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Parikh vectors
4JDT_1 2VEC_1 1RFP_1 Letter Amino acid
7 4 2 M Methionine
18 15 10 S Serine
20 10 6 V Valine
13 12 7 D Aspartic acid
26 11 6 K Lycine
7 12 11 R Arginine
36 9 14 N Asparagine
18 3 8 C Cysteine
15 18 3 Q Glutamine
29 20 12 G Glycine
11 8 3 F Phenylalanine
17 12 2 P Proline
29 11 7 T Threonine
14 25 12 A Alanine
7 10 3 Y Tyrosine
7 3 6 W Tryptophan
16 16 1 H Histidine
30 10 6 I Isoleucine
20 34 8 L Leucine
21 13 2 E Glutamic acid 

4JDT_1|Chain A[auth G]|gp120|Human Immunodeficiency Virus 1 (11676)
>2VEC_1|Chain A|PIRIN-LIKE PROTEIN YHAK|ESCHERICHIA COLI (562)
>1RFP_1|Chain A|LYSOZYME|Gallus gallus (9031)
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 \(w=f(c)\)
4JDT , Knot 151 361 0.82 40 206 334 
VWKDADTTLFCASDAKAHETECHNVWATHACVPTDPNPQEIHLENVTENFNMWKNNMVEQMQEDVISLWDQCLQPCVKLTGGSVIKQACPKISFDPIPIHYCTPAGYVILKCNDKNFNGTGPCKNVSSVQCTHGIKPVVSTQLLLNGSLAEEEIIIRSENLTNNAKTIIVHLNKSVEINCTRPSNGGSGSGGDIRKAYCEINGTKWNKVLKQVTEKLKEHFNNKTIIFQPPSGGDLEITMHHFNCRGEFFYCNTTQLFNNTCIGNETMKGCNGTITLPCKIKQIINMWQGTGQAMYAPPIDGKINCVSNITGILLTRDGGANNTSNETFRPGGGNIKDNWRSELYKYKVVQIEGSHHHHHH
2VEC , Knot 116 256 0.83 40 166 248 
MHHHHHHGTSLYKKAGSENLYFQGITTRTARQCGQADYGWLQARYTFSFGHYFDPKLLGYASLRVLNQEVLAPGAAFQPRTYPKVDILNVILDGEAEYRDSEGNHVQASAGEALLLSTQPGVSYSEHNLSKDKPLTRMQLWLDACPQRENPLIQKLALNMGKQQLIASPEGAMGSLQLRQQVWLHHIVLDKGESANFQLHGPRAYLQSIHGKFHALTHHEEKAALTCGDGAFIRDEANITLVADSPLRALLIDLPV
1RFP , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL

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 \(c\).

\(|P_{f(4JDT_1)}(2) \setminus P_{f(2VEC_1)}(2)|=106\), \(|P_{f(2VEC_1)}(2) \setminus P_{f(4JDT_1)}(2)|=66\). 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 1:1100100011010010100000001110010110010100101001000101100011001000110110001010101011011001010101011110000111011100000010101100010010000110111000111010110001110000100010011101000101000010011010110100100010100100110010001000100001110110110101010010001011000000110000110001010010101100100110110101011011110101001001011110001110000000101111010001000100001101010000000
Pair \(Z_2\) Length of longest common subsequence
4JDT_1,2VEC_1 172 6
4JDT_1,1RFP_1 186 3
2VEC_1,1RFP_1 178 3

Newick tree

 
[
	1RFP_1:92.63,
	[
		4JDT_1:86,2VEC_1:86
	]:6.63
]

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 \((0.85)(0.8)(\frac{617 }{\log_{20} 617}-\frac{256}{\log_{20}256})=101.\)
Status Protein1 Protein2 d d1/2
Query variables 4JDT_1 2VEC_1 126 108.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]