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Parikh vectors
9BYL_1 2HTO_1 4UOZ_1 Letter Amino acid
16 0 25 H Histidine
39 0 32 F Phenylalanine
49 0 40 S Serine
3 0 22 W Tryptophan
28 0 45 R Arginine
6 3 11 C Cysteine
33 2 45 G Glycine
42 0 20 Y Tyrosine
41 0 45 V Valine
48 0 60 D Aspartic acid
49 0 42 E Glutamic acid
27 0 13 M Methionine
17 0 35 P Proline
47 1 70 A Alanine
44 0 34 I Isoleucine
59 0 17 K Lycine
35 0 41 T Threonine
42 0 25 N Asparagine
25 0 20 Q Glutamine
50 0 53 L Leucine 

9BYL_1|Chains A, B|Ribonucleoside-diphosphate reductase subunit alpha|Bacillus subtilis (1423)
>2HTO_1|Chain A|DNA (5'-D(*DCP*DGP*DCP*DGP*DCP*DA)-3')|
>4UOZ_1|Chains A, B, C|BETA-GALACTOSIDASE|BIFIDOBACTERIUM ANIMALIS SUBSP. LACTIS BL-04 (580050)
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 \(w=f(c)\)
9BYL , Knot 271 700 0.84 40 286 659 
MSQNQVPKWIQLNNEIMIQKDGKFQFDKDKEAVHSYFVDYINQNTVFFHNLKEKLDYLVENQYYEEEFLSLYSFEDIKEVFKTAYAKKFRFPSFMSAFKFYNDYALKTNDKKKILERYEDRISIVALFFANGDTEKAKEYVNLMINQEYQPSTPTFLNAGRKRRGELVSCFLLEVNDSLNDISRAIDISMQLSKLGGGVSLNLSKLRAKGEAIKDVENATKGVVGVMKLLDNAFRYADQMGQRQGSGAAYLNIFHRDINDFLDTKKISADEDVRVKTLSIGVVIPDKFVELAREDKAAYVFYPHTIYKEYGQHMDEMDMNEMYDKFVDNPRVKKEKINPRKLLEKLAMLRSESGYPYIMFQDNVNKVHANNHISKVKFSNLCSEVLQASQVSSYTDYDEEDEIGLDISCNLGSLNILNVMEHKSIEKTVKLATDSLTHVSETTDIRNAPAVRRANKAMKSIGLGAMNLHGYLAQNGIAYESPEARDFANTFFMMVNFYSIQRSAEIAKEKGETFDQYEGSTYATGEYFDKYVSTDFSPKYEKIANLFEGMHIPTTEDWKKLKAFVAEHGMYHSYRLCIAPTGSISYVQSSTASVMPIMERIEERTYGNSKTYYPMPGLASNNWFFYKEAYDMDMFKVVDMIATIQQHIDQGISFTLFLKDTMTTRDLNRIDLYAHHRGIKTIYYARTKDTGQDSCLSCVV
2HTO , Knot 3 6 0.29 6 3 3 
CGCGCA
4UOZ , Knot 271 695 0.85 40 285 646 
MSASTQHRAHRWPQPLPGNDRKIWFGADYNPDQWPEDVQDEDIRLMKQAGVNIVSLAIFSWANIETSDGNFEFDWLDRVIDKLYKAGIAVDLASATASPPMWLTSAHPEVLRRDEQGHVIWPGARQHWRPTSPTFRTYALRLCREMAEHYKDNPAIVSWHVGNEYGCHNYFDYSDDAVQAFREWCRDRYGTIDKVNAAWGTNFWSQRLNSFEEILPPRYVGGEGNFTNPGRLLDFKHFCSDALKEFFCAERDVLSEVTPNIPLTTNFMVSASQNTLDYDDWAHEVDFVSNDHYFTPGSWHIDELAYSASLVDGISRKKPWFLMAQSTSAVNWREINPRKEPGELIRDSMLHLAMGADAICYFQWRQSRSGAEKFHSAMLPLAGEHSQIYRDVCALGADLDTLSDAGILRSKLSKARVAIVQDIQSEWATEHTATPTQHIREWTEPLDWFAAFANRGVTADVTPIHAQWDTYDAVVIPCVYLFSEEMAERLRTFVRNGGKAFVTYYSALADEHDRLHTEGWPGLIGDVVGVRIEEHCPLGTLFPGMLDHLDVSNGTVVHDLADVIDAIADDTTVLATFEADPATGMDGRAAITVHPYHEGGVAYIAGKLGRDGISQSLPEICAALGFELDADPRAGDVLRVVREQEDGAIFEFLFNRTRNTVTADRPAGDMLICSLATDSTDKVTLEPNGVLAFRR

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 \(c\).

\(|P_{f(9BYL_1)}(2) \setminus P_{f(2HTO_1)}(2)|=286\), \(|P_{f(2HTO_1)}(2) \setminus P_{f(9BYL_1)}(2)|=3\). 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 1:1000011011010001110001010100000110001100100001110010001001100000000110100100100110010100101101101101000011000000011000000101111111010000100010111000001001011011000010110011101000100100110101010011111010100101010110010010011111101100110010011000101110101100010011000010100010100101111110011011000011011010010000100100101001000110010100001010011001111000010101110001001010001001010010001101001000000000001110100011010110110000100010110001001000001001111001001100111111010101100111000101001100111110100100010110001001000010001010010001000101000011011011011000010010111100110000010111010100100001011111001000001000000111111000111000100101101101110100010011010111000100001001010100011001001000001000010011
Pair \(Z_2\) Length of longest common subsequence
9BYL_1,2HTO_1 289 1
9BYL_1,4UOZ_1 139 5
2HTO_1,4UOZ_1 284 2

Newick tree

 
[
	2HTO_1:16.47,
	[
		9BYL_1:69.5,4UOZ_1:69.5
	]:90.97
]

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 \((0.85)(0.8)(\frac{706 }{\log_{20} 706}-\frac{6}{\log_{20}6})=212.\)
Status Protein1 Protein2 d d1/2
Query variables 9BYL_1 2HTO_1 271 136.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]