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Parikh vectors
4UHT_1 2QOS_1 6FGG_1 Letter Amino acid
4 4 2 K Lycine
5 5 4 P Proline
7 9 4 T Threonine
6 9 12 E Glutamic acid
2 4 5 I Isoleucine
17 17 8 L Leucine
2 6 3 Y Tyrosine
8 12 9 R Arginine
2 1 4 N Asparagine
1 2 2 W Tryptophan
6 8 7 D Aspartic acid
5 15 2 Q Glutamine
6 17 4 V Valine
5 5 3 H Histidine
2 1 6 M Methionine
4 11 9 F Phenylalanine
6 10 9 S Serine
6 22 6 A Alanine
0 2 2 C Cysteine
8 13 5 G Glycine 

4UHT_1|Chains A, B|TRANSCRIPTIONAL REGULATORY PROTEIN CPXR|ESCHERICHIA COLI (83333)
>2QOS_1|Chain A[auth C]|Complement component 8, gamma polypeptide|Homo sapiens (9606)
>6FGG_1|Chain A|Bromodomain adjacent to zinc finger domain protein 2A|Homo sapiens (9606)
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 \(w=f(c)\)
4UHT , Knot 52 102 0.78 38 81 99 
SPTLEVDALVLNPGRQEASFDGQTLELTGTEFTLLYLLAQHLGQVVSREHLSQEVLGKRLTPFDHAIDMHISNLRRKLPDRKDGHPWFKTLRGRGYLMVSAA
2QOS , Knot 82 173 0.81 40 126 168 
ASPISTIQPKANFDAQQFAGTWLLVAVGSAARFLQEQGHRAEATTLHVAPQGTAMAVSTFRKLDGICWQVRQLYGDTGVLGRFLLQARGARGAVHVVVAETDYQSFAVLYLERAGQLSVKLYARSLPVSDSVLSGFEQRVQEAHLTEDQIFYFPKYGFCEAADQFHVLDEVRR
6FGG , Knot 56 106 0.82 40 89 103 
SMHSDLTFCEIILMEMESHDAAWPFLEPVNPRLVSGYRRIIKNPMDFSTMRERLLRGGYTSSEEFAADALLVFDNCQTFNEDDSEVGKAGHIMRRFFESRWEEFYQ

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 \(c\).

\(|P_{f(4UHT_1)}(2) \setminus P_{f(2QOS_1)}(2)|=42\), \(|P_{f(2QOS_1)}(2) \setminus P_{f(4UHT_1)}(2)|=87\). 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 1:010101011110110001010100101010010110111001101100001000111001011001101010010001100001011100101010111011
Pair \(Z_2\) Length of longest common subsequence
4UHT_1,2QOS_1 129 3
4UHT_1,6FGG_1 118 3
2QOS_1,6FGG_1 151 3

Newick tree

 
[
	2QOS_1:73.57,
	[
		4UHT_1:59,6FGG_1:59
	]:14.57
]

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 \((0.85)(0.8)(\frac{275 }{\log_{20} 275}-\frac{102}{\log_{20}102})=54.8\)
Status Protein1 Protein2 d d1/2
Query variables 4UHT_1 2QOS_1 68 53.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]