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Parikh vectors
3BLY_1 3COZ_1 5CWS_1 Letter Amino acid
40 50 21 A Alanine
9 0 5 C Cysteine
14 5 2 H Histidine
36 9 8 F Phenylalanine
44 21 12 P Proline
28 24 7 R Arginine
31 6 5 N Asparagine
29 9 16 Q Glutamine
41 28 12 G Glycine
26 9 7 I Isoleucine
41 31 16 V Valine
34 10 8 E Glutamic acid
17 7 3 M Methionine
45 17 18 T Threonine
39 21 9 D Aspartic acid
43 36 21 L Leucine
31 3 14 K Lycine
50 8 42 S Serine
10 0 2 W Tryptophan
15 7 9 Y Tyrosine 

3BLY_1|Chain A|Pyranose oxidase|Trametes multicolor (230624)
>3COZ_1|Chains A, B|Pantothenate synthetase|Mycobacterium tuberculosis (1773)
>5CWS_1|Chains A, G|sAB-158 Fab Light Chain|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 \(w=f(c)\)
3BLY , Knot 246 623 0.84 40 285 585 
MSTSSSDPFFNFAKSSFRSAAAQKASASSLPPLPGPDKKVPGMDIKYDVVIVGSGPIGCTYARELVGAGYKVAMFDIGEIDSGLKIGAHKKNTVEYQKNIDKFVNVIQGQLMSVSVPVNTLVVDTLSPTSWQASTFFVRNGSNPEQDPLRNLSGQAVTRVVGGMSTHWTCATPRFDREQRPLLVKDDADADDAEWDRLYTKAESYFQTGTDQFKESIRHNLVLNKLTEEYKGQRDFQQIPLAATRRSPTFVEWSSANTVFDLQNRPNTDAPEERFNLFPAVACERVVRNALNSEIESLHIHDLISGDRFEIKADVYVLTAGAVHNTQLLVNSGFGQLGRPNPANPPELLPSLGSYITEQSLVFCQTVMSTELIDSVKSDMTIRGTPGELTYSVTYTPGASTNKHPDWWNEKVKNHMMQHQEDPLPIPFEDPEPQVTTLFQPSHPWHTQIHRDAFSYGAVQQSIDSRLIVDWRFFGRTEPKEENKLWFSDKITDAYNMPQPTFDFRFPAGRTSKEAEDMMTDMCVMSAKIGGFLPGSWPQFMKPGLVLHLGGTHRMGFDEKEDNCCVNTDSRVFGFKNLFLGGCGNIPTAYGANPTLTAMSLAIKSCEYIKQNFTPSPFTSEAQ
3COZ , Knot 128 301 0.81 36 153 278 
AMAIPAFHPGELNVYSAPGDVADVSRALRLTGRRVMLVPTMGALHEGHLALVRAAKRVPGSVVVVSIFVNPMQFGAGGDLDAYPRTPDDDLAQLRAEGVEIAFTPTTAAMYPDGLRTTVQPGPLAAELEGGPRPTHFAGVLTVVLKLLQIVRPDRVFFGEKDYQQLVLIRQLVADFNLDVAVVGVPTVREADGLAMSSRNRYLDPAQRAAAVALSAALTAAAHAATAGAQAALDAARAVLDAAPGVAVDYLELRDIGLGPMPLNGSGRLLVAARLGTTRLLDNIAIEIGTFAGTDRPDGYR
5CWS , Knot 108 237 0.83 40 153 224 
MKSLLPTAAAGLLLLAAQPAMASDIQMTQSPSSLSASVGDRVTITCRASQSVSSAVAWYQQKPGKAPKLLIYSASSLYSGVPSRFSGSRSGTDFTLTISSLQPEDFATYYCQQSSSSLITFGQGTKVEIKRTVAAPSVFIFPPSDSQLKSGTASVVCLLNNFYPREAKVQWKVDNALQSGNSQESVTEQDSKDSTYSLSSTLTLSKADYEKHKVYACEVTHQGLSSPVTKSFNRGEC

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 \(c\).

\(|P_{f(3BLY_1)}(2) \setminus P_{f(3COZ_1)}(2)|=157\), \(|P_{f(3COZ_1)}(2) \setminus P_{f(3BLY_1)}(2)|=25\). 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 1:10000001110110001001110010100111111100011110100011111011110001001111100111101101001101110000010000010011011010110101110011100101001010011100100100011001010110011111000100101010000011110001010010100100010001001000100010001110010000010001001111100001011010010011010001000110001011111100011001100010010100110100101010101101111000011100111011010110110111011001000011100011000110010001010101101000100011100000101100010001100000111111001010100110100110001000110011100010001110101110001000001110001001001101010101111000001001100101101011111110110110111110111000111000000001000001111001111101011010110101011011100000100010101100010
Pair \(Z_2\) Length of longest common subsequence
3BLY_1,3COZ_1 182 4
3BLY_1,5CWS_1 184 5
3COZ_1,5CWS_1 154 4

Newick tree

 
[
	3BLY_1:95.84,
	[
		3COZ_1:77,5CWS_1:77
	]:18.84
]

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 \((0.85)(0.8)(\frac{924 }{\log_{20} 924}-\frac{301}{\log_{20}301})=168.\)
Status Protein1 Protein2 d d1/2
Query variables 3BLY_1 3COZ_1 218 159.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]