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Parikh vectors
1MJT_1 3SUY_1 7CTZ_1 Letter Amino acid
21 24 13 A Alanine
4 0 17 M Methionine
14 0 26 S Serine
23 0 38 Y Tyrosine
26 0 63 N Asparagine
3 22 17 C Cysteine
24 0 49 I Isoleucine
29 0 50 L Leucine
21 0 26 T Threonine
7 0 6 W Tryptophan
18 0 41 D Aspartic acid
13 0 12 H Histidine
28 0 61 K Lycine
17 0 34 F Phenylalanine
13 0 18 R Arginine
9 0 19 Q Glutamine
26 0 39 E Glutamic acid
18 35 25 G Glycine
16 0 19 P Proline
17 0 35 V Valine 

1MJT_1|Chains A, B|NITRIC-OXIDE SYNTHASE HOMOLOG|Staphylococcus aureus (1280)
>3SUY_1|Chain A[auth X]|riboswitch|Eubacterium siraeum V10Sc8a (717961)
>7CTZ_1|Chains A, B|Bifunctional dihydrofolate reductase-thymidylate synthase|Plasmodium falciparum (5833)
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 \(w=f(c)\)
1MJT , Knot 151 347 0.84 40 210 336 
HHLFKEAQAFIENMYKECHYETQIINKRLHDIELEIKETGTYTHTEEELIYGAKMAWRNSNRCIGRLFWDSLNVIDARDVTDEASFLSSITYHITQATNEGKLKPYITIYAPKDGPKIFNNQLIRYAGYDNCGDPAEKEVTRLANHLGWKGKGTNFDVLPLIYQLPNESVKFYEYPTSLIKEVPIEHNHYPKLRKLNLKWYAVPIISNMDLKIGGIVYPTAPFNGWYMVTEIGVRNFIDDYRYNLLEKVADAFEFDTLKNNSFNKDRALVELNYAVYHSFKKEGVSIVDHLTAAKQFELFERNEAQQGRQVTGKWSWLAPPLSPTLTSNYHHGYDNTVKDPNFFYKK
3SUY , Knot 32 101 0.48 8 16 51 
GGACAGAGUAGGUAAACGUGCGUAAAGUGCCUGAGGGACGGGGAGUUGUCCUCAGGACGAACACCGAAAGGUGGCGGUACGUUUACCGCAUCUCGCUGUUC
7CTZ , Knot 239 608 0.84 40 265 548 
MMEQVCDVFDIYAICACCKVESKNEGKKNEVFNNYTFRGLGNKGVLPWKCNSLDMKYFCAVTTYVNESKYEKLKYKRCKYLNKETVDNVNDMPNSKKLQNVVVMGRTSWESIPKKFKPLSNRINVILSRTLKKEDFDEDVYIINKVEDLIVLLGKLNYYKCFIIGGSVVYQEFLEKKLIKKIYFTRINSTYECDVFFPEINENEYQIISVSDVYTSNNTTLDFIIYKKTNNKMLNEQNCIKGEEKNNDMPLKNDDKDTCHMKKLTEFYKNVDKYKINYENDDDDEEEDDFVYFNFNKEKEEKNKNSIHPNDFQIYNSLKYKYHPEYQYLNIIYDIMMNGNKQSDRTGVGVLSKFGYIMKFDLSQYFPLLTTKKLFLRGIIEELLWFIRGETNGNTLLNKNVRIWEANGTREFLDNRKLFHREVNDLGPIYGFQWRHFGAEYTNMYDNYENKGVDQLKNIINLIKNDPTSRRILLCAWNVKDLDQMALPPCHILCQFYVFDGKLSCIMYQRSCDLGLGVPFNIASYSIFTHMIAQVCNLQPAQFIHVLGNAHVYNNHIDSLKIQLNRIPYPFPTLKLNPDIKNIEDFTISDFTIQNYVHHEKISMDMAA

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 \(c\).

\(|P_{f(1MJT_1)}(2) \setminus P_{f(3SUY_1)}(2)|=205\), \(|P_{f(3SUY_1)}(2) \setminus P_{f(1MJT_1)}(2)|=11\). 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 1:00110010111001000000000110001001010100010000000011011011100000011011100101101001000101100100010010001010101010110011011000110011000010110001001100111010100101111100110001010001001100111000001010010101011111001010111110101110110110011100110000001100110110100100001000011101001100010001101100101100101100001001001010101111110101000000100001001011000
Pair \(Z_2\) Length of longest common subsequence
1MJT_1,3SUY_1 216 2
1MJT_1,7CTZ_1 161 4
3SUY_1,7CTZ_1 269 4

Newick tree

 
[
	3SUY_1:13.95,
	[
		1MJT_1:80.5,7CTZ_1:80.5
	]:52.45
]

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 \((0.85)(0.8)(\frac{448 }{\log_{20} 448}-\frac{101}{\log_{20}101})=104.\)
Status Protein1 Protein2 d d1/2
Query variables 1MJT_1 3SUY_1 149 89.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]