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Parikh vectors
7GMD_1 4NEU_1 6MFR_1 Letter Amino acid
12 7 22 C Cysteine
26 18 53 G Glycine
7 9 32 H Histidine
11 21 48 I Isoleucine
13 13 58 P Proline
27 21 70 V Valine
11 11 57 R Arginine
17 10 39 F Phenylalanine
24 8 54 T Threonine
16 21 39 S Serine
3 3 6 W Tryptophan
17 16 57 A Alanine
17 20 41 D Aspartic acid
14 7 57 Q Glutamine
9 35 41 E Glutamic acid
11 31 53 K Lycine
10 18 21 M Methionine
21 19 22 N Asparagine
29 33 61 L Leucine
11 12 28 Y Tyrosine 

7GMD_1|Chains A, B|3C-like proteinase|Severe acute respiratory syndrome coronavirus 2 (2697049)
>4NEU_1|Chains A, B|Receptor-interacting serine/threonine-protein kinase 1|Homo sapiens (9606)
>6MFR_1|Chains A, B|Protein argonaute-2|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 \(w=f(c)\)
7GMD , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
4NEU , Knot 147 333 0.85 40 198 320 
MDYKDDDDKMQPDMSLNVIKMKSSDFLESAELDSGGFGKVSLCFHRTQGLMIMKTVYKGPNCIEHNEALLEEAKMMNRLRHSRVVKLLGVIIEEGKYSLVMEYMEKGNLMHVLKAEMSTPLSVKGRIILEIIEGMCYLHGKGVIHKDLKPENILVDNDFHIKIADLGLASFKMWSKLNNEEHNELREVDGTAKKNGGTLYYMAPEHLNDVNAKPTEKSDVYSFAVVLWAIFANKEPYENAICEQQLIMCIKSGNRPDVDDITEYCPREIISLMKLCWEANPEARPTFPGIEEKFRPFYLSQLEESVEEDVKSLKKEYSNENAVVKRMQSLQLD
6MFR , Knot 326 859 0.85 40 325 795 
MYSGAGPALAPPAPPPPIQGYAFKPPPRPDFGTSGRTIKLQANFFEMDIPKIDIYHYELDIKPEKCPRRVNREIVEHMVQHFKTQIFGDRKPVFDGRKNLYTAMPLPIGRDKVELEVTLPGEGKDRIFKVSIKWVSCVSLQALHDALSGRLPSVPFETIQALDVVMRHLPSMRYTPVGRSFFTASEGCSNPLGGGREVWFGFHQSVRPSLWKMMLNIDVSATAFYKAQPVIEFVCEVLDFKSIEEQQKPLTDSQRVKFTKEIKGLKVEITHCGQMKRKYRVCNVTRRPASHQTFPLQQESGQTVECTVAQYFKDRHKLVLRYPHLPCLQVGQEQKHTYLPLEVCNIVAGQRCIKKLTDNQTSTMIRATARSAPDRQEEISKLMRSADFNTDPYVREFGIMVKDEMTDVTGRVLQPPSILYGGRNKAIATPVQGVWDMRNKQFHTGIEIKVWAIACFAPQRQCTEVHLKSFTEQLRKISRDAGMPIQGQPCFCKYAQGADSVEPMFRHLKNTYAGLQLVVVILPGKTPVYAEVKRVGDTVLGMATQCVQMKNVQRTTPQTLSNLCLKINVKLGGVNNILLPQGRPPVFQQPVIFLGADVTHPPAGDGKKPSIAAVVGSMDAHPNRYCATVRVQQHRQEIIQDLAAMVRELLIQFYKSTRFKPTRIIFYRAGVSEGQFQQVLHHELLAIREACIKLEKDYQPGITFIVVQKRHHTRLFCTDKNERVGKSGNIPAGTTVDTKITHPTEFDFYLCSHAGIQGTSRPSHYHVLWDDNRFSSDELQILTYQLCHTYVRCTRSVSIPAPAYYAHLVAFRARYHLVDKEHDAAEGDHTDGQANGRDHQALAKAVQVHQDTLRTMYFA

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 \(c\).

\(|P_{f(7GMD_1)}(2) \setminus P_{f(4NEU_1)}(2)|=82\), \(|P_{f(4NEU_1)}(2) \setminus P_{f(7GMD_1)}(2)|=84\). 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 1:011001111010101011010010001011110011001001100000110100001110000001110110101011100100011010100101001000110101100101110001010110001101010101011010010111010000101001001011011011001010101111000010111000010101111101111010011100100010010111100000110000101111101001111101010100110011010011101110001011011000011010
Pair \(Z_2\) Length of longest common subsequence
7GMD_1,4NEU_1 166 4
7GMD_1,6MFR_1 181 4
4NEU_1,6MFR_1 177 4

Newick tree

 
[
	6MFR_1:91.57,
	[
		7GMD_1:83,4NEU_1:83
	]:8.57
]

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 \((0.85)(0.8)(\frac{639 }{\log_{20} 639}-\frac{306}{\log_{20}306})=92.5\)
Status Protein1 Protein2 d d1/2
Query variables 7GMD_1 4NEU_1 118 113.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]