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Parikh vectors
5WDP_1 7DJK_1 4UAL_1 Letter Amino acid
11 56 28 G Glycine
3 11 13 H Histidine
8 51 27 S Serine
12 45 21 A Alanine
4 35 15 N Asparagine
14 48 31 D Aspartic acid
11 41 18 T Threonine
0 6 7 W Tryptophan
9 21 16 Y Tyrosine
15 49 28 V Valine
3 5 9 C Cysteine
13 38 37 E Glutamic acid
11 32 21 I Isoleucine
10 64 45 L Leucine
8 45 22 K Lycine
3 34 15 P Proline
11 28 21 R Arginine
11 23 11 Q Glutamine
4 7 13 M Methionine
5 28 21 F Phenylalanine 

5WDP_1|Chain A|GTPase HRas|Homo sapiens (9606)
>7DJK_1|Chain A|Protein SUPPRESSOR OF QUENCHING 1, chloroplastic|Arabidopsis thaliana (3702)
>4UAL_1|Chain A|Serine/threonine-protein kinase MRCK beta|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 \(w=f(c)\)
5WDP , Knot 80 166 0.82 38 130 164 
MTEYKLVVVGAGGVGKSALTIQLIQNHFVDEYDPTIEDSYRKQVVIDGETCLLDILDTAGQEEYSAMRDQYMRTGEGFLCVFAINNTKSFEDIHQYREQIKRVKDSDDVPMVLVGNKCDAAARTVESRQAQDLARSYGIPYIETSAKTRQGVEDAFYTLVREIRQH
7DJK , Knot 256 667 0.83 40 266 619 
MGQTATTVPEFPSKLDWLNTAPLQFRRDLKGKVVILDFWTYSSINSMHVLPDLEFLEKKYKDMPFTVVGVHSAKFDNEKDLDAIRNAVLRYDISHPVVNDGDMYMWRELGINSWPTFAVVSPNGKVIAQIAGEGHRKDLDDVVAAALTYYGGKNVLDSTPLPTRLEKDNDPRLATSPLKFPGKLAIDTLNNRLFISDSNHNRIIVTDLEGNFIVQIGSSGEEGFQDGSFEDAAFNRPQGLAYNAKKNLLYVADTENHALREIDFVNERVQTLAGNGTKGSDYQGGRKGTKQLLNSPWDVCFEPVNEKVYIAMAGQHQIWEYSVLDGITRVFSGNGYERNLNGSTPQTTSFAQPSGISLGPDLKEAYIADSESSSIRALDLQTGGSRLLAGGDPYFSENLFKFGDNDGVGAEVLLQHPLGVLCANDGQIYLTDSYNHKIKKLDPVTKRVVTLAGTGKAGFKDGKVKGAQLSEPAGLAITENGRLFVADTNNSLIRYIDLNKGEDSEILTLELKGVQPPTPKAKSLKRLRKRASADTKIVKVDSVTSREGDLNLKISLPDGYHFSKEARSKFVVDVEPENAVAIDPTEGTLSPEGSTMLHFIQSSTSASVGKISCKVYYCKEDEVCLYQSVQFEVPFKVESELSASPTITFTVTPRAPDAGGLQLQGTR
4UAL , Knot 176 419 0.84 40 235 398 
GGSSAKVRLKKLEQLLLDGPWRNESALSVETLLDVLVCLYTECSHSALRRDKYVAEFLEWAKPFTQLVKEMQLHREDFEIIKVIGRGAFGEVAVVKMKNTERIYAMKILNKWEMLKRAETACFREERDVLVNGDCQWITALHYAFQDENHLYLVMDYYVGGDLLTLLSKFEDKLPEDMARFYIGEMVLAIDSIHQLHYVHRDIKPDNVLLDVNGHIRLADFGSCLKMNDDGTVQSSVAVGTPDYISPEILQAMEDGMGKYGPECDWWSLGVCMYEMLYGETPFYAESLVETYGKIMNHEERFQFPSHVTDVSEEAKDLIQRLICSRERRLGQNGIEDFKKHAFFEGLNWENIRNLEAPYIPDVSSPSDTSNFDVDDDVLRNTEILPPGSHTGFSGLHLPFIGFTFTTESCFSDRGSLKS

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 \(c\).

\(|P_{f(5WDP_1)}(2) \setminus P_{f(7DJK_1)}(2)|=27\), \(|P_{f(7DJK_1)}(2) \setminus P_{f(5WDP_1)}(2)|=163\). 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 1:1000011111111110011010110001100001010000000111010001101100110000011000010010111011110000010010000001001000001111111000011100100001001100011101000100001100110011001000
Pair \(Z_2\) Length of longest common subsequence
5WDP_1,7DJK_1 190 4
5WDP_1,4UAL_1 197 3
7DJK_1,4UAL_1 139 4

Newick tree

 
[
	5WDP_1:10.28,
	[
		7DJK_1:69.5,4UAL_1:69.5
	]:34.78
]

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 \((0.85)(0.8)(\frac{833 }{\log_{20} 833}-\frac{166}{\log_{20}166})=186.\)
Status Protein1 Protein2 d d1/2
Query variables 5WDP_1 7DJK_1 233 144.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]