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Parikh vectors
4DXC_1 7TVF_1 6GCR_1 Letter Amino acid
33 17 14 P Proline
19 13 10 Y Tyrosine
24 18 19 R Arginine
47 21 20 E Glutamic acid
61 39 27 L Leucine
48 22 16 K Lycine
40 23 12 D Aspartic acid
54 27 15 G Glycine
9 6 5 H Histidine
40 9 14 T Threonine
17 13 10 Q Glutamine
20 19 12 F Phenylalanine
12 3 5 W Tryptophan
33 13 16 V Valine
15 5 11 M Methionine
29 18 19 S Serine
67 15 16 A Alanine
29 15 9 N Asparagine
2 13 8 C Cysteine
38 20 18 I Isoleucine 

4DXC_1|Chain A|Maltose-binding periplasmic protein, Beta-lactamase TEM chimera|Escherichia coli (562)
>7TVF_1|Chains A[auth C], B[auth F]|Serine/threonine-protein phosphatase PP1-alpha catalytic subunit|Homo sapiens (9606)
>6GCR_1|Chain A|Focal adhesion kinase 1|Gallus gallus (9031)
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 \(w=f(c)\)
4DXC , Knot 245 637 0.82 40 265 589 
KIEEGKLVIWINGDKGYNGLAEVGKKFEKDTGIKVTVEHPDKLEEKFPQVAATGDGPDIIFWAHDRFGGYAQSGLLAEITPDKAFQDKLYPFTWDAVRYNGKLIAYPIAVEALSLIYNKDLLPNPPKTWEEIPALDKELKAKGKSALMFNLQEPYFTWPLIAADGGYAFKYENGKYDIKDVGVDNAGAKAGLTFLVDLIKNKHMNADTDYSIAEAAFNKGETAMTINGPWAWSNIDTSKVNYGVTVLPTFKGQPSKPFVGVLSAGINAASPNKELAKEFLENYLLTDEGLEAVNKDKPLGAVALKSYEEELAKDPRWFIADKSGAGERGSRGIIAALGPDGKPSRIVVIYTTGSQATMDERNRQIAEIGASLIKHWGSGGGHPETLVKVKDAEDQLGARVGYIELDLNSGKILESFRPEERFPMMSTFKVLLCGAVLSRIDAGQEQLGRRIHYSQNDLVEYSPVTEKHLTDGMTVRELCSAAITMSDNTAANLLLTTIGGPKELTAFLHNMGDHVTRLDRWEPELNEAIPNDERDTTMPVAMATTLRKLLTGELLTLASRQQLIDWMEADKVAGPLLRSALPAGSATMENAQKGEIMPNIPQMSAFWYAVRTAVINAASGRQTVDEALKDAQTRITK
7TVF , Knot 147 329 0.86 40 214 321 
SDSEKLNLDSIIGRLLEVQGSRPGKNVQLTENEIRGLCLKSREIFLSQPILLELEAPLKICGDIHGQYYDLLRLFEYGGFPPESNYLFLGDYVDRGKQSLETICLLLAYKIKYPENFFLLRGNHECASINRIYGFYDECKRRYNIKLWKTFTDCFNCLPIAAIVDEKIFCCHGGLSPDLQSMEQIRRIMRPTDVPDQGLLCDLLWSDPDKDVQGWGENDRGVSFTFGAEVVAKFLHKHDLDLICRAHQVVEDGYEFFAKRQLVTLFSAPNYCGEFDNAGAMMSVDETLMCSFQILKPADKNKGKYGQFSGLNPGGRPITPPRNSAKAKK
6GCR , Knot 125 276 0.84 40 185 266 
STRDYEIQRERIELGRCIGEGQFGDVHQGIYMSPENPAMAVAIKTCKNCTSDSVREKFLQEALTMRQFDHPHIVKLIGVITENPVWIIMELCTLGELRSFLQVRKFSLDLASLILYAYQLSTALAYLESKRFVHRDIAARNVLVSATDCVKLGDFGLSRYMEDSTYYKASKGKLPIKWMAPESINFRRFTSASDVWMFGVCMWEILMHGVKPFQGVKNNDVIGRIENGERLPMPPNCPPTLYSLMTKCWAYDPSRRPRFTELKAQLSTILEEEKLQ

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 \(c\).

\(|P_{f(4DXC_1)}(2) \setminus P_{f(7TVF_1)}(2)|=104\), \(|P_{f(7TVF_1)}(2) \setminus P_{f(4DXC_1)}(2)|=53\). 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 1:0100101111101001001110110010000110101001001000110111010110111110001110100111101010011000101101011000101110111101101100001110110010011110001010100111101001010111111011011000010001001110011101110111011000010100000110111001001101011111001000010011011101010100111111011101101000110011000110001101100001111111000000110010111100011100100111111110101001111000100101000000110111011001101110100110100100011101101010100101100101000111100101110111100101100011001000000110001100001001101001001110100001101110011110010111001100100100101010011100000001111110010011010110110000110110100111111001111101010010010111011010111011001110110100010011001000100
Pair \(Z_2\) Length of longest common subsequence
4DXC_1,7TVF_1 157 5
4DXC_1,6GCR_1 168 4
7TVF_1,6GCR_1 171 3

Newick tree

 
[
	6GCR_1:86.73,
	[
		4DXC_1:78.5,7TVF_1:78.5
	]:8.23
]

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 \((0.85)(0.8)(\frac{966 }{\log_{20} 966}-\frac{329}{\log_{20}329})=170.\)
Status Protein1 Protein2 d d1/2
Query variables 4DXC_1 7TVF_1 213 161
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]