
2 4 2 C Cysteine
22 7 23 S Serine
37 7 56 A Alanine
20 9 24 R Arginine
13 7 11 Q Glutamine
9 6 16 F Phenylalanine
6 0 2 W Tryptophan
26 12 34 V Valine
12 10 12 N Asparagine
21 8 33 D Aspartic acid
9 1 6 H Histidine
12 8 27 I Isoleucine
26 10 37 L Leucine
17 8 22 K Lycine
16 9 24 E Glutamic acid
38 10 39 G Glycine
25 10 29 T Threonine
10 9 9 Y Tyrosine
6 8 19 M Methionine
22 8 15 P Proline 

3WWX_1|Chain A|S12 family peptidase|Streptomyces sp. 82F2 (690348)
>6CNY_1|Chains A, B, C, D|Vivid PAS protein VVD|Neurospora crassa (5141)
>2TPT_1|Chain A|THYMIDINE PHOSPHORYLASE|Escherichia coli (83333)
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 \(w=f(c)\)
3WWX , Knot 147 349 0.82 40 207 332 
APAKPDHAATQQALEAAVADGVPGAVAQARDGRDRWTGTAGERGGDDRYRVGSITKTFTATVLLQLQAEGRIDLDDPVEKWLPGVVRGNGHDGRKITVRQLLNHTSGIYSYTEDPAFQAKVFGPGFLEHRYDTWTPKQLVAVAMAHEPDFTPGASWNYSNTNFVLAGMVIEKVTGRPYGKAVENRIIKPLKLRATTVPGTRSAMPEPSSPAYSKLSRDVNAPVHDVSTLNPSIAGAAGEMISDSRDLQTFYRALLQGRLLPKSALNEMTTTVQISPEYPNVGYGLGLMKDKLSCGVEVWGHGGGIHGSSSLAQVTRDGGHSLAGNFNADWAGDSQKVIEAEFCGTAPKK
6CNY , Knot 75 151 0.83 38 120 145 
MHTLYAPGGYDIMGYLIQIMNRPNPQVELGPVDTSCALILCDLKQKDTPIVYASEAFLYMTGYSNAEVLGRNCRFLQSPDGMVKPKSTRKYVDSNTINTMRKAIDRNAEVQVEVVNFKKNGQRFVNFLTMIPVRDETGEYRYSMGFQCETE
2TPT , Knot 185 440 0.85 40 217 414 
LFLAQEIIRKKRDGHALSDEEIRFFINGIRDNTISEGQIAALAMTIFFHDMTMPERVSLTMAMRDSGTVLDWKSLHLNGPIVDKHSTGGVGDVTSLMLGPMVAACGGYIPMISGRGLGHTGGTLDKLESIPGFDIFPDDNRFREIIKDVGVAIIGQTSSLAPADKRFYATRDITATVDSIPLITASILAKKLAEGLDALVMDVKVGSGAFMPTYELSEALAEAIVGVANGAGVRTTALLTDMNQVLASSAGNAVEVREAVQFLTGEYRNPRLFDVTMALCVEMLISGKLAKDDAEARAKLQAVLDNGKAAEVFGRMVAAQKGPTDFVENYAKYLPTAMLTKAVYADTEGFVSEMDTRALGMAVVAMGGGRRQASDTIDYSVGFTDMARLGDQVDGQRPLAVIHAKDENNWQEAAKAVKAAIKLADKAPESTPTVYRRISE

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 \(c\).

\(|P_{f(3WWX_1)}(2) \setminus P_{f(6CNY_1)}(2)|=133\), \(|P_{f(6CNY_1)}(2) \setminus P_{f(3WWX_1)}(2)|=46\). 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 1:1110100110001101111011111110100100010101100110000011010001010111010101010100110011111101010010010100110000110000001110101111111000000101001111111001010111010000001111111100101010101100011011010100111000111010011000100010111001001010111111011000001001001110101110011001000101010010110111110001001101110111101000110100011001110101011100001101010101100
Pair \(Z_2\) Length of longest common subsequence
3WWX_1,6CNY_1 179 3
3WWX_1,2TPT_1 144 4
6CNY_1,2TPT_1 165 5

Newick tree

 
[
	6CNY_1:90.27,
	[
		3WWX_1:72,2TPT_1:72
	]:18.27
]

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 \((0.85)(0.8)(\frac{500 }{\log_{20} 500}-\frac{151}{\log_{20}151})=102.\)
Status Protein1 Protein2 d d1/2
Query variables 3WWX_1 6CNY_1 130 92
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]