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Parikh vectors
6YPN_1 1XLS_1 3WIE_1 Letter Amino acid
16 7 13 H Histidine
30 33 21 L Leucine
23 13 32 K Lycine
26 14 35 V Valine
12 21 16 A Alanine
15 9 22 N Asparagine
22 20 20 E Glutamic acid
13 10 19 F Phenylalanine
23 4 13 Y Tyrosine
11 9 17 T Threonine
26 12 32 D Aspartic acid
9 5 7 Q Glutamine
14 11 31 G Glycine
10 7 10 M Methionine
14 13 16 S Serine
25 12 12 R Arginine
2 4 4 C Cysteine
19 12 29 I Isoleucine
14 14 18 P Proline
5 2 2 W Tryptophan 

6YPN_1|Chain A[auth B]|Casein kinase II subunit alpha|Homo sapiens (9606)
>1XLS_1|Chains A, B, C, D|Retinoic acid receptor RXR-alpha|Homo sapiens (9606)
>3WIE_1|Chains A, B, C, D|Glucose 1-dehydrogenase|Thermoplasma volcanium (273116)
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 \(w=f(c)\)
6YPN , Knot 148 329 0.87 40 195 315 
MSGPVPSRARVYTDVNTHRPREYWDYESHVVEWGNQDDYQLVRKLGRGKYSEVFEAINITNNEKVVVKILKPVKKKKIKREIKILENLRGGPNIITLADIVKDPVSRTPALVFEHVNNTDFKQLYQTLTDYDIRFYMYEILKALDYCHSMGIMHRDVKPHNVMIDHEHRKLRLIDWGLAEFYHPGQEYNVRVASRYFKGPELLVDYQMYDYSLDMWSLGCMLASMIFRKEPFFHGHDNYDQLVRIAKVLGTEDLYDYIDKYNIELDPRFNDILGRHSRKRWERFVHSENQHLVSPEALDFLDKLLRYDHQSRLTAREAMEHPYFYTVVK
1XLS , Knot 108 232 0.84 40 165 220 
NEDMPVERILEAELAVEPKTETYVEANMGLNPSSPNDPVTNICQAADKQLFTLVEWAKRIPHFSELPLDDQVILLRAGWNELLIASFSHRSIAVKDGILLATGLHVHRNSAHSAGVGAIFDRVLTELVSKMRDMQMDKTELGCLRAIVLFNPDSKGLSNPAEVEALREKVYASLEAYCKHKYPEQPGRFAKLLLRLPALRSIGLKCLEHLFFFKLIGDTPIDTFLMEMLEAP
3WIE , Knot 159 369 0.85 40 210 350 
MSTINAIVTDAPKGGVKYTKIDMPEPEKYEAKLKPVYIGICGTDRGEVAGALSFTYNPEGENFLVLGHEALLQVLDVSDNNYIKRGDFVVPLVRRPGKCVNCRIGRQDNCSIGDPDKHEAGITGLHGFMRDVIYDDIQNLVKVNDPDLGKIAVLTEPLKNVMKAFEVFDVVSKRSIFQNDDSTFIGKKMVVIGSGSEAFLYSFVGKDRGFDVTMVNRHDETENKMKMMDDFGVGFSNYLKDMPDKIDLLVDTSGDPSTIFKFVKKVNNNGVVILFGFNGKAPGYPVNGEDIDYIVERNITIAGSVDAAKIHYVQALDSLSNWYHRHPQTIKDIITYEAKPEETNIFFQKPKGEIKTVIKWPLEHHHHHH

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 \(c\).

\(|P_{f(6YPN_1)}(2) \setminus P_{f(1XLS_1)}(2)|=104\), \(|P_{f(1XLS_1)}(2) \setminus P_{f(6YPN_1)}(2)|=74\). 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 1:10111100101000100001000100000110110000001100110100001101101000001110110110000100010110010111011011011001100011111001000010010001000010101001101100000111100010100111000000101101111010011000010110001011011100010000101101101110111000111010000001101101110001000100001010101001110000001001100000011010110110011000000010100110010100110
Pair \(Z_2\) Length of longest common subsequence
6YPN_1,1XLS_1 178 3
6YPN_1,3WIE_1 181 4
1XLS_1,3WIE_1 169 4

Newick tree

 
[
	6YPN_1:91.43,
	[
		1XLS_1:84.5,3WIE_1:84.5
	]:6.93
]

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 \((0.85)(0.8)(\frac{561 }{\log_{20} 561}-\frac{232}{\log_{20}232})=93.7\)
Status Protein1 Protein2 d d1/2
Query variables 6YPN_1 1XLS_1 119 101.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]