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Parikh vectors
9FTN_1 7YHG_1 8PXK_1 Letter Amino acid
37 1 6 N Asparagine
19 4 16 Q Glutamine
34 1 14 I Isoleucine
65 2 46 L Leucine
18 0 5 M Methionine
59 3 16 S Serine
51 0 28 D Aspartic acid
28 1 6 H Histidine
31 5 4 C Cysteine
51 0 8 K Lycine
55 2 22 P Proline
47 4 24 T Threonine
43 4 8 Y Tyrosine
49 2 39 V Valine
26 1 64 A Alanine
48 0 34 R Arginine
49 0 18 E Glutamic acid
49 6 37 G Glycine
37 0 9 F Phenylalanine
11 0 4 W Tryptophan 

9FTN_1|Chain A|Isoform 2 of Ectonucleotide pyrophosphatase/phosphodiesterase family member 2|Rattus norvegicus (10116)
>7YHG_1|Chain A|Exoglucanase 1|Trichoderma reesei (51453)
>8PXK_1|Chain A|Ferredoxin reductase|Pseudomonas sp. KKS102 (307)
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 \(w=f(c)\)
9FTN , Knot 315 807 0.87 40 317 746 
GSCKGRCFELQEVGPPDCRCDNLCKSYSSCCHDFDELCLKTARGWECTKDRCGEVRNEENACHCSEDCLSRGDCCTNYQVVCKGESHWVDDDCEEIKVPECPAGFVRPPLIIFSVDGFRASYMKKGSKVMPNIEKLRSCGTHAPYMRPVYPTKTFPNLYTLATGLYPESHGIVGNSMYDPVFDASFHLRGREKFNHRWWGGQPLWITATKQGVRAGTFFWSVSIPHERRILTILQWLSLPDNERPSVYAFYSEQPDFSGHKYGPFGPEMTNPLREIDKTVGQLMDGLKQLRLHRCVNVIFVGDHGMEDVTCDRTEFLSNYLTNVDDITLVPGTLGRIRAKSINNSKYDPKTIIAALTCKKPDQHFKPYMKQHLPKRLHYANNRRIEDIHLLVDRRWHVARKPLDVYKKPSGKCFFQGDHGFDNKVNSMQTVFVGYGPTFKYRTKVPPFENIELYNVMCDLLGLKPAPNNGTHGSLNHLLRTNTFRPTMPDEVSRPNYPGIMYLQSEFDLGCTCDDKVEPKNKLEELNKRLHTKGSTKERHLLYGRPAVLYRTSYDILYHTDFESGYSEIFLMPLWTSYTISKQAEVSSIPEHLTNCVRPDVRVSPGFSQNCLAYKNDKQMSYGFLFPPYLSSSPEAKYDAFLVTNMVPMYPAFKRVWAYFQRVLVKKYASERNGVNVISGPIFDYNYDGLRDTEDEIKQYVEGSSIPVPTHYYSIITSCLDFTQPADKCDGPLSVSSFILPHRPDNDESCNSSEDESKWVEELMKMHTARVRDIEHLTGLDFYRKTSRSYSEILTLKTYLHTYESEI
7YHG , Knot 21 36 0.69 26 31 34 
TQCHYGQCGGIGYSGPTVCASGTTCQVLNPYYSQCL
8PXK , Knot 158 408 0.77 40 181 369 
MSQEALKAPVVVLGAGLASVSFVAELRQAGYQGLITVVGDEAERPYDRPPLSKDFMAHGDAEKIRLDCKRAPEVEWLLGVTAQSFDPQAHTVALSDGRTLPYGTLVLATGAAPRALPTLQGATMPVHTLRTLEDARRIQAGLRPQSRLLIVGGGVIGLELAATARTAGVHVSLVETQPRLMSRAAPATLADFVARYHAAQGVDLRFERSVTGSVDGVVLLDDGTRIAADMVVVGIGVLANDALARAAGLACDDGIFVDAYGRTTCPDVYALGDVTRQRNPLSGRFERIETWSNAQNQGIAVARHLVDPTAPGYAELPWYWSDQGALRIQVAGLASGDEEIVRGEVSLDAPKFTLIELQKGRIVGATCVNNARDFAPLRRLLAVGAKPDRAALADPATDLRKLAAAVAA

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 \(c\).

\(|P_{f(9FTN_1)}(2) \setminus P_{f(7YHG_1)}(2)|=291\), \(|P_{f(7YHG_1)}(2) \setminus P_{f(9FTN_1)}(2)|=5\). 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 1:100010010100111100000010000000000100101001011000000010100000100000001001000000011001000110000001011001111101111110101101001001001110100100010011010110100011010011011010001111001001110101010100010001111011110100011011011101011000011011011011000010101100001010100011111010011001000110110110010100010111110011001000000110001001001011110110101001000000100111110000100010101000110010010000100101110001011001101000101001101001100010010011110110100000111100101001100111101110010010100110000101011001001001111010001011000000101000100100010001000000110101111000000110000100100011111110000100010100110010001010101011100001100000010011111101000101000111100111101110011101001110001000011011011110000011000000100010100111100000110001010011000011101001111001000000000000001100110100101001001011010000000000110100010000001
Pair \(Z_2\) Length of longest common subsequence
9FTN_1,7YHG_1 296 3
9FTN_1,8PXK_1 192 4
7YHG_1,8PXK_1 176 3

Newick tree

 
[
	9FTN_1:13.77,
	[
		8PXK_1:88,7YHG_1:88
	]:46.77
]

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 \((0.85)(0.8)(\frac{843 }{\log_{20} 843}-\frac{36}{\log_{20}36})=234.\)
Status Protein1 Protein2 d d1/2
Query variables 9FTN_1 7YHG_1 304 158
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]