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Parikh vectors
9FPH_1 2BKL_1 7DCM_1 Letter Amino acid
41 37 7 T Threonine
5 35 1 Y Tyrosine
58 58 9 G Glycine
19 24 10 H Histidine
31 39 12 E Glutamic acid
27 17 5 N Asparagine
17 25 8 Q Glutamine
19 1 4 C Cysteine
43 14 8 I Isoleucine
34 44 5 K Lycine
18 11 3 M Methionine
30 38 12 P Proline
25 39 13 S Serine
80 67 20 A Alanine
26 39 15 R Arginine
11 25 2 F Phenylalanine
4 17 3 W Tryptophan
55 64 14 V Valine
33 48 9 D Aspartic acid
60 53 26 L Leucine 

9FPH_1|Chain A[auth X]|Carbon monoxide dehydrogenase 2|Carboxydothermus hydrogenoformans Z-2901 (246194)
>2BKL_1|Chains A, B|PROLYL ENDOPEPTIDASE|MYXOCOCCUS XANTHUS (34)
>7DCM_1|Chain A|Probable apo-citrate lyase phosphoribosyl-dephospho-CoA transferase|Escherichia coli (562)
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 \(w=f(c)\)
9FPH , Knot 245 636 0.83 40 256 566 
MARQNLKSTDRAVQQMLDKAKREGIQTVWDRYEAMKPQCGFGETGLCCRHCLQGPCRINPFGDEPKVGICGATAEVIVARGLDRSIAAGAAGHSGHAKHLAHTLKKAVQGKAASYMIKDRTKLHSIAKRLGIPTEGQKDEDIALEVAKAALADFHEKDTPVLWVTTVLPPSRVKVLSAHGLIPAGIDHEIAEIMHRTSMGCDADAQNLLLGGLRCSLADLAGCYMGTDLADILFGTPAPVVTESNLGVLKADAVNVAVHGHNPVLSDIIVSVSKEMENEARAAGATGINVVGICCTGNEVLMRHGIPACTHSVSQEMAMITGALDAMILDYQCIQPSVATIAECTGTTVITTMEMSKITGATHVNFAEEAAVENAKQILRLAIDTFKRRKGKPVEIPNIKTKVVAGFSTEAIINALSKLNANDPLKPLIDNVVNGNIRGVCLFAGCNNVKVPQDQNFTTIARKLLKQNVLVVATGCGAGALMRHGFMDPANVDELCGDGLKAVLTAIGEANGLGGPLPPVLHMGSCVDNSRAVALVAALANRLGVDLDRLPVVASAAEAMHEKAVAIGTWAVTIGLPTHIGVLPPITGSLPVTQILTSSVKDITGGYFIVELDPETAADKLLAAINERRAGLGLPW
2BKL , Knot 268 695 0.84 40 267 634 
MSYPATRAEQVVDTLHGVQVADPYRWLEDEKAPEVQTWMTAQNAHAREALAKFPGREALAARFKELFYTDSVSTPSRRNGRFFYVRTHKDKEKAILYWRQGESGQEKVLLDPNGWSKDGTVSLGTWAVSWDGKKVAFAQKPNAADEAVLHVIDVDSGEWSKVDVIEGGKYATPKWTPDSKGFYYEWLPTDPSIKVDERPGYTTIRYHTLGTEPSKDTVVHERTGDPTTFLQSDLSRDGKYLFVYILRGWSENDVYWKRPGEKDFRLLVKGVGAKYEVHAWKDRFYVLTDEGAPRQRVFEVDPAKPARASWKEIVPEDSSASLLSVSIVGGHLSLEYLKDATSEVRVATLKGKPVRTVQLPGVGAASNLMGLEDLDDAYYVFTSFTTPRQIYKTSVSTGKSELWAKVDVPMNPEQYQVEQVFYASKDGTKVPMFVVHRKDLKRDGNAPTLLYGYGGFNVNMEANFRSSILPWLDAGGVYAVANLRGGGEYGKAWHDAGRLDKKQNVFDDFHAAAEYLVQQKYTQPKRLAIYGGSNGGLLVGAAMTQRPELYGAVVCAVPLLDMVRYHLFGSGRTWIPEYGTAEKPEDFKTLHAYSPYHHVRPDVRYPALLMMAADHDDRVDPMHARKFVAAVQNSPGNPATALLRIEANAGHGGADQVAKAIESSVDLYSFLFQVLDVQGAQGGVAAQGRHHHHHH
7DCM , Knot 85 186 0.79 40 134 179 
GSHMHLLPELASHHAVSIPELLVSRDERQARQHVWLKRHPVPLVSFTVVAPGPIKDSEVTRRIFNHGVTALRALAAKQGWQIQEQAALVSASGPEGMLSIAAPARDLKLATIELEHSHPLGRLWDIDVLTPEGEILSRRDYSLPPRRCLLCEQSAAVCARGKTHQLTDLLNRMEALLNDVDACNVN

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 \(c\).

\(|P_{f(9FPH_1)}(2) \setminus P_{f(2BKL_1)}(2)|=71\), \(|P_{f(2BKL_1)}(2) \setminus P_{f(9FPH_1)}(2)|=82\). 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 1:110001000001100110010001100110000110100111001100000101100101110010111011010111101100011111110010100110010011010110011000001001100111100100000111011011110100000111110011110010110101111111000110110000110010100111111000110111001100110111101111100001111010110111010011100111010001000101111011011110001001110011110000100011110111011110000101011011000100110010100101100101100111001001101110010000101101101000111110001110110010100110111001101010110111100010110000100110011000111110101111110011101101001010110111011101011111111110110010000111111111001110100111110110110001111101110111100111111101011100110001001011011101010011001111100001111111
Pair \(Z_2\) Length of longest common subsequence
9FPH_1,2BKL_1 153 5
9FPH_1,7DCM_1 184 4
2BKL_1,7DCM_1 197 4

Newick tree

 
[
	7DCM_1:10.79,
	[
		9FPH_1:76.5,2BKL_1:76.5
	]:24.29
]

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 \((0.85)(0.8)(\frac{1331 }{\log_{20} 1331}-\frac{636}{\log_{20}636})=176.\)
Status Protein1 Protein2 d d1/2
Query variables 9FPH_1 2BKL_1 226 215.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]