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Parikh vectors
5ERU_1 1ZKD_1 8XRV_1 Letter Amino acid
24 27 48 E Glutamic acid
26 19 52 T Threonine
2 6 14 W Tryptophan
7 7 13 Y Tyrosine
10 11 21 F Phenylalanine
26 42 106 A Alanine
5 1 4 C Cysteine
16 15 23 Q Glutamine
9 19 20 H Histidine
19 19 30 S Serine
23 27 53 R Arginine
29 23 53 D Aspartic acid
40 41 77 L Leucine
32 24 49 P Proline
16 9 12 M Methionine
39 22 84 V Valine
14 3 10 N Asparagine
37 36 73 G Glycine
29 26 17 I Isoleucine
16 10 9 K Lycine 

5ERU_1|Chain A|Gephyrin|Rattus norvegicus (10116)
>1ZKD_1|Chains A, B|DUF185|Rhodopseudomonas palustris (258594)
>8XRV_1|Chains A, B, C, D, E, F|GH3 enzyme CcBgl3B|Cellulosimicrobium (157920)
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 \(w=f(c)\)
5ERU , Knot 171 419 0.82 40 225 395 
MSPFPLTSMDKAFITVLEMTPVLGTEIINYRDGMGRVLAQDVYAKDNLPPFPASVKDGYAVRAADGPGDRFIIGESQAGEQPTQTVMPGQVMRVTTGAPIPCGADAVVQVEDTELIRESDDGTEELEVRILVQARPGQDIRPIGHDIKRGECVLAKGTHMGPSEIGLLATVGVTEVEVNKFPVVAVMSTGNELLNPEDDLLPGKIRDSNRSTLLATIQEHGYPTINLGIVGDNPDDLLNALNEGISRADVIITSGGVSMGEKDYLKQVLDIDLHAQIHFGRVFMKPGLPTTFATLDIDGVRKIIFALPGNPVSAVVTCNLFVVPALRKMQGILDPRPTIIKARLSCDVKLDPRPEYHRCILTWHHQEPLPWAQSTGNQMSSRLMSMRSANGLLMLPPKTEQYVELHKGEVVDVMVIGRL
1ZKD , Knot 160 387 0.82 40 200 356 
MIDQTALATEIKRLIKAAGPMPVWRYMELCLGHPEHGYYVTRDPLGREGDFTTSPEISQMFGELLGLWSASVWKAADEPQTLRLIEIGPGRGTMMADALRALRVLPILYQSLSVHLVEINPVLRQKQQTLLAGIRNIHWHDSFEDVPEGPAVILANEYFDVLPIHQAIKRETGWHERVIEIGASGELVFGVAADPIPGFEALLPPLARLSPPGAVFEWRPDTEILKIASRVRDQGGAALIIDYGHLRSDVGDTFQAIASHSYADPLQHPGRADLTAHVDFDALGRAAESIGARAHGPVTQGAFLKRLGIETRALSLMAKATPQVSEDIAGALQRLTGEGRGAMGSMFKVIGVSDPKIETLVALSDDTDREAERRQGTHGLEHHHHHH
8XRV , Knot 281 768 0.81 40 252 658 
GSHMSTLPYLDPAVPVADRVEDLLARMTLPEKVGQMLQLDARDGVGPAVLEKHAGSLLHTSPENVLAAHELTGRTRLRIPLLLAEDCIHGHSFWVGATIFPTQLGMAATWDPALVEQVAHATAVEVAATGVHWTFSPVLCIARDLRWGRVDETFGEDPFLIGELASAMVRGYQGDGLSDPTGILATAKHFAGYSETQGGRDASEADISQRKLRSWFLPPFERVAREGCATFMLGYQSMDGVPVTVNGWLLDDVLRGEWGYTGTLVTDWDNVGRMVWEQHIQPDYVHASAAAVRAGNDMVMTTPRFFEGALEAVDRGLVEEAAIDAAVRRILTLKFRLGLFEDPRRPDVARQQAVIASAEHAAVNLEVARRSLVLLTNDGTLPFAGGLDRAAGTPDGRALAPAGAPARTIAVVGPNADDDHTQLGDWAGASGQADWLPDGHPREMTTTVLDGFRALAPEGWAVTHARGADILTLAPDPEGELFPDGQPRPQVVVPAAPDDALIAEAVAAARDADLAVAVVGDRIELVGEGRSTATLELVGGQVALLDALVATGTPVVVVVVASKPLVLPPSAHAAAAVVWAANPGMRGGQAVAELVLGLIEPEGRLPISFARHAGQQPTYYNVVRGQHGVRYADLTQSPAFAFGEGLSYTTVEYADLRVLGTEHGPDDVVRAEVTLTNTGSRPVRETVQVYVSDTVTSVTWAEKELKAYRKVDLAPGESATVGLEVPVADCTLVDAHGRRVVEPGEFELRVGPSSREDALLRASFTVAG

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 \(c\).

\(|P_{f(5ERU_1)}(2) \setminus P_{f(1ZKD_1)}(2)|=94\), \(|P_{f(1ZKD_1)}(2) \setminus P_{f(5ERU_1)}(2)|=69\). 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 1:10111100100111011010111100110000111011100101000111111010010110110111001111000110010001111011010011111011011101000011000001000101011101011001011100100100111010011100111110111001010011111110010011010001111010000000111010001010101111100100110110011001011100111011000010011010101010110111011110011010101100111111101101110001111111001011101010110101000101010100000110100001111100010010001101001011111110000010100101101111101
Pair \(Z_2\) Length of longest common subsequence
5ERU_1,1ZKD_1 163 4
5ERU_1,8XRV_1 143 5
1ZKD_1,8XRV_1 144 4

Newick tree

 
[
	1ZKD_1:78.61,
	[
		5ERU_1:71.5,8XRV_1:71.5
	]:7.11
]

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 \((0.85)(0.8)(\frac{806 }{\log_{20} 806}-\frac{387}{\log_{20}387})=113.\)
Status Protein1 Protein2 d d1/2
Query variables 5ERU_1 1ZKD_1 142 135.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]