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Parikh vectors
6LXG_1 5OZH_1 5GQM_1 Letter Amino acid
6 21 11 D Aspartic acid
1 9 9 Q Glutamine
4 5 15 E Glutamic acid
9 19 13 L Leucine
3 11 12 K Lycine
12 23 20 V Valine
0 5 3 W Tryptophan
2 13 22 Y Tyrosine
7 1 14 R Arginine
3 6 24 N Asparagine
0 2 2 C Cysteine
4 38 15 G Glycine
6 49 26 S Serine
5 47 4 T Threonine
9 29 16 A Alanine
2 20 8 F Phenylalanine
9 1 9 H Histidine
3 18 13 I Isoleucine
3 0 2 M Methionine
2 13 9 P Proline 

6LXG_1|Chains A, B|GTP pyrophosphokinase|Mycobacterium tuberculosis (1773)
>5OZH_1|Chain A|endothiapepsin|Cryphonectria parasitica (5116)
>5GQM_1|Chain A|Polyhedrin|Bombyx mori cypovirus 1 (110829)
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 \(w=f(c)\)
6LXG , Knot 44 90 0.73 36 68 84 
MKHHHHHHPMVLVAIQVEALDRHRLLSDVTRALADEKVNILSASVTTSGDRVAISRFTFEMGDPKHLGHLLNAVRNVEGVYDVYRVTSAA
5OZH , Knot 132 330 0.77 38 160 297 
STGSATTTPIDSLDDAYITPVQIGTPAQTLNLDFDTGSSDLWVFSSETTASEVDGQTIYTPSKSTTAKLLSGATWSISYGDGSSSSGDVYTDTVSVGGLTVTGQAVESAKKVSSSFTEDSTIDGLLGLAFSTLNTVSPTQQKTFFDNAKASLDSPVFTADLGYHAPGTYNFGFIDTTAYTGSITYTAVSTKQGFWEWTSTGYAVGSGTFKSTSIDGIADTGTTLLYLPATVVSAYWAQVSGAKSSSSVGGYVFPCSATLPSFTFGVGSARIVIPGDYIDFGPISTGSSSCFGGIQSSAGIGINIFGDVALKAAFVVFNGATTPTLGFASK
5GQM , Knot 111 248 0.82 42 170 242 
XADVAGTSNRDFRGREQRLFNSEQYNYNNSLNGEVSVWVYAYYSDGSVLVINKNSQYKVGISETFKALKEYREGQHNDSYDEYEVNQSIYYPNGGDARKFHSNAKPRAIQIIFSPSVNVRTIKMAKGNAVSVPDEYLQRSHPWEATGIKYRKIKRDGEIVGYSHYFELPHEYNSISLAVSGVHKNPSSYNVGSAHNVMDVFQSCDLALRFCNRYWAELELVNHYISPNAYPYLDINNHSYGVALSNRQ

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 \(c\).

\(|P_{f(6LXG_1)}(2) \setminus P_{f(5OZH_1)}(2)|=36\), \(|P_{f(5OZH_1)}(2) \setminus P_{f(6LXG_1)}(2)|=128\). 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 1:100000001111111010110000110010011100010110101000100111001010110100110110110010110010010011
Pair \(Z_2\) Length of longest common subsequence
6LXG_1,5OZH_1 164 3
6LXG_1,5GQM_1 162 3
5OZH_1,5GQM_1 178 4

Newick tree

 
[
	5OZH_1:87.04,
	[
		6LXG_1:81,5GQM_1:81
	]:6.04
]

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 \((0.85)(0.8)(\frac{420 }{\log_{20} 420}-\frac{90}{\log_{20}90})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 6LXG_1 5OZH_1 122 77.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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