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Parikh vectors
1IMO_1 1BXX_1 6HGR_1 Letter Amino acid
3 5 3 C Cysteine
3 19 13 E Glutamic acid
6 10 6 T Threonine
2 8 4 Y Tyrosine
8 19 12 R Arginine
7 11 9 D Aspartic acid
1 10 1 M Methionine
5 11 8 F Phenylalanine
6 15 11 P Proline
6 23 11 S Serine
2 4 1 W Tryptophan
7 11 15 A Alanine
3 29 8 K Lycine
4 17 17 G Glycine
1 9 2 H Histidine
3 23 7 I Isoleucine
9 20 29 L Leucine
7 23 13 V Valine
1 9 1 N Asparagine
4 9 7 Q Glutamine 

1IMO_1|Chain A|DNA LIGASE III|Homo sapiens (9606)
>1BXX_1|Chain A|PROTEIN (AP50)|Rattus norvegicus (10116)
>6HGR_1|Chains A, B|Adenine phosphoribosyltransferase|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 \(w=f(c)\)
1IMO , Knot 49 88 0.83 40 77 86 
GSADETLCQTKVLLDIFTGVRLYLPPSTPDFSRLRRYFVAFDGDLVQEFDMTSATHVLGSRDKNPAAQQVSPEWIWACIRKRRLVAPC
1BXX , Knot 128 285 0.84 40 183 276 
MHHHHHHQIGWRREGIKYRRNELFLDVLESVNLLMSPQGQVLSAHVSGRVVMKSYLSGMPECKFGMNDKIVIEKQGKGTADETSKSGKQSIAIDDCTFHQCVRLSKFDSERSISFIPPDGEFELMRYRTTKDIILPFRVIPLVREVGRTKLEVKVVIKSNFKPSLLAQKIEVRIPTPLNTSGVQVICMKGKAKYKASENAIVWKIKRMAGMKESQISAEIELLPTNDKKKWARPPISMNFEVPFAPSGLKVRYLKVFEPKLNYSDHDVIKWVRYIGRSGIYETRC
6HGR , Knot 83 178 0.80 40 120 173 
DSELQLVEQRIRSFPDFPTPGVVFRDISPVLKDPASFRAAIGLLARHLKATHGGRIDYIAGLDSRGFLFGPSLAQELGLGCVLIRKRGKLPGPTLWASYSLEYGKAELEIQKDALEPGQRVVVVDDLLATGGTMNAACELLGRLQAEVLECVSLVELTSLKGREKLAPVPFFSLLQYE

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 \(c\).

\(|P_{f(1IMO_1)}(2) \setminus P_{f(1BXX_1)}(2)|=40\), \(|P_{f(1BXX_1)}(2) \setminus P_{f(1IMO_1)}(2)|=146\). 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 1:1010001000011101101101011100101001000111101011001010010011100000111001010111101000011110
Pair \(Z_2\) Length of longest common subsequence
1IMO_1,1BXX_1 186 4
1IMO_1,6HGR_1 129 4
1BXX_1,6HGR_1 161 3

Newick tree

 
[
	1BXX_1:93.27,
	[
		1IMO_1:64.5,6HGR_1:64.5
	]:28.77
]

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 \((0.85)(0.8)(\frac{373 }{\log_{20} 373}-\frac{88}{\log_{20}88})=88.2\)
Status Protein1 Protein2 d d1/2
Query variables 1IMO_1 1BXX_1 113 73.5
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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