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Parikh vectors
6CPV_1 3DHH_1 6AEJ_1 Letter Amino acid
2 44 31 A Alanine
4 25 24 K Lycine
4 23 24 P Proline
10 22 26 V Valine
2 26 25 R Arginine
4 33 30 D Aspartic acid
0 4 13 C Cysteine
2 39 41 E Glutamic acid
1 15 22 H Histidine
0 21 10 M Methionine
10 24 15 F Phenylalanine
3 21 25 Q Glutamine
8 33 23 G Glycine
11 24 19 I Isoleucine
13 34 50 L Leucine
9 22 21 T Threonine
3 23 14 Y Tyrosine
3 17 16 N Asparagine
6 27 29 S Serine
1 23 12 W Tryptophan 

6CPV_1|Chains A, B|Potassium channel protein|Bacillus cereus (1396)
>3DHH_1|Chain A|toluene 4-monooxygenase hydroxylase alpha subunit|Pseudomonas mendocina (300)
>6AEJ_1|Chain A|Alpha-ketoglutarate-dependent dioxygenase FTO,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 \(w=f(c)\)
6CPV , Knot 50 96 0.79 36 77 93 
WKDKEFQVLFVLTILTLISGTIFYSTVEGLRPIDALYFSVVTLTTVGDGNFSPQTDFGKIFTILYIFIGIGLVFGFIHKLAVNVQLPSILSNLVPR
3DHH , Knot 210 500 0.87 40 266 483 
MAMHPRKDWYELTRATNWTPSYVTEEQLFPERMSGHMGIPLEKWESYDEPYKTSYPEYVSIQREKDAGAYSVKAALERAKIYENSDPGWISTLKSHYGAIAVGEYAAVTGEGRMARFSKAPGNRNMATFGMMDELRHGQLQLFFPHEYCKKDRQFDWAWRAYHSNEWAAIAAKHFFDDIITGRDAISVAIMLTFSFETGFTNMQFLGLAADAAEAGDYTFANLISSIQTDESRHAQQGGPALQLLIENGKREEAQKKVDMAIWRAWRLFAVLTGPVMDYYTPLEDRSQSFKEFMYEWIIGQFERSLIDLGLDKPWYWDLFLKDIDELHHSYHMGVWYWRTTAWWNPAAGVTPEERDWLEEKYPGWNKRWGRCWDVITENVLNDRMDLVSPETLPSVCNMSQIPLVGVPGDDWNIEVFSLEHNGRLYHFGSEVDRWVFQQDPVQYQNHMNIVDRFLAGQIQPMTLEGALKYMGFQSIEEMGKDAHDFAWADKCKPAMKKSA
6AEJ , Knot 201 470 0.87 40 260 454 
MGSSHHHHHHSSGLVPRGSHMTPKDDEFYQQWQLKYPKLILREASSVSEELHKEVQEAFLTLHKHGCLFRDLVRIQGKDLLTPVSRILIGNPGCTYKYLNTRLFTVPWPVKGSNIKHTEAEIAAACETFLKLNDYLQIETIQALEELAASNGQDEVDIKSRAAYNVTLLNFMDPQKMPYLKEEPYFGMGKMAVSWHHDENLVDRSAVAVYSYSCEGPEEESEDDSHLEGRDPDIWHVGFKISWDIETPGLAIPLHQGDCYFMLDDLNATHQHCVLAGSQPRFSSTHRVAECSTGTLDYILQRCQLALQNVCDDVDNDDVSLKSFEPAVLKQGEEIHNEVEFEWLRQFWFQGNRYRKCTDWWCQPMAQLEALWKKMEGVTNAVLHEVKREGLPVEQRNEILTAILASLTARQNLRREWHARCQSRIARTLPADQKPECRPYWEKDDASMPLPFDLTDIVSELRGQLLEAKP

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 \(c\).

\(|P_{f(6CPV_1)}(2) \setminus P_{f(3DHH_1)}(2)|=24\), \(|P_{f(3DHH_1)}(2) \setminus P_{f(6CPV_1)}(2)|=213\). 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 1:100001011111011011010110001011011011010110100110101010001101101101111111111110011101011011001110
Pair \(Z_2\) Length of longest common subsequence
6CPV_1,3DHH_1 237 3
6CPV_1,6AEJ_1 223 4
3DHH_1,6AEJ_1 162 3

Newick tree

 
[
	6CPV_1:12.34,
	[
		6AEJ_1:81,3DHH_1:81
	]:43.34
]

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 \((0.85)(0.8)(\frac{596 }{\log_{20} 596}-\frac{96}{\log_{20}96})=147.\)
Status Protein1 Protein2 d d1/2
Query variables 6CPV_1 3DHH_1 194 114.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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