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Parikh vectors
5XGC_1 1OCO_1 3HTQ_1 Letter Amino acid
15 8 10 R Arginine
23 19 25 N Asparagine
75 59 27 L Leucine
16 34 2 M Methionine
27 6 9 Q Glutamine
26 47 23 G Glycine
14 17 9 H Histidine
34 30 28 S Serine
51 40 19 A Alanine
23 16 12 D Aspartic acid
10 1 9 C Cysteine
42 9 17 E Glutamic acid
10 28 17 P Proline
2 17 7 W Tryptophan
2 19 15 Y Tyrosine
42 38 17 V Valine
30 37 18 I Isoleucine
35 9 19 K Lycine
11 42 10 F Phenylalanine
15 38 31 T Threonine

5XGC_1|Chain A|Rap1 GTPase-GDP dissociation stimulator 1|Homo sapiens (9606)
>1OCO_1|Chains A, N|CYTOCHROME C OXIDASE|Bos taurus (9913)
>3HTQ_1|Chain A|Hemagglutinin|Influenza A virus (A/WDK/JX/12416/2005(H1N1)) (666057)
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 \(w=f(c)\)
5XGC , Knot 200 503 0.82 40 227 461
GPLGSQSSCKAKVANIIAEVAKNEFMRIPCVDAGLISPLVQLLNSKDQEVLLQTGRALGNICYDSHSLQAQLINMGVIPTLVKLLGIHCQNAALTEMCLVAFGNLAELESSKEQFASTNIAEELVKLFKKQIEHDKREMIFEVLAPLAENDAIKLQLVEAGLVECLLEIVQQKVDSDKEDDITELKTGSDLMVLLLLGDESMQKLFEGGKGSVFQRVLSWIPSNNHQLQLAGALAIANFARNDANCIHMVDNGIVEKLMDLLDRHVEDGNVTVQHAALSALRNLAIPVINKAKMLSAGVTEAVLKFLKSEMPPVQFKLLGTLRMLIDAQAEAAEQLGKNVKLVERLVEWCEAKDHAGVMGESNRLLSALIRHSKSKDVIKTIVQSGGIKHLVTMATSEHVIMQNEALVALALIAALELGTAEKDLESAKLVQILHRLLADERSAPEIKYNSMVLICALMGSECLHKEVQDLAFLDVVSKLRSHENKSVAQQASLTEQRLTVES
1OCO , Knot 204 514 0.82 40 236 478
MFINRWLFSTNHKDIGTLYLLFGAWAGMVGTALSLLIRAELGQPGTLLGDDQIYNVVVTAHAFVMIFFMVMPIMIGGFGNWLVPLMIGAPDMAFPRMNNMSFWLLPPSFLLLLASSMVEAGAGTGWTVYPPLAGNLAHAGASVDLTIFSLHLAGVSSILGAINFITTIINMKPPAMSQYQTPLFVWSVMITAVLLLLSLPVLAAGITMLLTDRNLNTTFFDPAGGGDPILYQHLFWFFGHPEVYILILPGFGMISHIVTYYSGKKEPFGYMGMVWAMMSIGFLGFIVWAHHMFTVGMDVDTRAYFTSATMIIAIPTGVKVFSWLATLHGGNIKWSPAMMWALGFIFLFTVGGLTGIVLANSSLDIVLHDTYYVVAHFHYVLSMGAVFAIMGGFVHWFPLFSGYTLNDTWAKIHFAIMFVGVNMTFFPQHFLGLSGMPRRYSDYPDAYTMWNTISSMGSFISLTAVMLMVFIIWEAFASKREVLTVDLTTTNLEWLNGCPPPYHTFEEPTYVNLK
3HTQ , Knot 141 324 0.83 40 201 313
DTICIGYHANNSTDTVDTVLEKNVTVTHSVNLLEDNHNGKLCKLNGIAPLQLGKCNVAGWLLGNPECDLLLTANSWSYIIETSNSENGTCYPGEFIDYEELREQLSSVSSFERFEIFPKASSWPNHETTKGVTAACSYFGASSFYRNLLWITKKGTSYPKLSKSYTNNKGKEVLVLWGVHHPPTTSEQQTLYQNTDAYVSVGSSKYNRRFTPEIAARPKVRGQAGRMNYYWTLLDQGDTITFEATGNLIAPWYAFALNKGSDSGIITSDAPVHNCDTKCQTPHGAINSTLPFQNVHPITIGECPKYVKSTKLRMATGLRNIPSI

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 \(c\).

\(|P_{f(5XGC_1)}(2) \setminus P_{f(1OCO_1)}(2)|=88\), \(|P_{f(1OCO_1)}(2) \setminus P_{f(5XGC_1)}(2)|=97\). 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 1:11110000001011011101100011011010111101110110000001110010111010000001010110111110110111100001110010111110110100000011000110011011000100000011101111110001101011011110011011000100000001001001001111111100010011011010110011011100000101111111101100010010110011100110110001001010100111011001111110010110111001110110001111010111010111010101100110010110011010010001111100001101110000000110011001110011011000011100011111111111011010001001011011001110000110100001111011110001000100111101100100000001100101000010100
Pair \(Z_2\) Length of longest common subsequence
5XGC_1,1OCO_1 185 5
5XGC_1,3HTQ_1 174 4
1OCO_1,3HTQ_1 179 4

Newick tree

 
[
	1OCO_1:92.31,
	[
		5XGC_1:87,3HTQ_1:87
	]:5.31
]

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 \((0.85)(0.8)(\frac{1017 }{\log_{20} 1017}-\frac{503}{\log_{20}503})=134.\)
Status Protein1 Protein2 d d1/2
Query variables 5XGC_1 1OCO_1 174 171
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]