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Parikh vectors
6ZZI_1 5KCY_1 4NPA_1 Letter Amino acid
12 14 26 N Asparagine
12 22 37 E Glutamic acid
6 9 16 H Histidine
12 27 28 P Proline
13 20 24 S Serine
11 19 25 R Arginine
19 22 32 I Isoleucine
14 23 21 K Lycine
0 6 2 W Tryptophan
21 24 26 V Valine
3 18 15 Y Tyrosine
15 27 19 D Aspartic acid
0 6 5 C Cysteine
9 14 15 Q Glutamine
15 30 34 G Glycine
8 17 14 F Phenylalanine
22 31 37 A Alanine
24 56 47 L Leucine
4 8 18 M Methionine
21 26 15 T Threonine 

6ZZI_1|Chains A, B, C, D, E, F|Dihydrolipoyllysine-residue acetyltransferase component of pyruvate dehydrogenase complex|Corynebacterium glutamicum (strain ATCC 13032 / DSM 20300 / JCM 1318 / LMG 3730 / NCIMB 10025) (196627)
>5KCY_1|Chains A, B|aromatic prenyltransferase|Aspergillus terreus (33178)
>4NPA_1|Chains A, B, C, D|Putative uncharacterized protein|Vibrio cholerae O1 biovar El Tor (243277)
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 \(w=f(c)\)
6ZZI , Knot 106 241 0.80 36 162 230 
GSLRGTTQKVNRIREITAMKTVEALQISAQLTQLHEVDMTRVAELRKKNKPAFIEKHGVNLTYLPFFVKAVVEALVSHPNVNASFNAKTKEMTYHSSVNLSIAVDTPAGLLTPVIHDAQDLSIPEIAKAIVDLADRSRNNKLKPNDLSGGTFTITNIGSEGALSDTPILVPPQAGILGTGAIVKRPVVITEDGIDSIAIRQMVFLPLTYDHQVVDGADAGRFLTTIKDRLETANFEGDLQL
5KCY , Knot 177 419 0.85 40 231 400 
GPLGSRPWQILSQALGFPNYDQELWWQNTAETLNRVLEQCDYSVHLQYKYLAFYHKYILPSLGPFRRPGVEPEYISGLSHGGHPLEISVKIDKSKTICRLGLQAIGPLAGTARDPLNSFGDRELLKNLATLLPHVDLRLFDHFNAQVGLDRAQCAVATTKLIKESHNIVCTSLDLKDGEVIPKVYFSTIPKGLVTETPLFDLTFAAIEQMEVYHKDAPLRTALSSLKDFLRPRVPTDASITPPLTGLIGVDCIDPMLSRLKVYLATFRMDLSLIRDYWTLGGLLTDAGTMKGLEMVETLAKTLKLGDEACETLDAERLPFGINYAMKPGTAELAPPQIYFPLLGINDGFIADALVEFFQYMGWEDQANRYKDELKAKFPNVDISQTKNVHRWLGVAYSETKGPSMNIYYDVVAGNVARV
4NPA , Knot 190 456 0.85 40 246 439 
MSLIIQVSPAGSMDLLSQLEVERLKKTASSDLYQLYRNCSLAVLNSGSHTDNSKELLDKYKNFDITVMRRERGIKLELANPPEHAFVDGQIIKGIQEHLFSVLRDIVYVNMHLADSQRLNLTNATHITNLVFGILRNAGALIPGATPNLVVCWGGHSINEVEYQYTREVGHELGLRELNICTGCGPGAMEGPMKGAAVGHAKQRYSEYRYLGLTEPSIIAAEPPNPIVNELVIMPDIEKRLEAFVRMAHGIIIFPGGPGTAEELLYILGIMMHPENADQPMPIVLTGPKQSEAYFRSLDKFITDTLGEAARKHYSIAIDNPAEAARIMSNAMPLVRQHRKDKEDAYSFNWSLKIEPEFQLPFEPNHESMANLDLHLNQRPEVLAANLRRAFSGVVAGNVKAEGIREIERHGPFEMHGDPVLMKKMDQLLNDFVAQNRMKLPGGSAYEPCYKIVTEG

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 \(c\).

\(|P_{f(6ZZI_1)}(2) \setminus P_{f(5KCY_1)}(2)|=46\), \(|P_{f(5KCY_1)}(2) \setminus P_{f(6ZZI_1)}(2)|=115\). 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 1:1010100001001001011001011010101001001010011010000011110001101001111101110111001010101010000100000101011100111110111001001011011011101100000001010010110101001100111000111111011111011110011110001100111001111110000011011011011001000100101010101
Pair \(Z_2\) Length of longest common subsequence
6ZZI_1,5KCY_1 161 4
6ZZI_1,4NPA_1 158 4
5KCY_1,4NPA_1 159 6

Newick tree

 
[
	5KCY_1:80.33,
	[
		6ZZI_1:79,4NPA_1:79
	]:1.33
]

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 \((0.85)(0.8)(\frac{660 }{\log_{20} 660}-\frac{241}{\log_{20}241})=117.\)
Status Protein1 Protein2 d d1/2
Query variables 6ZZI_1 5KCY_1 149 115
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]