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Parikh vectors
5TNP_1 7PZQ_1 6KVW_1 Letter Amino acid
22 9 41 E Glutamic acid
24 26 39 G Glycine
18 7 16 H Histidine
19 13 26 P Proline
14 16 25 S Serine
17 11 24 R Arginine
12 17 29 D Aspartic acid
2 12 3 C Cysteine
19 27 24 V Valine
13 11 23 Y Tyrosine
8 21 22 N Asparagine
11 24 24 T Threonine
5 3 13 W Tryptophan
11 11 24 K Lycine
18 17 18 F Phenylalanine
28 29 26 L Leucine
9 10 10 M Methionine
29 17 37 A Alanine
13 14 7 Q Glutamine
9 11 29 I Isoleucine 

5TNP_1|Chains A, B, C, D|CFTR inhibitory factor|Pseudomonas aeruginosa (strain UCBPP-PA14) (208963)
>7PZQ_1|Chains A, B|3C-like proteinase nsp5|Severe acute respiratory syndrome coronavirus 2 (2697049)
>6KVW_1|Chains A, B|Sulfurtransferase|Chlorobium limicola (1092)
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 \(w=f(c)\)
5TNP , Knot 136 301 0.86 40 201 292 
AEEFPVPNGFESAYREVDGVKLHYVKGGQGPLVMLVHGFGQTWYEWHQLMPELAKRFTVIAPDLPGLGQSEPPKTGYSGEQVAVYLHKLARQFSPDRPFDLVAHDIGIWNTYPMVVKNQADIARLVYMQAPIPDARIYRFPAFTAQGESLVWHFSFFAADDRLAETLIAGKERFFLEHFIKSHASNTEVFSERLLDLYARSYAKPHSLNASFEYYRALNESVRQNAELAKTRLQMPTMTLAGGGHGGMGTFQLEQMKAYAEDVEGHVLPGCGHWLPEECAAPMNRLVIDFLSRGRHHHHHH
7PZQ , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
6KVW , Knot 188 460 0.83 40 243 421 
GSEFQNKNFRAPQSEAIGILYKLIETGSKHKNMYDHTEITTDSLLALLGSEKVKIIDVRSADAYNGWRMRGEVRGGHIKGAKSLPAKWLTDPEWLNIVRFKQIRPEDAIVLYGYTPEECEQTATRFKENGYNNVSVFHRFHPDWTGNDAFPMDRLEQYNRLVPAEWVNGLISGEEIPEYDNDTFIVCHAHYRNRDAYLSGHIPGATDMDTLALESPETWNRRTPEELKKALEEHGITASTTVVLYGKFMHPDNADEFPGSAAGHIGAIRLAFIMMYAGVEDVRVLNGGYQSWTDAGFAISKDDVPKTTVPEFGAPIPSRPEFAVDIDEAKEMLQSEDSDLVAVRSYPEYIGEVSGYNYIKKKGRIPGAIFAEAGSDAYHMENYRNHDHTTREYHEIEDIWAKSGIIPKKHLAFYCGTGWRGSEAWFNALLMGWPRVSVYDGGWFEWSNDPENPYETGVPK

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 \(c\).

\(|P_{f(5TNP_1)}(2) \setminus P_{f(7PZQ_1)}(2)|=89\), \(|P_{f(7PZQ_1)}(2) \setminus P_{f(5TNP_1)}(2)|=84\). 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 1:1001111011001000101101001011011111110111001001001110110010111101111100011001001001110100110010100110111001111000111100010110110101111010100111101010011101011110001100111100011100110001000011000110101000101001010100001100010001011000101101011111011110101001010100101011110101110001111001110110010000000
Pair \(Z_2\) Length of longest common subsequence
5TNP_1,7PZQ_1 173 3
5TNP_1,6KVW_1 174 4
7PZQ_1,6KVW_1 197 3

Newick tree

 
[
	6KVW_1:94.97,
	[
		5TNP_1:86.5,7PZQ_1:86.5
	]:8.47
]

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 \((0.85)(0.8)(\frac{607 }{\log_{20} 607}-\frac{301}{\log_{20}301})=85.5\)
Status Protein1 Protein2 d d1/2
Query variables 5TNP_1 7PZQ_1 111 109
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]