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Parikh vectors
3KFF_1 8CUM_1 7SWG_1 Letter Amino acid
10 17 2 N Asparagine
3 0 2 C Cysteine
7 16 13 A Alanine
1 6 0 W Tryptophan
8 12 5 S Serine
2 11 6 Q Glutamine
20 21 14 E Glutamic acid
14 22 11 L Leucine
4 10 8 M Methionine
10 8 5 F Phenylalanine
5 7 1 Y Tyrosine
10 6 14 D Aspartic acid
9 20 11 G Glycine
6 4 1 H Histidine
14 16 5 I Isoleucine
17 24 11 K Lycine
0 8 2 P Proline
9 15 4 T Threonine
8 15 9 V Valine
5 7 5 R Arginine 

3KFF_1|Chain A|Major urinary protein 4|Mus musculus (10090)
>8CUM_1|Chain A|Beta-lactamase|Acinetobacter baumannii (470)
>7SWG_1|Chain A|Troponin C, slow skeletal and cardiac muscles,Troponin I, cardiac muscle chimera|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 \(w=f(c)\)
3KFF , Knot 76 162 0.79 38 120 156 
EEATSKGQNLNVEKINGEWFSILLASDKREKIEEHGSMRVFVEHIHVLENSLAFKFHTVIDGECSEIFLVADKTEKAGEYSVMYDGFNTFTILKTDYDNYIMFHLINEKDGKTFQLMELYGRKADLNSDIKEKFVKLCEEHGIIKENIIDLTKTNRCLKARE
8CUM , Knot 112 245 0.83 38 161 230 
MHISSQQHEKAIKSYFDEAQTQGVIIIKEGKNLSTYGNALARANKEYVPASTFKMLNALIGLENHKATTNEIFKWDGKKRTYPMWEKDMTLGEAMALSAVPVYQELARRTGLELMQKEVKRVNFGNTNIGTQVDNFWLVGPLKITPVQEVNFADDLAHNRLPFKLETQEEVKKMLLIKEVNGSKIYAKSGWGMGVTPQVGWLTGWVEQANGKKIPFSLNLEMKEGMSGSIRNEITYKSLENLGII
7SWG , Knot 64 129 0.80 38 99 123 
GDDIYKAAVEQLTEEQKNEFKAAFDIFVLGAEDGCISTKELGKVMRMLGQNPTPEELQEMIDEVDEDGSGTVDFDEFLVMMVRCMKDDSKGAGAGVRISADAMMQALLGARAKESLDLRAHLKQVKKED

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 \(c\).

\(|P_{f(3KFF_1)}(2) \setminus P_{f(8CUM_1)}(2)|=65\), \(|P_{f(8CUM_1)}(2) \setminus P_{f(3KFF_1)}(2)|=106\). 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 1:001000100101001010110111100000010001010111001011000111010011010000111110000011000110011001011000000011101100001001011010100101000100011010000111000110100000010100
Pair \(Z_2\) Length of longest common subsequence
3KFF_1,8CUM_1 171 4
3KFF_1,7SWG_1 139 3
8CUM_1,7SWG_1 152 3

Newick tree

 
[
	8CUM_1:84.34,
	[
		3KFF_1:69.5,7SWG_1:69.5
	]:14.84
]

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 \((0.85)(0.8)(\frac{407 }{\log_{20} 407}-\frac{162}{\log_{20}162})=73.1\)
Status Protein1 Protein2 d d1/2
Query variables 3KFF_1 8CUM_1 94 77.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]