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Parikh vectors
1BDS_1 1HDI_1 4PKF_1 Letter Amino acid
0 21 71 E Glutamic acid
2 19 53 I Isoleucine
2 35 61 L Leucine
1 16 33 F Phenylalanine
5 17 41 P Proline
2 15 39 T Threonine
0 37 43 V Valine
2 57 76 A Alanine
1 5 26 H Histidine
2 24 59 S Serine
4 4 33 Y Tyrosine
2 10 46 R Arginine
0 7 33 Q Glutamine
7 40 61 G Glycine
2 39 62 K Lycine
2 4 15 W Tryptophan
2 21 35 N Asparagine
6 7 13 C Cysteine
0 13 26 M Methionine
1 22 52 D Aspartic acid 

1BDS_1|Chain A|BDS-I|Anemonia sulcata (6108)
>1HDI_1|Chain A|PHOSPHOGLYCERATE KINASE|SUS SCROFA (9823)
>4PKF_1|Chain A|TutD|Thauera aromatica (59405)
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 \(w=f(c)\)
1BDS , Knot 26 43 0.75 32 38 41 
AAPCFCSGKPGRGDLWILRGTCPGGYGYTSNCYKWPNICCYPH
1HDI , Knot 163 413 0.79 40 208 379 
NKLTLDKLNVKGKRVVMRVDFNVPMAAAQITNNARIKAAVPSIKFCLDDGAKSVVLMSHLGRPDGSPMPDKYSLQPVAAELKSALGKAVLFLKDCVGPAVEKACADPAAGSVILLENLRFHVEEEGKGKDASGNKAAGEPAKIKAFRASLSALGDVYVNDAFGTAHRAHSSMVGVNLPKKAGAFLMKKELNYFAAAAESPERPFLAILGGAKVADKIQLINNMLDKVNEMIIGGGMAFTFLKVLNNMEIGTSLFDEAGKKIVKNLMSKAAANGVKITLPVDFVTADKFDEQAKIGQATVASGIPAGWMGLDCGPKSSAKYSEAVARAKQIVWNGPVGVFEWEAFAQGTKALMDEVVKATSRGCITIIGGGDTATCCAKWNTEDNVSHVSTGGGASLELLEGKVLPGVDALSNV
4PKF , Knot 334 878 0.86 40 320 820 
MNDIVSAKVLEYKGKKLNFTPEDPAEETIPADELHEHLQKPSTARTKRLKERCRWKHASAGEFIEKSVTAGIERMRYLTEAHKASEGKPEAIRRALGLANVLNKSTLVLQEDEFIVGYHAEDPNMFPLYPELSHMAVQDYLRSDYSPQPADEAAAINEYWKPHSLQSKCQPYFDPADLGRMYQVSSMEAPSFASGYNSIVPPYETVLEDGLLARIKLAEKHIAEAQADMSTFPWNGTKGLDNIAKIDNWKAMVIACKAVISWARRQGRLCKIVAENFETDPKRQAELLEIADICQRIPAEPCKGLKDAMQAKFFTFLICHAIERYASGYAQKEDTLLWPYYKASVVDKKFQPMSHMDAVELVEMERLKISEHGAGKSRAYREIFPGSNDLFILTVGGTNAKGEDACNDMTDAILEAAKRIRTAEPSIVFRYSKKNREKTLRWVFECIRDGLGYPSIKHDEIGTEQMKEYAKFSLNGNGATDEEAHNWVNVLCMSPGIHGRRKTQKTRSEGGGSIFPAKLLEISLNDGYDWSYADMQLGPKTGDLSSLKSFEDVWEAFRKQYQYAINLCISTKDVSRYFEQRFLQMPFVSAIDDGCMELGMDACALSEQPNGWHNPITTIVAANSLVAIKKLVFEEKKYTLEQLSQALKANWEGFEEMRVDFKRAPKWGNDDDYADGIITRFYEEIIGGEMRKITNYSGGPVMPTGQAVGLYMEVGSRTGPTPDGRFGGEAADDGGISPYMGTDKKGPTAVLRSVSKVQKNQKGNLLNQRLSVPIMRSKHGFEIWNSYIKTWHDLNIDHVQFNVVSTDEMRAAQREPEKHHDLIVRVSGYSARFVDIPTYGQNTIIARQEQDFSASDLEFLNVEISGTGSGSSHHHHHH

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 \(c\).

\(|P_{f(1BDS_1)}(2) \setminus P_{f(1HDI_1)}(2)|=21\), \(|P_{f(1HDI_1)}(2) \setminus P_{f(1BDS_1)}(2)|=191\). 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 1:1110100101101011110100111010000000110100010
Pair \(Z_2\) Length of longest common subsequence
1BDS_1,1HDI_1 212 2
1BDS_1,4PKF_1 304 3
1HDI_1,4PKF_1 164 5

Newick tree

 
[
	1BDS_1:14.70,
	[
		1HDI_1:82,4PKF_1:82
	]:61.70
]

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 \((0.85)(0.8)(\frac{456 }{\log_{20} 456}-\frac{43}{\log_{20}43})=128.\)
Status Protein1 Protein2 d d1/2
Query variables 1BDS_1 1HDI_1 157 87
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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