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Parikh vectors
6FDZ_1 8GHR_1 8IAX_1 Letter Amino acid
13 69 44 G Glycine
16 44 39 I Isoleucine
15 69 37 K Lycine
5 15 2 W Tryptophan
18 28 49 A Alanine
22 37 28 R Arginine
16 49 35 D Aspartic acid
4 25 1 C Cysteine
14 61 46 V Valine
18 66 45 E Glutamic acid
10 36 11 H Histidine
7 16 17 M Methionine
7 48 8 Y Tyrosine
12 38 11 Q Glutamine
36 96 33 L Leucine
20 83 25 S Serine
6 56 24 N Asparagine
14 48 17 F Phenylalanine
17 77 14 P Proline
6 54 35 T Threonine 

6FDZ_1|Chain A[auth U]|Serine/threonine-protein kinase ULK3|Homo sapiens (9606)
>8GHR_1|Chains A, B|Ectonucleotide pyrophosphatase/phosphodiesterase family member 1, secreted form, Immunoglobulin gamma-1 heavy chain fusion|Homo sapiens (9606)
>8IAX_1|Chains A, B, C, D, E, F, G, H|Pyruvate kinase|Streptococcus pneumoniae R6 (171101)
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 \(w=f(c)\)
6FDZ , Knot 126 276 0.85 40 181 267 
AGPGWGPPRLDGFILTERLGSGTYATVYKAYAKKDTREVVAIKCVAKKSLNKASVENLLTEIEILKGIRHPHIVQLKDFQWDSDNIYLIMEFCAGGDLSRFIHTRRILPEKVARVFMQQLASALQFLHERNISHLDLKPQNILLSSLEKPHLKLADFGFAQHMSPWDEKHVLRGSPLYMAPEMVCQRQYDARVDLWSMGVILYEALFGQPPFASRSFSELEEKIRSNRVIELPLRPLLSRDCRDLLQRLLERDPSRRISFQDFFAHPWVDLEHMPS
8GHR , Knot 376 1015 0.85 40 321 911 
GEKSWVEEPCESINEPQCPAGFETPPTLLFSLDGFRAEYLHTWGGLLPVISKLKKCGTYTKNMRPVYPTKAFPNHYSIVTGLYPESHGIIDNKMYDPKMNASFSLKSKEKFNPEWYKGEPIWVTAKYQGLKSGTFFWPGSDVEINGIFPDIYKMYNGSVPFEERILAVLQWLQLPKDERPHFYTLYLEEPDSSGHSYGPVSSEVIKALQRVDGMVGMLMDGLKELNLHRCLNLILISDHGMEQGSCKKYIYLNKYLGDVKNIKVIYGPAARLRPSDVPDKYYSFNYEGIARNLSCREPNQHFKPYLKHFLPKRLHFAKSDRIEPLTFYLDPQWQLALNPSERKYCGSGFHGSDNVFSNMQALFVGYGPGFKHGIEADTFENIEVYNLMCDLLNLTPAPNNGTHGSLNHLLKNPVYTPKHPKEVHPLVQCPFTRNPRDNLGCSCNPSILPIEDFQTQFNLTVAEEKIIKHETLPYGRPRVLQKENTICLLSQHQFMSGYSQDILMPLWTSYTVDRNDSFSTEDFSNCLYQDFRIPLSPVHKCSFYKNNTKVSYGFLSPPQLNKNSSGIYSEALLTTNIVPMYQSFQVIWRYFHDTLLRKYAEERNGVNVVSGPVFDFDYDGRCDSLENLRQKRRVIRNQEILIPTHFFIVLTSCKDTSQTPLHCENLDTLAFILPHRTDNSESCVHGKHDSSWVEELLMLHRARITDVEHITGLSFYQQRKEPVSDILKLKTHLPTFSQEDTSSGGGGENLYFQSSGGGSGGGEPKSCDKTHTCPPCPAPELLGGPSVFLFPPKPKDTLMISRTPEVTCVVVDVSHEDPEVKFNWYVDGVEVHNAKTKPREEQYNSTYRVVSVLTVLHQDWLNGKEYKCKVSNKALPAPIEKTISKAKGQPREPQVYTLPPSRDELTKNQVSLTCLVKGFYPSDIAVEWESNGQPENNYKTTPPVLDSDGSFFLYSKLTVDKSRWQQGNVFSCSVMHEALHNHYTQKSLSLSPGKGGGGSGLNDIFEAQKIEWHEG
8IAX , Knot 203 521 0.81 40 229 477 
MGSSHHHHHHSSGLVPRGSHMNKRVKIVATLGPAVEIRGGKKFGEDGYWGEKLDVEASAKNIAKLIEAGANTFRFNFSHGDHQEQGERMATVKLAEKIAGKKVGFLLDTKGPEIRTELFEGEAKEYSYKTGEKIRVATKQGIKSTREVIALNVAGALDIYDDVEVGRQVLVDDGKLGLRVVAKDDATREFEVEVENDGIIAKQKGVNIPNTKIPFPALAERDNDDIRFGLEQGINFIAISFVRTAKDVNEVRAICEETGNGHVQLFAKIENQQGIDNLDEIIEAADGIMIARGDMGIEVPFEMVPVYQKMIIKKVNAAGKVVITATNMLETMTEKPRATRSEVSDVFNAVIDGTDATMLSGESANGKYPLESVTTMATIDKNAQALLNEYGRLDSDSFERNSKTEVMASAVKDATSSMDIKLVVTLTKTGHTARLISKYRPNADILALTFDELTERGLMLNWGVIPMLTDAPSSTDDMFEIAERKAVEAGLVESGDDIVIVAGVPVGEAVRTNTMRIRTVR

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 \(c\).

\(|P_{f(6FDZ_1)}(2) \setminus P_{f(8GHR_1)}(2)|=32\), \(|P_{f(8GHR_1)}(2) \setminus P_{f(6FDZ_1)}(2)|=172\). 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 1:111111110101111000110100101001010000001111001100010010100110010110110010110100101000010111010111010011000011100110111001101101100001001010100111001001010110111100101100001101011011101100000010101101111100111101111000100100010000110111011100000011001100010001010011101110100110
Pair \(Z_2\) Length of longest common subsequence
6FDZ_1,8GHR_1 204 4
6FDZ_1,8IAX_1 174 4
8GHR_1,8IAX_1 150 4

Newick tree

 
[
	6FDZ_1:10.53,
	[
		8IAX_1:75,8GHR_1:75
	]:25.53
]

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 \((0.85)(0.8)(\frac{1291 }{\log_{20} 1291}-\frac{276}{\log_{20}276})=267.\)
Status Protein1 Protein2 d d1/2
Query variables 6FDZ_1 8GHR_1 343 217.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]