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Parikh vectors
5SNK_1 6WQF_1 1TNZ_1 Letter Amino acid
10 7 9 H Histidine
34 16 22 S Serine
7 11 18 Y Tyrosine
24 27 22 V Valine
19 24 8 T Threonine
2 3 10 W Tryptophan
4 12 2 C Cysteine
52 26 12 G Glycine
21 11 21 I Isoleucine
36 29 37 L Leucine
18 17 9 F Phenylalanine
51 17 27 A Alanine
29 11 25 R Arginine
15 21 19 N Asparagine
22 9 36 E Glutamic acid
14 10 6 M Methionine
24 17 27 D Aspartic acid
7 14 26 Q Glutamine
8 11 15 K Lycine
17 13 26 P Proline 

5SNK_1|Chains A, B|3-oxoacyl-[acyl-carrier-protein] synthase 2|Pseudomonas aeruginosa (287)
>6WQF_1|Chain A|3C-like proteinase|Severe acute respiratory syndrome coronavirus 2 (2697049)
>1TNZ_1|Chains A, C, E, G, I, K|geranylgeranyltransferase type I alpha subunit|Rattus norvegicus (10116)
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 \(w=f(c)\)
5SNK , Knot 171 414 0.83 40 212 384 
MSRRRVVITGMGMLSPLGLDVPSSWEGILAGRSGIAPIEHMDLSAYSTRFGGSVKGFNVEEYLSAKEARKLDLFIQYGLAASFQAVRDSGLEVTDANRERIGVSMGSGIGGLTNIENNCRSLFEQGPRRISPFFVPGSIINMVSGFLSIHLGLQGPNYALTTAQTTGTHSIGMAARNIAYGEADVMVAGGSEMAACGLGLGGFGAARALSTRNDEPTRASRPWDRDRDGFVLSDGSGALVLEELEHARARGARIYAELVGFGMSGDAFHMTAPPEDGAGAARCMKNALRDAGLDPRQVDYINAHGTSTPAGDIAEIAAVKSVFGEHAHALSMSSTKSMTGHLLGAAGAVEAIFSVLALRDQVAPPTINLDNPDEGCDLDLVAHEAKPRKIDVALSNSFGFGGTNGTLVFRRFAD
6WQF , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
1TNZ , Knot 156 377 0.81 40 213 348 
MAATEGVGESAPGGEPGQPEQPPPPPPPPPAQQPQEEEMAAEAGEAAASPMDDGFLSLDSPTYVLYRDRAEWADIDPVPQNDGPSPVVQIIYSEKFRDVYDYFRAVLQRDERSERAFKLTRDAIELNAANYTVWHFRRVLLRSLQKDLQEEMNYIIAIIEEQPKNYQVWHHRRVLVEWLKDPSQELEFIADILNQDAKNYHAWQHRQWVIQEFRLWDNELQYVDQLLKEDVRNNSVWNQRHFVISNTTGYSDRAVLEREVQYTLEMIKLVPHNESAWNYLKGILQDRGLSRYPNLLNQLLDLQPSHSSPYLIAFLVDIYEDMLENQCDNKEDILNKALELCEILAKEKDTIRKEYWRYIGRSLQSKHSRESDIPASV

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 \(c\).

\(|P_{f(5SNK_1)}(2) \setminus P_{f(6WQF_1)}(2)|=91\), \(|P_{f(6WQF_1)}(2) \setminus P_{f(5SNK_1)}(2)|=75\). 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 1:100001110111110111101100101111100111110010101000011101011010001010010010111001111010110001101001000011101101111100100000011001100101111110110110111010111011001100100010001111100110101011111100111011111111110110000001001001100000111100101111100100101011010101111110101101011100111110010011001110100100101010001110110111100111001011010000010101111111101110111100011110101001001001011100101001011100011111001011100110
Pair \(Z_2\) Length of longest common subsequence
5SNK_1,6WQF_1 166 5
5SNK_1,1TNZ_1 167 4
6WQF_1,1TNZ_1 195 4

Newick tree

 
[
	1TNZ_1:93.21,
	[
		5SNK_1:83,6WQF_1:83
	]:10.21
]

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 \((0.85)(0.8)(\frac{720 }{\log_{20} 720}-\frac{306}{\log_{20}306})=114.\)
Status Protein1 Protein2 d d1/2
Query variables 5SNK_1 6WQF_1 142 124.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]