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Parikh vectors
5RFK_1 7ASS_1 8OPU_1 Letter Amino acid
11 6 13 Y Tyrosine
27 20 53 V Valine
21 16 18 N Asparagine
17 14 38 D Aspartic acid
10 9 20 M Methionine
3 10 5 W Tryptophan
11 19 41 I Isoleucine
11 18 45 K Lycine
16 16 43 S Serine
17 21 89 A Alanine
12 0 2 C Cysteine
9 15 50 E Glutamic acid
7 7 12 H Histidine
29 19 59 L Leucine
13 7 41 P Proline
24 12 41 T Threonine
11 13 31 R Arginine
14 14 24 Q Glutamine
26 16 81 G Glycine
17 13 30 F Phenylalanine 

5RFK_1|Chain A|3C-like proteinase|Severe acute respiratory syndrome coronavirus 2 (2697049)
>7ASS_1|Chains A, B, C, D|Beta-lactamase|Klebsiella pneumoniae (573)
>8OPU_1|Chains A, B|3-hydroxyacyl-CoA dehydrogenase|Mycobacterium tuberculosis H37Rv (83332)
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 \(w=f(c)\)
5RFK , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
7ASS , Knot 122 265 0.85 38 175 261 
MRVLALSAVFLVASIIGMPAVAKEWQENKSWNAHFTEHKSQGVVVLWNENKQQGFTNNLKRANQAFFPASTFKIPNSLIALDLGVVKDEHQVFKWDGQTRDIATWNRDHNLITAMKYSVVPVYQEFARQIGEARMSKMLHAFDYGNEDISGNVDSFWLDGGIRISATEQISFLRKLYHNKLHVSERSQRIVKQAMLTEANGDYIIRAKTGYSTRIEPKIGWWVGWVELDDNVWFFAMNMDMPTSDGLGLRQAITKEVLKQEKIIP
8OPU , Knot 267 736 0.79 40 257 640 
MGSSHHHHHHSQDPNSMPDNTIQWDKDADGIVTLTMDDPSGSTNVMNEAYIESMGKAVDRLVAEKDSITGVVVASAKKTFFAGGDVKTMIQARPEDAGDVFNTVETIKRQLRTLETLGKPVVAAINGAALGGGLEIALACHHRIAADVKGSQLGLPEVTLGLLPGGGGVTRTVRMFGIQNAFVSVLAQGTRFKPAKAKEIGLVDELVATVEELVPAAKAWIKEELKANPDGAGVQPWDKKGYKMPGGTPSSPGLAAILPSFPSNLRKQLKGAPMPAPRAILAAAVEGAQVDFDTASRIESRYFASLVTGQVAKNMMQAFFFDLQAINAGGSRPEGIGKTPIKRIGVLGAGMMGAGIAYVSAKAGYEVVLKDVSLEAAAKGKGYSEKLEAKALERGRTTQERSDALLARITPTADAADFKGVDFVIEAVFENQELKHKVFGEIEDIVEPNAILGSNTSTLPITGLATGVKRQEDFIGIHFFSPVDKMPLVEIIKGEKTSDEALARVFDYTLAIGKTPIVVNDSRGFFTSRVIGTFVNEALAMLGEGVEPASIEQAGSQAGYPAPPLQLSDELNLELMHKIAVATRKGVEDAGGTYQPHPAEAVVEKMIELGRSGRLKGAGFYEYADGKRSGLWPGLRETFKSGSSQPPLQDMIDRMLFAEALETQKCLDEGVLTSTADANIGSIMGIGFPPWTGGSAQFIVGYSGPAGTGKAAFVARARELAAAYGDRFLPPESLLS

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 \(c\).

\(|P_{f(5RFK_1)}(2) \setminus P_{f(7ASS_1)}(2)|=103\), \(|P_{f(7ASS_1)}(2) \setminus P_{f(5RFK_1)}(2)|=82\). 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
5RFK_1,7ASS_1 185 4
5RFK_1,8OPU_1 171 4
7ASS_1,8OPU_1 174 4

Newick tree

 
[
	7ASS_1:91.17,
	[
		5RFK_1:85.5,8OPU_1:85.5
	]:5.67
]

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 \((0.85)(0.8)(\frac{571 }{\log_{20} 571}-\frac{265}{\log_{20}265})=86.5\)
Status Protein1 Protein2 d d1/2
Query variables 5RFK_1 7ASS_1 111 105
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]