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Parikh vectors
6ACW_1 8AOB_1 3NKH_1 Letter Amino acid
13 24 15 D Aspartic acid
12 21 16 E Glutamic acid
10 28 19 I Isoleucine
20 20 1 P Proline
4 3 2 W Tryptophan
22 15 10 V Valine
20 19 10 R Arginine
3 7 1 C Cysteine
19 17 16 G Glycine
5 19 17 H Histidine
14 13 10 F Phenylalanine
19 19 17 N Asparagine
12 12 9 Q Glutamine
9 23 20 K Lycine
21 16 22 S Serine
32 14 13 T Threonine
25 26 9 A Alanine
15 43 21 L Leucine
11 10 10 M Methionine
12 19 6 Y Tyrosine 

6ACW_1|Chain A|VP1|Coxsackievirus A10 (42769)
>8AOB_1|Chain A|Mitogen-activated protein kinase 1|Homo sapiens (9606)
>3NKH_1|Chains A, B|Integrase|Staphylococcus aureus subsp. aureus USA300_TCH1516 (451516)
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 \(w=f(c)\)
6ACW , Knot 129 298 0.82 40 189 288 
GDPVEDIIHDALGNTARRAISSATNVESAANTTPSSHRLETGRVPALQAAETGATSNATDENMIETRCVVNRNGVLETTINHFFSRSGLVGVVNLTDGGTDTTGYATWDIDIMGFVQLRRKCEMFTYMRFNAEFTFVTTTENGGARPYMLQYMYVPPGAPKPTGRDAFQWQTATNPSVFVKLTDPPAQVSVPFMSPASAYQWFYDGYPTFGQHPETSNTTYGLCPNNMMGTFAVRVVSREASQLKLQTRVYMKLKHVRAWVPRPIRSQPYLLKNFPNYDSSKITNSARDRSSIKQANM
8AOB , Knot 160 368 0.85 40 236 353 
MAHHHHHHMAAAAAAGAGPEMVRGQVFDVGPRYTNLSYIGEGAYGMVCSAYDNVNKVRVAIKKISPFEHQTYCQRTLREIKILLRFRHENIIGINDIIRAPTIEQMKDVYIVQDLMETDLYKLLKTQHLSNDHICYFLYQILRGLKYIHSANVLHRDLKPSNLLLNTTCDLKICDFGLARVADPDHDHTGFLTEYVATRWYRAPEIMLNSKGYTKSIDIWSVGCILAEMLSNRPIFPGKHYLDQLNHILGILGSPSQEDLNCIINLKARNYLLSLPHKNKVPWNRLFPNADSKALDLLDKMLTFNPHKRIEVEQALAHPYLEQYYDPSDEPIAEAPFKFDMELDDLPKEKLKELIFEETARFQPGYRS
3NKH , Knot 110 244 0.82 40 162 236 
MHHHHHHSSGVDLGTENLYFQSNAYLELNEIESIIKDINTKAQKMHSGIHKRFYLFVALMTEFQALNGMRIGEMLAIQNEDIDFDNKSLNINGTIHWFHDESGGFGVKDTTKTESSYRTIGLSSRSCEILKKAILENKKDSKWNDGYLNRNFVFTNHKGNPMQTERFNKILREAAKDVGIDKEVSSHILRHSHISLLSQQGVSLKAIMDRVGHSDHRTTLSIYSHVTEQMDKDMMNKLEQVKLG

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 \(c\).

\(|P_{f(6ACW_1)}(2) \setminus P_{f(8AOB_1)}(2)|=71\), \(|P_{f(8AOB_1)}(2) \setminus P_{f(6ACW_1)}(2)|=118\). 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 1:1011001100111001001100100100110001000010010111101100110001000011000011000111000100110001111110100110000101010101111101000001100101010101100000111010110010111111010100110100100101110100111010111101101001100101011001000000011010011101110110001001010001010100101111011000101100110000001000100000100101
Pair \(Z_2\) Length of longest common subsequence
6ACW_1,8AOB_1 189 3
6ACW_1,3NKH_1 177 3
8AOB_1,3NKH_1 174 6

Newick tree

 
[
	6ACW_1:93.01,
	[
		3NKH_1:87,8AOB_1:87
	]:6.01
]

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 \((0.85)(0.8)(\frac{666 }{\log_{20} 666}-\frac{298}{\log_{20}298})=102.\)
Status Protein1 Protein2 d d1/2
Query variables 6ACW_1 8AOB_1 133 119.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]