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Parikh vectors
3EKN_1 6RCB_1 8PCO_1 Letter Amino acid
14 15 11 N Asparagine
22 15 22 G Glycine
7 16 2 H Histidine
15 11 5 M Methionine
27 26 15 V Valine
15 23 10 K Lycine
15 13 4 F Phenylalanine
18 26 15 R Arginine
16 27 13 D Aspartic acid
4 2 1 C Cysteine
10 19 11 I Isoleucine
17 14 13 P Proline
5 5 3 W Tryptophan
9 23 5 Y Tyrosine
17 13 35 A Alanine
9 10 18 Q Glutamine
26 22 11 E Glutamic acid
28 30 27 L Leucine
19 15 13 S Serine
14 11 29 T Threonine 

3EKN_1|Chain A|Insulin receptor|Homo sapiens (9606)
>6RCB_1|Chain A|Casein kinase II subunit alpha|Homo sapiens (9606)
>8PCO_1|Chain A|Beta-lactamase|Klebsiella pneumoniae (573)
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 \(w=f(c)\)
3EKN , Knot 135 307 0.84 40 184 293 
GVFPSSVYVPDEWEVSREKITLLRELGQGSFGMVYEGNARDIIKGEAETRVAVKTVNESASLRERIEFLNEASVMKGFTCHHVVRLLGVVSKGQPTLVVMELMAHGDLKSYLRSLRPEAENNPGRPPPTLQEMIQMAAEIADGMAYLNAKKFVHRNLAARNCMVAHDFTVKIGDFGMTRDIYETDYYRKGGKGLLPVRWMAPESLKDGVFTTSSDMWSFGVVLWEITSLAEQPYQGLSNEQVLKFVMDGGYLDQPDNCPERVTDLMRMCWQFNPNMRPTFLEIVNLLKDDLHPSFPEVSFFHSEENK
6RCB , Knot 150 336 0.86 40 198 321 
MSGPVPSRARVYTDVNTHRPREYWDYESHVVEWGNQDDYQLVRKLGRGKYSEVFEAINITNNEKVVVKILKPVKKKKIKREIKILENLRGGPNIITLADIVKDPVSRTPALVFEHVNNTDFKQLYQTLTDYDIRFYMYEILKALDYCHSMGIMHRDVKPHNVMIDHEHRKLRLIDWGLAEFYHPGQEYNVRVASRYFKGPELLVDYQMYDYSLDMWSLGCMLASMIFRKEPFFHGHDNYDQLVRIAKVLGTEDLYDYIDKYNIELDPRFNDILGRHSRKRWERFVHSENQHLVSPEALDFLDKLLRYDHQSRLTAREAMEHPYFYTVVKDQARMGS
8PCO , Knot 117 263 0.82 40 163 250 
QTSAVQQKLAALEKSSGGRLGVALIDTADNTQVLYRGDERFPMCSTSKVMAAAAVLKQSETQKQLLNQPVEIKPADLVNYNPIAEKHVNGTMTLAELSAAALQYSDNTAMNKLIAQLGGPGGVTAFARAIGDETFRLDRTEPTLNTAIPGDPRDTTTPRAMAQTLRQLTLGHALGETQRAQLVTWLKGNTTGAASIRAGLPTSWTVGDKTGSGDYGTTNDIAVIWPQGRAPLVLVTYFTQPQQNAESRRDVLASAARIIAEGL

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 \(c\).

\(|P_{f(3EKN_1)}(2) \setminus P_{f(6RCB_1)}(2)|=78\), \(|P_{f(6RCB_1)}(2) \setminus P_{f(3EKN_1)}(2)|=92\). 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 1:1111001011001010000101100110101111001010011010100011100100010100010110010110110000110111110010101111011101010001001010100011011101001101110110111010100110001110001110010101101110001000000001101111101111001001110000011011111101001100100110000110111011010010001001001101010101010101101101100010101101011000000
Pair \(Z_2\) Length of longest common subsequence
3EKN_1,6RCB_1 170 4
3EKN_1,8PCO_1 185 3
6RCB_1,8PCO_1 189 3

Newick tree

 
[
	8PCO_1:96.17,
	[
		3EKN_1:85,6RCB_1:85
	]:11.17
]

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 \((0.85)(0.8)(\frac{643 }{\log_{20} 643}-\frac{307}{\log_{20}307})=93.3\)
Status Protein1 Protein2 d d1/2
Query variables 3EKN_1 6RCB_1 124 117
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]