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Parikh vectors
3KXK_1 7UBE_1 2IRO_1 Letter Amino acid
17 2 0 V Valine
13 0 0 R Arginine
8 0 0 H Histidine
41 0 0 I Isoleucine
41 0 0 K Lycine
14 0 0 Y Tyrosine
16 0 0 N Asparagine
45 4 0 L Leucine
5 0 0 M Methionine
14 2 0 P Proline
0 0 0 W Tryptophan
0 0 2 C Cysteine
30 0 0 E Glutamic acid
16 0 0 F Phenylalanine
21 0 0 T Threonine
16 0 1 A Alanine
13 0 0 D Aspartic acid
12 0 0 Q Glutamine
16 0 4 G Glycine
26 0 0 S Serine 

3KXK_1|Chains A, B|GTP-binding protein (HflX)|Sulfolobus solfataricus (2287)
>7UBE_1|Chain A|Cyclic peptide D8.21 DVA-MLE-DPR-LEU-DVA-MLE-DPR-LEU|synthetic construct (32630)
>2IRO_1|Chains A, B|5'-R(P*GP*CP*GP*GP*AP*UP*GP*CP*U)-3'|
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 \(w=f(c)\)
3KXK , Knot 153 364 0.82 36 186 345 
MKTAALFVSKEFEEEAIALVEGANYKVTSIYKLPKSPNVKFYIQYDKLQQIKNDEEISTLIIFEQLKPRHFINIRRELKGKEVLDKILLLLEIFALHAGSKEAKMQIELARLKYELPIIKETYTKSKIGEQQGPLGAGTYGVESTIKFYKRRINKLMKELESIKIFKEKSIESNKRNNIPSIGIVGYTNSGKTSLFNSLTGLTQKVDTKLFTTMSPKRYAIPINNRKIMLVDTVPFIRGIPPQIVDAFFVTLSEAKYSDALILVIDSTFSENLLIETLQSSFEILREIGVSGKPILVTLNKIDKINGDLYKKLDLVEKLSKELYSPIFDVIPISALKRTNLELLRDKIYQLATQLSLEHHHHHH
7UBE , Knot 5 8 0.43 6 4 4 
VLPLVLPL
2IRO , Knot 6 9 0.48 8 7 7 
GCGGAUGCU

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 \(c\).

\(|P_{f(3KXK_1)}(2) \setminus P_{f(7UBE_1)}(2)|=182\), \(|P_{f(7UBE_1)}(2) \setminus P_{f(3KXK_1)}(2)|=0\). 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 1:1001111100010001111101100010010011001010101000010010000010011110010100110100010100110011111011110110001010101101000111100000000110001111110011000101000010011001001011000010000000110111110000100011001011000100011001010001111000011110011110111101101111010010000111111000100011100100010110011101011110100100101010001011001000100111011110110000101100010011001010000000
Pair \(Z_2\) Length of longest common subsequence
3KXK_1,7UBE_1 182 2
3KXK_1,2IRO_1 191 2
7UBE_1,2IRO_1 11 0

Newick tree

 
[
	3KXK_1:10.66,
	[
		7UBE_1:5.5,2IRO_1:5.5
	]:10.16
]

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 \((0.85)(0.8)(\frac{372 }{\log_{20} 372}-\frac{8}{\log_{20}8})=120.\)
Status Protein1 Protein2 d d1/2
Query variables 3KXK_1 7UBE_1 151 76.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]