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Parikh vectors
3KMQ_1 9DVR_1 4HOO_1 Letter Amino acid
34 45 25 G Glycine
18 60 18 Y Tyrosine
35 45 13 V Valine
7 11 6 C Cysteine
38 52 22 L Leucine
42 43 25 A Alanine
28 32 17 R Arginine
18 38 18 N Asparagine
31 38 13 D Aspartic acid
32 43 24 E Glutamic acid
18 22 12 H Histidine
25 51 20 I Isoleucine
25 33 18 F Phenylalanine
25 58 14 S Serine
9 31 14 Q Glutamine
25 41 19 K Lycine
14 11 8 M Methionine
21 34 18 P Proline
26 40 18 T Threonine
5 19 8 W Tryptophan 

3KMQ_1|Chain A|3D polymerase|Foot-and-mouth disease virus - type C (12116)
>9DVR_1|Chains A, B|Antiplasmin-cleaving enzyme FAP, soluble form|Homo sapiens (9606)
>4HOO_1|Chains A, B|Lysine-specific demethylase 4D|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 \(w=f(c)\)
3KMQ , Knot 201 476 0.86 40 252 456 
GLIVDTRDVEERVHVMRKTKLAPTVAHGVFNPEFGPAALSNKDPRLNEGVVLDEVIFSKHKSDTKMSAEDKALFRRCAADYASRLHSVLGTANAPLSIYEAIKGVDGLDAMEPDTAPGLPWALQGKRRGALIDFENGTVGPEVEAALKLMEKREYKFACQTFLKDEIRPMEKVRAGKTRIVDVLPVEHILYTRMMIGRFCAQMHSNNGPQIGSAVGCNPDVDWQRFGTHFAQYRNVWDVDYSAFDANHCSDAMNIMFEEVFRTEFGFHPNAEWILKTLVNTEHAYENKRITVEGGMPSGCSATSIINTILNNIYVLYALRRHYEGVELDTYTMISYGDDIVVASDYDLDFEALKPHFKSLGQTITPADKSDKGFVLGHSITDVTFLKRHFHMDYGTGFYKPVMASKTLEAILSFARRGTIQEKLISVAGLAVHSGPDEYRRLFEPFQGLFEIPSYRSLYLRWVNAVCGDAAALEHH
9DVR , Knot 293 747 0.86 40 310 694 
RPSRVHNSEENTMRALTLKDILNGTFSYKTFFPNWISGQEYLHQSADNNIVLYNIETGQSYTILSNRTMKSVNASNYGLSPDRQFVYLESDYSKLWRYSYTATYYIYDLSNGEFVRGNELPRPIQYLCWSPVGSKLAYVYQNNIYLKQRPGDPPFQITFNGRENKIFNGIPDWVYEEEMLATKYALWWSPNGKFLAYAEFNDTDIPVIAYSYYGDEQYPRTINIPYPKAGAKNPVVRIFIIDTTYPAYVGPQEVPVPAMIASSDYYFSWLTWVTDERVCLQWLKRVQNVSVLSICDFREDWQTWDCPKTQEHIEESRTGWAGGFFVSTPVFSYDAISYYKIFSDKDGYKHIHYIKDTVENAIQITSGKWEAINIFRVTQDSLFYSSNEFEEYPGRRNIYRISIGSYPPSKKCVTCHLRKERCQYYTASFSDYAKYYALVCYGPGIPISTLHDGRTDQEIKILEENKELENALKNIQLPKEEIKKLEVDEITLWYKMILPPQFDRSKKYPLLIQVYGGPCSQSVRSVFAVNWISYLASKEGMVIALVDGRGTAFQGDKLLYAVYRKLGVYEVEDQITAVRKFIEMGFIDEKRIAIWGWSYGGYVSSLALASGTGLFKCGIAVAPVSSWEYYASVYTERFMGLPTKDDNLEHYKNSTVMARAEYFRNVDYLLIHGTADDNVHFQNSAQIAKALVNAQVDFQAMWYSDQNHGLSGLSTNHLYTHMTHFLKQCFSLSDTGHHHHHHHHGGQ
4HOO , Knot 148 330 0.86 40 219 320 
QNPNCNIMIFHPTKEEFNDFDKYIAYMESQGAHRAGLAKIIPPKEWKARETYDNISEILIATPLQQVASGRAGVFTQYHKKAAAMTVGEYRHLANSKKYQTPPHQNFEDLERKYWKNRIYNSPIYGADISGSLFDENTKQWNLGHLGTIQDLLEKECGVVIEGVNTPYLYFGMWKTTFAWHTEDMDLYSINYLHLGEPKTWYVVPPEHGQRLERLARELFPGSSRGCGAFLRHKVALISPTVLKENGIPFNRITQEAGEFMVTFPYGYHAGFNHGFNCAEAINFATPRWIDYGKMASQCSCGEARVTFSMDAFVRILQPERYDLWKRGQD

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 \(c\).

\(|P_{f(3KMQ_1)}(2) \setminus P_{f(9DVR_1)}(2)|=46\), \(|P_{f(9DVR_1)}(2) \setminus P_{f(3KMQ_1)}(2)|=104\). 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 1:11110000100010110000111011011101011111100001010011110011100000000101000111000110010010011101011101001101101101101001111111101000111101001011101011101100000011000110001011001011000110111100110001111010101000011011011100101010011001100001101000110100000110111001100011101010111001100001000001010111101001001100110010110110000011010000110010011110000101011010100110010110000011111001001011000101001011001111000101110110010100011011111100110000011011011101100001010110110101111000
Pair \(Z_2\) Length of longest common subsequence
3KMQ_1,9DVR_1 150 4
3KMQ_1,4HOO_1 173 4
9DVR_1,4HOO_1 171 4

Newick tree

 
[
	4HOO_1:89.36,
	[
		3KMQ_1:75,9DVR_1:75
	]:14.36
]

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 \((0.85)(0.8)(\frac{1223 }{\log_{20} 1223}-\frac{476}{\log_{20}476})=193.\)
Status Protein1 Protein2 d d1/2
Query variables 3KMQ_1 9DVR_1 250 203
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]