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Parikh vectors
3USM_1 7RBT_1 9GIY_1 Letter Amino acid
14 20 3 N Asparagine
0 8 2 C Cysteine
13 8 8 M Methionine
50 20 7 F Phenylalanine
25 13 9 P Proline
17 14 7 Y Tyrosine
54 25 21 A Alanine
21 27 10 R Arginine
12 28 3 D Aspartic acid
6 18 6 Q Glutamine
19 23 6 K Lycine
18 18 12 S Serine
16 4 3 W Tryptophan
27 17 6 T Threonine
24 28 4 E Glutamic acid
45 14 8 G Glycine
6 7 2 H Histidine
54 24 4 I Isoleucine
61 30 10 L Leucine
37 27 5 V Valine 

3USM_1|Chain A|Transporter|Aquifex aeolicus (63363)
>7RBT_1|Chain A|Isoform Gnas-2 of Guanine nucleotide-binding protein G(s) subunit alpha isoforms short|Homo sapiens (9606)
>9GIY_1|Chain A|Mitochondrial pyruvate carrier 1-like protein|Homo sapiens (9606)
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 \(w=f(c)\)
3USM , Knot 204 519 0.82 38 220 457 
MEVKREHWATRLGLILAMAGNAVGLGNFLRFPVQAAENGGGAFMIPYIIAFLLVGIPLMWIEWAMGRYGGAQGHGTTPAIFYLLWRNRFAKILGVFGLWIPLVVAIYYVYIESWTLGFAIKFLVGLVPEPPPNATDPDSILRPFKEFLYSYIGVPKGDEPILKPSLFAYIVFLITMFINVSILIRGISKGIERFAKIAMPTLFILAVFLVIRVFLLETPNGTAADGLNFLWTPDFEKLKDPGVWIAAVGQIFFTLSLGFGAIITYASYVRKDQDIVLSGLTAATLNEKAEVILGGSISIPAAVAFFGVANAVAIAKAGAFNLGFITLPAIFSQTAGGTFLGFLWFFLLFFAGLTSSIAIMQPMIAFLEDELKLSRKHAVLWTAAIVFFSAHLVMFLNKSLDEMDFWAGTIGVVFFGLTELIIFFWIFGADKAWEEINRGGIIKVPRIYYYVMRYITPAFLAVLLVVWAREYIPKIMEETHWTVWITRFYIIGLFLFLTFLVFLAERRRNHESAGTLVPR
7RBT , Knot 164 373 0.86 40 228 361 
MGCTLSAEDKAAVERSKMIEKQLQKDKQVYRATHRLLLLGAGESGKSTIVKQMRILHVNGFNGDSEKATKVQDIKNNLKEAIETIVAAMSNLVPPVELANPENQFRVDYILSVMNVPDFDFPPEFYEHAKALWEDEGVRACYERSNEYQLIDCAQYFLDKIDVIKQADYVPSDQDLLRCRVLTSGIFETKFQVDKVNFHMFDVGGQRDERRKWIQCFNDVTAIIFVVASSSYNMVIREDNQTNRLQEALNLFKSIWNNRWLRTISVILFLNKQDLLAEKVLAGKSKIEDYFPEFARYTTPEDATPEPGEDPRVTRAKYFIRDEFLRISTASGDGRHYCYPHFTCAVDTENIRRVFNDCRDIIQRMHLRQYELL
9GIY , Knot 69 136 0.83 40 105 131 
MARMAVLWRKMRDNFQSKEFREYVSSTHFWGPAFSWGLPLAAFKDMKASPEIISGRMTTALILYSAIFMRFAYRVQPRNLLLMACHCTNVMAQSVQASRYLLYYYGGGGAEAKARDPPATAAAATSPGSQPPKQAS

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 \(c\).

\(|P_{f(3USM_1)}(2) \setminus P_{f(7RBT_1)}(2)|=82\), \(|P_{f(7RBT_1)}(2) \setminus P_{f(3USM_1)}(2)|=90\). 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 1:101000011001111111110111110110111011001111111101111111111111101111001110101001111011100011011111111111111100101001011111011111110111010010011011001100011110100111010111011111011101011101100110011011110111111111101111001010110110111010100100111111111011101011111110010010000011101101101000101111101011111111111011111011110111101111100011101111111111111110001111011111100010100001111011111101011111000100101111011111111001111111111001100100111101101000110010111111111111000110110000101110010111111110111111000000001101110
Pair \(Z_2\) Length of longest common subsequence
3USM_1,7RBT_1 172 4
3USM_1,9GIY_1 165 4
7RBT_1,9GIY_1 195 3

Newick tree

 
[
	7RBT_1:94.86,
	[
		3USM_1:82.5,9GIY_1:82.5
	]:12.36
]

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 \((0.85)(0.8)(\frac{892 }{\log_{20} 892}-\frac{373}{\log_{20}373})=139.\)
Status Protein1 Protein2 d d1/2
Query variables 3USM_1 7RBT_1 170 151.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]