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Parikh vectors
8ZUE_1 6FMK_1 3RSN_1 Letter Amino acid
36 7 13 K Lycine
7 3 6 M Methionine
19 8 10 P Proline
46 5 12 S Serine
25 9 14 T Threonine
15 0 10 N Asparagine
40 8 11 E Glutamic acid
26 1 5 H Histidine
7 0 4 W Tryptophan
27 7 6 A Alanine
10 2 7 C Cysteine
18 4 9 Q Glutamine
14 1 5 Y Tyrosine
24 5 10 G Glycine
41 5 7 I Isoleucine
21 7 9 F Phenylalanine
40 5 6 V Valine
39 7 6 R Arginine
54 10 15 D Aspartic acid
56 10 12 L Leucine 

8ZUE_1|Chain A|Anti-bacteriophage protein A|Escherichia coli K-12 (83333)
>6FMK_1|Chains A, D, G, J|Elongin-B|Homo sapiens (9606)
>3RSN_1|Chain A|Set1/Ash2 histone methyltransferase complex subunit ASH2|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 \(w=f(c)\)
8ZUE , Knot 224 565 0.83 40 262 531 
MGSSHHHHHHSSGLVPRGSHSDEVDAHMESNDSGGVAAKHGFLFQDCVAAYHVTRMLRDKTIRSVRCEVTDDIDIVSDGYIDFVQVKSTGKTRWNISDIVQNSKGADKKTIPCSSILHKSMQCESDLSLGRRYSIVTEEKVNKTLEYLTISPNARLDKPGRQELIDDLNKRTDNFLTDSGISVSDWIDAATWEVFSSLRELELLGIKNIRLASQDLHGVILSSETVAEDIWCRILDTVTRKGEHSRRIHSADDKSYLRPDLLEWFKQRVEDDQSRSGRKIYVKRDLPHILTPFRAPMASVCAKRKGQVLHQQYSLKKYRYKHIADNVCQWLDEVFLRPKEMSDIHKLTFIEKRERLKNSVFKSLHDVSEFLGRVLLHATIRQHHESQPIPCMLYVEKAGAEKILENVHIVRRDPEGDQLWIGFSELVTDINIAVRLPEIRDQLYEDISDCIDTARKKILDIKDDNYLLRHDIDEILDGSQPFDAHLDRFTFVLFVGYDSNLLTEPETPGFEDDLEKETAVLFEKFAADLIEDSPFANLCIHVFIYPAPSLERLTQLVDEKVREVV
6FMK , Knot 55 104 0.81 36 88 101 
MDVFLMIRRHKTTIFTDAKESSTVFELKRIVEGILKRPPDEQRLYKDDQLLDDGKTLGECGFTSQTARPQAPATVGLAFRADDTFEALCIEPFSSPPELPDVMK
3RSN , Knot 85 177 0.82 40 139 171 
SDTQAGSVDEENGRQLGEVELQCGICTKWFTADTFGIDTSSCLPFMTNYSFHCNVCHHSGNTYFLRKQANLKEMCLSALANLTWQSRTQDEHPKTMFSKDKDIIPFIDKYWECMTTRQRPGKMTWPNNIVKTMSKERDVFLVKEHPDPGSKDPEEDYPKFGLLDQDLSNIGPAYDNQ

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 \(c\).

\(|P_{f(8ZUE_1)}(2) \setminus P_{f(6FMK_1)}(2)|=192\), \(|P_{f(6FMK_1)}(2) \setminus P_{f(8ZUE_1)}(2)|=18\). 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 1:1100000000001111010000010101000001111100111100011100100110000100100010001011001010110100010001010011000011000011000110001000001011000011000010001001010101010011000110010000001100011010011011010110010010111100101100010111100001100110011001000100000100100000101011011000100000001001010001101101101111010100010110000010000000110010011001110100100100101100000100011001001001110111010100000001110110100111001100101100010100111110011001011101101000100010001001000110100000110001001101001101010010111111000011001001110001000011110011101100011101010111011101001001100010011
Pair \(Z_2\) Length of longest common subsequence
8ZUE_1,6FMK_1 210 3
8ZUE_1,3RSN_1 203 3
6FMK_1,3RSN_1 145 4

Newick tree

 
[
	8ZUE_1:11.65,
	[
		3RSN_1:72.5,6FMK_1:72.5
	]:39.15
]

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 \((0.85)(0.8)(\frac{669 }{\log_{20} 669}-\frac{104}{\log_{20}104})=163.\)
Status Protein1 Protein2 d d1/2
Query variables 8ZUE_1 6FMK_1 208 122
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]