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Parikh vectors
3KHJ_1 7WRF_1 7SYU_1 Letter Amino acid
3 26 498 C Cysteine
46 55 550 G Glycine
7 24 0 H Histidine
34 69 0 I Isoleucine
16 58 0 T Threonine
38 77 0 V Valine
13 66 0 N Asparagine
4 39 0 Q Glutamine
28 70 0 K Lycine
14 24 0 M Methionine
1 27 0 W Tryptophan
11 41 0 Y Tyrosine
12 54 0 D Aspartic acid
9 27 0 P Proline
20 59 418 A Alanine
32 77 0 E Glutamic acid
26 123 0 L Leucine
3 70 0 F Phenylalanine
31 80 0 S Serine
13 54 0 R Arginine 

3KHJ_1|Chains A, B, C, D, E, F, G, H|Inosine-5-monophosphate dehydrogenase|Cryptosporidium parvum (353152)
>7WRF_1|Chains A, B, C, D|Transient receptor potential cation channel subfamily M member 8|Mus musculus (10090)
>7SYU_1|Chain A[auth 2]|18S rRNA|Oryctolagus cuniculus (9986)
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 \(w=f(c)\)
3KHJ , Knot 149 361 0.81 40 187 331 
GSHMGTKNIGKGLTFEDILLVPNYSEVLPREVSLETKLTKNVSLKIPLISSAMDTVTEHLMAVGMARLGGIGIIHKNMDMESQVNEVLKVKNSGGLRVGAAIGVNEIERAKLLVEAGVDVIVLDSAHGHSLNIIRTLKEIKSKMNIDVIVGNVVTEEATKELIENGADGIKVGIGPGSICTTRIVAGVGVPQITAIEKCSSVASKFGIPIIADGGIRYSGDIGKALAVGASSVMIGSILAGTEESPGEKELIGDTVYKYYRGMGSVGAMKSGSGDRYFQEKRPENKMVPEGIEGRVKYKGEMEGVVYQLVGGLRSCMGYLGSASIEELWKKSSYVEITTSGLRESHVHDVEIVKEVMNYSK
7WRF , Knot 408 1120 0.85 40 337 1005 
MSFEGARLSMRSRRNGTMGSTRTLYSSVSRSTDVSYSDSDLVNFIQANFKKRECVFFTRDSKAMENICKCGYAQSQHIEGTQINQNEKWNYKKHTKEFPTDAFGDIQFETLGKKGKYLRLSCDTDSETLYELLTQHWHLKTPNLVISVTGGAKNFALKPRMRKIFSRLIYIAQSKGAWILTGGTHYGLMKYIGEVVRDNTISRNSEENIVAIGIAAWGMVSNRDTLIRSCDDEGHFSAQYIMDDFTRDPLYILDNNHTHLLLVDNGCHGHPTVEAKLRNQLEKYISERTSQDSNYGGKIPIVCFAQGGGRETLKAINTSVKSKIPCVVVEGSGQIADVIASLVEVEDVLTSSMVKEKLVRFLPRTVSRLPEEEIESWIKWLKEILESSHLLTVIKMEEAGDEIVSNAISYALYKAFSTNEQDKDNWNGQLKLLLEWNQLDLASDEIFTNDRRWESADLQEVMFTALIKDRPKFVRLFLENGLNLQKFLTNEVLTELFSTHFSTLVYRNLQIAKNSYNDALLTFVWKLVANFRRSFWKEDRSSREDLDVELHDASLTTRHPLQALFIWAILQNKKELSKVIWEQTKGCTLAALGASKLLKTLAKVKNDINAAGESEELANEYETRAVELFTECYSNDEDLAEQLLVYSCEAWGGSNCLELAVEATDQHFIAQPGVQNFLSKQWYGEISRDTKNWKIILCLFIIPLVGCGLVSFRKKPIDKHKKLLWYYVAFFTSPFVVFSWNVVFYIAFLLLFAYVLLMDFHSVPHTPELILYALVFVLFCDEVRQWYMNGVNYFTDLWNVMDTLGLFYFIAGIVFRLHSSNKSSLYSGRVIFCLDYIIFTLRLIHIFTVSRNLGPKIIMLQRMLIDVFFFLFLFAVWMVAFGVARQGILRQNEQRWRWIFRSVIYEPYLAMFGQVPSDVDSTTYDFSHCTFSGNESKPLCVELDEHNLPRFPEWITIPLVCIYMLSTNILLVNLLVAMFGYTVGIVQENNDQVWKFQRYFLVQEYCNRLNIPFPFVVFAYFYMVVKKCFKCCCKEKNMESNACCFRNEDNETLAWEGVMKENYLVKINTKANDNSEEMRHRFRQLDSKLNDLKSLLKEIANNIKLEGGSSGGWSHPQFEK
7SYU , Knot 343 1870 0.46 8 16 64 
