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Parikh vectors
5XGW_1 4KED_1 9LRC_1 Letter Amino acid
19 4 25 P Proline
14 6 28 R Arginine
19 13 32 D Aspartic acid
7 1 11 C Cysteine
41 10 52 G Glycine
40 11 71 L Leucine
15 11 39 K Lycine
8 1 14 M Methionine
18 16 73 S Serine
33 8 35 T Threonine
4 1 12 W Tryptophan
24 12 52 V Valine
17 6 28 N Asparagine
30 11 34 A Alanine
14 1 18 Q Glutamine
27 10 29 E Glutamic acid
13 1 28 H Histidine
29 5 39 I Isoleucine
17 8 45 F Phenylalanine
6 7 27 Y Tyrosine 

5XGW_1|Chains A, B|Isoaspartyl dipeptidase|Colwellia psychrerythraea (strain 34H / ATCC BAA-681) (167879)
>4KED_1|Chains A, B|Cofilin|Saccharomyces cerevisiae (559292)
>9LRC_1|Chain A[auth R]|Histamine H4 receptor,Genome polyprotein|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 \(w=f(c)\)
5XGW , Knot 165 395 0.83 40 220 378 
MNDSQTMLILLCNVNIYAPNPLGIKDVLIAGNKIAAIYDHGQGEITIPKQWPVKVINFDGAILTPGFIDSHAHITGGGGEAGFATQVPPVGLTEFTHAGVTTVVGLLGTDDTTRSTENLLSRVYGLREEGLSAYCWTGGYHFPLTTITGSAKSDIAFLEPVIGIGEFAISDHRSSQPTFEEVIRLASETHVAGLITGKAGVIHFHLGDGERRLELIERAIRETELPARVFNPTHVNRNKPLFEDSCKLLSKGCHIDLTAFPAGTAQPGWEACDAIEMAVERQLPLEQITLSSDGGGCLPCFDPQGELQHMDFGRASTLGETLVATLNKGLSLETVLPMLTSNVANILRFKNKGQIAVGFDADLLVMNEKYEITDVMAQGVWHKQNNQTMIKGTFE
4KED , Knot 70 143 0.81 40 109 138 
GSRSGVAVADESLTAFNDLKLGKKYKFILFGLNDAKTEIVVKETSTDPSYDAFLEKLPENDCLYAIYDFEYEINGNEGKRSKIVFFTWSPDTAPVRSKMVYASSKDALRRALNGVSTDVQGTDFSEVSYDSVLEDVSRGAGSH
9LRC , Knot 266 692 0.83 40 292 620 
MKTIIALSYIFCLVFADYKDDDDKGSGGSSGGGSPDTNSTINLSLSTRVTLAFFMSLVAFAIMLGNALVILAFVVDKNLRHRSSYFFLNLAISDFFVGVISIPLYIPHTLFEWDFGKEICVFWLTTDYLLCTASVYNIVLISYDRYLSVSNAVSYRTQHTGVLKIVTLMVAVWVLAFLVNGPMILVSESWKDEGSECEPGFFSEWYILAITSFLEFVIPVILVAYFNMNIYWSLWKRDHLSRCQSHPGLTAVSSNICGHSFRGRLSSRRSLSASTEVPASFHSERQRRKSSLMFSSRTKMNSNTIASKMGSFSQSDSVALHQREHVELLRARRLAKSLAILLGVFAVCWAPYSLFTIVLSFYSSATGPKSVWYRIAFWLQWFNSFVNPLLYPLCHKRFQKAFLKIFCIKKQPLPSQHSRSVSSGGSGGGSGGSSGGGSLEVLFQGPVSKGEELFTGVVPILVELDGDVNGHKFSVSGEGEGDATYGKLTLKFICTTGKLPVPWPTLVTTLTYGVQCFSRYPDHMKQHDFFKSAMPEGYVQERTIFFKDDGNYKTRAEVKFEGDTLVNRIELKGIDFKEDGNILGHKLEYNYNSHNVYIMADKQKNGIKVNFKIRHNIEDGSVQLADHYQQNTPIGDGPVLLPDNHYLSTQSKLSKDPNEKRDHMVLLEFVTAAGITLGMDELYKHHHHHHHH

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 \(c\).

\(|P_{f(5XGW_1)}(2) \setminus P_{f(4KED_1)}(2)|=143\), \(|P_{f(4KED_1)}(2) \setminus P_{f(5XGW_1)}(2)|=32\). 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 1:10000011111001010110111100111110011110001010101100111011010111101111000101011110111100111111001001110011111100000000001100101100011010010110011100101010001111011111101110000000101001101100001111101011110101101000101100110000111011010010000111000001100100101011111010111010011011100011100101000111011010101010010110100110011101001101001111100011011010001011111010111100000100111011100000001101010
Pair \(Z_2\) Length of longest common subsequence
5XGW_1,4KED_1 175 4
5XGW_1,9LRC_1 176 4
4KED_1,9LRC_1 207 4

Newick tree

 
[
	9LRC_1:98.75,
	[
		5XGW_1:87.5,4KED_1:87.5
	]:11.25
]

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 \((0.85)(0.8)(\frac{538 }{\log_{20} 538}-\frac{143}{\log_{20}143})=115.\)
Status Protein1 Protein2 d d1/2
Query variables 5XGW_1 4KED_1 147 97.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]