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Parikh vectors
5QDL_1 9JKQ_1 4ATE_1 Letter Amino acid
19 28 13 R Arginine
30 32 15 E Glutamic acid
14 17 18 G Glycine
18 23 8 I Isoleucine
29 30 16 L Leucine
25 17 18 S Serine
18 31 16 D Aspartic acid
14 21 9 Q Glutamine
11 7 5 M Methionine
22 14 17 P Proline
9 14 14 Y Tyrosine
13 28 15 A Alanine
11 24 17 N Asparagine
6 4 8 W Tryptophan
17 26 17 V Valine
4 8 0 C Cysteine
11 7 11 H Histidine
23 26 20 K Lycine
14 20 17 F Phenylalanine
13 17 12 T Threonine 

5QDL_1|Chain A|Tyrosine-protein phosphatase non-receptor type 1|Homo sapiens (9606)
>9JKQ_1|Chain A|Guanine nucleotide-binding protein G(s) subunit alpha isoforms short|Homo sapiens (9606)
>4ATE_1|Chain A|BETA-PORPHYRANASE A|ZOBELLIA GALACTANIVORANS (63186)
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 \(w=f(c)\)
5QDL , Knot 141 321 0.84 40 205 309 
MEMEKEFEQIDKSGSWAAIYQDIRHEASDFPSRVAKLPKNKNRNRYRDVSPFDHSRIKLHQEDNDYINASLIKMEEAQRSYILTQGPLPNTVGHFWEMVWEQKSRGVVMLNRVMEKGSLKCAQYWPQKEEKEMIFEDTNLKLTLISEDIKSYYTVRQLELENLTTQETREILHFHYTTWPDFGVPESPASFLNFLFKVRESGSLSPEHGPVVVHCSAGIGRSGTFCLADTCLLLMDKRKDPSSVDIKKVLLEMRKFRMGLIQTADQLRFSYLAVIEGAKFIMGDSSVQDQWKELSHEDLEPPPEHIPPPPRPPKRILEPHN
9JKQ , Knot 169 394 0.85 40 227 378 
MGCLGNSKTEDQRNEEKAQREANKKIEKQLQKDKQVYRATHRLLLLGAGESGKNTIVKQMRILHVNGFNGEGGEEDPQAARSNSDGEKATKVQDIKNNLKEAIETIVAAMSNLVPPVELANPENQFRVDYILSVMNVPDFDFPPEFYEHAKALWEDEGVRACYERSNEYQLIDCAQYFLDKIDVIKQADYVPSDQDLLRCRVLTSGIFETKFQVDKVNFHMFDVGAQRDERRKWIQCFNDVTAIIFVVASSSYNMVIREDNQTNRLQAALKLFDSIWNNKWLRDTSVILFLNKQDLLAEKVLAGKSKIEDYFPEFARYTTPEDATPEPGEDPRVTRAKYFIRDEFLRISTASGDGRHYCYPHFTCAVDTENIRRVFNDCRDIIQRMHLRQYELL
4ATE , Knot 121 266 0.84 38 183 256 
HHHHHHAQLPSPTNGKKWEKVEQLSDEFNGNSIDTNKWYDYHPFWEGRAPSNFKKGNAFVSDGFLNLRSTLRKEPSSVQDPFKDIWVDAAAAVSKTKAQPGYYYEARFKASSLSMTSSFWFRVGQFSEIDVIEHIGNPSKENRQDDLPYQYHVNTHYYGKHAGLQPLGTEYKMPGRGRDNFYTYGFWWKSPNELLFYFNGKQVMRIVPRVPLDEELRMIFDTEVFPFATAGVANIGLPKPENLRDNSKNTMKVDWVRVYKLVDGTA

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 \(c\).

\(|P_{f(5QDL_1)}(2) \setminus P_{f(9JKQ_1)}(2)|=73\), \(|P_{f(9JKQ_1)}(2) \setminus P_{f(5QDL_1)}(2)|=95\). 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 1:101000100100010111100010001001100110110000000000101100001010000000101011010010000110011110011011011100000111110011001010010011000000111000010101100010000010010100100000001101000011011110011011011101000101010011111000111100101011000111100000100101001110100101111001001010011110110111100010001001000010111001111101100110100
Pair \(Z_2\) Length of longest common subsequence
5QDL_1,9JKQ_1 168 4
5QDL_1,4ATE_1 176 5
9JKQ_1,4ATE_1 156 4

Newick tree

 
[
	5QDL_1:88.53,
	[
		9JKQ_1:78,4ATE_1:78
	]:10.53
]

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 \((0.85)(0.8)(\frac{715 }{\log_{20} 715}-\frac{321}{\log_{20}321})=108.\)
Status Protein1 Protein2 d d1/2
Query variables 5QDL_1 9JKQ_1 140 125
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]