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Parikh vectors
1SJH_1 3UFH_1 2BGP_1 Letter Amino acid
8 55 3 K Lycine
17 57 15 V Valine
8 20 3 R Arginine
7 78 13 G Glycine
6 36 8 H Histidine
13 44 6 P Proline
6 72 15 S Serine
3 18 3 Q Glutamine
15 57 12 L Leucine
3 10 3 M Methionine
8 36 4 I Isoleucine
4 4 4 W Tryptophan
4 29 6 Y Tyrosine
9 44 8 N Asparagine
12 50 3 D Aspartic acid
19 41 6 E Glutamic acid
12 75 12 T Threonine
9 96 14 A Alanine
2 26 0 C Cysteine
15 26 2 F Phenylalanine 

1SJH_1|Chain A|HLA class II histocompatibility antigen, DR alpha chain|Homo sapiens (9606)
>3UFH_1|Chain A|UndA|Shewanella sp. HRCR_06 (1043181)
>2BGP_1|Chain A|ENDO-B1,4-MANNANASE 5C|CELLVIBRIO JAPONICUS (155077)
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 \(w=f(c)\)
1SJH , Knot 85 180 0.81 40 140 176 
EEHVIIQAEFYLNPDQSGEFMFDFDGDEIFHVDMAKKETVWRLEEFGRFASFEAQGALANIAVDKANLEIMTKRSNYTPITNVPPEVTVLTNSPVELREPNVLICFIDKFTPPVVNVTWLRNGKPVTTGVSETVFLPREDHLFRKFHYLPFLPSTEDVYDCRVEHWGLDEPLLKHWEFDA
3UFH , Knot 320 874 0.82 40 283 758 
MSKKLLSVLFGASLAALALSPTAFAADQGGSDGKDGEDGKPGPVGLDISQATTLKATLEDVKIDNGTVSVDIVLTNANGVPVTGLEQYAQINAIGLGIAKLTPESGKGYKTPQWVSYINSVKAADPARSLANYSYTDGKDSAGNPITKEVKFTPGDAIQANIESSCKTTCLTVVDSGVYRYTFQTNLSTLPAIEGLDLTYDPTLIHRITLELQTDGSKDAKLVNSHIDFLPSDNFRVAKETETRTVVDLEANCIKCHSTNYSDTSSTAKPLALHGGRRIGIANCQVCHTSYSKDPETGSPLDMGAMVHAIHKGTYAMVGYSGTAYDFSGTMAKAAAESGYPQYREGKDVSERVTLPVSIGNCQSCHSTDDKGPVDAASFKHHKGLACASCHMSGFNPVDNSEWLTPPEGQKDRGFVGNYFHYYATPEIDGIPGVNLVHVFQNGGCASCHAEQGEEGSAKYHLAKANATKLLRTEYAYKLENGTFDVAKGELTFTVNWHSDVAPHQDPKVKEFWVSLTAFNGTEYTMGPRPSNGTLGRSENRISVNLAKVETNANLTAVPNGSKVTYTLTGIKAVIGTSSVPYKQIVSIGKGFMDGKLLICANSAELDPTMDAAIDCSNTEAPIYEVIVGSNKASFSADASNVTARSIVISEAKCANCHGEKADFSASHALTHAADKPDNSCGTCHSAVPNTAVALADGSCVACHNGAPAHSKKPFERGFDFKVMIHQIHADTRSVRRLTTDAATFPENPANCAACHDKGQLSLATLGNKPAFLASTGEYSPTVAACASCHATTATDSAVIGHFETNGGVYNAAAGTYTPGSETCATCHGEGKSFGVDKVHPVKYKGELKLEGKPIPNPLLGLDSTRTGHHHHHH
2BGP , Knot 68 140 0.80 38 103 133 
MAVPEGNSWTYTAASASITAPAQLVGNVGELQGAGSAVIWNVDVPVTGEYRINLTWSSPYSSKVNTLVMDGTALSYAFAEATVPVTYVQTKTLSAGNHSFGVRVGSSDWGYMNVHSLKLELLGGLTIRSPANLEHHHHHH

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 \(c\).

\(|P_{f(1SJH_1)}(2) \setminus P_{f(3UFH_1)}(2)|=25\), \(|P_{f(3UFH_1)}(2) \setminus P_{f(1SJH_1)}(2)|=168\). 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 1:000111010101010001011101010011010110000110100110110101011110111001010110000000110011101011000110100101110110010111101011001011001100011110000110010011111000010000100111001110010101
Pair \(Z_2\) Length of longest common subsequence
1SJH_1,3UFH_1 193 4
1SJH_1,2BGP_1 155 4
3UFH_1,2BGP_1 212 6

Newick tree

 
[
	3UFH_1:10.15,
	[
		1SJH_1:77.5,2BGP_1:77.5
	]:30.65
]

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 \((0.85)(0.8)(\frac{1054 }{\log_{20} 1054}-\frac{180}{\log_{20}180})=237.\)
Status Protein1 Protein2 d d1/2
Query variables 1SJH_1 3UFH_1 296 177
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]