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Parikh vectors
4PLI_1 5ZKP_1 4BPK_1 Letter Amino acid
3 16 7 N Asparagine
5 26 7 D Aspartic acid
10 23 13 E Glutamic acid
4 10 2 H Histidine
0 7 4 M Methionine
4 34 9 F Phenylalanine
2 37 14 A Alanine
3 37 4 I Isoleucine
0 14 3 P Proline
2 20 10 R Arginine
6 49 14 L Leucine
10 18 5 K Lycine
0 25 5 T Threonine
4 40 12 V Valine
2 15 1 C Cysteine
4 13 10 Q Glutamine
5 30 13 G Glycine
4 26 13 S Serine
4 9 5 W Tryptophan
3 17 6 Y Tyrosine 

4PLI_1|Chains A, B|At1g02740|Arabidopsis thaliana (3702)
>5ZKP_1|Chain A|Platelet-activating factor receptor,Flavodoxin,Platelet-activating factor receptor|Homo sapiens (9606)
>4BPK_1|Chains A, B|BCL-2-LIKE PROTEIN 1|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 \(w=f(c)\)
4PLI , Knot 42 75 0.80 34 61 71 
GSHFEEGERVLAKHSDCFYEAKVLKVEFKDNEWKYFVHYIGWNKSWDEWIRLDCLLKHSDENIEKQKEQGLKQQG
5ZKP , Knot 194 466 0.85 40 239 450 
GAPEPHDSSHMDSEFRYTLFPIVYSIIFVLGVIANGYVLWVFARLYPCKKFNEIKIFMVNLTMADMLFLITLPLWIVYYQNQGNWILPKFLCNVAGCLFFINTYCSVAFLGVITYNRYQAVTRPIKTAQANTRKRGISLSLVIWVAIVGAASYFLILDSTNTVPDSAGSGDVTRCFEHYEKGSVPVLIIHIFIVFSFFLVFLIILFCNLVIIRTLLMQAKALIVYGSTTGNTEYTAETIARELADAGYEVDSRDAASVEAGGLFEGFDLVLLGCSTWGDDSIELQDDFIPLFDSLEETGAQGRKVACFGCGDSSWEYFCGAVDAIEEKLKNLGAEIVQDGLRIDGDPRAARDDIVGWAHDVRGAIAEVKRRDLWMACTVLAVFIICFVPHHVVQLPWTLAELGFQDSKFHQAINDAHQVTLCLLSTNCVLNPVIYCFLTKKFRKHLTEKFYSMRSSRKEFLEVLFQ
4BPK , Knot 78 157 0.83 40 125 155 
GPLGMSQSNRELVVDFLSYKLSQKGYSWSQMAAVKQALREAGDEFELRYRRAFSDLTSQLHITPGTAYQSFEQVVNELFRDGVNWGRIVAFFSFGGALCVESVDKEMQVLVSRIAAWMATYLNDHLEPWIQENGGWDTFVELYGNNAAAESRKGQER

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 \(c\).

\(|P_{f(4PLI_1)}(2) \setminus P_{f(5ZKP_1)}(2)|=27\), \(|P_{f(5ZKP_1)}(2) \setminus P_{f(4PLI_1)}(2)|=205\). 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 1:100100100111000001001011010100001001100111000100110100110000001000000110001
Pair \(Z_2\) Length of longest common subsequence
4PLI_1,5ZKP_1 232 3
4PLI_1,4BPK_1 148 2
5ZKP_1,4BPK_1 192 3

Newick tree

 
[
	5ZKP_1:11.27,
	[
		4PLI_1:74,4BPK_1:74
	]:41.27
]

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 \((0.85)(0.8)(\frac{541 }{\log_{20} 541}-\frac{75}{\log_{20}75})=139.\)
Status Protein1 Protein2 d d1/2
Query variables 4PLI_1 5ZKP_1 182 105
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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