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Parikh vectors
8JLO_1 5GSC_1 2IPZ_1 Letter Amino acid
20 10 0 F Phenylalanine
26 24 7 V Valine
17 36 1 G Glycine
24 12 2 N Asparagine
8 0 0 C Cysteine
21 16 1 Q Glutamine
26 18 5 K Lycine
28 13 2 R Arginine
31 12 1 D Aspartic acid
23 15 0 I Isoleucine
30 32 6 L Leucine
7 11 1 M Methionine
14 22 0 P Proline
18 29 1 S Serine
17 16 0 T Threonine
27 46 1 A Alanine
7 16 1 H Histidine
4 3 0 W Tryptophan
14 18 1 Y Tyrosine
32 17 4 E Glutamic acid 

8JLO_1|Chain A|Guanine nucleotide-binding protein G(s) subunit alpha isoforms short|Homo sapiens (9606)
>5GSC_1|Chains A, B|Beta-lactamase|Enterobacter aerogenes (548)
>2IPZ_1|Chains A, B, C, D|General control protein GCN4|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
8JLO , Knot 169 394 0.85 40 227 378 
MGCLGNSKTEDQRNEEKAQREANKKIEKQLQKDKQVYRATHRLLLLGAGESGKNTIVKQMRILHVNGFNGEGGEEDPQAARSNSDGEKATKVQDIKNNLKEAIETIVAAMSNLVPPVELANPENQFRVDYILSVMNVPDFDFPPEFYEHAKALWEDEGVRACYERSNEYQLIDCAQYFLDKIDVIKQADYVPSDQDLLRCRVLTSGIFETKFQVDKVNFHMFDVGAQRDERRKWIQCFNDVTAIIFVVASSSYNMVIREDNQTNRLQAALKLFDSIWNNKWLRDTSVILFLNKQDLLAEKVLAGKSKIEDYFPEFARYTTPEDATPEPGEDPRVTRAKYFIRDEFLRISTASGDGRHYCYPHFTCSVDTENIRRVFNDCRDIIQRMHLRQYELL
5GSC , Knot 155 366 0.83 38 192 342 
MHHHHHHGEASPVDPLRPVVDASIQPLLKEHRIPGMAVAVLKDGKAHYFNYGVANRESGAGVSEQTLFEIGSVSKTLTATLGAYAVVKGAMQLDDKASRHAPWLKGSAFDSITMGELATYSAGGLPLQFPEEVDSSEKMRAYYRQWAPVYSPGSHRQYSNPSIGLFGHLAASSLKQPFAPLMEQTLLPGLGMHHTYVNVPKQAMASYAYGYSKEDKPIRVNPGMLADEAYGIKTSSADLLRFVKANIGGVDDKALQQAISLTHQGHYSVGGMTQGLGWESYAYPVTEQTLLAGNSAKVILEANPTAAPRESGSQVLFNKTGSTNGFGAYVAFVPARGIGIVMLANRNYPIEARIKAAHAILAQLAG
2IPZ , Knot 19 34 0.65 28 28 32 
MKVKQLVDKVEELLSKNYHLVNEVARLVKLVGER

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 \(c\).

\(|P_{f(8JLO_1)}(2) \setminus P_{f(5GSC_1)}(2)|=102\), \(|P_{f(5GSC_1)}(2) \setminus P_{f(8JLO_1)}(2)|=67\). 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 1:1101100000000000010001000100010000010010001111111001000110010110101101011000101100000100100100100010011001111100111110110100010100110110110101110100010111000110100000000011001001100101100100110000110001100111000101001010110111000000011001001011111110000011100000000101110110011000110000111110000111001111000100011011000010010101100101001001100011010010101000001010001000010011000001100101000011
Pair \(Z_2\) Length of longest common subsequence
8JLO_1,5GSC_1 169 4
8JLO_1,2IPZ_1 207 4
5GSC_1,2IPZ_1 184 2

Newick tree

 
[
	2IPZ_1:10.00,
	[
		8JLO_1:84.5,5GSC_1:84.5
	]:17.50
]

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 \((0.85)(0.8)(\frac{760 }{\log_{20} 760}-\frac{366}{\log_{20}366})=107.\)
Status Protein1 Protein2 d d1/2
Query variables 8JLO_1 5GSC_1 138 131.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]