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Parikh vectors
1ODN_1 1KER_1 9ARJ_1 Letter Amino acid
12 11 15 R Arginine
2 1 7 C Cysteine
10 14 10 H Histidine
15 23 74 I Isoleucine
20 24 68 K Lycine
20 17 61 S Serine
19 22 61 N Asparagine
19 27 40 G Glycine
4 3 13 M Methionine
20 13 27 P Proline
22 22 43 V Valine
25 25 66 L Leucine
5 7 12 W Tryptophan
20 17 24 Y Tyrosine
28 30 31 A Alanine
24 26 44 D Aspartic acid
17 8 31 Q Glutamine
18 25 43 E Glutamic acid
16 11 35 F Phenylalanine
15 22 43 T Threonine 

1ODN_1|Chain A|ISOPENICILLIN N SYNTHASE|Emericella nidulans (strain FGSC A4 / ATCC 38163 / CBS 112.46 / NRRL 194 / M139) (227321)
>1KER_1|Chains A, B|dTDP-D-glucose 4,6-dehydratase|Streptococcus suis (1307)
>9ARJ_1|Chain A[auth C]|Toxin|Clostridium botulinum E1 str. 'BoNT E Beluga' (536233)
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 \(w=f(c)\)
1ODN , Knot 149 331 0.87 40 206 319 
MGSVSKANVPKIDVSPLFGDDQAAKMRVAQQIDAASRDTGFFYAVNHGINVQRLSQKTKEFHMSITPEEKWDLAIRAYNKEHQDQVRAGYYLSIPGKKAVESFCYLNPNFTPDHPRIQAKTPTHEVNVWPDETKHPGFQDFAEQYYWDVFGLSSALLKGYALALGKEENFFARHFKPDDTLASVVLIRYPYLDPYPEAAIKTAADGTKLSFEWHEDVSLITVLYQSNVQNLQVETAAGYQDIEADDTGYLINCGSYMAHLTNNYYKAPIHRVKWVNAERQSLPFFVNLGYDSVIDPFDPREPNGKSDREPLSYGDYLQNGLVSLINKNGQT
1KER , Knot 154 348 0.86 40 199 331 
MSQFKNIIVTGGAGFIGSNFVHYVYNNHPDVHVTVLDKLTYAGNKANLEAILGDRVELVVGDIADAELVDKLAAKADAIVHYAAESHNDNSLNDPSPFIHTNFIGTYTLLEAARKYDIRFHHVSTDEVYGDLPLREDLPGHGEGPGEKFTAETNYNPSSPYSSTKAASDLIVKAWVRSFGVKATISNCSNNYGPYQHIEKFIPRQITNILAGIKPKLYGEGKNVRDWIHTNDHSTGVWAILTKGRMGETYLIGADGEKNNKEVLELILEKMGQPKDAYDHVTDRAGHDLRYAIDASKLRDELGWTPQFTDFSEGLEETIQWYTDNQDWWKAEKEAVEANYAKTQEVIK
9ARJ , Knot 284 748 0.83 40 278 668 
MTNLKPYIIYDWKETILKNSKDNYSINESIPKIFSKKICGGRFFNSTLSGNWKSWTLTDEGEGPHPVLKCTIDNGYLEIYSNTSSEKHSLKDIEIKVCMSIKPNSDGTHSLCKNSFYIKTNSLKLSEDRLILSHCLDKLILAWFKDNHKYIELFINRSRIQTRVEGDLSLLGWDIESSVSYKTMNEFIKKDNLYEKKFHQYMEVRRNEYTIDGEFGPWQMTTGADGQNIRFLCPIKSATYKINDDVYIAKPDNFIIIQVDLKYFDSKTTIIDPSGLNNGQQFNLKVKTDSTDEINAVILVGSRITDVNEDLYPGDDVSLEIVFKTWFNANIQKFTQIFSYILLNETSKIPEYQWLKPTQISYGSASVTMPDPSNPNKELSNLDASTFAAMAMVENHKNDRPNHAVDNRFLELSKTPAAFAISMPEFLKHFLVTGLQAMQIDNLDAFEVSSENLVITNKKKINFGKIQDQNRQVDALIEPNNFKLAIQNNQVVVEIVDATWQQVVGVTGHFGYRQAYNLILKNENNVYKPMLEESGDVTISYMVTEEAWKTTQDAIISATVGLVVGTIIGTAFSKLSDKLYKFLKSKFIVKNKKASLKISGKDINEVIEMSDISKPQLLSIKKANAKISTEEVGLISQNGSTSLENLAIFKNKPRPIGERVQILGLKLVSGLITTFGWSIGFVLPDILKDVINANINNNFEVLPGIQQFTQQCIGSIQWPDNSELKIDFAKLQGVYLLGGNLVKIPESN

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 \(c\).

\(|P_{f(1ODN_1)}(2) \setminus P_{f(1KER_1)}(2)|=81\), \(|P_{f(1KER_1)}(2) \setminus P_{f(1ODN_1)}(2)|=74\). 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 1:1101001011010101111000110101100101100001110110011010010000001010101000101110100000000101100101110011001001010101001010100100010111000001110011000010111100111010111110000111001010001101111001010101011100110100101010001011011000010010100111000101000101100100110100000011100101101000011111011000110110100101000001100100100111011000100
Pair \(Z_2\) Length of longest common subsequence
1ODN_1,1KER_1 155 3
1ODN_1,9ARJ_1 164 4
1KER_1,9ARJ_1 161 4

Newick tree

 
[
	9ARJ_1:82.46,
	[
		1ODN_1:77.5,1KER_1:77.5
	]:4.96
]

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 \((0.85)(0.8)(\frac{679 }{\log_{20} 679}-\frac{331}{\log_{20}331})=95.9\)
Status Protein1 Protein2 d d1/2
Query variables 1ODN_1 1KER_1 120 119
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]