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Parikh vectors
8TRN_1 8IZL_1 7GGH_1 Letter Amino acid
28 25 27 V Valine
23 19 17 D Aspartic acid
7 27 14 Q Glutamine
34 21 9 E Glutamic acid
9 6 7 H Histidine
19 26 13 P Proline
25 31 17 A Alanine
5 1 12 C Cysteine
15 12 10 M Methionine
12 8 17 F Phenylalanine
22 31 24 T Threonine
4 0 3 W Tryptophan
15 4 11 Y Tyrosine
11 14 21 N Asparagine
26 28 11 I Isoleucine
27 25 29 L Leucine
23 25 11 K Lycine
24 35 16 S Serine
17 17 11 R Arginine
30 36 26 G Glycine 

8TRN_1|Chains A, B, C|Actin|Toxoplasma gondii (5811)
>8IZL_1|Chains A[auth B], B[auth C], C[auth D], D[auth E]|Phosphoprotein|Mumps orthorubulavirus (2560602)
>7GGH_1|Chain A|3C-like proteinase|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
8TRN , Knot 159 376 0.83 40 207 360 
MADEEVQALVVDNGSGNVKAGVAGDDAPRAVFPSIVGKPKNPGIMVGMEEKDCYVGDEAQSKRGILTLKYPIEHGIVTNWDDMEKIWHHTFYNELRVAPEEHPVLLTEAPLNPKANRERMTQIMFETFNVPAMYVAIQAVLSLYSSGRTTGIVLDSGDGVSHTVPIYEGYALPHAIMRLDLAGRDLTEYMMKILHERGYGFTTSAEKEIVRDIKEKLCYIALDFDEEMKAAEDSSDIEKSYELPDGNIITVGNERFRCPEALFQPSFLGKEAAGVHRTTFDSIMKCDVDIRKDLYGNVVLSGGTTMYEGIGERLTKELTSLAPSTMKIKVVAPPERKYSVWIGGSILSSLSTFQQMWITKEEYDESGPSIVHRKCF
8IZL , Knot 162 391 0.82 38 197 358 
MDQFIKQDETGDLIETGMNVANHFLSTPIQGTNSLSKASILPGVAPVLIGNPEQKNIQHPTASHQGSKTKGRGSGVRSIIVSPSEAGNGGTQIPEPLFAQTGQGGIVTTVYQDPTIQPTGSYRSVELAKIGKERMINRFVEKPRTSTPVTEFKRGGPGAAAQGQTIQEEGIDGNGASAGSKERSGSLSGATLYAHLSLPQQDSTPANVGIAPQSAISANEIMDLLRGMDARLQHLEQKVDKVLAQGSMVTQIKNELSTVKTTLATIEGMMATVKIMDPGNPTGVPVDELRRSFSDHVTIVSGPGDVSFSSSEKPTLYLDELARPVSKPRPAKQTKSQPVKDLAGQKVMITKMITDCVANPQMKQAFEQRLAKASTEDALNDIKRDIIRSAI
7GGH , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ

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 \(c\).

\(|P_{f(8TRN_1)}(2) \setminus P_{f(8IZL_1)}(2)|=79\), \(|P_{f(8IZL_1)}(2) \setminus P_{f(8TRN_1)}(2)|=69\). 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 1:1100010111100101010111110011011110111010011111110000001100100001110100110011100100100110001000101110001111001110101000010011100101111011101110100010001111001011000111001011101110101110010001101100010110001000110010001001110100010110000010000011010110110001001011101011100111100001001100010100010101110110010011100100010011100101011111000001111101100100100111000000001101100001
Pair \(Z_2\) Length of longest common subsequence
8TRN_1,8IZL_1 148 4
8TRN_1,7GGH_1 175 3
8IZL_1,7GGH_1 169 4

Newick tree

 
[
	7GGH_1:89.66,
	[
		8TRN_1:74,8IZL_1:74
	]:15.66
]

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 \((0.85)(0.8)(\frac{767 }{\log_{20} 767}-\frac{376}{\log_{20}376})=106.\)
Status Protein1 Protein2 d d1/2
Query variables 8TRN_1 8IZL_1 135 132
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]