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Parikh vectors
6PFB_1 6DAI_1 7CRP_1 Letter Amino acid
9 10 1 C Cysteine
10 8 2 H Histidine
46 27 15 L Leucine
33 33 12 K Lycine
15 0 0 M Methionine
19 19 17 A Alanine
28 3 18 R Arginine
33 17 1 N Asparagine
23 11 6 P Proline
7 10 0 W Tryptophan
25 9 3 Y Tyrosine
32 20 8 G Glycine
24 21 10 T Threonine
32 22 4 D Aspartic acid
17 6 8 Q Glutamine
38 8 7 E Glutamic acid
24 21 6 V Valine
45 19 7 I Isoleucine
26 11 4 F Phenylalanine
35 38 6 S Serine 

6PFB_1|Chains A, B, C, D, E|Bifunctional dihydrofolate reductase-thymidylate synthase|Cryptosporidium hominis (237895)
>6DAI_1|Chain A|WD repeat-containing protein 5|Homo sapiens (9606)
>7CRP_1|Chains A[auth E], E[auth M]|Histone H3|Xenopus laevis (8355)
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 \(w=f(c)\)
6PFB , Knot 210 521 0.84 40 260 489 
MSEKNVSIVVAASVLSSGIGINGQLPWSISEDLKFFSKITNNKCDSNKKNALIMGRKTWDSIGRRPLKNRIIVVISSSLPQDEADPNVVVFRNLEDSIENLMNDDSIENIFVCGGESIYRDALKDNFVDRIYLTRVALEDIEFDTYFPEIPETFLPVYMSQTFCTKNISYDFMIFEKQEKKTLQNCDPARGQLKSIDDTVDLLGEIFGIRKMGNRHKFPKEEIYNTPSIRFGREHYEFQYLDLLSRVLENGAYRENRTGISTYSIFGQMMRFDMRESFPLLTTKKVAIRSIFEELIWFIKGDTNGNHLIEKKVYIWSGNGSKEYLERIGLGHREENDLGPIYGFQWRHYNGEYKTMHDDYTGVGVDQLAKLIETLKNNPKDRRHILTAWNPSALSQMALPPCHVLSQYYVTNDNCLSCNLYQRSCDLGLGSPFNIASYAILTMMLAQVCGYEPGELAIFIGDAHIYENHLTQLKEQLSRTPRPFPQLKFKRKVENIEDFKWEDIELIGYYPYPTIKMDMAV
6DAI , Knot 122 313 0.74 38 162 269 
SATQSKPTPVKPNYALKFTLAGHTKAVSSVKFSPNGEWLASSSADKLIKIWGAYDGKFEKTISGHKLGISDVAWSSDSNLLVSASDDKTLKIWDVSSGKCLKTLKGHSNYVFCCNFNPQSNLIVSGSFDESVRIWDVKTGKCLKTLPAHSDPVSAVHFNRDGSLIVSSSYDGLCRIWDTASGQCLKTLIDDDNPPVSFVKFSPNGKYILAATLDNTLKLWDYSKGKCLKTYTGHKNEKYCIFANFSVTGGKWIVSGSEDNLVYIWNLQTKEIVQKLQGHTDVVISTACHPTENIIASAALENDKTIKLWKSDC
7CRP , Knot 64 135 0.77 36 101 127 
ARTKQTARKSTGGKAPRKQLATKAARKSAPATGGVLKPHRYRPGTVALREIRRYQKSTELLIRKLPFQRLVREIAQDFKTDLRFQSSAVLALQEASEAYLVGLFEDTNLCGIHAKRVTILPKDIQLARRIRGERA

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 \(c\).

\(|P_{f(6PFB_1)}(2) \setminus P_{f(6DAI_1)}(2)|=141\), \(|P_{f(6DAI_1)}(2) \setminus P_{f(6PFB_1)}(2)|=43\). 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 1:10000101111101100111101011101000101100100000000000111110001001100110001111100011000101011110010001001100001001110110010001100011001010011100101000110110011110100010000100011110000000100001101010010001011101111001100001100010001010110000010010110011001100000011000011101101010001111000011100110011111010001001100010110101000010011110000001111011010000100001000001111001101100100010000011011010110011111001100001000001000100000011110110110011101111010100110111111010100001001000100010111010100010010010100101110010101010111
Pair \(Z_2\) Length of longest common subsequence
6PFB_1,6DAI_1 184 4
6PFB_1,7CRP_1 203 3
6DAI_1,7CRP_1 165 4

Newick tree

 
[
	6PFB_1:10.20,
	[
		6DAI_1:82.5,7CRP_1:82.5
	]:18.70
]

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 \((0.85)(0.8)(\frac{834 }{\log_{20} 834}-\frac{313}{\log_{20}313})=141.\)
Status Protein1 Protein2 d d1/2
Query variables 6PFB_1 6DAI_1 184 140.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]