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Parikh vectors
4WOR_1 8IJM_1 6IHE_1 Letter Amino acid
3 42 11 F Phenylalanine
5 64 23 S Serine
9 77 27 V Valine
14 82 25 A Alanine
6 42 6 Q Glutamine
12 53 19 E Glutamic acid
4 13 5 H Histidine
1 13 3 W Tryptophan
7 28 8 Y Tyrosine
6 43 10 N Asparagine
5 98 30 I Isoleucine
11 88 17 L Leucine
10 66 13 T Threonine
4 24 11 M Methionine
6 35 14 P Proline
8 50 20 D Aspartic acid
0 13 0 C Cysteine
10 63 24 G Glycine
23 53 28 K Lycine
5 39 9 R Arginine 

4WOR_1|Chain A|Thermonuclease|Staphylococcus aureus (1280)
>8IJM_1|Chain A|Sodium/potassium-transporting ATPase subunit alpha|Rattus (10114)
>6IHE_1|Chains A, B|Malate dehydrogenase (NAD)|Metallosphaera sedula (43687)
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 \(w=f(c)\)
4WOR , Knot 71 149 0.79 38 110 141 
ATSTKKLHKEPATLIKAIDGDTVKLMYKGQPMTFRLLLVDTPETKHPKKGVEKYGPEASAFTKKMVENAKKIEVEFDKGQRTDKYGRGLAYIYADGKMVNEALVRQGLAKVAYVYKPNNTHEQHLRKSEAQAKKEKLNIWSEDNADSGQ
8IJM , Knot 365 986 0.85 40 318 886 
GMDLDDHRLSNTELEQKYGTNIIQGLSSVRATELLARDGPNTLTPPKQTPEIIKFLKQMVGGFSILLWIGAALCWIAFVIQYVNNSASLDNVYLGAILVLVVILTGIFAYYQEAKSTNIMASFSKMIPQQALVIRDAEKKVISAEQLVVGDVVEIKGGDQIPADIRLVFSQGCKVDNSSLTGESEPQARSTEFTHENPLETKNIGFYSTTCLEGTATGIVINTGDRTIIGRIASLASGVGSEKTPIAIEIEHFVHIVAGVAVSIDIIFFITAVCMKYYVLDAIIFLISIIVANVPEGLLATVTVTLSLTAKRMAKKNCLVKNLEAVETLGSTSIICSDKTGTLTQNRMTVAHLWFDNQIFVADTSENQTKQAFDQSSGTWASLSKIITLCNRAEFRPGQESVPIMKRTVVGDASETALLKFSEVILGDVMGIRKRNHKVAEIPFNSTNKFQLSIHETEDPNNKRFLVVMKGAPERILEKCSTIMINGQEQPLDKSSADSFHTAYMELGGLGERVLGFCHLYLPAEQFPQSYIFDVDSVNFPTSNFCFVGLLSMIDPPRSTVPDAVSKCRSAGIKVIMVTGDHPITAKAIAKSVGIISANNETVEDIAKRRNIAVEQVNKREAKAAVVTGMELKDMTPEQLDELLTNYQEIVFARTSPQQKLIIVEGCQRQDAIVAVTGDGVNDSPALKKADIGIAMGIAGSDAAKNAADMVLLDDNFASIVTGVEEGRLIFDNLKKTIAYTLTANIAELCPFLIYIVAGLPLPIGTITILFIDLGTDIIPSIALAYEKAESDIMNRKPRHKKKDRLVNTQLAIYSYLHIGLMQALGGFLVYFTVYAQQGFWPTSLINLRVAWETDDINDLEDSYGQEWTRYQRKYLEWTGSTAFFVAIMIQQIADLIIRKTRRNSIFQQGLFRNKVIWVGIASQVIVALILSYGLGSVPALSFTMLRVQYWFVAVPHAILIWVYDEMRKLFIRLYPGSWWDKNMYY
6IHE , Knot 130 303 0.81 38 187 290 
AKVGFIGAGKIGQTIAYSALVSGAVDEAVIYDIIPELPDKFEHELRHAFATKGIKANVLGTNSLDDVSGMDIVVISAGKPRKPGMSRRDLFVDNAKIMIDLAQKLPSKNPGAIYLMVANPVDMMASVFMKYSKQFTISAGDQVETMRMRSFIAKKLKIPVTSVDGFVGGEHGEDAVVLWSTVKIKGKPVDEFNINKDEVSDYVKKIPGEIIRVIGGTTWGPGTIIADIIKSIAFSENRVMSIATPKEYEKEIIHVSAPTVVGSSIGPSLESLLDEKDRWHLNSAMKDFYEAYKENLKQLEQAT

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 \(c\).

\(|P_{f(4WOR_1)}(2) \setminus P_{f(8IJM_1)}(2)|=10\), \(|P_{f(8IJM_1)}(2) \setminus P_{f(4WOR_1)}(2)|=218\). 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 1:10000010001101101101001011001011010111100100001001100011010110001100100101010010000001011101010101100111001110110100100000001000010100001011000010010
Pair \(Z_2\) Length of longest common subsequence
4WOR_1,8IJM_1 228 3
4WOR_1,6IHE_1 161 3
8IJM_1,6IHE_1 169 4

Newick tree

 
[
	8IJM_1:10.13,
	[
		4WOR_1:80.5,6IHE_1:80.5
	]:25.63
]

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 \((0.85)(0.8)(\frac{1135 }{\log_{20} 1135}-\frac{149}{\log_{20}149})=268.\)
Status Protein1 Protein2 d d1/2
Query variables 4WOR_1 8IJM_1 339 193.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]