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Parikh vectors
5KCR_1 4FZL_1 7BVX_1 Letter Amino acid
639 2 0 C Cysteine
0 8 10 E Glutamic acid
913 23 25 G Glycine
0 19 24 S Serine
0 18 11 N Asparagine
0 10 28 Q Glutamine
0 10 15 H Histidine
0 24 22 L Leucine
0 5 15 K Lycine
0 16 9 V Valine
761 21 27 A Alanine
0 15 13 I Isoleucine
0 3 1 W Tryptophan
0 6 12 Y Tyrosine
0 11 2 R Arginine
0 4 5 M Methionine
0 10 8 F Phenylalanine
0 13 12 P Proline
0 15 35 T Threonine
0 14 18 D Aspartic acid 

5KCR_1|Chain A[auth 1A]|23S Ribosomal RNA|Escherichia coli (strain K12) (83333)
>4FZL_1|Chains A, B|Bacteriocin|Pseudomonas syringae pv. tomato (223283)
>7BVX_1|Chain A|Pilus assembly protein|Lactobacillus rhamnosus (47715)
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 \(w=f(c)\)
5KCR , Knot 487 2904 0.44 8 16 64 
GGUUAAGCGACUAAGCGUACACGGUGGAUGCCCUGGCAGUCAGAGGCGAUGAAGGACGUGCUAAUCUGCGAUAAGCGUCGGUAAGGUGAUAUGAACCGUUAUAACCGGCGAUUUCCGAAUGGGGAAACCCAGUGUGUUUCGACACACUAUCAUUAACUGAAUCCAUAGGUUAAUGAGGCGAACCGGGGGAACUGAAACAUCUAAGUACCCCGAGGAAAAGAAAUCAACCGAGAUUCCCCCAGUAGCGGCGAGCGAACGGGGAGCAGCCCAGAGCCUGAAUCAGUGUGUGUGUUAGUGGAAGCGUCUGGAAAGGCGCGCGAUACAGGGUGACAGCCCCGUACACAAAAAUGCACAUGCUGUGAGCUCGAUGAGUAGGGCGGGACACGUGGUAUCCUGUCUGAAUAUGGGGGGACCAUCCUCCAAGGCUAAAUACUCCUGACUGACCGAUAGUGAACCAGUACCGUGAGGGAAAGGCGAAAAGAACCCCGGCGAGGGGAGUGAAAAAGAACCUGAAACCGUGUACGUACAAGCAGUGGGAGCACGCUUAGGCGUGUGACUGCGUACCUUUUGUAUAAUGGGUCAGCGACUUAUAUUCUGUAGCAAGGUUAACCGAAUAGGGGAGCCGAAGGGAAACCGAGUCUUAACUGGGCGUUAAGUUGCAGGGUAUAGACCCGAAACCCGGUGAUCUAGCCAUGGGCAGGUUGAAGGUUGGGUAACACUAACUGGAGGACCGAACCGACUAAUGUUGAAAAAUUAGCGGAUGACUUGUGGCUGGGGGUGAAAGGCCAAUCAAACCGGGAGAUAGCUGGUUCUCCCCGAAAGCUAUUUAGGUAGCGCCUCGUGAAUUCAUCUCCGGGGGUAGAGCACUGUUUCGGCAAGGGGGUCAUCCCGACUUACCAACCCGAUGCAAACUGCGAAUACCGGAGAAUGUUAUCACGGGAGACACACGGCGGGUGCUAACGUCCGUCGUGAAGAGGGAAACAACCCAGACCGCCAGCUAAGGUCCCAAAGUCAUGGUUAAGUGGGAAACGAUGUGGGAAGGCCCAGACAGCCAGGAUGUUGGCUUAGAAGCAGCCAUCAUUUAAAGAAAGCGUAAUAGCUCACUGGUCGAGUCGGCCUGCGCGGAAGAUGUAACGGGGCUAAACCAUGCACCGAAGCUGCGGCAGCGACGCUUAUGCGUUGUUGGGUAGGGGAGCGUUCUGUAAGCCUGCGAAGGUGUGCUGUGAGGCAUGCUGGAGGUAUCAGAAGUGCGAAUGCUGACAUAAGUAACGAUAAAGCGGGUGAAAAGCCCGCUCGCCGGAAGACCAAGGGUUCCUGUCCAACGUUAAUCGGGGCAGGGUGAGUCGACCCCUAAGGCGAGGCCGAAAGGCGUAGUCGAUGGGAAACAGGUUAAUAUUCCUGUACUUGGUGUUACUGCGAAGGGGGGACGGAGAAGGCUAUGUUGGCCGGGCGACGGUUGUCCCGGUUUAAGCGUGUAGGCUGGUUUUCCAGGCAAAUCCGGAAAAUCAAGGCUGAGGCGUGAUGACGAGGCACUACGGUGCUGAAGCAACAAAUGCCCUGCUUCCAGGAAAAGCCUCUAAGCAUCAGGUAACAUCAAAUCGUACCCCAAACCGACACAGGUGGUCAGGUAGAGAAUACCAAGGCGCUUGAGAGAACUCGGGUGAAGGAACUAGGCAAAAUGGUGCCGUAACUUCGGGAGAAGGCACGCUGAUAUGUAGGUGAGGUCCCUCGCGGAUGGAGCUGAAAUCAGUCGAAGAUACCAGCUGGCUGCAACUGUUUAUUAAAAACACAGCACUGUGCAAACACGAAAGUGGACGUAUACGGUGUGACGCCUGCCCGGUGCCGGGAGGUUAAUUGAUGGGGUUAGCGCAAGCGAAGCUCUUGAUCGAAGCCCCGGUAAACGGCGGCCGUAACUAUAACGGUCCUAAGGUAGCGAAAUUCCUUGUCGGGUAAGUUCCGACCUGCACGAAUGGCGUAAUGAUGGCCAGGCUGUCUCCACCCGAGACUCAGUGAAAUUGAACUCGCUGUGAAGAUGCAGUGUACCCGCGGCAAGACGGAAAGACCCCGUGAACCUUUACUAUAGCUUGACACUGAACAUUGAGCCUUGAUGUGUAGGAUAGGUGGGAGGCUUUGAAGUGUGGACGCCAGUCUGCAUGGAGCCGACCUUGAAAUACCACCCUUUAAUGUUUGAUGUUCUAACGUUGACCCGUAAUCCGGGUUGCGGACAGUGUCUGGUGGGUAGUUUGACUGGGGCGGUCUCCUCCUAAAGAGUAACGGAGGAGCACGAAGGUUGGCUAAUCCUGGUCGGACAUCAGGAGGUUAGUGCAAUGGCAUAAGCCAGCUUGACUGCGAGCGUGACGGCGCGAGCAGGUGCGAAAGCAGGUCAUAGUGAUCCGGUGGUUCUGAAUGGAAGGGCCAUCGCUCAACGGAUAAAAGGUACUCCGGGGAUAACAGGCUGAUACCGCCCAAGAGUUCAUAUCGACGGCGGUGUUUGGCACCUCGAUGUCGGCUCAUCACAUCCUGGGGCUGAAGUAGGUCCCAAGGGUAUGGCUGUUCGCCAUUUAAAGUGGUACGCGAGCUGGGUUUAGAACGUCGUGAGACAGUUCGGUCCCUAUCUGCCGUGGGCGCUGGAGAACUGAGGGGGGCUGCUCCUAGUACGAGAGGACCGGAGUGGACGCAUCACUGGUGUUCGGGUUGUCAUGCCAAUGGCACUGCCCGGUAGCUAAAUGCGGAAGAGAUAAGUGCUGAAAGCAUCUAAGCACGAAACUUGCCCCGAGAUGAGUUCUCCCUGACCCUUUAAGGGUCCUGAAGGAACGUUGAAGACGACGACGUUGAUAGGCCGGGUGUGUAAGCGCAGCGAUGCGUUGAGCUAACCGGUACUAAUGAACCGUGAGGCUUAACCUU
4FZL , Knot 114 247 0.84 40 163 239 
GMQNPVATVLLLQGDLYCSPNCLATFQDQARRDSFGIQSKVALKTFAAADQREAEGRDLRTAYNEIATDIGRSQQINENIIKYPPGNHVLSGGLMTPFHALAHGMFGLGAPLTFPIQNVGLNVDIRGIPDVMNVIQSARPVGTSSLDVNFAYDVGKDSNASWLTLGNITLRLVGTIDKNASGAWTFSGEIRAFNDVYDANPSNHRGWLGENLTSLLSAVPFTSYSIEIPGSLPVTVSGNLEHHHHHH
7BVX , Knot 119 292 0.77 38 153 271 
MGSSHHHHHHSSGLVPRGSHMTNQQYGFQFQKKTTDGTDLSADQLKAMQFNLTQYSDNSFQQASKTNAITSTDLQALAPGYYGIQEAAAPTGYQLDGTTYLFQLTSDGQWQYHGTKDNVTSGSVINGQQTLNPVGDKSDDFTVTGDHQQILTLTKYDEPKPSMTLRVIKQDNQSQYLAGAAFTLQPSAGEAETITSSATSEGQAFATKLVADGTYTMSETKAPDGYQSNPAKIAIQVATTGKEATVTIDGEALKPGESKNGYTLAIDGSTITLQAINQPLAILPLEHHHHHH

