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Parikh vectors
3DDA_1 7EGV_1 3IEK_1 Letter Amino acid
2 2 1 C Cysteine
27 36 37 E Glutamic acid
26 59 50 G Glycine
26 54 38 V Valine
24 45 16 D Aspartic acid
10 31 12 Q Glutamine
35 42 12 K Lycine
30 37 12 T Threonine
23 11 12 Y Tyrosine
22 34 15 S Serine
19 73 39 A Alanine
12 26 18 H Histidine
29 36 10 I Isoleucine
35 47 64 L Leucine
32 23 17 F Phenylalanine
13 33 33 R Arginine
38 21 5 N Asparagine
7 30 7 M Methionine
18 41 31 P Proline
2 7 2 W Tryptophan 

3DDA_1|Chain A|Botulinum neurotoxin A light chain|Clostridium botulinum (1491)
>7EGV_1|Chains A, B|Acetolactate synthase|Trichoderma harzianum (5544)
>3IEK_1|Chains A, B, C, D|Ribonuclease TTHA0252|Thermus thermophilus (300852)
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 \(w=f(c)\)
3DDA , Knot 172 430 0.80 40 212 395 
MPFVNKQFNYKDPVNGVDIAYIKIPNAGQMQPVKAFKIHNKIWVIPERDTFTNPEEGDLNPPPEAKQVPVSYYDSTYLSTDNEKDNYLKGVTKLFERIYSTDLGRMLLTSIVRGIPFWGGSTIDTELKVIDTNCINVIQPDGSYRSEELNLVIIGPSADIIQFECKSFGHEVLNLTRNGYGSTQYIRFSPDFTFGFEESLEVDTNPLLGAGKFATDPAVTLAHELIHAGHRLYGIAINPNRVFKVNTNAYYEMSGLEVSFEELRTFGGHDAKFIDSLQENEFRLYYYNKFKDIASTLNKAKSIVGTTASLQYMKNVFKEKYLLSEDTSGKFSVDKLKFDKLYKMLTEIYTEDNFVKFFKVLNRKTYLNFDKAVFKINIVPKVNYTIYDGFNLRNTNLAANFNGQNTEINNMNFTKLKNFTGLFEHHHHHH
7EGV , Knot 266 688 0.84 40 275 625 
MHHHHHHSSGLVPRGSGMKETAAAKFERQHMDSPDLGTDDDDKAMADIGSMRPVPSPAFNTADNDRSHVQPLVNPQRPDMDESFIGKSGGEIFHEMMLRQGVKHIFGYPGGAILPVFDAIYNSKHFDFILPRHEQGAGHMAEGYARASGKPGVVVVTSGPGATNVITPMQDALSDGTPMVVFCGQVPTAAIGSDAFQEADVVGISRACTKWNVMVKSVAELPRRINEAFEIATSGRPGPVLVDLPKDVTAGILRRAIPTDTALPTLPSAATRAAKELSTQQLNASIKRAADLINMGKKPVIYAGQGVIQSEGGPELLKELADKASIPVTTTLHGLGAFDELDEKSLHMLGMHGAAYANMAMQQADVIIALGSRFDDRVTGVVSKFAPAARQAAAEGRGGIIHFEIMPKNINKVVQATEAVEGDVGANMKLLIPQVKAKTMEDRKEWFDAIKGWKKKYPLSHYQRAERTGLIKPQTVMEEISNLTADRKDKTYIATGVGQHQMWVAQHFRWRHPRSMITSGGLGTMGYGLPAAIGAKVAQPDALVIDVDGDASFNMTLTELSTAAQFNIGVKVVVLNNEEQGMVTQWQNLFYEDRYSHTHQRNPDFMKLADAMGIQHQRVAEPDKLVDALKWLINTDGPALLEVVTDKKVPVLPMVPAGSALHEFLVFDPEKDKQRRELMKERTKGVHS
3IEK , Knot 172 431 0.80 40 193 395 
MRIVPFGAAREVTGSAHLLLAGGRRVLLDCGMFQGKEEARNHAPFGFDPKEVDAVLLTHAHLDHVGRLPKLFREGYRGPVYATRATVLLMEIVLEDALKVMDEPFFGPEDVEEALGHLRPLEYGEWLRLGALSLAFGQAGHLPGSAFVVAQGEGRTLVYSGDLGNREKDVLPDPSLPPLADLVLAEGTYGDRPHRPYRETVREFLEILEKTLSQGGKVLIPTFAVERAQEILYVLYTHGHRLPRAPIYLDSPMAGRVLSLYPRLVRYFSEEVQAHFLQGKNPFRPAGLEVVEHTEASKALNRAPGPMVVLAGSGMLAGGRILHHLKHGLSDPRNALVFVGYQPQGGLGAEIIARPPAVRILGEEVPLRASVHTLGGFSGHAGQDELLDWLQGEPRVVLVHGEEEKLLALGKLLALRGQEVSLARFGEGVPV

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 \(c\).

\(|P_{f(3DDA_1)}(2) \setminus P_{f(7EGV_1)}(2)|=50\), \(|P_{f(7EGV_1)}(2) \setminus P_{f(3DDA_1)}(2)|=113\). 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 1:1111000100001101101101011011010110110100011111000010010010101110100111000000010000000001011001100100001101110011011111110010001011000010110101000000101111110101101000011001101000101000010101010111000101000111111011001110110011011001011110100110100010001011010100100111001011001000010100000100110010010011100101001001100001100000101010010100100110010000011011011000001010011101011101000100110100001110101000010010100100101110000000
Pair \(Z_2\) Length of longest common subsequence
3DDA_1,7EGV_1 163 6
3DDA_1,3IEK_1 165 4
7EGV_1,3IEK_1 152 4

Newick tree

 
[
	3DDA_1:83.90,
	[
		7EGV_1:76,3IEK_1:76
	]:7.90
]

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 \((0.85)(0.8)(\frac{1118 }{\log_{20} 1118}-\frac{430}{\log_{20}430})=180.\)
Status Protein1 Protein2 d d1/2
Query variables 3DDA_1 7EGV_1 231 185
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]