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Parikh vectors
6HNU_1 1QMV_1 5ZMD_1 Letter Amino acid
47 17 48 L Leucine
6 0 10 M Methionine
17 10 25 R Arginine
27 16 30 D Aspartic acid
18 4 23 Q Glutamine
36 11 42 E Glutamic acid
23 10 19 I Isoleucine
28 10 20 T Threonine
35 14 33 A Alanine
4 3 14 C Cysteine
12 3 15 H Histidine
30 11 23 P Proline
34 10 26 S Serine
14 7 17 N Asparagine
36 18 22 G Glycine
29 12 16 F Phenylalanine
7 2 12 W Tryptophan
37 14 26 K Lycine
17 7 15 Y Tyrosine
34 18 27 V Valine 

6HNU_1|Chains A, B|Aromatic amino acid aminotransferase I|Candida albicans WO-1 (294748)
>1QMV_1|Chains A, B, C, D, E, F, G, H, I, J|PEROXIREDOXIN-2|HOMO SAPIENS (9606)
>5ZMD_1|Chains A, C, E, G|Alpha-ketoglutarate-dependent dioxygenase FTO|Homo sapiens (9606)
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 \(w=f(c)\)
6HNU , Knot 204 491 0.85 40 250 461 
MTSDTKPQAKDLTHLLSNESKARQTSPLKGIFKYYKQPGITFLGGGLPLSDYFPFEKVTADIPTPSFSGGIGAPIEGENKTTIEVFKKAADNVPDQIELARSLQYGSTFGLPEFLQFIKEHTDMVHKVPYENWDVIVSVGNTEAWDSTLRTFCSKGDTILVEEYTFSSALESANGQGVNTVPVTMDEFGIIPEKLEELMSRWVGNKPKFLYTICTGQNPTGSSLSAERRKQIYDIACKYDFLIIEDEPYYFLQMETYTKDKAAREGKAVHDHDEFLKALVPSFISLDVEGRVVRLDSFSKVLAPGLRLGWIVGQKDLLERYVRLHEVSVQNPSGFSEALANALLRKWGHSGYLDWLIGLRAEYTHKRDVAIDALDQFVPKEVSSFNPPVAGMFFTVTLDASKHPKYKEFLEDPLKVEAAVHEQAIKQGCLLAPGSWFKAEGQSSPPQKNLPANPSHKTHIFFRGTYAAVPLDQLVVGLEKFGKAVRAEFGL
1QMV , Knot 93 197 0.83 38 143 191 
ASGNARIGKPAPDFKATAVVDGAFKEVKLSDYKGKYVVLFFYPLDFTFVCPTEIIAFSNRAEDFRKLGCEVLGVSVDSQFTHLAWINTPRKEGGLGPLNIPLLADVTRRLSEDYGVLKTDEGIAYRGLFIIDGKGVLRQITVNDLPVGRSVDEALRLVQAFQYTDEHGEVCPAGWKPGSDTIKPNVDDSKEYFSKHN
5ZMD , Knot 198 463 0.87 40 262 450 
EFYQQWQLKYPKLILREASSVSEELHKEVQEAFLTLHKHGCLFRDLVRIKGKDLLTPVSRILIGNPGCTYKYLNTRLFTVPWPVKGSNIKHTEAEIAAACETFLKLNDYLQIETIQALEELAAKEKANEDAVPLCMSADFPRVGMGSSYNGQDEVDIKSRAAYNVTLLNFMDPQKMPYLKEEPYFGMGKMAVSWHHDENLVDRSAVAVYSYSCEGPEEESEDDSHLEGRDPDIWHVGFKISWDIETPGLAIPLHQGDCYFMLDDLNATHKHCVLAGSQPRFSSTHRVAECSTGTLDYILQRCQLALQNVCDDVDNDDVSLKSFEPAVLKQGEEIHNEVEFEWLRQFWFQGNRYRKCTDWWCQPMAQLEALWKKMEGVTNAVLHEVKREGLPVEQRNEILTAILASLTARQNLRREWHARCQSRIARTLPADQKPECRPYWEKDDASMPLPFDLTDIVSELRGQ

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 \(c\).

\(|P_{f(6HNU_1)}(2) \setminus P_{f(1QMV_1)}(2)|=137\), \(|P_{f(1QMV_1)}(2) \setminus P_{f(6HNU_1)}(2)|=30\). 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 1:10000010100100110000010000110111000001110111111110001110010101101010111111101000001011001100110010110010010011110110110000011001100010111011000110001001000100111000010011001010110011101001111100100110011100101100100100101001010000010011000011110001001101000000011001011000001101111011010101011010010011111101111110001100010100101001011001110111001100101011111010000000111011001110010010111111110101010001000011001101011100011001011111011010100011000111010000011101001111100111110011011010111
Pair \(Z_2\) Length of longest common subsequence
6HNU_1,1QMV_1 167 4
6HNU_1,5ZMD_1 162 4
1QMV_1,5ZMD_1 185 3

Newick tree

 
[
	1QMV_1:90.36,
	[
		6HNU_1:81,5ZMD_1:81
	]:9.36
]

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 \((0.85)(0.8)(\frac{688 }{\log_{20} 688}-\frac{197}{\log_{20}197})=138.\)
Status Protein1 Protein2 d d1/2
Query variables 6HNU_1 1QMV_1 177 121
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]