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 \(c\).

\(|P_{f(5KCR_1)}(2) \setminus P_{f(4FZL_1)}(2)|=13\), \(|P_{f(4FZL_1)}(2) \setminus P_{f(5KCR_1)}(2)|=160\). 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 1:110011101100111010101011011101000011011001111101101111110101001100010110111010011011110110101110010010110011011000001110111111100011010100001101010010010011001110001011100110111101110011111110011110100011101000011111111111100110011110000000110110110111011101111110110001111000111001101010101001101111101000111111101010110101111011011000010101011111010101010010111000110111011110111101010110100001000111010111111100100000011110011101000001100110011011011100110100101111111111011111111000011011111111011111111100011110010101010101110110111110101000111010101100101010000001010110111001101100010100001011011110011001110111111100111111111001110000110011101001110010111101011100011110001101100011001011101110011111001110110100110011111100111001100110100111111001101110110001011001111101111110011001110011111101100110000000011111001000111011010000101110001000001111101111010010000110111111100100001100010011000110101110010111010011111101001001011111101010110111010011010001001011111111111011000111001001100111100001111001011001110111111011010111111100011101100111101001100011111011001001000111111110101101100010011001110011000101011111101011011110011100101010011110010110110110100010101001001110111111101000010111000101111101010010111101010011111010011111010111010011010111011011011110111011111100010001001111110011111000001000110100110011110111101110011000001111011110011111101011001101111110111001101000001010001101001001011111111110111111110010100110011101101100100001100011101010111001100000011101110001111110011110011110101101101111010010110100111101101110100001000001111111100000111010011101101001110010100001110011010111011001110111111010011110100011111110001110111111100111011110110100101100001111111110101001101010111011110000001011101111001111001100111110100110011001011001000100111110101101001010111010111110111010101011010110100010001101001111110011001101111001101011101111000001100111100001101110110110010110010110110000111101101111000000100111011100001100010101110110101101101100111001000001000111100011011110011100010010111110101101010001011011110111111100001011100000100101100011010011101001110000110101011110111011111100001111010111010011000101011110011000011110100100000011010001101000011010011000101100011100101110110100011011101100011001111011000000000111111011011111110101111100110011000011001110100111111001101011011010111001100011001011101011011010111011101011111011100101101100011011000011101111111001001000110111011111101000011111011011100110100100011111000101001101101101000110100001101001100010010100001111001111011100001111101011001000100100011110110101011100111000111101001011110110001100000100010010111010011111100111111110010000011010111111100111101110101001001101000111001001010011011010010001101100111010111111110111010011111010001110101111000100001111011100000000110000001111100001111111010011111011011010011011100111010101110101101101010011100110011010011011100101111000110000
Pair \(Z_2\) Length of longest common subsequence
5KCR_1,4FZL_1 173 3
5KCR_1,7BVX_1 163 4
4FZL_1,7BVX_1 146 8

Newick tree

 
[
	5KCR_1:87.40,
	[
		7BVX_1:73,4FZL_1:73
	]:14.40
]

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 \((0.85)(0.8)(\frac{3151 }{\log_{20} 3151}-\frac{247}{\log_{20}247})=705.\)
Status Protein1 Protein2 d d1/2
Query variables 5KCR_1 4FZL_1 485 297
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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