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Parikh vectors
5JOS_1 6KIJ_1 4QPL_1 Letter Amino acid
12 17 7 P Proline
10 10 7 S Serine
17 6 5 T Threonine
19 13 8 V Valine
11 15 13 R Arginine
20 14 8 D Aspartic acid
8 13 7 I Isoleucine
17 17 16 L Leucine
17 17 12 G Glycine
9 3 2 M Methionine
3 8 4 W Tryptophan
11 5 7 Y Tyrosine
17 25 10 A Alanine
7 5 5 N Asparagine
9 11 4 Q Glutamine
16 19 13 E Glutamic acid
0 3 8 C Cysteine
11 12 4 F Phenylalanine
9 5 3 H Histidine
24 9 12 K Lycine 

5JOS_1|Chain A|Cyclohexadienyl dehydratase|synthetic construct (32630)
>6KIJ_1|Chain A[auth B]|SOS response-associated protein|Escherichia coli (562)
>4QPL_1|Chains A, C|E3 ubiquitin-protein ligase RNF146|Mus musculus (10090)
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 \(w=f(c)\)
5JOS , Knot 107 247 0.79 38 166 233 
MRGSHHHHHHIAASRLDEIMERGTLRVGTTGDYKPFSYRDPDGQYTGFDIDVAKSLAKSLGVKVEFVPTTWPTLMSDLQADKFDIAMGGVTVTPERQKKADFSDPYMTFGKTPLVRKEDADKFKSLEDINRPDVRVAVNPGGTNEKFAREHLKKAKITVYENNVEIFQEVASGRADVMITDTVEALYYAKKHPGLAAVLVDKPFTHSEKGYMMPKGDQEFLNYVNQWLDQMKQQGTYEKLYEKWLKL
6KIJ , Knot 105 227 0.83 40 157 221 
CGRFAQSQTREDYLALLAEDIERDIPYDPEPIGRYNVAPGTKVLLLSERDEHLHLDPVFWGYAPGWWDKPPLINARVETAATSRMFKPLWQHGRAICFADGWFEWKKEGDKKQPFFIYRADGQPIFMAAIGSTPFERGDEAEGFLIVTAAADQGLVDIHDRRPLVLSPEAAREWMRQEISGKEASEIAASGCVPANQFSWHPVSRAVGNVKNQGAELIQPVLEVLFQ
4QPL , Knot 78 155 0.84 40 126 153 
GSLTVPECAICLQTCVHPVSLPCKHVFCYLCVKGASWLGKRCALCRQEIPEDFLDKPTLLSPEELKAASRGNGEYAWYYEGRNGWWQYDERTSRELEDAFSKGKKNTEMLIAGFLYVADLENMVQYRRNEHGRRRKIKRDIIDIPKKGVAGLRLD

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 \(c\).

\(|P_{f(5JOS_1)}(2) \setminus P_{f(6KIJ_1)}(2)|=88\), \(|P_{f(6KIJ_1)}(2) \setminus P_{f(5JOS_1)}(2)|=79\). 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 1:1010000000111001001100101011001000110000101000110101100110011101011100110110010100101111110101000001010010101100111000010010010010010101110111000011000100101010000101100110101011100010110010001111111100110000010111010001100100110010001000010001101
Pair \(Z_2\) Length of longest common subsequence
5JOS_1,6KIJ_1 167 4
5JOS_1,4QPL_1 164 4
6KIJ_1,4QPL_1 159 4

Newick tree

 
[
	5JOS_1:83.80,
	[
		4QPL_1:79.5,6KIJ_1:79.5
	]:4.30
]

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 \((0.85)(0.8)(\frac{474 }{\log_{20} 474}-\frac{227}{\log_{20}227})=71.4\)
Status Protein1 Protein2 d d1/2
Query variables 5JOS_1 6KIJ_1 87 85.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]