UACCUGGUUGAUCCUGCCAGUAGCAUAUGCUUGUCUCAAAGAUUAAGCCAUGCAUGUCUAAGUACGCACGGCCGGUACAGUGAAACUGCGAAUGGCUCAUUAAAUCAGUUAUGGUUCCUUUGGUCGCUCGCUCCUCUCCUACUUGGAUAACUGUGGUAAUUCUAGAGCUAAUACAUGCCGACGGGCGCUGACCCCCUUCGCGGGGGGGAUGCGUGCAUUUAUCAGAUCAAAACCAACCCGGUCAGCCUCCCCCCGGCCCCGGCCGGGGGGGUGGGCGCCGGCGGCUUUGGUGACUCUAGAUAACCUCGGGCCGAUCGCACGCCCUCCGUGGCGGCGACGACCCAUUCGAACGUCUGCCCUAUCAACUUUCGAUGGUAGUCGCCGUGCCUACCAUGGUGACCACGGGUGACGGGGAAUCAGGGUUCGAUUCCGGAGAGGGAGCCUGAGAAACGGCUACCACAUCCAAGGAAGGCAGCAGGCGCGCAAAUUACCCACUCCCGACCCGGGGAGGUAGUGACGAAAAAUAACAAUACAGGACUCUUUCGAGGCCCUGUAAUUGGAAUGAGUCCACUUUAAAUCCUUUAACGAGGAUCCAUUGGAGGGCAAGUCUGGUGCCAGCAGCCGCGGUAAUUCCAGCUCCAAUAGCGUAUAUUAAAGUUGCUGCAGUUAAAAAGCUCGUAGUUGGAUCUUGGGAGCGGGCGGGCGGUCCGCCGCGAGGCGAGCCACCGCCCGUCCCCGCCCCUUGCCUCUCGGCGCCCCCUCGAUGCUCUUAGCUGAGUGUCCCGCGGGGCCCGAAGCGUUUACUUUGAAAAAAUUAGAGUGUUCAAAGCAGGCCCGAGCCGCCUGGAUACCGCAGCUAGGAAUAAUGGAAUAGGACCGCGGUUCUAUUUUGUUGGUUUUCGGAACUGAGGCCAUGAUUAAGAGGGACGGCCGGGGGCAUUCGUAUUGCGCCGCUAGAGGUGAAAUUCUUGGACCGGCGCAAGACGGACCAGAGCGAAAGCAUUUGCCAAGAAUGUUUUCAUUAAUCAAGAACGAAAGUCGGAGGUUCGAAGACGAUCAGAUACCGUCGUAGUUCCGACCAUAAACGAUGCCGACCGGCGAUGCGGCGGCGUUAUUCCCAUGACCCGCCGGGCAGCUUCCGGGAAACCAAAGUCUUUGGGUUCCGGGGGGAGUAUGGUUGCAAAGCUGAAACUUAAAGGAAUUUGGCGAAGGGCACCACCAGGAGUGGAGCCUGCGGCUUAAUUUGACUCAACACGGGAAACCUCACCCGGCCCGGACACGGACAGGAUUGACAGAUUGAUAGCUCUUUCUCGAUUCCGUGGGUGGUGGUGCAUGGCCGUUCUUAGUUGGUGGAGCGAUUUGUCUGGUUAAUUCCGAUAACGAACGAGACUCUGGCAUGCUAACUAGUUACGCGACCCCCGAGCGGUCGGCGUCCCCCAACUUCUUAGAGGGACAAGUGGCGUUCAGCCACCCGAGAUUGAGCAAUAACAGGUCUGUGAUGCCCUUAGAUGUCCGGGGCUGCACGCGCGCUACACUGACUGGCUCAGCGUGUGCCUACCCUACGCCGGCAGGCGCGGGUAACCCGUUGAACCCCAUUCGUGAUGGGGAUCGGGGAUUGCAAUUAUUCCCCAUGAACGAGGAAUUCCCAGUAAGUGCGGGUCAUAAGCUUGCGUUGAUUAAGUCCCUGCCCUUUGUACACACCGCCCGUCGCUACUACCGAUUGGAUGGUUUAGUGAGGCCCUCGGAUCGGCCCCGCCGGGGUCGGCCCACGGCCCUGGCGGAGCGCUGAGAAGACGGUCGAACUUGACUAUCUAGAGGAAGUAAAAGUCGUAACAAGGUUUCCGUAGGUGAACCUGCGGAAGGAUCAUUA

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 \(c\).

\(|P_{f(3KHJ_1)}(2) \setminus P_{f(7WRF_1)}(2)|=14\), \(|P_{f(7WRF_1)}(2) \setminus P_{f(3KHJ_1)}(2)|=164\). 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 1:1001100011011010011111000011100101000100010101111001100100011111110111111100010100010011010001110111111100100101110111011110010100101100100100010101111011000100011001101101111110100001111111101011000001100111111101110001011011111100111101111000011000111001000001110111100101000100001000111011010100010101110011111000110110101001100000101000110000100101100110000
Pair \(Z_2\) Length of longest common subsequence
3KHJ_1,7WRF_1 178 4
3KHJ_1,7SYU_1 195 3
7WRF_1,7SYU_1 337 3

Newick tree

 
[
	7SYU_1:15.41,
	[
		3KHJ_1:89,7WRF_1:89
	]:61.41
]

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 \((0.85)(0.8)(\frac{1481 }{\log_{20} 1481}-\frac{361}{\log_{20}361})=288.\)
Status Protein1 Protein2 d d1/2
Query variables 3KHJ_1 7WRF_1 368 239
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]