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Parikh vectors
7KSX_1 6QEO_1 7QGD_1 Letter Amino acid
4 2 15 M Methionine
5 7 5 W Tryptophan
5 11 23 Y Tyrosine
30 15 12 Q Glutamine
27 4 23 E Glutamic acid
15 6 22 H Histidine
45 18 34 L Leucine
27 25 22 G Glycine
22 27 30 V Valine
6 10 21 I Isoleucine
12 3 23 K Lycine
22 22 25 S Serine
27 12 26 R Arginine
3 16 17 N Asparagine
13 8 27 D Aspartic acid
7 8 2 C Cysteine
28 17 22 A Alanine
18 3 13 F Phenylalanine
21 7 24 P Proline
19 19 13 T Threonine 

7KSX_1|Chain A|DNA-directed DNA/RNA polymerase mu|Homo sapiens (9606)
>6QEO_1|Chain A|Chymotrypsin-like elastase family member 1|Sus scrofa (9823)
>7QGD_1|Chain A|Casein kinase II subunit alpha|Homo sapiens (9606)
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 \(w=f(c)\)
7KSX , Knot 147 356 0.80 40 198 344 
GSAAAPLSPAWMPAYACQRPTPLTHHNTGLSEALEILAEAAGFEGSEGRLLTFCRAASVLKALPSPVTTLSQLQGLPHFGEHSSRVVQELLEHGVCEEVERVRRSERYQTMKLFTQIFGVGVKTADRWYREGLRTLDDLREQPQKLTQQQKAGLQHHQDLSTPVLRSDVDALQQVVEEAVGQALPGATVTLTGGFRRGKLQGHDVDFLITHPKEGQEAGLLPRVMCRLQDQGLILYHQHQHSCCESPTRLAQQSHMDAFERSFCIFRLPQPGSWKAVRVDLVVAPVSQFPFALLGWTGSKLFQRELRRFSRKEKGLWLNSHGLFDPEQKTFFQAASEEDIFRHLGLEYLPPEQRNA
6QEO , Knot 107 240 0.81 40 163 230 
VVGGTEAQRNSWPSQISLQYRSGSSWAHTCGGTLIRQNWVMTAAHCVDRELTFRVVVGEHNLNQNDGTEQYVGVQKIVVHPYWNTDDVAAGYDIALLRLAQSVTLNSYVQLGVLPRAGTILANNSPCYITGWGLTRTNGQLAQTLQQAYLPTVDYAICSSSSYWGSTVKNSMVCAGGDGVRSGCQGDSGGPLHCLVNGQYAVHGVTSFVSRLGCNVTRKPTVFTRVSAYISWINNVIASN
7QGD , Knot 172 399 0.86 40 218 376 
MSGPVPSRARVYTDVNTHRPREYWDYESHVVEWGNQDDYQLVRKLGRGKYSEVFEAINITNNEKVVVKILKPVKKKKIKREIKILENLRGGPNIITLADIVKDPVSRTPALVFEHVNNTDFKQLYQTLTDYDIRFYMYEILKALDYCHSMGIMHRDVKPHNVMIDHEHRKLRLIDWGLAEFYHPGQEYNVRVASRYFKGPELLVDYQMYDYSLDMWSLGCMLASMIFRKEPFFHGHDNYDQLVRIAKVLGTEDLYDYIDKYNIELDPRFNDILGRHSRKRWERFVHSENQHLVSPEALDFLDKLLRYDHQSRLTAREAMEHPYFYTVVKDQARMGSSSMPGGSTPVSSANMMSGISSVPTPSPLGPLAGSPVIAAANPLGMPVPAAAGAQQLEHHHHHH

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 \(c\).

\(|P_{f(7KSX_1)}(2) \setminus P_{f(6QEO_1)}(2)|=105\), \(|P_{f(6QEO_1)}(2) \setminus P_{f(7KSX_1)}(2)|=70\). 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 1:10111110111111010001011000001100110111011110100101101001101101110110010010111011000001100110011000100100000000101100111111001001000110010010001001000001110000010011100010110011001110111110101011100101010010111001001001111101100100011110000000000010011000010110001011011011010110101111110011111111010011000100100000111100011101000011011000011001110011100001
Pair \(Z_2\) Length of longest common subsequence
7KSX_1,6QEO_1 175 4
7KSX_1,7QGD_1 180 4
6QEO_1,7QGD_1 191 3

Newick tree

 
[
	7QGD_1:94.48,
	[
		7KSX_1:87.5,6QEO_1:87.5
	]:6.98
]

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 \((0.85)(0.8)(\frac{596 }{\log_{20} 596}-\frac{240}{\log_{20}240})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 7KSX_1 6QEO_1 128 107
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]