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Parikh vectors
5UPF_1 5SZL_1 6TVK_1 Letter Amino acid
5 3 10 C Cysteine
15 21 34 Q Glutamine
3 7 23 M Methionine
18 11 30 F Phenylalanine
21 29 29 P Proline
29 29 29 S Serine
28 10 28 Y Tyrosine
30 30 37 D Aspartic acid
14 20 37 R Arginine
26 23 23 N Asparagine
38 31 49 E Glutamic acid
49 14 27 K Lycine
30 30 30 T Threonine
7 0 9 W Tryptophan
38 36 45 V Valine
25 20 51 A Alanine
36 27 63 G Glycine
29 25 35 I Isoleucine
45 42 64 L Leucine
13 19 36 H Histidine 

5UPF_1|Chains A, B|Nicotinamide phosphoribosyltransferase|Homo sapiens (9606)
>5SZL_1|Chains A, B, C, D|PROTOCADHERIN GAMMA A1 EXTRACELLULAR CADHERIN DOMAINS 1-4, Protein Pcdhga1|Mus musculus (10090)
>6TVK_1|Chain A[auth AAA]|Alpha-L-fucosidase|Paenibacillus thiaminolyticus (49283)
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 \(w=f(c)\)
5UPF , Knot 203 499 0.84 40 241 472 
MNPAAEAEFNILLATDSYKVTHYKQYPPNTSKVYSYFECREKKTENSKLRKVKYEETVFYGLQYILNKYLKGKVVTKEKIQEAKDVYKEHFQDDVFNEKGWNYILEKYDGHLPIEIKAVPEGFVIPRGNVLFTVENTDPECYWLTNWIETILVQSWYPITVATNSREQKKILAKYLLETSGNLDGLEYKLHDFGYRGVSSQETAGIGASAHLVNFKGTDTVAGLALIKKYYGTKDPVPGYSVPAAEHSTITAWGKDHEKDAFEHIVTQFSSVPVSVVSDSYDIYNACEKIWGEDLRHLIVSRSTQAPLIIRPDSGNPLDTVLKVLEILGKKFPVTENSKGYKLLPPYLRVIQGDGVDINTLQEIVEGMKQKMWSIENIAFGSGGGLLQKLTRDLLNCSFKCSYVVTNGLGINVFKDPVADPNKRSKKGRLSLHRTPAGNFVTLEEGKGDLEEYGQDLLHTVFKNGKVTKSYSFDEIRKNAQLNIELEAAHHDYKDDDDK
5SZL , Knot 176 427 0.83 38 218 392 
GNIRYSVPEETDKGSFVGSIAKDLGLETRELMERGIRIVSRGRSQLFSLNPRSGSLVTAGRIDREELCAQSTPCVVSFNILMEDEMKLLPIEVEIIDINDNTPQFQLEELELKMSEITTPGTRIPLPLGQDLDVGINSLQSYQLSANPHFSLDVQQGPEGPQQPEMVLQRPLDREKDAVHYLVLTASDGGSPIHSGTLQIHVQVVDVNDNPPAFTKAEYHVSVPENVPLGTRLLKVNATDPDEGANGRVTYSFHKVDHSVVRKFQLDAYTGELSNKEPLDFEEYKVYPMEIQAQDGAGLMARAKVLVTVLDVNDNAPEVGITSVTNTVPENFPPGTTIALISVHDQDADNNGHITCSIPGNLPFKLEKLVDNYYRLVTERTLDREQSSRHNITITATDQGTPPLSTQAHISLLVTDINDHHHHHHHH
6TVK , Knot 270 689 0.85 40 298 642 
MGHHHHHHHHHHSSGHIEGRHGENLYFQGMRYRQVHLDFHTSEHISDIGRNFSKKNFQEMLQLGHVNSITVFAKCHHGWAYFPSATNEIHPRLDFDLLGAQIEAAHEIGVKVPIYISVGFDEKLAWEKPQWLMRDEADRMNWVDSFMKPGYHQFCLNTPYLDLVIEQVQEVVRKYDGDGIFLDIVGERTCYCTTCLKQMQADGLDPHNKEDVIANGRRIYANYTTRIREAIDAIKPGLPVFHNAGHIHQGRRDLMGMNSHLELESLPTGGWGYDHFPLSARYAQPTGFHFLGMTGKFHTFWGEFGGYKHPNALRYETALSLANGARCSIGDQLHPGGQMDRATYELIGKAYAEVEAKEAWCVNAVNLADVALLTVEAAGVQQESGAMYSGKVDMGAVRMLLEGKILFDIVDLESDWSGYKVLILPDSIVMKDTILPKVEAFLAAGGKVLASGRSGLNVELTRQMLPLGFTDSGLNPFRPDYFRPLCDGMANLGEAAYVMYGDGRRIELTDGTELGRREDPYFNRQAFRFCSHQHAPSSEQEGGPGMVESAQGIYIAWNVFEDYATKGSLILKEMVLFALRRLLGEQITLKTTLPAQGVTTLQHQAAERRYINHLLYASPVKRGERVEIIEDMIPLQQVEVQLQLPVTDVKRVYLAPQMTEIEFKASGGDVQFTVPQLECHQMVVVEYNE

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 \(c\).

\(|P_{f(5UPF_1)}(2) \setminus P_{f(5SZL_1)}(2)|=77\), \(|P_{f(5SZL_1)}(2) \setminus P_{f(5UPF_1)}(2)|=54\). 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 1:1011101010111100000100000011000010001000000000001001000001101100110001010110000100100100001000110001100110000101110101110111110101110100001000110011001110010110110000000011100110001010110001001100110000011111010110101000111111100001000111100111100001011100000011001100100111011000001001000111001001110000011111010010110011011011100111000001001111010110101101001001101100011010011110111110010001100010000110011110110011101000000101010001110110100101010001001100110010100000100100010101010110000000000
Pair \(Z_2\) Length of longest common subsequence
5UPF_1,5SZL_1 131 4
5UPF_1,6TVK_1 133 4
5SZL_1,6TVK_1 148 8

Newick tree

 
[
	6TVK_1:71.89,
	[
		5UPF_1:65.5,5SZL_1:65.5
	]:6.39
]

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 \((0.85)(0.8)(\frac{926 }{\log_{20} 926}-\frac{427}{\log_{20}427})=132.\)
Status Protein1 Protein2 d d1/2
Query variables 5UPF_1 5SZL_1 163 150
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]