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Parikh vectors
3VXB_1 9JGD_1 6YRV_1 Letter Amino acid
35 25 49 L Leucine
48 9 70 A Alanine
28 13 35 D Aspartic acid
17 25 23 E Glutamic acid
16 1 25 Q Glutamine
28 17 44 V Valine
15 21 22 I Isoleucine
24 13 30 P Proline
32 6 42 S Serine
11 2 7 W Tryptophan
4 1 6 C Cysteine
48 10 67 G Glycine
3 8 11 H Histidine
11 6 9 M Methionine
21 8 17 F Phenylalanine
21 7 31 T Threonine
12 8 17 Y Tyrosine
11 16 32 R Arginine
19 11 17 N Asparagine
32 22 24 K Lycine 

3VXB_1|Chain A|Putative sugar-binding lipoprotein|Streptomyces thermoviolaceus (1952)
>9JGD_1|Chain A|Ribosomal RNA small subunit methyltransferase Nep1|Pyrococcus horikoshii OT3 (70601)
>6YRV_1|Chain A[auth AAA]|Fatty acid Photodecarboxylase|Chlorella variabilis (554065)
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 \(w=f(c)\)
3VXB , Knot 181 436 0.84 40 225 406 
MQSYSRRWFLGAGATTLISAAGLTACGSGSGSGGSGGTITAMVYGDDAVKVQDKAASRFNASAEAKKANAKVKMERIPASDYPAKLRTAMGSPNAPDIFFNWGGGSIKAYKEAGQLVDLTDVIKSDEVLSTGFLPSVVAAGSLDGHEYGIPMRGMQPVLLFYNKSVFAEHKLTPPTTWDQLLDNVAKLKKAGVTPFALGGVEIWPELMWLEYLVDRIGGPQVFDKIRNGDASGWGDPAVLKAAQTVKQLVDEGAFGKGFSSVSYNNGGAPALLAKGKAGMHLMGSWEYSTQLGKFPDFAKKDLGWCAFPSFEGGAGDIRNVVGNPCNYWSVNARTGNKDGAIAFLRDCASEAYTKDLIDNGDVPTTTIAENMLDSSPNPEFAKFQYQLVQKAPNFTLSWDQAVDPDWQQPMLTEINKLFVGKSSPEQFVSALKGLK
9JGD , Knot 103 229 0.81 40 154 218 
MEEKKRLHLIIADAELETVPPEILDHPAIVNYAKRRKKRPEKIILDSTYHHAALRQLEDGERRGRPDIVHICLLNALDSILNKEDRLRVYVHTRNDYVIYIKPETRLPRNYNRFIGLMENLFEKGAVPEDLELLRMEKKTLNELIEEINPDVVFIMHEEGELMIPKNFGKLLDKFKKPTVIVGGFPHGDFKSKVDGVKISLYREPLMAWTIVNEVIVSYEWEVIKKFKS
6YRV , Knot 228 578 0.83 40 256 529 
SPVAGQKYDYILVGGGTAACVLANRLSADGSKRVLVLEAGPDNTSRDVKIPAAITRLFRSPLDWNLFSELQEQLAERQIYMARGRLLGGSSATNATLYHRGAAGDYDAWGVEGWSSEDVLSWFVQAETNADFGPGAYHGSGGPMRVENPRYTNKQLHTAFFKAAEEVGLTPNSDFNDWSHDHAGYGTFQVMQDKGTRADMYRQYLKPVLGRRNLQVLTGAAVTKVNIDQAAGKAQALGVEFSTDGPTGERLSAELAPGGEVIMCAGAVHTPFLLKHSGVGPSAELKEFGIPVVSNLAGVGQNLQDQPACLTAAPVKEKYDGIAISDHIYNEKGQIRKRAIASYLLGGRGGLTSTGCDRGAFVRTAGQALPDLQVRFVPGMALDPDGVSTYVRFAKFQSQGLKWPSGITMQLIACRPQSTGSVGLKSADPFAPPKLSPGYLTDKDGADLATLRKGIHWARDVARSSALSEYLDGELFPGSGVVSDDQIDEYIRRSIHSSNAITGTCKMGNAGDSSSVVDNQLRVHGVEGLRVVDASVVPKIPGGQTGAPVVMIAERAAALLTGKATIGASAAAPATVAA

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 \(c\).

\(|P_{f(3VXB_1)}(2) \setminus P_{f(9JGD_1)}(2)|=133\), \(|P_{f(9JGD_1)}(2) \setminus P_{f(3VXB_1)}(2)|=62\). 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 1:1000000111111100110111101010101011011010111010011010001100101010100101010100111000110100111010110111011110101000110110100110000110011110111110101000111101101111100001110001011001001100110100111011111110111011110011001111011001001010111011110110010011001111011001000011111111010111011101000001101101100011101110101111010011101000101010010001111110001001000011001011000110011000101011010001100110101010011010100111001001111000100110110110
Pair \(Z_2\) Length of longest common subsequence
3VXB_1,9JGD_1 195 3
3VXB_1,6YRV_1 147 5
9JGD_1,6YRV_1 176 3

Newick tree

 
[
	9JGD_1:98.48,
	[
		3VXB_1:73.5,6YRV_1:73.5
	]:24.98
]

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 \((0.85)(0.8)(\frac{665 }{\log_{20} 665}-\frac{229}{\log_{20}229})=122.\)
Status Protein1 Protein2 d d1/2
Query variables 3VXB_1 9JGD_1 158 120
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]