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Parikh vectors
3AGK_1 5BTW_1 5RGM_1 Letter Amino acid
30 4 17 A Alanine
15 1 17 D Aspartic acid
2 2 12 C Cysteine
13 1 14 Q Glutamine
39 12 9 E Glutamic acid
28 11 26 G Glycine
30 12 27 V Valine
20 12 11 I Isoleucine
30 16 11 K Lycine
15 10 17 F Phenylalanine
14 10 13 P Proline
3 1 3 W Tryptophan
10 2 11 Y Tyrosine
23 0 11 R Arginine
7 11 21 N Asparagine
3 11 7 H Histidine
45 12 29 L Leucine
14 1 10 M Methionine
17 10 16 S Serine
15 7 24 T Threonine 

3AGK_1|Chain A|Peptide chain release factor subunit 1|Aeropyrum pernix (56636)
>5BTW_1|Chains A, B|Uncharacterized protein|Legionella pneumophila (272624)
>5RGM_1|Chain A|3C-like proteinase|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
3AGK , Knot 156 373 0.82 40 201 349 
MSQGRLEERLTISKRELARLLKELKKWSAPATVLLSLYIPPGRPLSDVMTLLRQEYSITDNIKLKRTRQAVKRALSAAMDRLQMLTSTPPNGLVLFCGEDMSTGKFECFMFSPPEPIRVFYYRTDKRFITDFLEDMVEDNNAIGIIIVERDQATIGLLKGARLEVLKELEGFVPGKHKMGGQSQRRYERIIEQMVDEFFKKVGEEASNLLVPLAEKGVLKGVIVAGPGLAKQEFVEGNYLDYRLKKILAPELVDVAYQGLQGLKEAVMKAEKVVEAQMYRDAVNAMEEFKLHLAKGTGMIVYGEKDVEAALEMGAVKTLLIHESREDLEEWVEKAKSSGAQVIVVPESLAEAEWFLKTFGGLAGILRFRISTV
5BTW , Knot 67 146 0.76 38 107 139 
MVTKIIWVSNNGKPNLKIEFVSEEEKSNFFKEVKKKASELGLNFPLVQGSGNSLLIEASNYPINPCGCYISPGGKLAINFGKVELSHFILPKVGVKTEHAEIFKDHNTIFFHKHKLPGVNSELTFIPTGTPVIVPVTKLEHHHHHH
5RGM , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ

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 \(c\).

\(|P_{f(3AGK_1)}(2) \setminus P_{f(5BTW_1)}(2)|=129\), \(|P_{f(5BTW_1)}(2) \setminus P_{f(3AGK_1)}(2)|=35\). 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 1:1001010001010000110110010010111011101011110110011011000001000101000001100110111001011000110111110100100101001110110110110000000110011001100001111111000010111101101011001011111000111000000001100110011001100100111111001110111111111100011010010001001111011011001101100111010011010100011011001010110101111010001011101111001110000001001100100011011111001101011100111111110101001
Pair \(Z_2\) Length of longest common subsequence
3AGK_1,5BTW_1 164 4
3AGK_1,5RGM_1 183 4
5BTW_1,5RGM_1 177 4

Newick tree

 
[
	5RGM_1:92.52,
	[
		3AGK_1:82,5BTW_1:82
	]:10.52
]

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 \((0.85)(0.8)(\frac{519 }{\log_{20} 519}-\frac{146}{\log_{20}146})=109.\)
Status Protein1 Protein2 d d1/2
Query variables 3AGK_1 5BTW_1 136 93
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]