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Parikh vectors
1EJR_1 6HHG_1 6IPA_1 Letter Amino acid
47 27 13 V Valine
37 24 13 D Aspartic acid
8 6 5 C Cysteine
17 36 27 K Lycine
14 25 4 F Phenylalanine
23 18 10 S Serine
7 7 1 W Tryptophan
24 28 6 R Arginine
19 15 5 Q Glutamine
55 30 15 G Glycine
44 18 12 I Isoleucine
41 27 5 T Threonine
9 17 4 Y Tyrosine
29 44 18 E Glutamic acid
21 12 6 H Histidine
41 41 21 L Leucine
69 26 10 A Alanine
18 13 18 N Asparagine
16 14 3 M Methionine
28 18 8 P Proline 

1EJR_1|Chain A[auth C]|UREASE ALPHA SUBUNIT|Klebsiella aerogenes (548)
>6HHG_1|Chain A|RAC-alpha serine/threonine-protein kinase|Homo sapiens (9606)
>6IPA_1|Chains A, B, C, D|aminoacyl-tRNA synthetase-interacting multifunctional protein p43|Plasmodium vivax (5855)
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 \(w=f(c)\)
1EJR , Knot 222 567 0.82 40 250 522 
MSNISRQAYADMFGPTVGDKVRLADTELWIEVEDDLTTYGEEVKFGGGKVIRDGMGQGQMLAADCVDLVLTNALIVDHWGIVKADIGVKDGRIFAIGKAGNPDIQPNVTIPIGAATEVIAAEGKIVTAGGIDTHIHWICPQQAEEALVSGVTTMVGGGTGPAAGTHATTCTPGPWYISRMLQAADSLPVNIGLLGKGNVSQPDALREQVAAGVIGLKIHEAWGATPAAIDCALTVADEMDIQVALHSDTLNESGFVEDTLAAIGGRTIHTFHTEGAGGGHAPDIITACAHPNILPSSTNPTLPYTLNTIDEHLDMLMVCHHLDPDIAEDVAFAESRIRRETIAAEDVLHDLGAFSLTSSDSQAMGRVGEVILRTWQVAHRMKVQRGALAEETGDNDNFRVKRYIAKYTINPALTHGIAHEVGSIEVGKLADLVVWSPAFFGVKPATVIKGGMIAIAPMGDINASIPTPQPVHYRPMFGALGSARHHCRLTFLSQAAAANGVAERLNLRSAIAVVKGCRTVQKADMVHNSLQPNITVDAQTYEVRVDGELITSEPADVLPMAQRYFLF
6HHG , Knot 188 446 0.85 40 240 426 
GSDVAIVKEGWLHKRGEYIKTWRPRYFLLKNDGTFIGYKERPQDVDQREAPLNNFSVAQCQLMKTERPRPNTFIIRCLQWTTVIERTFHVETPEEREEWTTAIQTVADGLKKQAAAEMDFRSGSPSDNSGAEEMEVSLAKPKHRVTMNEFEYLKLLGKGTFGKVILVKEKATGRYYAMKILKKEVIVAKDEVAHTLTENRVLQNSRHPFLTALKYSFQTHDRLCFVMEYANGGELFFHLSRERVFSEDRARFYGAEIVSALDYLHSEKNVVYRDLKLENLMLDKDGHIKITDFGLCKEGIKDGATMKTFCGTPEYLAPEVLEDNDYGRAVDWWGLGVVMYEMMCGRLPFYNQDHEKLFELILMEEIRFPRTLGPEAKSLLSGLLKKDPKQRLGGGSEDAKEIMQHRFFAGIVWQHVYEKKLSPPFKPQVTSETDTRYFDEEFTAQM
6IPA , Knot 91 204 0.79 40 138 193 
GAMADKNAKNENGDNEKGKKKNNAQNKNAQKKKVEEPKNLDDITRLNIIVGYVEEVEIHPDADTLYCLKINVGEEKSRDICSGLRLKKNSEDLLHKYVLVLANLKEKSLRGRKSHGMVLCGSFGEQIELLAPPDGVNVGERIICENMDVNKLPDKTLSFDKEKNPFFHIQPHLLVKNGVAHYKDAKWLSSKGEITCPLEQGTIS

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 \(c\).

\(|P_{f(1EJR_1)}(2) \setminus P_{f(6HHG_1)}(2)|=89\), \(|P_{f(6HHG_1)}(2) \setminus P_{f(1EJR_1)}(2)|=79\). 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 1:100100010101111011001011000111010001000100101111011001110101111001011100111100111101011100101111101101010101011111100111101011011110001011010010011101100111110111110010000111101001101100111011111010100101100011111111010011110111100110110010101110000100011100011111100100100011111011011010101011100001011001001000101111000101011001111000100001110011001111010000001110110111001011001010011110001000010100011000101110011100110101101101111011111101101101111111111010101101011000111111101000001011001111011100101001111101000100101100010101010100001010101100011011111000111
Pair \(Z_2\) Length of longest common subsequence
1EJR_1,6HHG_1 168 5
1EJR_1,6IPA_1 188 4
6HHG_1,6IPA_1 186 3

Newick tree

 
[
	6IPA_1:96.46,
	[
		1EJR_1:84,6HHG_1:84
	]:12.46
]

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 \((0.85)(0.8)(\frac{1013 }{\log_{20} 1013}-\frac{446}{\log_{20}446})=149.\)
Status Protein1 Protein2 d d1/2
Query variables 1EJR_1 6HHG_1 190 171.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]