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Parikh vectors
6ZOF_1 5MDM_1 3FPU_1 Letter Amino acid
97 22 3 V Valine
89 35 10 G Glycine
103 35 5 L Leucine
43 28 5 K Lycine
53 15 2 F Phenylalanine
67 30 8 T Threonine
106 20 4 A Alanine
47 23 9 D Aspartic acid
2 12 8 C Cysteine
51 23 7 E Glutamic acid
17 15 6 H Histidine
9 12 1 W Tryptophan
27 17 5 Y Tyrosine
38 14 6 R Arginine
39 12 7 N Asparagine
72 23 3 I Isoleucine
42 6 1 M Methionine
62 37 2 S Serine
48 12 2 Q Glutamine
45 28 6 P Proline

6ZOF_1|Chains A, B, C|Multidrug efflux pump subunit AcrB|Escherichia coli K-12 (83333)
>5MDM_1|Chains A, B[auth C], C[auth E], D[auth F]|Glycoprotein|Chandipura virus (11272)
>3FPU_1|Chain A|Evasin-1|Rhipicephalus sanguineus (34632)
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 \(w=f(c)\)
6ZOF , Knot 376 1057 0.82 40 302 940
MPNFFIDRPIFAWVIAIIIMLAGGLAILKLPVAQYPTIAPPAVTISASYPGADAKTVQDTVTQVIEQNMNGIDNLMYMSSNSDSTGTVQITLTFESGTDADIAQVQVQNKLQLAMPLLPQEVQQQGVSVEKSSSSFLMVVGVINTDGTMTQEDISDYVAANMKDAISRTSGVGDVQLFGSQYAMRIWMNPNELNKFQLTPVDVITAIKAQNAQVAAGQLGGTPPVKGQQLNASIIAQTRLTSTEEFGKILLKVNQDGSRVLLRDVAKIELGGENYDIIAEFNGQPASGLGIKLATGANALDTAAAIRAELAKMEPFFPSGLKIVYPYDTTPFVKISIHEVVKTLVEAIILVFLVMYLFLQNFRATLIPTIAVPVVLLGTAAVLAAFGFSINTLTMFGMVLAIGLLVDDAIVVVENVERVMAEEGLPPKEATRKSMGQIQGALVGIAMVLSAVFVPMAFFGGSTGAIYRQFSITIVSAMALSVLVALILTPALCATMLKPIAKGDHGEGKKGFFGWFNRMFEKSTHHYTDSVGGILRSTGRYLVLYLIIVVGMAYLFVRLPSSFLPDEDQGVFMTMVQLPAGATQERTQKVLNEVTHYYLTKEKNNVESVFAVNGFGFAGRGQNTGIAFVSLKDWADRPGEENKVEAITMRATRAFSQIKDAMVFAFNLPAIVELGTATGFDFELIDQAGLGHEKLTQARNQLLAEAAKHPDMLTSVRPNGLEDTPQFKIDIDQEKAQALGVSINDINTTLGAAWGGSYVNDFIDRGRVKKVYVMSEAKYRMLPDDIGDWYVRAADGQMVPFSAFSSSRWEYGSPRLERYNGLPSMEILGQAAPGKSTGEAMELMEQLASKLPTGVGYDWTGMSYQERLSGNQAPSLYAISLIVVFLCLAALYESWSIPFSVMLVVPLGVIGALLAATFRGLTNDVYFQVGLLTTIGLSAKNAILIVEFAKDLMDKEGKGLIEATLDAVRMRLRPILMTSLAFILGVMPLVISTGAGSGAQNAVGTGVMGGMVTATVLAIFFVPVFFVVVRRRFSRKNEDIEHSHTVDHHLEHHHHHH
5MDM , Knot 176 419 0.84 40 233 398
YLSIAFPENTKLDWKPVTKNTRYCPMGGEWFLEPGLQEESFLSSTPIGATPSKSDGFLCHAAKWVTTCDFRWYGPKYITHSIHNIKPTRSDCDTALASYKSGTLVSLGFPPESCGYASVTDSEFLVIMITPHHVGVDDYRGHWVDPLFVGGECDQSYCDTIHNSSVWIPADQTKKNICGQSFTPLTVTVAYDKTKEIAAGGIVFKSKYHSHMEGARTCRLSYCGRNGIKFPNGEWVSLDVKTRIQEKHLLPLFKECPAGTEVRSTLQSDGAQVLTSEIQRILDYSLCQNTWDKVERKEPLSPLDLSYLASKSPGKGLAYTVINGTLSFAHTRYVRMWIDGPVLKEPKGKRESPSGISSDIWTQWFKYGDMEIGPNGLLKTAGGYKFPWHLIGMGIVDNELHELSEANPLDHPQLPHAQS
3FPU , Knot 50 100 0.76 40 80 94
EDDEDYGDLGGCPFLVAENKTGYPTIVACKQDCNGTTETAPNGTRCFSIGDEGLRRMTANLPYDCPLGQCSNGDCIPKETYEVCYRRNWRDKKNHHHHHH

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 \(c\).

\(|P_{f(6ZOF_1)}(2) \setminus P_{f(5MDM_1)}(2)|=114\), \(|P_{f(5MDM_1)}(2) \setminus P_{f(6ZOF_1)}(2)|=45\). 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 1:1101110011111111111111111111011110010111111010100111010010001001100010110011010000000101010101001001011010100010111111100100011010000001111111100010100001000111010011000011101011100011011101001001010110110110100101111011101110100101011100010000011011101000100111001101011100001110101011011110110110110011110101101011110110110100001110101001100110111111111011100101011101111111110111111111010010111111111111001111100100111001111001000011010111111111101111111111100111000101011011110111111101110101101110100101001111110011000000000011111000100111011111111011101100111000011110110111110000000110010000100000010011110111111010001111101001100110000101101010011001001111110111110110101101011001111000100100011101100101100101011000101010100001011110100100011111110010011001010010110010001110011010101101011110110000100101010000111010111011110001011011001100110111001011000001010011010110111111011110001011101111111111111111010110001010111100111010011111011001100010111010101101010111100111111111111001110110011101111111010111111111111110001000000100000100010000000
Pair \(Z_2\) Length of longest common subsequence
6ZOF_1,5MDM_1 159 4
6ZOF_1,3FPU_1 260 6
5MDM_1,3FPU_1 223 3

Newick tree

 
[
	3FPU_1:13.09,
	[
		6ZOF_1:79.5,5MDM_1:79.5
	]:52.59
]

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 \((0.85)(0.8)(\frac{1476 }{\log_{20} 1476}-\frac{419}{\log_{20}419})=270.\)
Status Protein1 Protein2 d d1/2
Query variables 6ZOF_1 5MDM_1 342 238
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]