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Parikh vectors
6HWT_1 9QDT_1 6ZRA_1 Letter Amino acid
19 17 16 R Arginine
23 27 16 E Glutamic acid
10 7 5 M Methionine
13 16 6 F Phenylalanine
16 32 31 G Glycine
17 27 17 P Proline
22 23 25 V Valine
4 6 6 C Cysteine
5 6 2 W Tryptophan
25 40 31 A Alanine
14 18 9 N Asparagine
27 28 10 D Aspartic acid
15 20 10 Q Glutamine
13 9 2 H Histidine
22 20 13 I Isoleucine
43 37 26 L Leucine
19 29 8 K Lycine
21 17 14 S Serine
19 21 17 T Threonine
15 17 6 Y Tyrosine 

6HWT_1|Chain A|Mitogen-activated protein kinase 14|Homo sapiens (9606)
>9QDT_1|Chain A|Probable cytochrome c peroxidase|Escherichia coli K-12 (83333)
>6ZRA_1|Chain A|17-beta-hydroxysteroid dehydrogenase 14|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 \(w=f(c)\)
6HWT , Knot 159 362 0.86 40 224 348 
GSHSQERPTFYRQELNKTIWEVPERYQNLSPVGSGAYGSVCAAFDTKTGLRVAVKKLSRPFQSIIHAKRTYRELRLLKHMKHENVIGLLDVFTPARSLEEFNDVYLVTHLMGADLNNIVKCQKLTDDHVQFLIYQILRGLKYIHSADIIHRDLKPSNLAVNEDCELKILDFGLARHTDDEMTGYVATRWYRAPEIMLNWMHYNQTVDIWSVGCIMAELLTGRTLFPGTDHIDQLKLILRLVGTPGAELLKKISSESARNYIQSLTQMPKMNFANVFIGANPLAVDLLEKMLVLDSDKRITAAQALAHAYFAQYHDPDDEPVADPYDQSFESRDLLIDEWKSLTYDEVISFVPPPLDQEEMES
9QDT , Knot 178 417 0.85 40 237 397 
ENNKVLGFLREKGCDYCHTPSAELPAYYYIPGAKQLMDYDIKLGYKSFNLEAVRAALLADKPVSQSDLNKIEWVMQYETMPPTRYTALHWAGKVSDEERAEILAWIAKQRAEYYASNDTAPEHRNEPVQPIPQKLPTDAQKVALGFALYHDPRLSADSTISCAHCHALNAGGVDGRKTSIGVGGAVGPINAPTVFNSVFNVEQFWDGRAATLQDQAGGPPLNPIEMASKSWDEIIAKLEKDPQLKTQFLEVYPQGFSGENITDAIAEFEKTLITPDSPFDKWLRGDENALTAQQKKGYQLFKDNKCATCHGGIILGGRSFEPLGLKKDFNFGEITAADIGRMNVTKEERDKLRQKVPGLRNVALTAPYFHRGDVPTLDGAVKLMLRYQVGKELPQEDVDDIVAFLHSLNGVYTPYMQ
6ZRA , Knot 118 270 0.81 40 170 259 
MATGTRYAGKVVVVTGGGRGIGAGIVRAFVNSGARVVICDKDESGGRALEQELPGAVFILCDVTQEDDVKTLVSETIRRFGRLDCVVNNAGHHPPPQRPEETSAQGFRQLLELNLLGTYTLTKLALPYLRKSQGNVINISSLVGAIGQAQAVPYVATKGAVTAMTKALALDESPYGVRVNCISPGNIWTPLWEELAALMPDPRASIREGMLAQPLGRMGQPAEVGAAAVFLASEANFCTGIELLVTGGAELGYGCKASRSTPVDAPDIPS

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 \(c\).

\(|P_{f(6HWT_1)}(2) \setminus P_{f(9QDT_1)}(2)|=73\), \(|P_{f(9QDT_1)}(2) \setminus P_{f(6HWT_1)}(2)|=86\). 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 1:10000001010000100011011000001011101101010111000011011100100110011010000001011001000011111011011001001001011001111010011000010000101110011011001001011000101001110000010110111100000010101100100110111011000001011011011101101001111000100101110111011101100100001000100100110101101111101111011001111000001011011101011000010001110100001000011100100100001101111110000100
Pair \(Z_2\) Length of longest common subsequence
6HWT_1,9QDT_1 159 4
6HWT_1,6ZRA_1 172 4
9QDT_1,6ZRA_1 163 4

Newick tree

 
[
	6ZRA_1:85.15,
	[
		6HWT_1:79.5,9QDT_1:79.5
	]:5.65
]

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 \((0.85)(0.8)(\frac{779 }{\log_{20} 779}-\frac{362}{\log_{20}362})=113.\)
Status Protein1 Protein2 d d1/2
Query variables 6HWT_1 9QDT_1 147 135
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]