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Parikh vectors
7UYY_1 4OBT_1 2PMQ_1 Letter Amino acid
25 9 21 P Proline
8 6 7 W Tryptophan
42 34 23 V Valine
7 4 8 C Cysteine
47 23 38 G Glycine
9 3 8 M Methionine
54 16 38 L Leucine
18 9 9 F Phenylalanine
25 11 19 T Threonine
26 11 22 D Aspartic acid
8 8 9 Q Glutamine
19 13 22 I Isoleucine
17 3 18 H Histidine
66 31 44 A Alanine
38 8 23 R Arginine
13 12 7 N Asparagine
8 4 9 Y Tyrosine
39 19 25 E Glutamic acid
12 18 8 K Lycine
33 15 19 S Serine 

7UYY_1|Chains A, B|Aldehyde dehydrogenase|Pseudomonas aeruginosa PAO1 (208964)
>4OBT_1|Chains A, B|Triosephosphate isomerase, cytosolic|Arabidopsis thaliana (3702)
>2PMQ_1|Chains A, B|Mandelate racemase/muconate lactonizing enzyme|Roseovarius sp. (314265)
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 \(w=f(c)\)
7UYY , Knot 202 514 0.81 40 226 458 
MGSSHHHHHHSSGLVPRGMTELLSREEYAAIAATLDLPQRAFIDGGFRDACGGRTFASTNPATGELLAQVAACDAEDVGHAVASARRAFEDGRWRSRTPAERKAALLRLAELLEEHLLELAVMESLDSGKPIRECQHTDLPETINTLRWHAELIDKIYDSTAPVGSAALAMVVREPIGVVGLVLPWNFPLLMLAWKIGPALAAGCSVVVKPAPETSLTALRVAELASQAGIPAGVFNVVPGGGREAGEPLGRHPDVAMVSFTGSTATGRLFLKYAAESNLKRVVLECGGKNPAVVMNDVEDLDLVAQHVVNGAFWNMGENCSASSRLIVHAEVREALLERIGAQLREWRMGDPLDPRNRLGALVSPAHFEKVRAYLDQARTERLAVRFGGATEAGIFVEPTVVDGVSPHSRLFREEIFGPLLSVTSFDDIDEAIALANDTVYGLAASAYTGSLRHALRLSREIRAGIVTVNCFGEGDASTPFGGYKESGFGGRDKSVWAHDQYTELKTIWIDAS
4OBT , Knot 113 257 0.81 40 160 238 
GPHMARKFFVGGNWKCNGTAEEVKKIVNTLNEAQVPSQDVVEVVVSPPYVFLPLVKSTLRSDFFVAAQNCWVKKGGAFTGEVSAEMLVNLDIPWVILGHSERRAILNESSEFVGDKVAYALAQGLKVIACVGETLEEREAGSTMDVVAAQTKAIADRVTNWSNVVIAYEPVWAIGTGKVASPAQAQEVHDELRKWLAKNVSADVAATTRIIYGGSVNGGNCKELGGQADVDGFLVGGASLKPEFIDIIKAAEVKKSA
2PMQ , Knot 156 377 0.81 40 205 353 
MSLKIAEIQLFQHDLPVVNGPYRIASGDVWSLTTTIVKIIAEDGTIGWGETCPVGPTYAEAHAGGALAALEVLASGLAGAEALPLPLHTRMDSLLCGHNYAKSALDIAVHDLWGKRLGVPVHELLGGALTDSVSSYYSLGVMEPDEAARQALEKQREGYSRLQVKLGARPIEIDIEAIRKVWEAVRGTGIALAADGNRGWTTRDALRFSRECPDIPFVMEQPCNSFEDLEAIRPLCHHALYMDEDGTSLNTVITAAATSLVDGFGMKVSRIGGLQHMRAFRDFCAARNLPHTCDDAWGGDIVSAACTHIASTVLPRLMEGAWLAQPYVAEHYDAENGVRIEGGRIRVPQGPGLGLTIDPERFGPPLFSAEGHHHHHH

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 \(c\).

\(|P_{f(7UYY_1)}(2) \setminus P_{f(4OBT_1)}(2)|=114\), \(|P_{f(4OBT_1)}(2) \setminus P_{f(7UYY_1)}(2)|=48\). 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 1:1100000000001111011001100000111110101100111011100101100110001101011101110010011011101001100101000011000111101101100011011110010010110000000110010010101011001000011110111111100111111111110111111110111111110011101110001011011011001111111101111110011011100101111010100101011100110001001110011001111100100101110011011110110000100011101010011100111010010110110100011111011010010101001000011101111001111101011011010001100011111101001001001111100010111101001010011010001011110100110101001111000011110000111000000100111010
Pair \(Z_2\) Length of longest common subsequence
7UYY_1,4OBT_1 162 4
7UYY_1,2PMQ_1 145 6
4OBT_1,2PMQ_1 167 4

Newick tree

 
[
	4OBT_1:85.26,
	[
		7UYY_1:72.5,2PMQ_1:72.5
	]:12.76
]

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 \((0.85)(0.8)(\frac{771 }{\log_{20} 771}-\frac{257}{\log_{20}257})=141.\)
Status Protein1 Protein2 d d1/2
Query variables 7UYY_1 4OBT_1 177 131.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]